partialdifferentialequationsfor …...in order to price the option before maturity, some assumptions...

PARTIAL DIFFERENTIAL EQUATIONS FOR

OPTION PRICING

Yves Achdou1

Olivier Pironneau 2

August 10, 2011

1UFR Mathematiques, Universite Paris 7, Case 7012, 75251 PARIS

Cedex 05, France and Laboratoire Jacques-Louis Lions, Universite Paris 6.

[email protected] Jacques-Louis Lions, Universite Pierre et Marie Curie, Boıte courrier

187, 75252 Paris Cedex 05. France. [email protected]

Contents

3

4 CONTENTS

Introduction

Option pricing is one of the many problems of financial mathematics or financialengineering as it is called now. It all started in the seventies with the celebratedmodel of Merton, Black and Scholes [81, 18] and their Nobel consecration later.Some of these ideas were already in the thesis of Bachelier[13] in 1900, buteveryone forgot because the era of fast electronic transactions had not come.Today financial assets (stocks, bonds, commodities, etc...) are used as a base forthousands of more complex financial products known as financial derivatives.The simplest example may be the European put option on a given asset whichenables its holder to sell the asset at a future time T , the maturity, for a price K,the strike. If the asset is worth ST at time T , the option will be exercised onlyif K > ST , generating a profit K−ST . In the other case, the option will not beexercised, and the profit will be 0. Therefore, the profit generated by the optionat T will be (K−ST )+. Assuming that the market is liquid and that arbitrage isnot possible (one cannot make an instantaneous benefit without taking a risk),the price of the option at T will be (K − ST )

+. Option pricing at time t < Tis more difficult because ST is not known. The Black-Scholes model makes theabove assumptions and supposes furthermore that the market is made of twoassets, the previously mentioned risky asset and a riskless asset whose priceevolves with a known interest rate r. It allows for pricing the above mentionedput option as the expectation of (K − ST )

+ discounted at the interest rate r.Another assumption of the Black-Scholes model is that St+δt evolves from Stwith a mean tendency µ and a random fluctuation of intensity σ, the volatility:

St+δt = St(1 + µδt) + σStN(0, δt),

where N(0, v) is a normal distribution with mean zero and variance v, and thatSt+δt − St is independent from the events before t. A very simple set of ideasindeed! But as the model should not depend on the time increment δt, oneshould use continuous-time stochastic processes: therefore the European put Ptis priced by

Pt = e−r(T−t)E(K − ST )

+, with dSt = St(µdt+ σdBt), (1)

where Bt is a Brownian motion. Since St is a Markov process, there exists a twovariable function P , called the pricing function, such that Pt = P (St, t), and P

1

2 CONTENTS

solves the partial differential equation (PDE):

∂P

∂t+σ2S2

2

∂2P

∂S2+ rS

∂P

∂S− rP = 0, (2)

for t ∈ [0, T ) and S > 0.There are three important classes of numerical methods in financial engineering:Monte-Carlo methods, tree based methods and deterministic methods based onthe PDE (2). The goal of the present paper is to focus on the latter.Of course, it may seem that numerical methods for (2) are now well known,but there are special difficulties in finance, because traders require a quick andaccurate response. Furthermore, there are much more complicated contractsthan the one described above, and the PDE may become much more complex:for example, American option pricing involves variational inequalities, stochas-tic volatility models lead to multidimensional PDEs; for e.g. basket optionsthe problem can become numerically formidable because set in a space whosedimension is the number of assets in the basket; finally models involving moregeneral Levy processes lead to partial integro-differential equations, [30].The diversity of the models for financial derivatives has grown to such a pointthat it is not possible to discuss all of them in one book chapter. Here only localvolatility (the volatility may depend on time and on the price of the underlyingasset) and stochastic volatility models will be considered, and models based onjump process will just be briefly described. Similarly the types of contracts onthe markets are too numerous and we will mostly deal with European, Americanoptions with or without barriers, possibly multidimensional (basket options forexample).Our objective in this paper is numerical: what is a good method for the com-puter implementations?

One may choose between finite difference methods, [98], finite element meth-ods, [28, 27, 113], finite volume methods, [42], spectral methods, [92, 17], etc...We have decided to work with the Finite Element Method (FEM) because it isvery flexible on the one hand and supported by a strong theory on the otherhand. In financial engineering, the PDEs often have a parabolic character (thefirst order terms in the equations are usually not dominant) which makes FEMswell adapted.In this restricted context, what is the best way to implement the methods,namely what polynomial degree, what mesh, what linear system solver, etc...?As usual the answer goes through a mathematical analysis of the variationalformulation of the problem, on which one can prove existence, uniqueness andqualitative properties of the solution. Then error estimates, especially a pos-teriori estimates are most useful and the FEM is best suited for such analysis.The outcome of such a study is that one can guarantee the precision of thecalculations, a property appreciated in banking where, in view of the large sumsinvolved, an error greater than 0. 1% is often unacceptable.For clarity we present the material by increasing order of mathematical complex-ity. The first chapter deals with plain vanilla European options. The variationalformulation is given and studied and the posteriori estimates derived in [6] are

CONTENTS 3

restated. A C++ implementation on an arbitrary mesh is given at the end.The second chapter deals with higher dimensional models. We consider in par-ticular European and American options on baskets and stochastic volatilitymodels. In this context, we discuss the variational analysis of the boundaryvalue problems. When the dimension of the problem is rather small (≤ 4),FEMs are competitive; we describe several techniques concerning the solutionprocedure, in particular for American options. We also review on a promisingand new class of methods which may be used for parabolic PDEs when thedimension of the problem lies between 4 and 20, the sparse grid and sparseGalerkin methods.In the third chapter, we recall the method given in [6] for computing the sensi-tivity of the solution with respect to the parameters of the problem, the Greeks(called so because practitioners have used Greek letters). The method is basedon automatic differentiation of computer programs, [54, 57], a very powerfultechnique particularly appropriate to financial engineering. The operator over-loading feature of the C++ language makes it easy to implement this approach.It is also useful in the context of parameters calibration, where the gradients ofthe least square functionals with respect to the model parameters are needed.This brings us to another important topic of financial engineering: better modelsor better parameters? As an example of calibration, we consider the calibrationof local volatility . We discuss Dupire’s equation [37] and the use of least squaresmethods for calibration.

4 CONTENTS

Chapter 1

One Dimensional Partial

Differential Equations For

Option Pricing

1.1 The Partial Differential Equation

A European vanilla call (respectively put) option is a contract giving its ownerthe right to buy (respectively sell) a share of a specific common stock at afixed price K at a certain date T . The specific stock is called the underlyingasset. The fixed price K is termed the strike and T is called the maturity. Theterm vanilla is used to notify that this kind of option is the simplest amongpossibly complicated contracts. The price of the underlying asset at time twill be referred to as the spot price and will be noted St. Assuming that themarket rules out arbitrage (the possibility to make an instantaneous risk-freebenefit), it is easy to see that the price of a call (resp. put) option at maturity isC0(ST ) = (ST −K)+, (resp. P0(ST ) = (K−ST )+). The payoff of the option atmaturity is a function of ST , called the payoff function. Naturally, other payofffunctions than the ones mentioned above are possible and used in practice.In order to price the option before maturity, some assumptions have to be madeon the spot price St: the Black-Scholes model assumes the existence of a risk-

free asset whose price at time t is S0t = S0

0 exp(∫ t

0 r(s)ds), where r(t) is the

interest rate; the model assumes that the price of the risky asset satisfies thefollowing stochastic differential equation,

dSt = St(µdt+ σtdBt), (1.1)

where Bt is a standard Brownian motion on a probability space (Ω,A,P). Hereσt is a positive number called the volatility. With the Black-Scholes assump-

5

6 CHAPTER 1. ONE DIMENSIONAL PDES FOR OPTION PRICING

tions, it is possible to prove that the option’s price at time t is given by

Pt = exp

(∫ T

t

r(s)ds

)E∗ (P0(ST )|Ft) , (1.2)

where the expectation E∗ is taken with respect to the so-called risk-neutralprobability P∗ (equivalent to P and under which dSt = St(rdt + σtdWt), Wt

being a standard Brownian motion under P∗ and Ft being the natural filtrationof Wt).From (1.2) and since St is a Markov process, it can be shown that the option’sprice Pt is a function of t and of St, i.e., that there exists a two variable functionP , called the pricing function, such that Pt = P (St, t).Assuming that σt = σ(St, t), where σ is a smooth enough function, it can beseen that the pricing function P solves the backward in time parabolic partialdifferential equation:

∂P

∂t+σ2(S, t)S2

2

∂2P

∂S2+ r(t)S

∂P

∂S− r(t)P = 0 (1.3)

for t ∈ [0, T ) and S > 0, and satisfies the final time condition

P (S, t = T ) = P0(S) (1.4)

for S > 0. Problem (1.3), (1.4) is called a final value problem.The volatility is the difficult parameter of the Black-Scholes model. It is con-venient to take it to be constant but then the computed options’ prices do notmatch the prices given by the market. There are essentially three ways forimproving on the Black-Scholes model with a constant volatility:

• Use a local volatility, i.e., assume that the volatility is a function of timeand of the stock price. Then one has to calibrate the volatility from themarket data, i.e., to find a volatility function which permits to recover theprices of the options available on the market.

• assume that the volatility is itself a stochastic process, see for example[58, 44] and §2.4.

• generalize the Black-Scholes model by assuming that the spot price is forexample a Levy process, see [30] and references therein.

There is much discussion among specialists in finance on comparing the meritsof the three kinds of models above. In this paragraph, we will focus on the firstone.

1.1.1 Changes of Variables

Several changes of variables and of unknown functions can be used.

1.1. THE PARTIAL DIFFERENTIAL EQUATION 7

Step 1 Consider the function v such that P (S, t) = v(S, t)e−λ(t), then (1.3)can be written

∂P

∂t= −λ′(t)e−λ(t)v + e−λ(t)

∂v

∂t,

∂P

∂S= e−λ(t)

∂v

∂S,∂2P

∂S2= e−λ(t)

∂2v

∂S2.

Choosing λ(t) = −∫ Tt r(s)ds leads to

∂v

∂t+ rS

∂v

∂S+σ2S2

2

∂2v

∂S2= 0.

Step 2 Now set x = logS and check that ∂v∂S = 1

S∂v∂x and ∂2v

∂S2 = − 1S2

∂v∂x +

1S2

∂2v∂x2 . We also set τ = T − t and w(x, τ) = v(ex, T − τ). Calling r and σ the

functions defined by r(τ) = r(t) and σ(x, τ) = σ(ex, t), we have

∂w

∂τ− σ2(x, τ)

2

∂2w

∂x2+ (r(τ) − σ2(x, τ)

2)∂w

∂x= 0, in R× (0, T ). (1.5)

Step 3 When σ depends on t only, one may use the change of variable (x, τ) 7→(y, τ), where y = x −

∫ τ0 (r(θ) −

σ2(θ)2 )dθ and set W (y, τ) = w(x, τ): it is easy

to see that

∂W

∂τ(y, τ)− σ2(τ)

2

∂2W

∂y2(y, τ) = 0, in R× (0, T ),

and that W (y, 0) = w(y, 0). When σ is a positive constant, this equation is theheat equation.A similar idea can be used if x 7→ σ2(x, τ) is Lipschitz continuous uniformlywith respect to τ : we call X(θ;x, τ) the solution of the ordinary differentialequation

d

dθX(θ;x, τ) = (r(θ)− σ2(X(θ;x, τ), θ)

2) θ ∈ (0, T ), X(τ ;x, τ) = x.

Assuming that (x, θ) 7→ X(θ;x, τ) is regular enough and introducing W (x, θ) =w(X(θ;x, τ), θ), we obtain

∂W

∂θ(x, θ) =

∂w

∂t(X(θ;x, τ), θ) + (r(θ)− σ2(X(θ;x, τ), θ)

2)∂w

∂x(X(θ;x, τ), θ)

and

∂W

∂x(x, θ) =

∂w

∂x(X(θ;x, τ), θ)

∂X(θ;x, τ)

∂x,

∂2W

∂x2(x, θ) =

∂2w

∂x2(X(θ;x, τ), θ)

(∂X(θ;x, τ)

∂x

)2

+∂w

∂x(X(θ;x, τ), θ)

∂2X(θ;x, τ)

∂x2.

Taking θ = τ−δt for δt small, we have that ∂X(τ ;x,τ−δt)∂x ∼ 1 and ∂2X(τ ;x,τ−δt)

∂x2 ∼0. Then using (1.5), we obtain the following semi-discrete scheme:

1

δt(W (x, τ) −W (x, τ − δt))− σ2(x, τ)

2

∂2W

∂x2(x, τ) ∼ 0,


i.e.

1

δt(w(x, τ) − w(X(τ − δt;x, τ), τ − δt))− σ2(x, τ)

2

∂2w

∂x2(x, τ) ∼ 0,

which is known as the method of characteristics and often used in fluid mechan-ics.

1.1.2 The Black-Scholes Formulas

Calling P (S, t) the price of an option with maturity T and payoff function P0

and assuming that r and σ > 0 are constant, the Black-Scholes formula is

P (S, t) = e−r(T−t)E∗(P0(Se

r(T−t)eσ(WT−Wt)− σ2

2 (T−t))), (1.6)

and since under P ∗, WT −Wt is a centered Gaussian distribution with varianceT − t,

P (S, t) =1√2πe−r(T−t)

∫

R

P0(Se(r−σ2

2 )(T−t)+σx√T−t)e−

x2

2 dx. (1.7)

When the option is a vanilla European option, noting C the price of the calland P the price of the put), a more explicit formula can be deduced from (1.2).Take for example a call:

C(S, t) =1√2π

∫ +∞

−d2

(Se−

σ2

2 (T−t)+σx√T−t −Ke−r(T−t)

)e−

x2

2 dx

=1√2π

∫ d2

−∞

(Se−

σ2

2 (T−t)−σx√T−t −Ke−r(T−t)

)e−

x2

2 dx,

(1.8)

where

d1 =log( SK ) + (r + σ2

2 )(T − t)

σ√T − t

and d2 = d1 − σ√T − t. (1.9)

Finally, introducing the upper tail of the Gaussian function

N(d) =1√2π

∫ d

−∞e−

x2

2 dx, (1.10)

and using (1.8), (1.9), we obtain the Black-Scholes formula:

Proposition 1.1 When σ and r are constant, the price of the call is given by

C(S, t) = SN(d1)−Ke−r(T−t)N(d2), (1.11)

and the price of the put is given by

P (S, t) = −SN(−d1) +Ke−r(T−t)N(−d2), (1.12)

where d1 and d2 are given by (1.9) and N is given by (1.10).

Remark 1.1 If r is a function of time (1.9) must be replaced by

d1 =log( SK ) +

∫ Ttr(τ)dτ + σ2

2 (T − t)

σ√T − t

and d2 = d1 − σ√T − t. (1.13)


1.1.3 Classical Solutions

In the previous paragraph, we have seen that if the coefficients are constant(with a positive volatility), then (1.3) (1.4) has a solution given by (1.7).In this paragraph, we give a classical existence and uniqueness result for the finalvalue problem (1.3) (1.4) in the general case when r = r(t) and σ = σ(S, t). Itis necessary to restrict the growth of the solutions when S → 0 or S → +∞.Here, we will impose that the solution is bounded but this restriction can berelaxed (for example, depending on P0, one can look for solutions with lineargrowth as S → +∞).

Definition 1.1 We fix a positive number ρ0. Let α be a real number such that0 < α < 1. We call C0,α(Rd) the space of continuous real valued functionsv ∈ C0(Rd) such that

‖v‖C0,α(Rd) = supx∈Rd

|v(x)|+ supx,y∈Rd,|x−y|≤ρ0

|v(x) − v(y)||x− y|α < +∞.

The space C0,α(Rd) endowed with the norm ‖ · ‖C0,α(Rd) is a Banach space.

We call Cα,α/2(Rd × [0, T ]) the space of continuous real valued functions v ∈C0(Rd × [0, T ]) such that

‖v‖Cα,α/2(Rd×[0,T ]) =

sup(x,t)∈Rd×[0,T ]

|v(x, t)|+ sup(x, t), (y, s) ∈ Rd × [0, T ],|x− y|+ |t− s| ≤ ρ0

|v(x, t) − v(y, s)|(|x− y|2 + |t− s|)α

2< +∞.

The space Cα,α/2(Rd × [0, T ]) endowed with the norm ‖ · ‖Cα,α/2(Rd×[0,T ]) is aBanach space.

Theorem 1.1 Under the following assumptions on the coefficients and the finalvalue:

1. the real valued function defined on R× [0, T ], (x, t) 7→ σ2(ex, t) belongs toCα,α/2(R× [0, T ]),

2. the function t 7→ r(t) belongs to Cα/2([0, T ]),

3. the function P0 defined on R by P0(x) = P0(ex) is such that P0,

∂P0

∂x , and∂2P0

∂x2 belong to Cα(R),

4. there exists a positive constant σ such that, for all x ∈ R, t ∈ [0, T ],σ(t, ex) ≥ σ,

the final value problem (1.3) (1.4) has a unique solution P such that, calling P

the function defined on [0, T ] × R by P (x, t) = P (ex, t), the functions P , ∂P∂x ,

∂P∂t ,

∂2P∂x2 belong to Cα,α/2(R× [0, T ]).


Under the previous assumptions except the one on P0 and assuming that P0 is abounded function, (1.3), (1.4) has a unique solution P such that for all τ < T ,

the functions P , ∂P∂x ,

∂P∂t ,

∂2P∂x2 belong to Cα,α/2(R× [0, τ ]).

This theorem is proved in [73], see also [45, 72].

1.1.4 Variational Framework

Weighted Sobolev Norms

The theory of variational formulations of parabolic equations is well known,see the work of Lions and Magenes [76]. It is particularly useful when strongsolutions do not exist either because of some singularity in the data or thedomain boundary or the coefficients or nonlinearity. Such situations are veryfrequent in physics and engineering. Even when the boundary value problemhas a classical solution, the variational theory is interesting for several reasons:

• it provides global estimates, often called energy estimates.

• It has strong connections with the finite element method which will beadvocated below.

• it is the most natural way to study obstacle problems, see the section § 2.3devoted to American options on baskets.

Note that there are other theories of weak solutions, in particular the theory ofviscosity solutions, which may also be quite useful in the context of quantitativefinance. We will not discuss viscosity solutions here, and we refer the reader to[35, 14, 43].Almost all the proofs of the results below are omitted for brevity; they can befound in [6].Variational formulations of parabolic PDE rely on suitable Sobolev spaces. Weare going to introduce the Sobolev space useful for the initial value problem(1.3) (1.4) posed in the price variable S.We denote by L2(R+) the Hilbert space of square integrable functions on R+

endowed with the norm ‖v‖L2(R+) = (∫R+v(S)2dS)

12 and the inner product

(v, w)L2(R+) =∫R+v(S)w(S)dS. Calling D(R+) the space of the smooth func-

tions with compact support in R+, we know that D(R+) is dense in L2(R+).Let us introduce the space

V =

v ∈ L2(R+) : S

dv

dS∈ L2(R+)

, (1.14)

where the derivative must be understood in the sense of the distributions onR+. A natural scalar product for V is (v, w)V = (v, w)+(S dv

dS , SdwdS ); The space

V endowed with the norm ‖v‖V =√(v, v)V is a Hilbert space.

We have the following properties (see [6])


Theorem 1.2 • The space D(R+) is dense in V .

• (Poincare’s inequality) If v ∈ V , then

‖v‖L2(R+) ≤ 2‖S dvdS

‖L2(R+). (1.15)

so the semi-norm |v|V = ‖S dvdS‖L2(R+) is also a norm on V , equivalent to

‖.‖V .

• For any w ∈ L2(R+) the function S → v(S) = 1S

∫ S0 w(s)ds belongs to V ,

and ‖v‖V ≤ C‖w‖L2(R+) for some positive constant C independent of w.

We denote by V ′ the topological dual space of V , and for w ∈ V ′, ‖w‖V ′ =

supv∈V \0(w,v)|v|V .

The Weak Formulation of the Black-Scholes Equation

Consider a vanilla put option with maturity T and payoff function u0. Let ube the pricing function, i.e., the price of the option at time T − t and when thespot price is S is u(S, t). The function u solves the initial value problem

∂u

∂t−σ

2S2

2

∂2u

∂S2−rS ∂u

∂S+ru = 0 in R+×(0, T ), u(S, 0) = u0(S) in R+. (1.16)

Let us multiply (1.16) by a smooth real valued function w defined on R+ andintegrate in the variable S on R+. Assuming that integrations by part areallowed, we obtain

d

dt

(∫

R+

u(S, t)w(S)dS

)+ at(v, w) = 0,

where the bilinear form at is defined by

at(v, w) =

∫

R+

(1

2S2σ2(S, t)

∂v

∂S

∂w

∂S+ r(t)vw

)dS

+

∫

R+

(−r(t) + σ2(S, t) + Sσ(S, t)

∂σ

∂S(S, t)

)S∂v

∂Sw dS.

(1.17)

Assume that the coefficient r ≥ 0 is bounded and that σ is sufficiently regularso that the following makes sense:

Assumption 1.1 1. There exists two positive constants, σ and σ such thatfor all t ∈ [0, T ] and all S ∈ R+,

0 < σ ≤ σ(S, t) ≤ σ. (1.18)

2. There exists a positive constant Cσ such that for all t ∈ [0, T ] and allS ∈ R+,

|S ∂σ∂S

(S, t)| ≤ Cσ. (1.19)


Lemma 1.1 Under Assumption 1.1, the bilinear form at is continuous on V ,i.e. there exists a positive constant µ such that for all v, w ∈ V ,

|at(v, w)| ≤ µ|v|V |w|V . (1.20)

It also satisfies Garding’s inequality : there exists a non negative constant λsuch that for all v ∈ V ,

at(v, v) ≥σ2

4|v|2V − λ‖v‖2L2(R+). (1.21)

One associates with the bilinear form at the continuous linear operator At:V → V ′; for all v, w ∈ V , (Atv, w) = at(v, w). The interpretation of At is asfollows:

Atv = −1

2σ2(S, t)S2 ∂

2v

∂S2− r(t)S

∂v

∂S+ r(t)v.

We define C0([0, T ];L2(R+)) as the space of continuous functions on [0, T ] withvalues in L2(R+), and L2(0, T ;V ) as the space of square integrable functionson (0, T ) with values in V . Assuming that u0 ∈ L2(R+), and following [76], itis possible to write a weak formulation for (1.16):

Weak formulation of (1.16)

Find u ∈ C0([0, T ];L2(R+)) ∩ L2(0, T ;V ) with ∂u∂t ∈ L2(0, T ;V ′), and

u|t=0 = u0 in R+, and for a.e. t ∈ (0, T ), (1.22)

∀v ∈ V,

(∂u

∂t(t), v

)+ at(u(t), v) = 0. (1.23)

Theorem 1.3 under Assumption 1.1 and if u0 ∈ L2(R+), the weak formulation(1.22) (1.23) has a unique solution, and we have the estimate, for all t, 0 < t <T

e−2λt‖u(t)‖2L2(R+) +1

2σ2

∫ t

0

e−2λτ |u(τ)|2V dτ ≤ ‖u0‖2L2(R+). (1.24)

Note that Theorem 1.3 does not apply to a European call option because thepayoff is not a function of L2(R+); one must either use the put-call parity, see§ 1.1.4, an deduce the price of the call from that of the put or work with adifferent Sobolev space with a weight decaying at infinity.

Regularity of the Weak Solutions

If the interest rate, the volatility, and the payoff are smooth enough, then itis possible to prove additional regularity for the solution to (1.22) (1.23). Inparticular for all t ∈ [0, T ] and for λ given in Lemma 1.1, the domain of At + λis

D = v ∈ V ;S2 ∂2v

∂S2∈ L2(R+). (1.25)

Let us assume the following:


Assumption 1.2 There exists a positive constant C and 0 ≤ α ≤ 1 such thatfor all t1, t2 ∈ [0, T ] and S ∈ R+,

|r(t1)− r(t2)|+ |σ(S, t1)− σ(S, t2)|+ S

∣∣∣∣∂σ

∂S(S, t1)−

∂σ

∂S(S, t2)

∣∣∣∣ ≤ C|t1 − t2|α.(1.26)

Theorem 1.4 Under Assumptions 1.1 and 1.2, for all s, 0 < t ≤ T , the so-lution u of (1.22) (1.23) satisfies u ∈ C0([t, T ];D) and ∂u

∂t ∈ C0([t, T ];L2(R+)),and there exists a constant C such that for all t, 0 < t ≤ T ,

‖Atu(t)‖L2(R+) ≤C

t.

If u0 ∈ D, then the solution u of (1.22) (1.23) belongs to C0([0, T ];D) and∂u∂t ∈ C0([0, T ];L2(R+)).Furthermore if u0 ∈ V then the solution to (1.22) (1.23) belongs to C0([0, T ];V )∩L2(0, T ;D), ∂u∂t ∈ L2(0, T ;L2(R+)), and there exists a non negative constant λsuch that

e−2λt‖S ∂u∂S

(t)‖2L2(R+) +σ2

2

∫ t

0

e−2λτ |S ∂u∂S

(τ)|2V dτ ≤ ‖S∂u0∂S

‖2L2(R+). (1.27)

The Maximum Principle for weak solutions

We refer to [91] for a monograph on the maximum principle. The solutionsof (1.16) may not vanish for S → +∞, therefore, we are going to state themaximum principle for a class of functions much larger than V , i.e.,

V = v : ∀ǫ > 0, v(S)e−ǫ log2(S+2) ∈ V . (1.28)

Note that the polynomial functions belong to V .

Theorem 1.5 (Weak Maximum Principle) Let u(S, t) be such that for allpositive number ǫ,

• ue−ǫ log2(S+2) ∈ C0([0, T ];L2(R+)) ∩ L2(0, T ;V ),

• u|t=0 ≥ 0 a.e.,

• ∂u∂t +Atu ≥ 0, (in the sense of distributions)

then u ≥ 0 almost everywhere.

Various Bounds The maximum principle is an extremely powerful tool forproving estimates on the solutions of elliptic and parabolic partial differentialequations. Here we give easy examples of its application to option pricing.


Proposition 1.2 Under Assumption 1.1, let u be the weak solution to (1.16),with u0 ∈ L2(R+) being a bounded positive function, i.e., 0 ≤ u0 ≤ u0(S) ≤ u0.Then, a.e.

u0e−

∫t0r(τ)dτ ≤ u(S, t) ≤ u0e

−∫

t0r(τ)dτ . (1.29)

Proof. We know that u0e−

∫t0r(τ)dτ and u0e

−∫

t0r(τ)dτ are two solutions of

(1.16). Therefore, we can apply the maximum principle to u − u0e−

∫t0r(τ)dτ

and to u0e−

∫t0r(τ)dτ − u.

Remark 1.2 In the case of a vanilla put option: u0(S) = (K−S)+, Proposition1.2 just says that 0 ≤ u(S, t) ≤ Ke−

∫ t0r(τ)dτ , which is certainly not a surprise.

For the vanilla put option as in Remark 1.2, we have more information:

Proposition 1.3 Under Assumption 1.1, let u be the weak solution to (1.16),with u0(S) = (K − S)+, then

(Ke−∫ t0r(τ)dτ − S)+ ≤ u(S, t) ≤ Ke−

∫ t0r(τ)dτ . (1.30)

Proof. Observe that Ke−∫

t0r(τ)dτ − S is a solution to (1.16) and apply the

maximum principle to u(S, t)−(Ke−∫ t0r(τ)dτ−S). We have Ke−

∫ t0r(τ)dτ−S ≤

u(S, t). Then (1.30) is obtained by combining this estimate with the one given

in Remark 1.2. Note that (1.30) yields u(0, t) = Ke−∫

t0r(τ)dτ for all t ≤ T .

The Put-Call Parity Let u be the pricing function of a vauilla put optionwith strike K, and consider C(S, t) given by:

C(S, t) = S −Ke−∫

t0r(τ)dτ + u(S, t). (1.31)

From the fact that u and S − Ke−∫

t0r(τ)dτ satisfy (1.16),it is clear that C is

a solution to (1.16) with the Cauchy condition C(S, 0) = (S − K)+. Thisis precisely the boundary value problem for the European vanilla call option.Furthermore, from the maximum principle, we know that a well behaved solutionof this boundary value problem (in the sense of Theorem 1.5) is unique.

Convexity of u in the Variable S

Assumption 1.3 There exists a positive constant C such that

|S2 ∂2σ

∂S2(S, t)| ≤ C, a.e. (1.32)

Proposition 1.4 Under Assumptions 1.1 and 1.3, let u be the weak solution to

(1.16) where u0 ∈ V is a convex function such that ∂2u0

∂S2 has a compact support.Then, for all t > 0, u(S, t) is a convex function of S.


As a consequence, we see that under Assumptions 1.1 and 1.3, the price of avanilla European put option is convex with respect to S, and thanks to thecall-put parity, this is also true for the vanilla European call.

More Bounds

We focus on a vanilla put with a local volatility σ. By using Proposition 1.4, itis possible to compare u with the pricing function of vanilla puts with constantvolatilities:

Proposition 1.5 Under Assumptions 1.1, we have for all t ∈ [0, T ] and for allS > 0,

u(S, t) ≤ u(S, t) ≤ u(S, t), (1.33)

where u, (respectively u) is the solution to (1.16) with σ = σ, (respectively σ).

Localization Again, we focus on a vanilla put. For a numerical approximationto u, one has to limit the domain in the variable S, i.e. to consider only S ∈ (0, S)for S large enough and to impose some artificial boundary condition at S = S.Imposing that the new function vanishes on the artificial boundary, we obtainthe new boundary value problem:

∂u

∂t− 1

2σ2S2 ∂

2u

∂S2− rS

∂u

∂S+ ru = 0, t ∈ (0, T ], S ∈ (0, S),

u(S, t) = 0, t ∈ (0, T ],

(1.34)

with the Cauchy data u(S, 0) = (K−S)+ in (0, S). The theory of Lions-Magenesapplies to this new boundary value problem, but one has to work in the newSobolev space:

V = v, S ∂v∂S

∈ L2((0, S)), v(S) = 0.

The theory of weak solutions can be applied to problem (1.34). The question isto estimate the error between u and u.

Proposition 1.6 Under Assumptions 1.1, the error maxt∈[0,T ],S∈[0,S] |u(S, t)−u(S, t)| decays faster than any negative power of S as S → ∞, i.e., faster thanS−η for any positive number η.

Proof. From the maximum principle applied to weak solutions of (1.16) in(0, S)×(0, T ], we immediately see that u ≥ u in (0, S)×(0, T ), because u(S, t) ≥u(S, t) = 0. On the other hand, from Proposition 1.5, u ≤ u, which impliesthat u(S, t) ≤ u(S, t). Call π(S) = maxt∈[0,T ] u(S, t). The maximum principleapplied to the function E(S, t) = π(S)−u(S, t)+ u(S, t) yields that π(S) ≥ u− uin [0, S]× [0, T ]. At this point, we have proved that

0 ≤ u− u ≤ π(S), in [0, S]× [0, T ].

But π(S) can be computed semi-explicitly by the Black-Scholes formula (1.12),and it is easy to see that for all η > 0, limS→∞ π(S)Sη = 0.


Therefore, maxt∈[0,T ],S∈[0,S] |u(S, t)− u(S, t)| decays faster than any power S−η

as S → ∞.

1.2 A Finite Element Method

1.2.1 Description of the Method

Consider the boundary value problem

∂u

∂t− 1

2σ2(S, t)S2 ∂

2u

∂S2− α(t)S

∂u

∂S+ β(t)u = 0, t ∈ (0, T ), S ∈ (0, S),

u(S, 0) = u0(S) S ∈ (0, S), u(S, t) = 0 t ∈ (0, T ]. (1.35)

This generalization of (1.34) is used for pricing European puts with possiblycontinuously paid dividends: this corresponds to the choice α = r(t)− q(t) andβ = r(t), where r is the interest rate and q is the dividend yield. A problem ofthe form (1.35) also arises when one looks for the option’s price as a function ofthe maturity and the strike at a fixed spot price (the partial differential equationis known as Dupire’s equation [36, 37, 6], see § 3.2.5): this corresponds to thechoice α = −r(t) + q(t) and β = q(t).To apply the Finite Element Method of degree one we start with the variationalformulation introduced in §1.1.4, given in (1.22), (1.23).We introduce a partition of the interval [0, S] into subintervals κi = [Si−1, Si],1 ≤ i ≤ N + 1, such that 0 = S0 < S1 < · · · < SN < SN+1 = S. We callhi = Si−Si−1 and h = maxi=1,...,N+1 hi. We define the mesh Th of [0, S] as theset κ1, . . . , κN+1. In what follows, we will assume that the strike K coincideswith some node of Th, i.e., there Sk0 = K for some admissible k0.

We define the discrete space Vh by

Vh =vh ∈ C0([0, S]), vh(S) = 0; ∀κ ∈ Th, vh|κ is affine

. (1.36)

The assumption on the mesh ensures that u0 ∈ Vh when u0 = (K − S)+. Thediscrete problem obtained by applying the Euler implicit scheme in time reads:

find (umh )1≤m≤M , umh ∈ Vh with u0h(Si) = u0(Si), i = 0...N + 1, and

for m = 1...M , ∀vh ∈ Vh,(umh − um−1

h , vh)+ δtmatm(umh , vh) = 0, (1.37)

where

at(v, w) =

∫ S

0

1

2S2σ2(S, t)

∂v

∂S

∂w

∂S

+

∫ S

0

(−α(t) + σ2(S, t) + Sσ(S, t)

∂σ

∂S(S, t)

)S∂v

∂Sw + β(t)

∫ S

0

vw.

(1.38)

1.2. A FINITE ELEMENT METHOD 17

Note that we have a simpler expression for at(v, w) for v, w ∈ Vh when σ iscontinuous with respect to S:

at(v, w) = −N∑

i=1

1

2S2i σ

2(Si, t)[∂v

∂S](Si)w(Si)− α(t)

∫ S

0

S∂v

∂Sw + β(t)

∫ S

0

vw,

(1.39)

where [·] denotes the jump

[∂v

∂S](Si) =

∂v

∂S(S+i )−

∂v

∂S(S−i ). (1.40)

For i = 0, ..., N + 1, let wi be the piecewise linear function on the mesh whichtakes the value 1 at Si and 0 at Sj, j 6= i, j = 0, ..., N +1. Then (wi)i=0,...N isthe nodal basis of Vh, and

umh (S) =

N∑

0

umh (Si)wi(S). (1.41)

Let M and Am in RN×N be respectively the mass and stiffness matricesdefined by M i,j = (wi, wj), Am

i,j = atm(wj , wi), 0 ≤ i, j ≤ N . Calling

um = (umh (S0), . . . , umh (SN ))T (1.37) is equivalent to

(M + δtmAm)um = Mum−1, (1.42)

The shape functions wi corresponding to vertex Si is supported in [Si−1, Si+1].This implies that the matricesM and Am are tridiagonal because when |i−j| >1, the intersection of the supports of wi and wj has measure 0. Furthermore,for i ≤ N ,

wi(S) =S − Si−1

hi,

∂wi∂S

=1

hi, ∀S ∈ (Si−1, Si),

wi(S) =Si+1 − S

hi+1,

∂wi∂S

= − 1

hi+1, ∀S ∈ (Si, Si+1), (1.43)

giving,

∫ S

0

wi−1wi =hi6,

∫ S

0

Swi∂wi−1

∂S= −Si−1

6− Si

3,

∫ S

0

wiwi =hi + hi+1

3,

∫ S

0

Swi∂wi∂S

= −1

2

∫ S

0

w2i = −hi + hi+1

6, if i > 0,

∫ S

0

w0w0 =h13

∫ S

0

Sw0∂w0

∂S= −1

2

∫ S

0

w20 = −h1

6,

∫ S

0

wi+1wi =hi+1

6,

∫ S

0

Swi∂wi+1

∂S=Si+1

6+Si3.


From this, a few calculations show that the entries of Am are

Ami,i−1 = −S

2i σ

2(Si, tm)

2hi+α(tm)Si

2+ (β(tm)− α(tm))

hi6, 1 ≤ i ≤ N,

Ami,i =

S2i σ

2(Si, tm)

2(1

hi+

1

hi+1) +

α(tm)

2(hi+1 + hi) + (β(tm)− α(tm))

hi + hi+1

3, 1 ≤ i ≤ N,

Am0,0 =

α(tm)

2h1 + (β(tm)− α(tm))

h13,

Ami,i+1 = −S

2i σ

2(Si, tm)

2hi+1− α(tm)Si

2+ (β(tm)− α(tm))

hi+1

6, 0 ≤ i ≤ N − 1.

When the mesh is uniform, this matrix is close (but not proportional) to thestiffness matrix obtained by using the finite difference method with a centeredscheme, see [6]. The entries of M are

M i,i−1 = hi

6 , 1 ≤ i ≤ N,

M i,i =hi+hi+1

3 , 1 ≤ i ≤ N, M 0,0 = h1

3 ,

M i,i+1 = hi+1

6 0 ≤ i ≤ N − 1.

Remark 1.3 The value of u at S = 0 is known for all time, because the equation

degenerates into ∂u∂S + β(t)u = 0. Therefore u(0, t) = u0(0) exp

(−∫ t0 β(s)ds

).

hence, it is possible to impose that

um0 = u0(0) exp

(−∫ tm

0

β(s)ds

),

and plug this into (1.42). In this case, since um0 is known, (1.42) can be rewrittenas: ∀i = 1, ..., N

N∑

j=1

(M i,j + δtmAmi,j)u

mj =

N∑

j=0

M i,jum−1j − (M i,0 + δtmAm

i,0)um0 . (1.44)

1.2.2 A C++ Implementation

The following is a simple C++ implementation of the above for a put optionwith dividend d(t) on a general mesh which may vary at each time step. It canalso solve the Dupire equation (see section § 3.2.5).The boundary condition at zero is implemented as in Remark 1.3. There aretwo classes, one for the mesh and one for the put option problem. The meshclass has a simple constructor for a mesh that can be refined near the strike andat the origin in time. The calling program is

int main()

VarMesh m(50,100,0.5,300.,1.05,0.9,100.,1.02);


Option p(1,&m,100.,0.05,0.3);

p.calc();

ofstream result("u.txt");

for(int i = 0; i < m.nT; i++)

for(int j = 0; j < m.nX[i]; j++)

ff << m.x[i][j] << "\t" << m.t[i] << "\t" <<p.u[i][j] <<endl;

ff << endl;

return 0;

The mesh is called m; it has 50 time steps and at most 100 mesh pointsmaximum over (0, 300)× (0, 0.5). At each time level the time step is increasedby a factor 1.05 and the number of mesh points Nm is decreased by a factor1.02, namely

δtm = 1.05δtm−1, Nm = int(Nm−1/1.02), m = 1, ..., 49.

Finally the mesh is not uniform in space, it is refined near the strike xS = 100by a factor 0.9, namely on the left of xS , δxi = 0.9δxi−1 and on the right of xS ,δxi−1 = 0.9δxi.The put option is called p; it is solution of Black-Schole’s PDE (hence the1 as first parameter in the constructor, 0 being for Dupire’s equation). Thestrike is 100, the maturity is obtained from the second parameter in the meshconstructor; the interest rate is constant here and equal to 0.05 and so is thevolatility which is here 0.3. The function calc solves Black-Scholes or Dupire’sequation by FEM with LU factorization at each time step. The results arein m.u and written in the file u.txt in a format readable by gnuplot. Thefollowing gnuplot statement splot"u.txt"w l displays something like figure1.1. File optionhb.hpp contains the definitions and implementation of theclasses VarMesh and Option.


Inappropriate ioctl for device

using namespace std;

// # include "ddouble.h"

typedef double ddouble; // for automatic differentiation

class VarMesh

public:

const int nT; // nb time step

int *nX; // nb of vertices at each time step

double *t, **x; // mesh points x at times t

double T, xmax;

double *xx, *vxx; // holds vertices of 2 mesh levels; holds function val

int kk, *ixx; // total nb vertices; holds origin of vertex (<0 if from

level k-1)

VarMesh(const int nt, const int nx, const double T1, const double xmax1,

const double tscal=1,const double xc=0, const double xS=1,

const double xsize=1); // defaults=>uniform mesh

void interpol(ddouble* v, const int k, ddouble *w); // w=interpol of v[]

on x[k-1], to x[k]

void intersect(const int k); // intersect level k and k-1, result in

xx,ixx,kk

void integral(ddouble* v, const int k, ddouble* w); // even if v and w

are on != mesh

;

class Option

public:

const int s; // s=0 for Dupire and =1 for B&S

VarMesh *msh; // mesh

const double S,T; // spot price and Maturity

double *r, *d; // interest rate and dividend

ddouble **u, **sigma; // solution and volatility(function of K and T)

ddouble *w, *am, *bm, *cm; // working arrays for M.u and Gauss fact.

ddouble u0(const double x); // initial condition

void factLU(const int nX1);

void solveLU(const int nX1, ddouble* z);

void calc();

Option(const int s1, VarMesh* msh1, const double S1,

double r1, double sigma1, double d1=0);

~Option();

;

Option::Option(const int s1, VarMesh* msh1, const double S1,

double r1, double sigma1, double d1): s(s1), msh(msh1), S(S1), T(msh1->T)

VarMesh& m = *msh;

r = new double[m.nT];

d = new double[m.nT];

u = new ddouble*[m.nT];

sigma = new ddouble*[m.nT];

int nXmax =0;


if(nXmax<m.nX[i] ) nXmax=m.nX[i];

u[i] = new ddouble[m.nX[i]];

sigma[i] = new ddouble[m.nX[i]];


r[i]=r1; d[i] = d1;

for(int j=0;j<m.nX[i];j++)sigma[i][j]=sigma1;

am = new ddouble[nXmax];

bm = new ddouble[nXmax];

cm = new ddouble[nXmax];

w = new ddouble[nXmax];

ddouble Option::u0(const double x1) return x1<S ? (S-x1) : 0;

void Option::factLU(const int nX1)

cm[1] /= bm[1];

for(int i=2;i<nX1-1;i++)

bm[i] -= am[i]*cm[i-1];

cm[i] /= bm[i];

void Option::solveLU(const int nX1, ddouble *z)

z[1] /= bm[1];

for(int i=2;i<nX1-1;i++)

z[i] = (z[i] - am[i]*z[i-1])/bm[i];

for(int i=nX1-2;i>0;i--)

z[i] -= cm[i]*z[i+1];

void Option::calc( )

VarMesh& m = *msh;

for(int i=0;i<m.nX[0];i++) u[0][i] = u0(m.x[0][i]);

const double nml=1./6.; // no mass lumping = 1./6., mass lumping =0

for(int j=1;j<m.nT;j++) // time loop

double dt = m.t[j]-m.t[j-1];

double aux = (r[j]*(4*s-1)+d[j]*(3-4*s))/3, auy = (r[j]*(1-s)+d[j]*s)*dt/6;

m.integral(u[j-1],j,w); // rhs of PDE

for(int i=1;i<m.nX[j]-1;i++)

double hi = m.x[j][i]-m.x[j][i-1], hi1 = m.x[j][i+1]-m.x[j][i];

double xss = m.x[j][i]*sigma[j][i]*sigma[j][i];

bm[i] =(hi+hi1)*(0.5-nml +dt*(m.x[j][i]*xss/hi/hi1+aux)/2); // FEM

matrix

am[i] = nml*hi - dt*m.x[j][i]*(xss/hi - (2*s-1)*(r[j]-d[j]))/2 + auy*hi;

cm[i] = nml*hi1- dt*m.x[j][i]*(xss/hi1 + (2*s-1)*(r[j]-d[j]))/2 + auy*hi1;

u[j][m.nX[j]-1]=0; // C.L.

u[j][0]=u[j-1][0]*exp(((1-s)*d[j]-s*r[j])*dt); // C.L.

double hi = m.x[j][1]-m.x[j][0];

w[1] -= u[j][0]*(nml*hi - dt*m.x[j][1]*

( m.x[j][1]*sigma[j][1]*sigma[j][1]/hi- (2*s-1)*(r[j]-d[j]))/2 + auy*hi);

factLU(m.nX[j]);

solveLU(m.nX[j],w);

for(int i=1;i<m.nX[j]-1;i++) u[j][i]=w[i];


0 50

100 150

200 250

300 0

0.1

0.2

0.3

0.4

0.5

-20

0

20

40

60

80

100

PDE solutionBlack-Scholes formula

Figure 1.1: Result of the program of Section 1.2.2 and comparison with theBlack-Scholes formula at the mesh points (crosses).

1.3 Mesh Adaptivity

Here, we discuss a strategy in order to separately adapt the grids in the variablest and S. Moreover, the mesh in the variable S may vary in time. This methodwas originally proposed by Bernardi et al [16]. The goal is to find local errorindicators which can be explicitly computed from the solution of the discreteproblem, and such that their Hilbertian sum is equivalent to the global error.These indicators are said to be optimal if the constants of the norm-equivalenceinequalities are independent of the error. We consider two families of errorindicators, both of residual type. The first family is global with respect to thespot price variable and local with respect to time: it gives relevant informationin order to refine the mesh in time. The second family is local with respect toboth spot and time variables, and provides an efficient tool for mesh adaptivityin the price variable at each time step.All the results below are proved in [6].

1.3. MESH ADAPTIVITY 23

1.3.1 Multiple meshes and integration

Consider a parabolic problem in the variables x and t, where t is the timevariable. Mesh adaptivity may lead to situations when the mesh used for thevariable x varies between two time steps; assume that mesh adaption is per-formed at time tm: in this case, um is defined on the mesh TH used at timetm and has to be integrated against functions defined on the mesh Th used attime tm+1. Let wi be a nodal basis function associated with the mesh Th. Tocompute the integral of umwi we intersect the meshes TH and Th; the result isa new mesh on which both functions um and wi are piecewise linear; therefore,the integral can be computed without error.

1.3.2 Error Indicators for the Semi-Discrete Problem in

Time

Our goal is Theorem 1.6 below, but we need first to define appropriate norms.We go back to problem (1.34). From assumption 1.1 and from Garding’s in-equality (1.21)), it is possible to prove that (1.34) admits a unique solution thatwe write u for simplicity. Moreover, introducing the norm

[[v]](t) =

(e−2λt‖v(t)‖2 + 1

2σ2

∫ t

0

e−2λτ |v(τ)|2V dτ) 1

2

, (1.45)

where ‖ · ‖ stands for the norm in L2(0, S), we have, by multiplying (1.34) byu(t)e−2λt and integrating in (0, S)× (0, T ),

[[u]](t) ≤ ‖u0‖. (1.46)

We introduce a partition of the interval [0, T ] into subintervals [tn−1, tn], 1 ≤n ≤ N , such that 0 = t0 < t1 < · · · < tN = T . We denote by δtn the lengthtn − tn−1, and by δt the maximum of the δtn, 1 ≤ n ≤ N . We also define theregularity parameter ρδt:

ρδt = max2≤n≤N

δtnδtn−1

. (1.47)

Given u0 = u0 ∈ L2(0, S), the semi-discrete problem arising from an implicitEuler scheme is: find (un)1≤n≤N ∈ V N satisfying

∀n, 1 ≤ n ≤ N,(un − un−1, v

)+ δtnatn(u

n, v) = 0, ∀v ∈ V, (1.48)

where V is the space

V =

v ∈ L2(0, S) : S

∂v

∂S∈ L2(0, S), v(S) = 0

. (1.49)

For δt smaller than 1/(2λ), (where λ is the constant appearing in Garding’sinequality (1.21)), the existence and uniqueness of (un)1≤n≤N is a consequenceof the Lax-Milgram lemma. We call uδt the function which is affine on each


interval [tn−1, tn], and such that uδt(tn) = un, 0 ≤ n ≤ N .From the standard identity (a−b, a) = 1

2 |a|2+ 12 |a−b|2− 1

2 |b|2, a few calculationsshow that

(1 − 2λδtn)‖un‖2 +1

2δtnσ

2|un|2V ≤ ‖un−1‖2. (1.50)

Consider the discrete norm for the sequence (vm)1≤m≤n:

[[(vm)]]n =

((n∏

i=1

(1− 2λδti)

)‖vn‖2 + σ2

2

n∑

m=1

δtm

(m−1∏

i=1

(1− 2λδti)

)|vm|2V

) 12

(1.51)

Multiplying equation (1.50) by∏n−1i=1 (1− 2λδti) and summing the equations on

n, we obtain[[(um)]]2n ≤ ‖u0‖2. (1.52)

We will need an equivalence relation between [[(um)]]n and [[uδt]](tn):

Lemma 1.2 There exists a positive real number α ≤ 12 such that the following

equivalence property holds for δt ≤ αλ and for any family (vn)0≤n≤N in V N+1

0 ,

1

8[[(vm)]]2n ≤ [[vδt]]

2(tn) ≤ max(2, 1 + ρδt)[[(vm)]]2n +

1

2σ2δt1|v0|2V . (1.53)

From (1.52) and (1.53), we deduce that for all n, 1 ≤ n ≤ N ,

[[uδt]](tn) ≤ c(u0) (1.54)

where

c(u0) =

(max(2, 1 + ρδt)‖u0‖2 +

1

2σ2δt1|u0|2V

) 12

(1.55)

To evaluate [[u− uδt]](tn) we make a further assumption on the coefficients:

Assumption 1.4 The function r is Lipschitz continuous in [0, T ]. The func-tions σ and S ∂σ∂S are Lipschitz continuous with respect to t uniformly in thevariable S.

With Assumption 1.4 and the previous set of assumptions there exists constantsL1, L2 and L3 such that, for all t and t′ in [0, T ],∥∥σ2(., t)− σ2(., t′)

∥∥L∞(0,S)

≤ L1|t′ − t|, |r(t) − r(t′)| ≤ L3|t′ − t| (1.56)

and a similar inequality for ‖− r(t)+ 12σ

2(., t)+Sσ(., t) ∂σ∂S (., t)‖L∞(0,S) with L2

on the right hand side.

Lemma 1.3 Assume that u0 ∈ V0h. Then, there exists a constant α ≤ 12

such that if δt ≤ αλ , the following a posteriori error estimate holds between the

solutions of (1.34) and (1.48):

[[u− uδt]](tn) ≤ cb

with b =

(L

σ2c(u0)δt+

µ

σ2(1 + ρδt)[[uδt − uh,δt]](tn) +

µ

σ2(

n∑

m=1

η2m)12

),

(1.57)


where

η2m = δtme−2λtm−1

σ2

2|umh − um−1

h |2V , (1.58)

and c is a positive constant, L = 4L1+2L2+L3 where L1, L2, L3 are given by(1.56), c(u0) is given by (1.55). Furthermore if the assumptions of Proposition1.3 are satisfied, the following a posteriori error estimate holds :

‖ ∂∂t

(u − uδt)‖L2(0,tn,V ′) ≤ c(µ+ σ2

σ)b. (1.59)

where µ is the continuity constant of a in (1.20).

1.3.3 Error Indicators for the Fully Discrete Problem

The fully discrete problem has already been defined in (1.37). For each timeinterval let (Tnh) be the mesh of Ω. Let h(n) denote the maximal size of theintervals in Tnh. For a given element ω ∈ Tnh, let hω be the diameter of ω andlet Smin(ω), Smax(ω) be the endpoints of ω. We assume that there exists aconstant ρh such that, for two adjacent elements ω and ω′ of (Tnh),

hω ≤ ρhhω′ . (1.60)

For each h, we define the discrete spaces by

Vnh =vh ∈ V, ∀ω ∈ Tnh, vh|ω ∈ P1

. (1.61)

In this study the grids Tnh for different values of n are not independent: indeed,each triangulation Tnh is derived from Tn−1,h by cutting some elements of Tn−1,h

into a limited number of smaller intervals or on the contrary by gluing togethera limited number of elements of Tn−1,h. This permits (wn−1

h , vnh) to be evaluatedexactly if wn−1

h ∈ Vn−1,h and vnh ∈ Vnh.Assuming that u0 ∈ V0h, the fully discrete problem reads:find (unh)1≤n≤N , u

nh ∈ Vnh satisfying

∀vh ∈ Vnh,(unh − un−1

h , vh)+ δtnatn(u

nh, vh) = 0, (1.62)

where u0h = u0. As above, for δt smaller than 1/(2λ), the existence and unique-ness of (unh)0≤n≤N is a consequence of the Lax-Milgram lemma, and we havethe stability estimate

[[(umh )]]n ≤ ‖u0‖. (1.63)

We call uh,δt the function which is affine on each interval [tn−1, tn], and suchthat uh,δt(tn) = unh.We wish to bound the error [[u − uh,δt]](tn), 1 ≤ n ≤ N ,by indicators com-putable from uh,δt. First let us separate the time discretization error from thespace discretization by the triangular inequality

[[u− uh,δt]](tn) ≤ [[u − uδt]](tn) + [[uδt − uh,δt]](tn).


Lemma 1.4 Assume that u0 ∈ V0h. Then the following a posteriori error esti-mate holds between the solution (un)1≤n≤N of problem (1.48) and the solution(unh)1≤n≤N of problem (1.62): there exists a constant c such that, for all tn,1 ≤ n ≤ N ,

[[(uδt − uh,δt)]]2(tn) ≤

c

σ2max(2, 1 + ρδt)

n∑

m=1

δtm

m−1∏

i=1

(1− 2λδti)∑

ω∈Tmh

η2m,ω,

(1.64)where

ηm,ω =

(hω

Smax(ω)‖u

mh − um−1

h

δtm− rS

∂umh∂S

+ rumh ‖L2(ω)

). (1.65)

Combining Lemmas 1.3 and 1.4 leads to the full a posteriori error estimate

Theorem 1.6 Assume that u0 ∈ V0h and that λδt ≤ α is as in Lemma 1.2.Then the following a posteriori error estimate holds between the solution u ofproblem (1.34) and the solution uh,δt of problem (1.62): there exists a constantc such that, for all tn, 1 ≤ n ≤ N ,

[[u− uh,δt]](tn)

≤c

L

σ2c(u0)δt

+µ

σ2

(n∑

m=1

η2m +δtmσ2

g(ρδt)

m−1∏

i=1

(1− 2λδti)∑

ω∈Tmh

η2m,ω

) 12

,

(1.66)

where L = 4L1 + 2L2 + L3, L1, L2, L3 are given by (1.56), c(u0) is given by(1.55), ηm is given by (1.58), and ηm,ω is given by (1.65), and

g(ρδt) = (1 + ρδt)2 max(2, 1 + ρδt).

Conclusion In (1.57), (1.59) and (1.64), we have bounded the norm of theerror produced by the finite element method by a Hilbert sum involving the errorindicators ηm and ηm,ω given in (1.58) and (1.65), which are respectively localin t and local in t and S. Conversely, in Propositions 1.7 and 1.8 below, we willsee that the error indicators can be bounded by local norms of the error. Thisshows that the error indicators are both reliable and efficient, or in other wordsthat the error produced by the method is well approximated by these indicators.Furthermore, since the indicators are local, they tell us where the mesh shouldbe refined. It is now possible to build a computer program which adapts themesh so as to reduce the error to a given number ǫ; from the result of an initialcomputation uh,δt, we can adapt separately the meshes in the variables t and Sso as to decrease the Hilbert sum in (1.66). The process may be repeated untilthe desired accuracy is obtained.


1.3.4 Upper Bounds for the Error Indicators

We now investigate the efficiency of the indicators in (1.58) and (1.65). Forthat, we introduce the notation [[vn]], for (vn)1≤n≤N , vn ∈ V :

[[vn]]2 =σ2

2δtn

n−1∏

i=1

(1− 2λδti)|vn|2V . (1.67)

Proposition 1.7 Assume that u0 belong to V , and that λδt ≤ α as in Lemma1.2. The following estimate holds for the indicator ηn, 2 ≤ n ≤ N ,

ηn ≤ c

[[un − unh]] +√ρδt[[u

n−1 − un−1h ]]

+e−λtn−1

σ(‖ ∂∂t

(u− uδt)‖L2(tn−1,tn;V ′) + ‖u− uδt‖L2(tn−1,tn;V ))

+( Lσ2 (max(1, ρδt))12 + λµ

σ2 )δtn||u0||

,

(1.68)and

η1 ≤ c

[[u1 − u1h]] +1

σ(‖ ∂∂t

(u − uδt)‖L2(0,t1;V ′) + ‖u− uδt‖L2(0,t1;V ))

+L+λµσ2 δt1‖u0‖+ L

σ (δt1)32 |u0|V

,

(1.69)where c is a positive constant.

The most important property of estimate (1.68) is that, up to the last termwhich depends on the data, all the terms in the right hand side of (1.68) arelocal in time. More precisely, they involve the solution in the interval [tn−1, tn].

We need to define a few notations before stating the upper bound result forηn,ω.For ω ∈ Tn,h, letKω be the union of ω and the element that shares a node with ω,and V0(Kω) be the closure of D(Kω) in V (Kω) = v ∈ L2(Kω);S

∂v∂S ∈ L2(Kω)

endowed with the norm ‖v‖V (Kω) = (∫Kω

v2(S)+S2( ∂v∂S (S))2)

12 . We also define

‖v‖V0(Kω) = (∫Kω

S2( ∂v∂S (S))2)

12 , for v ∈ V0(Kω). We denote by V ′

0(Kω) the

dual space of V0(Kω), endowed with dual norm. We also need the assumptionthat the meshes do not vary too much between two time steps:

Assumption 1.5 For n = 1, . . . , N , there exists a family of grids (T ∗n,h)h with

property (1.60) such that for all h and n each element of Tn,h and of Tn−1,h isthe union of at most s elements of T ∗

n,h, (where s is bounded independently of hand n).

Proposition 1.8 Under Assumption 1.5, the following estimate holds for theindicator ηn,ω defined in (1.65), for all ω ∈ Tn,h, 1 ≤ n ≤ N ,

ηn,ω ≤ C

(‖u

n−1 − un−1h − un + unhδtn

‖V ′0(Kω) + µ‖S∂(u

n − unh)

∂S‖L2(Kω)

).

(1.70)


1.3.5 Computation of the Bounds

To compute ρδt and ηn,ω we may do as follows %verbatimvoid errorindic(Option& p, double*

rho, double** etamw)

int nT = p.msh->nT;

double sigma_m=p.sigma[0][0];

for(int k=1;k<nT;k++)

for(int i=0;i<p.msh->nX[k];i++)

if(sigma_m>p.sigma[k][i]) sigma_m= p.sigma[k][i];

rho[k]=0;

double aux = 0.5*(p.msh->t[k]-p.msh->t[k-1])*sqr(sigma_m);

for(int i=1;i<p.msh->nX[k];i++)

rho[k] += aux*(

p.msh->x[k][i]*sqr((p.u[k][i]-p.u[k-1][i]-p.u[k][i-1]+p.u[k-1][i-1])

/(p.msh->x[k][i]-p.msh->x[k][i-1])) + sqr(p.u[k][i]-p.u[k-1][i]));

etamw[k][i] = fabs((p.u[k][i]-p.u[k-1][i])/(p.msh->t[k]-p.msh->t[k-1]))

*(p.msh->x[k][i]-p.msh->x[k][i-1])/p.msh->x[k][i];

% verbatimThe graphs below compare these two indicators with the true error.


0 50

100 150

200 250

300 0

0.1

0.2

0.3

0.4

0.5

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

"u.txt" using 1:2:5

0 50

100 150

200 250

300 0

0.1

0.2

0.3

0.4

0.5

0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008

"u.txt" using 1:2:6

0 50

100 150

200 250

300 0

0.1

0.2

0.3

0.4

0.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

"u.txt" using 1:2:7

Figure 1.2: The first graph displays the error between the computed solutionon a uniform mesh and one computed with Black-Scholes formula. The secondgraph show ρδt; the third graphs shows ηn,ω as a function of x and t. Theparameters are : K = 100, T = 0.5, r = 0, σ = 0.3, 50 time steps and 100 meshpoints.

Chapter 2

Multidimensional Partial

Differential Equations For

Option Pricing

2.1 European Basket Options

We consider d risky assets whose prices at time t are called Si,t, i = 1, . . . , d. Weassume that for all i, 1 ≤ i ≤ d, Si,t satisfies the stochastic differential equation

dSi,t = Si,t (µidt+ σidWi,t) . (2.1)

Here

• (Wi,t), 1 ≤ i ≤ d, are possibly correlated standard Brownian motions ona probability space (Ω,A,P). We call ρi,j the correlation factor of (Wi,t)and (Wj,t). We have −1 ≤ ρi,j ≤ 1. Of course ρi,i = 1.

• the volatilities σi, 1 ≤ i ≤ d, are positive constants.

The Black-Scholes model assumes the existence of a risk-free asset whose interestrate r may be supposed constant for simplicity, yet, it is possible to generalizewhat follows to the case when the volatilities are sufficiently regular functionsof t and of the prices Sj,t, j = 1, . . . , d and when r is a nonnegative function oft.In what follows, the notation S is used for the vector (S1, . . . , Sd). Considera European option on a basket containing the above mentioned d risky assets,whose maturity is T and whose payoff function is S 7→ P(S). As for an optionon a single asset, it is possible to find a risk neutral probability P∗ under whichthe price of the option at time t is

Pt = e−r(T−t)E∗(P(S1,T , . . . , Sd,T )|Ft). (2.2)

31

32 CHAPTER 2. MULTIDIMENSIONAL PDES FOR OPTION PRICING

The payoff functions actually used in finance may be quite complicated. Amongthe simplest ones, we can list

• the payoff is a function of a weighted sum of the assets’prices:

– call option on a weighted sum: P(S) =(∑d

i=1 αiSi −K)+

.

– put option on a weighted sum: P(S) =(K −∑d

i=1 αiSi

)+.

Calling Pt the price of the put option and Ct the price of the call option,the put-call parity Ct−Pt =

∑di=1 αiSi,t−Ke−r(T−t) can be proved either

by using (2.2) or by arguments on the partial differential equations usedfor pricing the options, see § 2.1.1.

• the payoff is a function of maxi=1,...,d Si: these options are called best-ofoptions.

– best-of call option: P(S) = (maxi=1,...,d Si −K)+.

– best-of put option: P(S) = (K −maxi=1,...,d Si)+.

In contrast with the previous case, there is not put-call parity for thesetwo options. Payoff functions depending on mini=1,...,d Si are used as well.

2.1.1 The Partial Differential Equation

Definition 2.1 Let Ω be an open subset of Rd. A function f : Ω × (0, T ) →R, continuous and such that its partial derivatives ∂f

∂t ,∂f∂Si

and ∂2f∂Si∂Sj

, i, j =

1, . . . , d, exist and are continuous on Ω × (0, T ) is said to belong to the classC2,1(Ω × (0, T )). If furthermore f and the above mentioned partial derivativeshave continuous extensions in Ω× [0, T ), it is said that f ∈ C2,1(Ω× [0, T )).

It is possible to relate the option’s price to the solution of a parabolic partial dif-ferential equation with 1+d variables. The partial differential operator appearsnaturally in the following result:

Proposition 2.1 Call L the partial differential operator with variable coeffi-cients:

Lf =1

2

d∑

i=1

d∑

j=1

Ξi,jSiSj∂2f

∂Si∂Sj+ r

d∑

i=1

Si∂f

∂Si, (2.3)

withΞi,j = σiσjρi,j . (2.4)

For any function u : (S, t) 7→ u(S, t), u ∈ C2,1(Rd+ × [0, T )) and such that

|Si ∂u∂Si| ≤ C(1 + |Si|), i = 1, . . . , d, with C independent of t, the process

Mt = e−rtu(S1,t . . . , Sd,t, t)−∫ t

0

e−rτ(∂u

∂t+ Lu− ru

)(S1,τ , . . . , Sd,τ , τ)dτ

2.1. EUROPEAN BASKET OPTIONS 33

is a martingale under P∗. The partial differential operator L is called the in-finitesimal generator of the Markov family (S1,t, . . . , Sd,t).

Proof. The proof uses the multidimensional Ito’s formula. See [69, 74]

Theorem 2.1 Consider a continuous function P ∈ C2,1(Rd+× [0, T )), such that

|Si ∂P∂Si| ≤ C(1 + Si) with C independent of t. Assume that P satisfies

(∂P

∂t+ LP − rP

)(S, t) = 0, t < T, S ∈ R

d+ (2.5)

andP (S, T ) = P(S), S ∈ R

d+, (2.6)

then the price of the European option given by (2.2) satisfies

Pt = P (S1,t . . . , Sd,t, t). (2.7)

Remark 2.1 All the preceding results can be generalized to the case when r is abounded and continuous function on [0, T ), and when the drifts and volatilitiesare functions of the variables S and t such that

1. For all i = 1, . . . , d, (S, t) 7→ µi(S, t) and (S, t) 7→ σi(S, t) are boundedand continuous functions.

2. For all i = 1, . . . , d, S 7→ µi(S, t)Si and S 7→ σi(S, t)Si are Lipschitzcontinuous with a Lipschitz constant C independent of t.

In this case, Ξi,j and r in (2.3) are functions instead of parameters: Ξi,j(S, t) =ρi,jσi(S, t)σj(S, t) and r = r(t).

A First Change of Variables It is possible to make the change of variablexi = log(Si), i = 1, . . . , d; calling x the vector (x1, . . . , xd) and P the function:

P (x, t) = P (S, t), the final value problem (2.5) (2.6) becomes(∂P

∂t+ LP − rP

)(x, t) = 0, t < T, x ∈ R

d,

P (x, T ) = P(ex1 , . . . , exd), x ∈ R

d,

(2.8)

where

Lf =1

2

d∑

i=1

d∑

j=1

Ξi,j∂2f

∂xi∂xj+

d∑

i=1

(r − σ2i

2)∂f

∂xi. (2.9)

One sees that when the volatilities and the interest rate are constant, the op-erator L has constant coefficients. In that case, calling v the vector of Rd suchthat vi = σ2

i /2− r and P (y, t) = ertP (y + tv, T − t), we have

∂P

∂t− 1

2

d∑

i=1

d∑

j=1

Ξi,j∂2P

∂yi∂yj= 0, 0 < t ≤ T, y ∈ R

d,

P (y, 0) = P(ey1 , . . . , eyd), y ∈ R

d,

(2.10)


We make the following assumption: there exists a unitary matrix Θ such that

ΘTΞΘ = A is a diagonal positive matrix:

A = Diag(a1, . . . , ad).(2.11)

Consider the change of variables

z =√2A− 1

2ΘTy,

then the function P (z, t) = P (y, t) satisfies the heat equation

∂P

∂t−∆P = 0, 0 < t ≤ T, z ∈ R

d. (2.12)

From this, calling G the fundamental solution of the heat equation,

G(z, t) = (4πt)−d2 e−

|z|24t ,

we have

er(T−t)P (S, t) =√√√√2d

d∏

i=1

ai

∫

Rd

G(√

2A− 12ΘT

(log(S)− z − (T − t)v

), T − t

)P(e

z1 , . . . , ezd)dz,

(2.13)

where log(S) = (log(S1), . . . , log(Sd))T .

Assume that P has a compact support contained in the rectangular domain[0, S)d, with S > 1, and that there exists a constant K such that 0 ≤ P ≤ K.The identity (2.13) gives information on the decay of P (S, t) when t ∈ [0, T ]and maxi=1,...,d Si tends to ∞:let us introduce

a = max(a1, . . . , ad), (2.14)

and let us assume that maxi=1,...,d Si ≥ S ≫ 1, where S is a positive number.Let z ∈ Rd be such that (ez1 , . . . , ezd) belongs to the support of P. Then−∞ < zi ≤ log(S), and

P (S, t) ≤ K

√√√√2dd∏

i=1

ai

d∏

i=1

∫ log(S)

−∞

e−2(log(Si)−zi−(T−t)vi)

2

4a(T−t)

(4π(T − t))12

dzi

≤ Kadd∏

i=1

∫ +∞

log(Si)−log(S)−(T−t)vi√a(T−t)

e−u2

2√2π

du

Let us assume that

log(S) > log(S) + T |v|∞. (2.15)


From the assumptions on S, there exists at least one index ℓ, 1 ≤ ℓ ≤ d, suchthat Sℓ ≥ S; therefore

P (S, t) ≤ Kad∫ +∞

log(S)+(t−t)vℓ−log(Sℓ)√a(T−t)

e−u2

2√2π

du

≤ Kad∫ +∞

log(S)−log(S)−T |v|∞√aT

e−u2

2√2π

du

≤ K

2ade−

(log(S/S)−T |v|∞)2

2aT .

(2.16)

Solutions of (2.5) (2.6) In the previous paragraph, we have seen that if thecoefficients are constant and if assumption (2.11) is satisfied, then (2.5) (2.6)has a solution given by (2.13).In this paragraph, we give a classical existence and uniqueness result for the finalvalue problem (2.5), (2.6) in the general case when r = r(t) and σi = σi(S, t),i = 1, . . . , d. It is necessary to restrict the growth of the solutions when Si → 0 orSi → +∞. Here, we will impose that the solution is bounded but this restrictioncan be relaxed (for example, depending on P, one can look for solutions withlinear growth as Si → +∞).

Theorem 2.2 Under the following regularity assumptions on the coefficientsand the final value:

1. for i, j = 0, . . . , d, the real valued function Ξi,j defined on Rd × [0, T ]

Ξi,j(x1, . . . , xd, t) = Ξi,j(ex1 , . . . , exd , t)

belongs to Cα,α/2(Rd × [0, T ]),

2. the function t 7→ r(t) belongs to Cα/2([0, T ]),

3. the function P defined on Rd by P(x1, . . . , xd) = P(ex1 , . . . , exd) is such

that P,∂P∂xi

, ∂2P∂xi∂xj

, i, j = 1, . . . , d belong to Cα(Rd),

and under the following ellipticity assumption:there exists a positive constant c such that, for all (x1, . . . , xd) ∈ Rd, t ∈ [0, T ],ξ ∈ R

d,d∑

i=1

d∑

j=1

Ξi,j(x1, . . . , xd, t)ξiξj ≥ c|ξ|2,

the final value problem (2.8) has a unique solution P such that the functions P ,∂P∂xi

, ∂P∂t ,∂2P

∂xi∂xj, i, j = 1, . . . , d, belong to Cα,α/2(Rd × [0, T ]).

Under the previous assumptions except the one on P and assuming that P isa bounded function, (2.8) has a unique solution P ∈ C2,1(Rd × [0, T )) such that

for all τ < T , the functions P , ∂P∂xi

, ∂P∂t ,∂2P

∂xi∂xjbelong to Cα,α/2(Rd × [0, τ ]).

This theorem is proved in [73], see also [45, 72].


A Second Change of Variables For basket options with a payoff dependingon the weighted sum

∑di=1 αiSi, the following change of variables has been

proposed by Reisinger[95, 97]:

y1 =

d∑

i=1

αiSi ∈ R+,

yi =αi−1Si−1∑dk=i−1 αkSk

∈ [0, 1], 2 ≤ i ≤ d.

(2.17)

This mapping is a C1 diffeomorphism from Rd+ onto R+× (0, 1)d−1. The inversechange of variables is

S1 = y1y2/α1,

Si = y1yi+1/αi

i∏

k=2

(1− yk), 2 ≤ i ≤ d− 1,

Sd = y1/αd

d∏

k=2

(1− yk).

(2.18)

After some calculus, one sees that

Si∂

∂Si= y1f1,i(y)

∂

∂y1−

d∑

j=2

yj(1− yj)fj,i(y)∂

∂yj+ yi+1(1− yi+1)

∂

∂yi+1, 1 ≤ i < d,

Sd∂

∂Sd= y1f1,d(y)

∂

∂y1−

d∑

j=2

yj(1− yj)fj,d(y)∂

∂yj,

(2.19)where

fj,i(y) = yi+1

∏ik=j+1(1− yk), j < i < d,

fj,d(y) =∏dk=j+1(1− yk), j < d,

fi,i(y) = yi+1, i < d,fd,d(y) = 1,fj,i(y) = 0, i < j.

(2.20)

The identitiesd∑

k=1

fi,k =

d∑

k=i

fi,k = 1, ∀i = 1, . . . , d, (2.21)

are true as well. One can verify that

d∑

i=1

Si∂

∂Si=

∂

∂y1. (2.22)

Playing with the identities (2.19), (2.20), (2.21) and (2.22), one obtains that inthe new variables y1, . . . , yd the partial differential equation (2.5) becomes(∂P

∂t+ LP − rP

)(y1, . . . , yd, t) = 0, t < T, (y1, . . . , yd) ∈ R+ × (0, 1)d−1,

(2.23)


where

Lf =1

2

d∑

i=1

d∑

j=1

Ξi,j(y)yiyj∂2f

∂yi∂yj+

d∑

i=1

βi(y)yi∂f

∂yi, (2.24)

with

Ξ1,1(y) =

d∑

k,l=1

Ξk,lf1,k(y)f1,l(y),

Ξ1,j(y) = Ξj,1(y) = (1− yj)

d∑

k,l=1

(Ξk,j−1 − Ξk,l)f1,k(y)fj,l(y), 1 < j ≤ d

Ξi,j(y) = (1− yi)(1− yj)

d∑

k,l=1

((Ξk,l + Ξi−1,j−1

−Ξi−1,l − Ξk,j−1

)fi,k(y)fj,l(y)

), 1 < i, j ≤ d,

(2.25)and

β1(y) = r,

βi(y) = (1− yi)

d∑

k,l=1

((2(1− yi)Ξk,l − 2yiΞi−1,i−1

+(2yi − 1)(Ξi−1,l + Ξk,i−1)

)fi,k(y)fi,l(y)

), 1 < i ≤ d,

(2.26)No boundary conditions are necessary on y1 = 0 and on yi = 0 or 1, 2 ≤ i ≤ d,because the partial differential equation is degenerate on these parts of thedomain’s boundary. The same remark can be made for the boundaries yi = 1,i = 2, . . . , d, because Ξi,j and Ξi,j vanish, for j = 1, . . . , d.The final condition becomes

P (y1, . . . , yd, T ) = P(y1).

For numerical computations (with d small enough), the main interest of sucha change of variables is the following: if the payoff function has a singularityon the hyperplane

∑di=1 αiSi = K, then one needs to refine the mesh in the

neighborhood of this hyperplane. This is incompatible with the use of Cartesiangrids in the S variables. After the change of variables S 7→ y, one can use aCartesian grid in the y1, . . . , yd variables, with a refinement in the neighborhoodof y1 = K. The price to pay is that the operator in the PDE has variablecoefficients with rather complicated expressions. Of course, a more general androbust alternative is to use finite elements with adaptive mesh refinement.

Consequences of the Maximum Principle The maximum principle maybe applied to (2.5). This has many consequences:the super-replication principle holds for basket options: take two European putoptions on the above mentioned basket with the same maturity and differentpayoff functions P and Q. Call P and Q their respective pricing functions,which both satisfy (2.5). We assume that P and Q have reasonable growth at

infinity, i.e., lim|S|→∞ max(P(S), Q(S))e−ǫ log2(|S|+2) = 0 for all ǫ > 0. This


is always satisfied in practice. Applying the maximum principle to (2.5), onesees that if P(S) ≤ Q(S) for all S, then for all t ≤ T and S, P (S, t) ≤ Q(S, t).For example, if Pt is the price of the best-of put option with maturity T andpayoff function (K −maxj=1,...,d Sj)

+, then for all i, 1 ≤ i ≤ d, and for all time

t < T , we haveQi,t ≥ Pt ≥ Qt, (2.27)

where Qt is the price of the put option on the weighted sum with the same

maturity T and payoff function(K −∑d

j=1 Sj

)+, and where Qi,t is the price of

the put option on the single asset indexed by i, with maturity and with payofffunction (K − Si)

+.

The maximum principle also gives information on the behavior of the pric-ing function at the boundary of Rd+ where the operator is degenerate. It can be

proved that, for a put option on a weighted sum, (i.e. P(S) =(K −∑d

i=1 αiSi

)+),

the pricing function satisfies

P (S1, . . . , Si−1, 0, Si+1, . . . , Sd, t) = Q(S1, . . . , Si−1, Si+1, . . . , Sd, t),

where Q is the pricing function of the put option on the basket containing allthe previous assets but the one indexed by i, with payoff function

Q(S1, . . . , Si−1, Si+1, . . . , Sd) =

K −d∑

j=1,j 6=iαjSj

+

.

The same is true for the best-of put option with payoff function (P(S) =(K −maxi=1,...,d Si)

+): we have

P (S1, . . . , Si−1, 0, Si+1, . . . , Sd, t) = Q(S1, . . . , Si−1, Si+1, . . . , Sd, t),

whereQ is the pricing function of the best-of put option on the basket containingall the previous assets but the one indexed by i, with payoff function

Q(S1, . . . , Si−1, Si+1, . . . , Sd) =

(K − d

maxj=1,j 6=i

Sj

)+

.

2.1.2 Variational Formulation

Here, we assume that the volatilities are functions of the variables S, t and thatr is a function of time.In order to obtain both global energy estimates and a nice framework for study-ing more complex situations (i.e. for American options on the same basket), weaim at finding a variational formulation for (2.5) (2.6), assuming that the payofffunction is square integrable. For more general payoff functions, variational for-mulation can be proposed as well with suitable weighting functions as |S| → ∞.Variational formulations are also the cornerstone for the finite element method,


see § 2.2.2 below.The change of variable T − t→ t yields a forward in time parabolic initial valueproblem:

(∂P∂t − LP + rP

)(S, t) = 0, 0 < t ≤ T, S ∈ Rd+,

P (S, 0) = P(S), S ∈ Rd+,(2.28)

where L is given by (2.3) with Ξi,j(S, t) = ρi,jσi(S, t)σj(S, t). Assuming thatthe coefficients are regular enough (this will be made clear below), we write theoperator in divergence form:

Lu =1

2

d∑

i=1

∂

∂Si

d∑

j=1

Ξi,jSiSj∂u

∂Sj

+

d∑

j=1

(r(t)Sj −

1

2

d∑

i=1

∂

∂Si(Ξi,jSiSj)

)∂u

∂Sj.

(2.29)Multiplying −Lu+ ru by a test function v, integrating on Rd+ and performingsuitable integrations by part, one obtains the bilinear form

at(u, v) =1

2

d∑

i=1

d∑

j=1

∫

Rd+

Ξi,jSiSj∂u

∂Sj

∂v

∂Si

−d∑

j=1

∫

Rd+

(r(t)Sj −

1

2

d∑

i=1

∂

∂Si(Ξi,jSiSj)

)∂u

∂Sjv + r(t)

∫

Rd+

uv.

(2.30)We introduce the Hilbert space

V =

v : v ∈ L2(Rd+), Si

∂v

∂Si∈ L2(Rd+), i = 1, . . . , d

, (2.31)

with the norm

‖v‖V =

(‖v‖2L2(Rd

+) +

d∑

i=1

‖Si∂v

∂Si‖2L2(Rd

+)

) 12

, (2.32)

one can check the following properties

• The space D(Rd+) of smooth and compactly supported functions in Rd+ isdense in V

• V is separable.

• The semi-norm |.|V defined by |v|2V =∑di=1 ‖Si ∂v∂Si

‖2L2(Rd

+)is in fact a

norm on V equivalent to ‖.‖V because ‖v‖L2(Rd+) ≤ 2|v|V .

We make the following assumptions

• The functions Ξi,j , 1 ≤ i, j ≤ d, are bounded by a positive constant σ2

independent of S and t:

‖Ξi,j‖L∞(Rd+)) ≤ σ2. (2.33)


• There exists a positive constant σ such that for all q ∈ Rd,

d∑

i=1

d∑

j=1

Ξi,jqiqj ≥ σ2|q|2. (2.34)

• There exists a positive constant M such that, for i = 1, . . . , d,

∥∥∥∥∥

d∑

i=1

∂

∂Si(Ξi,jSi)

∥∥∥∥∥L∞(Rd

+))

≤M. (2.35)

• The function r is nonnegative and bounded by a constant.

Lemma 2.1 With the assumptions above, the bilinear form at is continuous onV ×V and there exists a constant c independent of t such that, for any v, w ∈ V ,

at(v, w) ≤ c|v|V |w|V . (2.36)

Moreover, there is a uniform Garding’s inequality for at: there exist a positiveconstant c and a nonnegative constant λ such that, for any v ∈ V ,

at(v, v) ≥ c|v|2V − λ‖v‖2L2(Rd+). (2.37)

From Lemma 2.1, we deduce the existence and uniqueness of a weak solution tothe initial value problem (2.28).

Theorem 2.3 Under the assumptions above, for any P ∈ L2(Rd+), there exists

a unique P in L2(0, T ;V )∩C0([0, T ];L2(Rd+)), with∂P∂t ∈ L2(0, T ;V ′) such that,

for any smooth function φ ∈ D(0, T ), for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Rd+

P (t)v

)dt+

∫ T

0

φ(t)at(P, v)dt = 0 (2.38)

andP (t = 0) = P. (2.39)

The mapping P 7→ P is continuous from L2(Rd+) to L2(0, T ;V )∩C0([0, T ];L2(Rd+)),

and we have the energy estimate

e−2λt‖P (t)‖2L2(Rd+) + 2c

∫ t

0

e−2λτ |P (τ)|2V dτ ≤ ‖P‖2L2(Rd+). (2.40)

Naturally, there may be barrier options on baskets. For a barrier independentof time, pricing the option then amounts to solving the boundary value problem


)(S, t) = 0, 0 < t ≤ T, S ∈ Ω,

P (S, t) = 0, 0 < t ≤ T, S ∈ ∂Ω s.t. Si > 0, i = 1, . . . , d,P (S, 0) = P(S), S ∈ Ω,

(2.41)

2.2. NUMERICAL METHODS FOR EUROPEAN BASKET OPTIONS 41

for a domain Ω of Rd+. We restrict ourselves to domains whose boundaries arelocally the graph of Lipschitz continuous functions. Then, the Sobolev space towork with is the closure of D(Ω) in the space v ∈ L2(Ω); Si

∂v∂Si

∈ L2(Ω), i =1, . . . , d equipped with the norm

(‖v‖2L2(Ω) +

d∑

i=1

‖Si∂v

∂Si‖2L2(Ω)

) 12

.

2.2 Numerical Methods for European Basket Op-

tions

2.2.1 Localization

Consider the initial value problem (2.28) for pricing the European basket optiondiscussed in § 2.1. For computing a numerical approximation to P , one may

• truncate the domain in the variables S: the pricing function will be com-puted for Si ∈ (0, S), with S large enough

• impose some boundary condition at the artificial boundaries Si = S. Nat-urally, the choice of these boundary conditions has to depend on the payofffunction P.

We introduce the rectangular domain Ω = (0, S)d and Γ0 the part of the bound-ary of Ω given by the following

Γ0 = S ∈ ∂Ω; maxi=1,...,d

Si = S. (2.42)

Let us assume that P has a compact support contained in the rectangulardomain [0, S)d, and that there exists K ∈ R+ such that 0 ≤ P(S) ≤ Kfor S ∈ Rd+. This is the case for European put options on weighted sums or

for best-of European put options. We choose S such that S > S. From theestimate (2.16), a natural choice of boundary condition is P = 0 on Γ0. Thenew boundary value problem becomes


)(S, t) = 0, 0 < t ≤ T, S ∈ Ω,P (S) = 0, 0 < t ≤ T, S ∈ Γ0,

P (S, 0) = P(S), S ∈ Ω,(2.43)

with L given by (2.3).One can introduce a variational formulation for (2.43): the only modificationto bring to the content of § 2.1.2 is the choice of the space V , which must takeinto account the change in the domain and the new boundary conditions. Weintroduce the Hilbert space

V =

v : v ∈ L2(Ω), Si

∂v

∂Si∈ L2(Ω), i = 1, . . . , d

, (2.44)


with the norm ‖v‖V =(‖v‖2L2(Ω) +

∑di=1 ‖Si ∂v∂Si

‖2L2(Ω)

) 12

, and V as the com-

pletion in V of the space of smooth functions with compact support in Ω . Itcan be proved that there is a continuous trace operator from V to L2(Γ0): for

a function v ∈ V , we call v|Γ0 ∈ L2(Γ0) its trace on Γ0. The identity

V = v ∈ V ; v|Γ0 = 0 (2.45)

is a consequence of the continuity of the trace operator. Making the sameassumptions on the volatilities σi and on r as in § 2.1.2, we introduce thebilinear form at on V × V :

at(u, v) =1

2

d∑

i=1

d∑

j=1

∫

Ω

Ξi,jSiSj∂u

∂Sj

∂v

∂Si

−d∑

j=1

∫

Ω

(r(t)Sj −

1

2

d∑

i=1

∂

∂Si(Ξi,jSiSj)

)∂u

∂Sjv + r(t)

∫

Ω

uv.

(2.46)

It can be proved that at is continuous on V and that there is a Garding’sinequality in V as in (2.37). The variational formulation for (2.43) is to findP in L2(0, T ;V ) ∩ C0([0, T ];L2(Ω)) with ∂P

∂t ∈ L2(0, T ;V ′) such that, for anysmooth function φ ∈ D(0, T ), for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Ω

P (t)v

)dt+

∫ T

0

φ(t)at(P, v)dt = 0 (2.47)

and

P (t = 0) = P. (2.48)

This problem has a unique solution.

Localization error Of course, the artificial boundary conditions produce anerror because the solution Pexact of (2.28) does not vanish on Γ0.The maximum principle can be used for proving that

P (S, t) < Pexact(S, t), for S ∈ Ω, and 0 < t ≤ T.

It can also be proved, at least for the above mentioned two examples of options,that the maximum of Pexact − P in Ω× [0, T ] is reached on Γ0 × (0, T ].On the other hand, if the volatilities and the interest rate r are constant, wecall v the vector of Rd whose components are vi = σ2

i /2 − r, and we define aby (2.14), (2.11); if S satisfies (2.15), then Pexact satisfies (2.16) for all S ∈ Γ0.Thus

‖P − Pexact‖L∞(Ω×(0,T )) ≤ ‖Pexact‖L∞(Γ0×(0,T )) ≤K

2ade−

(log(S/S)−T |v|∞)2

2aT .

(2.49)


We see that the error between P and Pexact can be made arbitrarily small byletting S tend to infinity. In particular, the choice

S ≥ S exp

T |v|∞ +

√

2aT log(Kad

ǫ)

guarantees an error smaller than ǫ.For nonconstant coefficients, obtaining such an accurate result is not as easy.Yet, the formula above may be used for a reasonable choice of S.For compactly supported payoff functions, the Neumann boundary conditions

∂P

∂Si= 0, on

(Γ0 ∩ Si = S

)

can be used as well. Accurate error bounds may be obtained for put options ona weighted sum and for best-of put options if the coefficients are constant. Thevariational formulation with the Neumann boundary conditions is (2.47) (2.48)

but now, V = V .For a call option on the weighted sum

∑di=1 αiSi, the following conditions can

be used

• Dirichlet conditions: P (S, t) =∑d

i=1 αiSi −Ke−rt on Γ0 × (0, T ). With

V given by (2.45), the variational formulation is to find P in L2(0, T ; V )∩C0([0, T ];L2(Ω)) with ∂P

∂t ∈ L2(0, T ;V ′) such that, for any smooth func-tion φ ∈ D(0, T ), for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Ω

P (t)v

)dt+

∫ T

0

φ(t)at(P, v)dt = 0, (2.50)

P (t)|Γ0 =∑d

i=1 αiSi −Ke−rt for a.e. t, and (2.48).

• Neumann conditions:∑d

j=1 Ξi,jSj∂P∂Sj

=∑d

j=1 Ξi,jSjαj on(Γ0 ∩ Si = S

)×

(0, T ). The variational formulation is to find P in L2(0, T ; V )∩C0([0, T ];L2(Ω)),

with ∂P∂t ∈ L2(0, T ; V ′) such that, for any smooth function φ ∈ D(0, T ),

for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Ω

P (t)v

)dt+

∫ T

0

φ(t)at(P, v)dt

=S

2

∫ T

0

φ(t)

d∑

i=1

∫

Γ0∩Si=S

d∑

j=1

Ξi,jSjαj

v

dt.

(2.51)

and (2.48).

The error due to artificial boundary conditions can be accurately estimated inthe case of constant coefficients, by using the previously obtained estimates for


the corresponding put options and the put-call parity.

For a best-of call option, finding reasonable boundary condition near the regionsSi = Sj = S, i 6= j is much more difficult. One may have to use an alterna-tive option to artificial boundary conditions, i.e., a change of variables whichmaps the unbounded domain to a bounded one: one obtains a new boundaryvalue problem in a bounded domain. The partial differential equation becomesdegenerate on the part of the boundary which is sent to infinity by the inversemapping, thus no boundary condition is needed there. An example of such aprogram is given in § 2.4.2 below in the context of option pricing with stochasticvolatility.

2.2.2 Finite Element Methods

Conforming FEM (Finite Element Methods) are numerical approximations closelylinked to the theory of variational or weak formulations presented in § 2.1.2. Thefirst finite element method can be attributed to R. Courant, [34].Conforming finite element methods have the same framework in any dimensionof space d: for a weak formulation posed in an infinite dimensional functionspace V , for example (2.51) (2.48), it consists of choosing a finite dimensionalsubspace Vh of V , for instance the space of continuous piecewise affine functionson a triangulation of Ω, and of solving the problem with test and trial functionsin Vh instead of V . We speak of conforming methods because Vh ⊂ V . Non-conforming methods, i.e., Vh 6⊂ V are possible too but we will not consider thistopic here. In the simpler finite element methods, the construction of the spaceVh is done as follows:

• The domain is partitioned into nonoverlapping cells (elements) whoseshapes are simple and fixed: for example, intervals in one dimension, trian-gles or quadrilaterals in two dimensions, tetrahedra, prisms or hexahedrain three dimensions. The set of the elements is in general an unstructuredmesh called a triangulation.

• The maximal degree k of the polynomial approximation in the elementsis chosen.

• Vh is made of continuous functions of V whose restriction to the elementsare polynomial of degree less than k.

Programming the method is also somewhat similar in any dimension, but meshgeneration is very much dimension dependent. A nice survey on the finite ele-ment method, both on the theoretical and practical viewpoints, is proposed in[41].There is a very well understood theory on error estimates for finite elements.It is possible to distinguish a priori and a posteriori error estimates: in a prioriestimates, the error is bounded by some quantity depending on the solution ofthe continuous problem (which is unknown, but for which estimates are avail-able), whereas, in a posteriori estimates, the error is bounded by some quantity


depending on the solution of the discrete problem, which is available.For a priori error estimates, one can see the books of Raviart and Thomas [93],Strang and Fix [104], Braess [19], Brenner and Scott [23], Ciarlet [28, 27], andThomee [105] for parabolic problems. By and large, deriving error estimates forfinite elements method consists of

1. establishing the stability of the discretization with respect to some normsrelated to ‖.‖V .

2. Once this is done, one sees (at least in simple cases) that the error dependson some distance of the solution of the continuous problem to the space Vh.This quantity cannot be computed exactly since the solution is unknown.However, it can be estimated from a priori knowledge on the regularity ofthe solution.

When sharp results on the solution of the continuous problem are available,the a priori estimates give very valuable information on how to choose the dis-cretization a priori, see the nice papers by C. Schwab et al [102, 110], in thecase of homogeneous parabolic problems with smooth coefficients.A posteriori error estimates are a precious tool since they give practical in-

formation that can be used to refine the mesh when needed. The bibliographyon a posteriori error estimates for finite element methods is quite rich: onecan see the book of Verfurth [107] and the references therein. For time depen-dent problems, a posteriori error estimates and mesh adaption for space-timefinite element problems have been investigate by Eriksson et al [38, 39, 40].Another strategy based on decoupled space and time error indicators can beimplemented, see [16] and §1.3 for an example with one space variable.When the space variable is multidimensional, very anisotropic meshes may beuseful. A trend in mesh adaptivity consists of building anisotropic meshes byimposing regularity and quasi-uniformity with respect to a new metric con-structed from the a posteriori error estimates, see [47]. Below, we show someexamples of anisotropic meshes generated with the open source software BAMGof F. Hecht et al.

Example: the Case of a Put Option on a Basket of Two Assets Inthe sequel, we deal with a simple implementation of the finite element methodfor approximating the pricing function of an option on a basket containing twoassets. Therefore d = 2.

The Time Semi-Discrete Problem We introduce a partition of the interval[0, T ] into subintervals [tm−1, tm], 1 ≤ m ≤ M , such that 0 = t0 < t1 < · · · <tm = T . We denote by δtm the length tm − tm−1, and by δt the maximum ofthe δtm, 1 ≤ m ≤M .For simplicity, we assume that P ∈ V , where V is given by (2.45) with d = 2.We discretize (2.47) by means of an implicit Euler scheme, i.e. we look for


Pm ∈ V , m = 0, . . . ,M , such that P 0 = P, and for m = 1, . . . ,M , ∀v ∈ V ,

1

δtm(Pm − Pm−1, v)L2(Ω) + atm(Pm, v) = 0, (2.52)

where atm is given by (2.46). This scheme is first order.

Remark 2.2 If P does not belong to V , then we first have to approximate Pby a function in V , at the cost of an additional error.

The Full Discretization: Lagrange Finite Elements Discretization withrespect to S1, S2 consists of replacing V with a finite dimensional subspaceVh ⊂ V . For example, one may choose Vh as a space of continuous piecewisepolynomial functions on a triangulation of Ω: for a positive real number h,consider a partition Th of Ω into nonoverlapping closed triangles, (Th is the setof all the triangles forming the partition) such that

• Ω = ∪K∈ThK.

• For all K 6= K ′ two triangles of Th, K∩K ′ is either empty, either a vertexof both K and K ′, or a whole edge of both K and K ′.

Remark 2.3 If Ω is not polygonal but has a smooth boundary, it is possible tofind a set Th of nonoverlapping triangles of diameters less than h such that thedistance between Ω and ∪K∈Th

K scales like h2.

For a positive integer k, we introduce the spaces

Wh = wh ∈ C0(Ω) : wh|K ∈ Pk, ∀K ∈ Th, Vh = vh ∈Wh, vh|Γ0 = 0.(2.53)

Hereafter, we focus on the case when k = 1, i.e. the functions in Wh arepiecewise affine. It is clear that Vh is a finite dimensional subspace of V .Assuming that P ∈ Vh, the full discretization of the variational formulationconsists of finding Pmh ∈ Vh, m = 0, . . . ,M , such that P 0

h = P, and

∀vh ∈ Vh,1

δtm(Pmh − Pm−1

h , vh)L2(Ω) + atm(Pmh , vh) = 0. (2.54)

Here, for simplicity, we assume that atm(uh, vh) can be computed algebraicallyfor uh, vh ∈ Vh, which is the case when the volatilities do not depend on S1, S2,for example. If this is not the case, then quadrature formulas have to be usedwhich induces an additional but controlled source of error.

The Discrete Problem in Matrix Form A basis of Vh is chosen, (wi)i=1,...,N .Then, for 1, . . . ,M , umh can be written

umh (S1, S2) =

N∑

1

umj wj(S1, S2), (2.55)


and, using (2.55) in (2.54) with vh = wi, we obtain a system of linear equationsfor Um = (umj )Tj=1,...,N :

M (Um −Um−1) + δtmAmUm = 0, (2.56)

where M and A are matrices in RN×N : assuming that the volatilities do notdepend on S1, S2,

M ij =

∫

Ω

wiwj

Ami,j = a(wj , wi) =

1

2

2∑

ℓ=1

2∑

k=1

∫

Ω

Ξℓ,k(tm)SℓSk∂wj∂Sk

∂wi∂Sℓ

−2∑

k=1

∫

Ω

(r(tm)Sk −

1

2

2∑

ℓ=1

∂

∂Sℓ(Ξℓ,k(tm)SℓSk)

)∂wj∂Sk

wi + r(tm)

∫

Ω

wjwi.

(2.57)

The matrix M is called the mass matrix and Am is called the stiffness matrix.It can be proved thanks to estimates (2.36) (2.37) that if δt is small enough,

then M + δtmAm is invertible, and it is possible to solve (2.54).

The Nodal Basis On each triangle K ∈ Th, noting by qi, i = 1, 2, 3 thevertices of K, we define for S ∈ R

2 the barycentric coordinates of S, i.e. thesolution of ∑

i=1,2,3

λKi (S)qi = S,∑

i=1,2,3

λKi (S) = 1.

This 3× 3 system of linear equations is never singular because its determinantis twice the area of K. It is obvious that the barycentric coordinates λKi areaffine functions of S. Furthermore,

• when S ∈ K, λKi ≥ 0, i = 1, 2, 3,

• if K = [qi1 , qi2 , qi3 ] and S is aligned with qi1 , qi2 then λKi3 = 0.

Let vh be a function in Vh: it is easy to check that, on each triangle K ∈ Th,

vh(S) =∑

j=1,2,3

vh(qij )λKij (S) ∀S ∈ K.

Therefore, a function in Vh is uniquely defined by its values at the nodes of Thnot located on Γ0.Call (qi)i=1,...,N the nodes of Th not located on Γ0, and let wi be the uniquefunction in Vh such that wi(qj) = δi,j , ∀j = 1, . . . , N . For a triangle K suchthat qi is a vertex of K, it is clear that wi coincides in K with one of the threebarycentric coordinates attached to triangle K. Therefore, we have the identity

vh =

N∑

i=1

vh(qi)wi, (2.58)


which shows that (wi)i=1,...,N is a basis of Vh. As shown in Figure 2.1, thesupport of wi is the union of the triangles of Th containing the node qi, so it isvery small when the mesh is fine, and the support of two basis functions wi andwj intersect if and only if qi and qj are the vertices of a same triangle of Th.Therefore, the matrices M and Am constructed with this basis are sparse. Thisdramatically reduces the complexity when solving properly (2.56). The basis(wi)i=1,...,N is often called the nodal basis of Vh. The shape functions wi aresometimes called hat functions. For vh ∈ Vh, the values vi = vh(q

i) are calledthe degrees of freedom of vh. If K = [qi1 , qi2 , qi3 ], and if bi1 is the point aligned

Sj

Figure 2.1: The shape function wj

with qi2 and qi3 and such that ~bi1qi1 ⊥ ~qi2qi3 , then

∇λKi1 =1

| ~bi1qi1 |2~bi1qi1 , (2.59)

and calling ni1 the unit vector orthogonal to ~qi2qi3 and pointing to qi1 , i.e.

ni1 = 1

| ~bi1qi1 |~bi1qi1 and Ei1 the length of the edge of K opposite to qi1 , and

using the well known identity Ei1 | ~bi1qi1 | = 2|K|, we obtain

∇λKi1 =Ei1

2|K|ni1 . (2.60)

The following integration formula is very important for the numerical imple-mentation of the finite element method:

Proposition 2.2 Calling λi i = 1, 2, 3 the barycentric coordinates of the tri-angle K, and ν1, ν2, ν3 three nonnegative integers and |K| the measure of K,

∫

K

(λK1 )ν1(λK2 )ν2 (λK3 )ν3 = 2|K| ν1!ν2!ν3!

(ν1 + ν2 + ν3 + 2)!. (2.61)

Remark 2.4 It may be useful to use other bases than the nodal basis, for ex-ample bases related to wavelet decompositions, in particular for speeding up thesolution of (2.56), see [79, 109].

Remark 2.5 The integral of a quadratic function on a triangle K is one thirdthe sum of the values of the function on the mid-edges times |K|, therefore (2.61)


is simpler when ν1 + ν2 + ν3 = 2:∫

K

λKi λKj =

|K|12

(1 + δij). (2.62)

When the system (2.56) becomes large, iterative methods such as gradient meth-ods, GMRES or BICG-stab become attractive. We refer to [12, 49, 100, 50, 82]for good books on this topic . Iterative methods do not need the matrixM + δtmAm but only a function which implements U → (M + δtmAm)U ,i.e. which computes

∑

j

uj((wj , wi)L2(Ω) + δtmatm(wj , wi)

).

Let us show how AmU should be computed, (we take AmU instead of (M +δtmAm)U only for simplicity). We use the fact that

AmU =∑

K

Am,KU ,

where Am,KU is the vector whose entries are∑

j ujaKtm(wj , wi), i = 1, . . . , N

and where

aKt (u, v) =1

2

2∑

ℓ=1

2∑

k=1

∫

K

Ξℓ,k(t)SℓSk∂u

∂Sk

∂v

∂Sℓ

−2∑

k=1

∫

K

(r(t)Sk −

1

2

2∑

ℓ=1

∂

∂Sℓ(Ξℓ,k(t)SℓSk)

)∂u

∂Skv + r(t)

∫

K

uv.

(2.63)Hence

(Am,KU)i =∑

j

uj∑

K

Am,Kij . (2.64)

For simplicity only, let us only consider the first term in (2.63), so aKt becomes

aKt (u, v) =1

2

2∑

ℓ=1

2∑

k=1

∫

K

Ξℓ,k(t)SℓSk∂u

∂Sk

∂v

∂Sℓ,

and

Am,Kij =

1

2

2∑

ℓ=1

2∑

k=1

∫

K

Ξℓ,k(tm)SℓSk∂wj

∂Sk

∂wi

∂Sℓ.

But ∇wi,∇wj are constant on K, and Sk =∑3ν=1 Sk,νλ

Kν , so from (2.62),

Am,Ki,j =

1

2

2∑

k,ℓ=1

Ξℓ,k(tm)∂wi

∂Sℓ

∂wj

∂Sk

3∑

ν1=1

3∑

ν2=1

Sℓ,ν1Sk,ν2

∫

K

λKν1λKν2

=|K|24

2∑

k,ℓ=1

Ξℓ,k(tm)∂wi

∂Sℓ

∂wj

∂Sk

3∑

ν1=1

3∑

ν2=1

Sℓ,ν1Sk,ν2(1 + δν1ν2).

(2.65)


The summation (2.64) should not be programmed directly like

for i = 1..Nfor j = 1..Nfor K ∈ Th

(AmU)i + = Am,Kij uj , (2.66)

because the numerical complexity of this loop is of the order of N2NT , whereNT is the number of triangles in Th. One should rather notice that the sumscommute, i.e.

for K ∈ Thfor j = 1..Nfor i = 1..N

(AmU)i + = Am,Kij uj , (2.67)

and then see that Am,Kij is zero when qi or qj are not in K. The loop

for K ∈ Thfor jloc = 1, 2, 3for iloc = 1, 2, 3

(AmU)iiloc + = Am,Kiilocijloc

uijloc (2.68)

has a complexity of the order of O(NT ). This technique is called assembling.It has brought up the fact that vertices of triangle K have global indices (theirposition in the array that store them) and local indices, their position in thetriangle K i.e. 1,2 or 3. The notation iiloc refers to the map from local toglobal.The convergence of iterative methods for solving linear systems depends onthe spectral properties of the matrix: for example, if the matrix is symmetricand positive definite, the convergence rate of the conjugate gradient methoddepends on the condition number of the matrix; for a general matrix, the speedof convergence of the GMRES method depends on the numerical range of thematrix. For the linear systems arising from the discretization of parabolic PDE,it is observed that the convergence deteriorates when the size of the systemsincreases. Therefore, when solving for example the linear system (2.56), onehas better solve instead

B−1(M + δtmAm)Um = B−1MUm−1, (2.69)

where B is a matrix such that

• the spectral properties of B−1(M + δtmAm) are better than those ofM + δtmAm. This means that B is in some sense close to M + δtmAm.

• the solution of a linear system of the form BV = G can be achieved at areasonable computational cost.


Such a matrix B is called a preconditioner for (2.56), and the iterative methodapplied to (2.69) is called a preconditioned iterative method . The constructionof good preconditioners is an important topic in numerical analysis. We againrefer to [12, 49, 100, 50, 82].

Remark 2.6 (Mass Lumping for piecewise linear triangular elements)Let f be a smooth function and consider the following approximation for the in-tegral of f over Ω = ∪K∈Th

K, where Th is a triangulation of Ω:

∫

Ω

f =∑

K∈Th

∫

K

f ≈∑

K∈Th

|K|3

3∑

i=1

f(qKi ),

where qK1 , qK2 , q

K3 are the 3 vertices of K. If f is affine this formula is exact,

otherwise it computes the integral with an error O(h2).This approximation is called mass lumping: for two function uh, vh ∈ Vh, wecall U and V the vectors of their coordinates in the nodal basis: applying masslumping, one approximates

∫Ω uhvh by UTMV , where M is a diagonal matrix

with positive diagonal entries.

Results The discrete method discussed above has been applied to computethe pricing function of a best-of put option on a two assets basket, P(S1, S2) =(100−max(S1, S2))

+. The artificial boundary Γ0 is max(S1, S2) = S = 200.Homogeneous Dirichlet conditions have been imposed on Γ0. Such a choice ofS may not be enough for a good accuracy; in fact S = 200 was chosen to obtainfigures with nice proportions.The parameters of the Black-Scholes model are

σ1 = 0.2, σ2 = 0.1, r = 0.05.

The correlation factor is either of −0.3 (Figure 2.2) or of −0.9 (Figure 2.3).The first order implicit Euler scheme has been used with a uniform time stepof 1/250 year. Mesh adaption in the (S1, S2) variable has been performed every1/10 year. For mesh adaption, we have used the software BAMG, see [47].In Figure 2.2, the adapted mesh and the contours of the pricing function areplotted 0.2 year to maturity (top) and one year to maturity (bottom). Themesh is refined near the lines where the payoff function exhibits singularities.As time to maturity grows, the mesh becomes coarser in these regions. In fact,such a large number of mesh adaptions are not necessary.It is clearly seen that the pricing function diffuses more in the S1 variable, whichis not surprising, because the volatility of the first asset is higher.

Multigrid Methods

Geometric multigrid method can be applied if there is a hierarchy of nestedmeshes Thi , i = 1, . . . , q, in such a way that the corresponding finite elementspaces satisfy V1 ⊂ . . . Vi ⊂ Vi+1 ⊂ . . . Vq . The dimensions of these spaces


Figure 2.2: The adapted mesh and the contours of P , at the times to maturity0.2 year (top) and 1 year (bottom). σ1 = 0.2, σ2 = 0.1, ρ = −0.3.


Figure 2.3: The contours of P , 1 year to maturity. σ1 = 0.2, σ2 = 0.1, ρ = −0.9.

are N1 < · · · < Ni < Ni+1 < · · · < Nq. The heuristics supporting multigridmethods for elliptic problem is as follows: the first observation is that withcommon iterative solvers like Jacobi or SOR, see [82, 24], the components ofthe error corresponding to high frequencies are usually decreased much fasterthan those associated with the lower frequencies. This also explains why theconvergence rate of such methods deteriorates as the number of unknowns grows.To summarize, the iterative solver makes the error smooth and the smooth partof the error has a slow decay. An iterative solver with this property is calleda smoother. The second observation is that for a given function f ∈ Vi+1, itsprojection on Vi will be decreased faster by the smoother at level i (operating onVi) than by the smoother at level i+ 1 (operating on Vi+1), because it appearsless smooth to the first operator. From these observations, an efficient procedurecan be designed by combining the iterations of the smoother with coarse levelcorrections. If this idea is also applied to the coarse level correction, the resultis a recursive algorithm.Assume that a Galerkin method is applied for approximating the solution ofa boundary value problem by a function in Vq: the system of linear equation

readsA(q)u(q) = f (q). Note that it is also possible to define the similar Galerkindiscretizations at the lower levels: using the nodal basis, the correspondingsystem reads

A(i)u(i) = f (i), 1 ≤ i ≤ q. (2.70)

For simplicity, assume that A(i) are symmetric and positive definite. Denoteby S(i) the smoother at level i: the vector obtained by performing ν iterationsof the smoother at level i for solving (2.70) starting from the initial guess w is

written S(q)(f (i),w, ν). An ingredient of the method is the canonical injection


from V i to V i+1: let Ii+1i stand for its matrix in the nodal bases. Another

ingredient is the restriction operator from Vi+1 to Vi, whose matrix is Iii+1: apossible choice is to take the Galerkin projection, i.e.,

(A(i)Iii+1u,w) = (A(i+1)u, Ii+1i w), ∀u ∈ R

Ni+1 , ∀w ∈ RNi . (2.71)

We denote by MG(f (i),w, i) one iteration of the multigrid method at level ifor solving (2.70) starting from w.One of the most commonly used multigrid algorithm is the V-cycle:

One V-cycle : MG(f (i),w, i) → w

If i = 1, solve the system (2.70) with a direct method, and let w be the solution.Else

1. Perform ν1 iterations of the smoother at level i: S(i)(f (i),w, ν1) → w.

2. Compute the residual r ∈ RNi−1 on level i− 1 by

(r, z) = (f (i) −A(i)w, Iii−1z), ∀z ∈ RNi−1 .

Note that r can be expressed in terms of A(i−1)Ii−1i w and of the projec-

tion of f i.

3. Apply the multigrid method at level i− 1: MG(r, 0, i− 1) → w.

4. Add the coarse level correction to w: w + Iii−1w → w.

5. Perform another ν2 iterations of the smoother at level i: S(i)(f (i),w, ν2) →w.

The iterative method consists of computing the sequencewn+1 =MG(f (q),wn, q)until the residual norm becomes smaller than some tolerance ǫ. Under some rea-sonable assumptions on the elliptic equation, on the mesh and on the smoother,see e.g. [20, 111], it can be proved that the norm ‖uq −wn‖A decays like ρn

where ρ < 1 does not depend on the mesh parameters.A very nice introduction to multigrid methods is given in [24]. Multigrid meth-ods can also be used in the construction of preconditioners, see [21, 111]. Finally,the ideas above have been generalized in the so-called algebraic multigrid meth-ods, when there is no hierarchy of grids, see [99]. Algebraic multigrid methodsare among the most robust and efficient for solving the linear systems arisingfrom the discretization of elliptic and parabolic PDEs. Open source libraries areavailable, like the library hypre, see http://www.llnl.gov/CASC/linear solvers/.

2.2.3 Sparse Methods

Consider a boundary value problem in the hypercube Ω = (0, 1)d. One canthink of a Poisson problem ∆u = −f with the Dirichlet boundary condi-tions u = 0 on ∂Ω. For the variational formulation, we need to use the


space H1(Ω), equipped with the norm ‖v‖H1(Ω) =√‖v‖2L2(Ω) + |v|2H1(Ω) where

|v|2H1(Ω) =∑d

i=1 ‖ ∂v∂xi‖2L2(Ω), and H1

0 (Ω), the completion in H1(Ω) of the sub-space of smooth functions compactly supported in Ω. The previous ellipticproblem has a weak or variational formulation in H1

0 (Ω): find u ∈ H10 (Ω) such

that∫Ω∇u · ∇v =

∫ωfv, for all v ∈ H1

0 (Ω).Assume that the solution of the Poisson problem is approximated by a con-

forming multilinear finite element method on a Cartesian mesh, more preciselywith piecewise linear functions of total degree ≤ d. This is the lowest orderfinite element method on this mesh. Assume that the mesh is uniform and thateach element is a cube of size n−1. It is easy to see that the dimension of theapproximation space is of the order of nd: the algorithmic complexity growsexponentially with d, which actually forbids the use of this method for d > 4.This too rapid growth in complexity is known as the curse of dimensionality.Yet, quite recent developments have shown that it may be possible to use de-terministic Galerkin methods or grid based methods for elliptic or parabolicproblems in dimension d, for 4 ≤ d ≤ 20: these methods are based either onsparse grids [112, 53, 51] or on sparse tensor product approximation spaces[52, 109].In this paragraph, we aim at rapidly describing the principle of sparse approx-imations. This presentation heavily relies on the review article by H.J. Bun-gartz and M.Griebel [25]. We concentrate on the previously mentioned Dirich-let boundary value problem in Ω. The solution u will be approximated by aGalerkin method, i.e., a variational problem posed in a finite dimensional ap-proximation space Vn instead ofH1

0 (Ω). The goal is to use approximation spacesVn whose dimensions do not grow too rapidly with d.The results below are proved in [25].

Notations and Preliminary Results In this section, bold letters will standfor d-uples: for example, x = (x1, . . . , xd) and α = (α1, . . . , αd). We set 1 =(1, . . . , 1) ∈ Rd and 0 = (0, . . . , 0) ∈ Rd. Take a sufficiently smooth function fdefined on [0, 1]d; if α ∈ N

d, we call Dαf the partial derivative

Dαf =∂|α|f

∂xα11 . . . ∂xαd

d

,

where |α| =∑di=1 αi. For two multi-indices α and β and a scalar λ, we define

α · β =

d∑

i=1

αiβi, λα = (λα1, . . . , λαd), 2α = (2α1 , . . . , 2αd).

We say that α ≤ β if αi ≤ βi, i = 1, . . . , d, and that α < β if α ≤ β andα 6= β.Let us introduce the function spaces Xq,r(Ω), for r ∈ N and q ∈ [1,+∞]:

Xq,r(Ω) = u ∈ Lq(Ω), ∀α s.t. α ≤ r1, Dαu ∈ Lq(Ω), (2.72)


which are endowed with the semi-norms:

|u|q,α =

(∫

Ω

|Dαu|q) 1

q

, α ≤ r1, if q <∞,

|u|∞,α = ‖Dαu‖L∞(Ω), α ≤ r1, if q = ∞.

Note that Xq,r(Ω) is imbedded in the more usual Sobolev space W q,r(Ω) =u ∈ Lq(Ω), ∀α s.t. |α| ≤ r, Dαu ∈ Lq(Ω).For a multi-index ℓ, consider the Cartesian meshes Tℓ of Ω with mesh stepshℓ = 2−ℓ = (2−ℓ1 , . . . , 2−ℓd). The grid nodes of Tℓ are the points xi = i · hℓ,0 ≤ i ≤ 2ℓ.We note by φ the mother hat function:

φ(x) =

1− |x| if |x| < 1,0 if |x| ≥ 1,

and φℓ,i the d-dimensional hat function:

φℓ,i(x) =

d∏

k=1

φ(2ℓkxk − ik). (2.73)

We call VℓVℓ = span

(φℓ,i, 1 ≤ i ≤ 2ℓ − 1

)(2.74)

We also consider the wavelet subspaces:

Wk = spanφk,i,1 ≤ i ≤ 2k − 1, ij odd , 1 ≤ j ≤ d

. (2.75)

We haveVℓ =

⊕

1≤k≤ℓ

Wk.

The basis of Vℓ obtained by assembling the previously mentioned bases of Wk

1 ≤ k ≤ ℓ is called the hierarchical basis of Vℓ. Calling Iℓ = i ≤ 2ℓ − 1 :ij odd , 1 ≤ j ≤ d, the hierarchical basis of Vℓ is φk,i, i ∈ Ik, k ≤ ℓ. Notethat the completion of

⊕1≤kWk with respect to the H1(Ω) norm is exactly

H10 (Ω).

Rescaling the φk,i as follows

ψk,i = −2−(k+1)·1φk,i, i ∈ Ik, (2.76)

we obtain another basis of Wk.If a function u is smooth enough, then the coefficients of its expansion in thehierarchical basis are obtained by a simple integral formula:

Lemma 2.2 If u ∈ H10 (Ω) ∩X1,2(Ω), then

u =∑

k≥1

∑

i∈Ik

uk,iφk,i, where uk,i =

∫

Ω

D2u · ψk,i. (2.77)


By using Lemma 2.2, one may evaluate the contribution uk of a subspace Wk

to the hierarchical expansion of u:

Lemma 2.3 If u ∈ H10 (Ω) ∩ X2,2(Ω), then the component uk ∈ Wk of the

expansion of u in the hierarchical representation is such that

‖uk‖L2(Ω) ≤ 2−2|k|3−d|u|2,2,

|uk|H1(Ω) ≤ 2−2|k|3−d+12

d∑

j=1

22kj

12

|u|2,2.(2.78)

Sparse Galerkin Methods It is clear that the dimension of Vℓ is∏dj=1(2

ℓj −1). In particular, dim(Vn1) = (2n − 1)d. As already mentioned, the full tensorproduct space Vn1 is often too large for practical use when d > 4.Let us give an example of a sparse Galerkin method: the discrete space is chosento be

Vn =⊕

1≤k,|k|≤n+d−1

Wk (2.79)

instead of the full tensor product space Vn1 =⊕

1≤k≤n1Wk. One may provethat

dim(Vn) = 2n(

nd−1

(d− 1)!+O(nd−2)

). (2.80)

Therefore dim(Vn) is much smaller than dim(Vn1). It can be seen that a Galerkinmethod with Vn is feasible for d of the order of 10. On Figure 2.4, we displaythe bases of Vn1 and Vn.Consider the discretization of the Dirichlet problem in Ω: the discretizationerror of the Galerkin method with the approximation space Vn (resp. Vn1)is of the same order as the best fit error when approximating the solution ofthe continuous problem by a function of Vn (resp. Vn1). Let us assume thatu is smooth. We know that infv∈Vn1

‖v − u‖H1(Ω) ≤ C2−n|u|W 2,2(Ω), where|u|2W 2,2(Ω) =

∑|α|=2 ‖Dαu‖2L2(Ω). Since Vn is much smaller than Vn1, a similar

estimate is not true for infv∈Vn ‖v − u‖H1(Ω). Griebel et al have proved thefollowing theorem:

Theorem 2.4 If u ∈ H10 (Ω)∩X2,2(Ω), and if un ∈ Vn is the component of the

expansion of u in the hierarchical representation,

‖u− un‖L2(Ω) ≤(2−2n+1

12d

d−1∑

k=0

(n+ d− 1

k

))|u|2,2 = O(2−2nnd−1)|u|2,2,(2.81)

|u− un|H1(Ω) ≤(

2−nd√3 6d−1

)|u|2,2 = O(2−n)|u|2,2. (2.82)

Theorem 2.4 says that under the assumption that u ∈ H10 (Ω) ∩X2,2(Ω) (which

is a rather strong regularity assumption, much stronger than the assumption


Figure 2.4: The case d = 2: each entry of this array corresponds to a pair ofinteger k = (k1, k2), 1 ≤ k1, k2 ≤ 4, and contains the grid corresponding to Wk.Each space Wk is the tensor product of two spaces whose bases are plotted onthe sides of the array. The full tensor space Vn1 is given by Vn1 =

⊕1≤k≤n1Wk

whereas the sparse tensor space Vn is given by Vn =⊕

1≤k,|k|≤n+d−1Wk, (onlythe spaces Wk corresponding to the entries above the diagonal are used toconstruct Vn)


u ∈ H10 (Ω)∩W 2,2(Ω) required when the full tensor product space is used), then

using the sparse approximation space Vn instead of the full tensor space Vn1does not deteriorate the accuracy, at least with respect to the H1 semi-norm.There is a moderate deterioration for the L2 norm of the error.In our presentation, we have focused on sparse methods based on tensorizingone dimensional hierachical bases made of hat functions. This technique canbe generalized to other classes of bases functions, for example higher orderpiecewise polynomial functions or wavelets as in Figure 2.5.

Figure 2.5: An example of wavelets

Sparse Grids Before defining finite difference methods on sparse grids, weneed to introduce new notations and concepts.Consider the one variable shape functions: φℓ,i(x) = φ(2ℓx− i), ℓ ≥ 1, 1 ≤ i ≤2ℓ − 1, and call Vℓ the space spanned by (φℓ,i)1≤i≤2ℓ−1. Call Wℓ the subspaceof Vℓ spanned by (φℓ,2i−1)1≤i≤2ℓ−1 . We have Vℓ =Wℓ ⊕ Vℓ−1. We have alreadyseen that V1 ⊂ . . . Vℓ ⊂ Vℓ+1 ⊂ . . . is a multiresolution analysis of H1

0 ((0, 1)).For a function u ∈ C0([0, 1]) s.t. u(0) = u(1) = 0, we have

u =

∞∑

ℓ=1

2ℓ−1∑

i=1

uℓ,iφℓ,2i−1,

and the projection of u on Vℓ is

2ℓ−1∑

i=1

uℓ,iφℓ,i =

ℓ∑

k=1

2k−1∑

i=1

uk,iφk,2i−1.

The change of coordinates (uℓ,i)i=1,...,2ℓ−1 7→ (uk,i)k=1,...ℓ,i=1,...,2k−1 is called Tℓ.

We call Uℓ and U ℓ the column vectors: U ℓ = (uℓ,1, . . . , uℓ,2ℓ−1) ∈ R2ℓ−1 and

Uℓ = (uℓ,1, . . . , uℓ,2ℓ−1) ∈ R2ℓ−1

. We have

TℓUℓ =

U1

...Uℓ

.


We denote by P ℓ the restriction operator

P ℓ : C0([0, 1]) → R2ℓ−1, P ℓu = U ℓ. (2.83)

Note that T−1ℓ is the representation of the operator P ℓ in the wavelet basis, i.e.,

P ℓ

∑

k≤ℓ

2k−1∑

i=1

uk,iφk,2i+1

= T−1ℓ

U1

...Uℓ

.

We introduce the interpolation operator Iℓ:

Iℓ : R2ℓ−1 → C0([0, 1]), IℓU =

2ℓ−1∑

i=1

uiφℓ,i. (2.84)

We also denote by Dℓ the finite difference operator for the discretization of d2

dx2 :

Dℓ :R2ℓ−1 → R2ℓ−1,

∀U, V ∈ R2ℓ−1 (DℓU, V ) = 2ℓ

∫ 1

0

(IℓU)′(IℓV )′.(2.85)

We consider the uniform grids of (0, 1): ωℓ = 2−ℓ1, . . . , 2ℓ − 1. For ℓ ∈Nd, 1 ≤ ℓ, we introduce the Cartesian grid of Ω: Ωℓ =

∏di=1 ω

ℓi . A gridfunction on Ωℓ is a mapping from Ωℓ to R. The space of the grid functions on

Ωℓ is exactly∏di=1 R

2ℓi−1. The mapping (ui)1≤i≤2ℓ−1 7→ u =∑

1≤i≤2ℓ−1uiφℓ,i

is an isomorphism from the space of the grid functions on Ωℓ onto Vℓ definedin (2.74). Moreover, the function u can be written on the wavelet basis u =∑

1≤k≤ℓ

∑i∈Ik

uk,iφk,i. Calling Uk the vector (uk,i)i∈Ik, the grid function will

be represented by the family (Uk)1≤k≤ℓ.For a positive integer n, we define the sparse grid Ωn as follows:

Ωn = ∪1≤ℓ,|ℓ|≤n+d−1Ωℓ ⊂ Ωn1. (2.86)

An example of a sparse grid in dimension d = 2 is presented in Figure 2.6.A grid function on Ωn is a mapping from Ωn to R. The space of the gridfunctions on Ωn is isomorphic to Vn defined in (2.79). As for the full ten-sor grid, a grid function on Ωn can be represented on the wavelet basis by∑

1≤k,|k|≤n∑

i∈Ikuk,iφk,i. Calling Uk the vector (uk,i)i∈Ik

, the sparse grid

function will be represented by the family (Uk)1≤k,|k|≤n+d−1.

We now define the sparse finite difference discretization of ∂2

∂x21: given the vectors

k = (k2, . . . , . . . kd) ∈ Nd−1, ı ∈ Ik and a sparse grid function represented by

(Uk)1≤k,|k|≤n+d−1, let k be the positive integer k = n+d−1−|k|; we introduceUk by

Uk,ı =

U(1,k,ı)

...U(k,k,ı)

where U(j,k,ı) =

(u(j,k),(m,ı)

)Tm odd, 1≤m≤2j−1

.


Figure 2.6: An example of a sparse grid for d = 2, n+ 1 = 8


The sparse grid discretization of the operator ∂2

∂x21is

(Uk)1≤k,|k|≤n+d−1 7→ (Vk)1≤k,|k|≤n+d−1

such that Vk,ı = TkDkT−1

kUk,ı, ∀k, i ∈ Ik.

(2.87)

The sparse grid discretization of the operators ∂2

∂x2j, ∂∂xj

, j = 1, . . . , d, can be

done in a similar way.

It is natural to define the restriction operator P ℓ : u 7→ u|Ωℓ and the inter-

polation operator Iℓ = Iℓ1 ⊗ · · · ⊗ Iℓd :∏di=1 R

2ℓi−1 → C0(Ω). The finitedifference approximation of ∂2x1

u on the grid Ωℓ is (Iℓ (Dℓ ⊗ Id) P ℓ)(u). Ithas been proved by Koster, see [71], that the sparse grid approximation of ∂2x1

ucan be written in terms of these finite difference operators:

Theorem 2.5 For a function u ∈ C0(Ω) s.t. u = 0 on ∂Ω, we note Dn(u) thefunction of Vn whose expansion in the wavelet basis is given by (Vk)1≤k,|k|≤n+d−1

in (2.87), where (Uk)1≤k,|k|≤n+d−1 is the expansion on the wavelet basis of theprojection of u on Vn. Then

Dn(u) =

∑

1≤k,|k|≤n+d−1

f(k)Ik (Dk1 ⊗ Id) Pk

(u), (2.88)

where f(k) is recursively defined by

f(k) = 0, if |k| > n+ d− 1 or k < 1,

f(k) = 1−∑

ℓ:k<ℓ

f(ℓ), if |k| ≤ n+ d− 1 and k ≥ 1. (2.89)

Before stating a consistency estimate, let us introduce some Holder spaces: let αbelong to Rd+. Call [α] the vector of Nd whose ith component is the integer partof αi. Call α = α− [α]. We note Cα(Ω) the space of continuous functions usuch that for all β ≤ [α], Dβu is continuous and

sup

|D[α]u(x+ h)−D[α](x)||h1|α1 . . . |hd|αd ,x,x+ h ∈ Ω, |hi| > 0, i = 1, . . . , d

< +∞.

The last quantity corresponds to a semi-norm on Cα(Ω), which we call |u|Cα(Ω).Theorem 2.5 is the key to the following consistency estimate, obtained in [71]:

Theorem 2.6 Assume that u ∈ Cα(Ω), where α1 > 2, αi > 0, i = 2, . . . , d,and that u = 0 on ∂Ω. Let Pn be the restriction operator on the sparse grid Ωn:Pn(u) = u(Ωn). We have the consistency error estimate

‖Pn(∂2u

∂x21)− Pn Dn(u)‖∞ ≤ Cnd−12−nmin(α1−2,α2,...,αd,2)|u|Cα(Ω). (2.90)


Similarly, for the sparse discretization of the Laplace operator, the consistencyerror may be bounded by Cnd−12−nmin(α1−2,α2−2,...,αd−2,2)|u|Cα(Ω) if u ∈ Cα(Ω)with αi > 2, i = 1, . . . , d.We see that the sparse grid discretization of ∆ is consistent and that the consis-tency error is almost of the same order (up to the factor nd−1) as the consistencyerror obtained with a full tensor grid.We are left with studying the stability of the sparse grid discretization. As faras we know, there is unfortunately no theoretical stability estimates. There iseven no proof that the matrix D arising in the discrete problem is invertible.Indeed D does not fall into the well studied classes of matrices: in particular,D is neither a symmetric nor a M matrix. No discrete maximum principle isavailable. Nevertheless, numerical tests were done in [101], indicating that thestability constant , i.e., ‖D−1‖∞ is bounded by Cnd−1.If such a stability estimate is true, we see that the sparse grid discretization ofthe Poisson problem is convergent, with an error of the order of

n2d−22−nmin(α1−2,α2−2,...,αd−2,2),

if u ∈ Cα(Ω) with αi > 2, i = 1, . . . , d.

The combination technique For a linear PDE in Ω with say Dirichlet con-ditions, there is an alternative technique which consists of separately computingthe approximations of the solution with standard finite difference schemes onall the Cartesian grids Ωℓ, 1 ≤ ℓ, |ℓ| ≤ n+ d− 1, and suitably combining thesesolutions, see [112, 53, 97, 94] : the discrete solution is

∑

1≤ℓ,|ℓ|≤n+d−1

f(ℓ)uℓ =

n+d−1∑

ℓ=n

aℓ−n∑

ℓ,|ℓ|=ℓuℓ,

where uℓ is the discrete solution computed with the standard finite differencescheme on Ωℓ, f(ℓ) is defined in (2.89) and where

aj = (−1)d−1−j(d− 1j

), 0 ≤ j ≤ d− 1.

The choice of the coefficients aj comes from

• performing a multidimensional Taylor expansion of the error between thesolution of the continuous problem and its approximation by a linear finitedifference scheme on a Cartesian grid of steps (h1, . . . , hd) with respect toh1, . . . , hd.

• combining the discrete solutions on the Cartesian grids Ωℓ, 1 ≤ ℓ, |ℓ| ≤n+d−1, in order to cancel the larger terms in the above mentioned Taylorexpansions.


Doing so, there is an approximation error (and not only a consistency estimate),see [97, 94]: for a second order scheme and a sufficently smooth u, the error inmaximum norm is bounded by

Cd(n+ 2(d− 1))d−12−2n.

Applications to Option Pricing Sparse methods have been applied forpricing derivatives by several authors, in particular Reisinger [95], Schwab et al[109] with wavelets.For option pricing, one of the main difficulty is that the payoff function is gener-ally not smooth and furthermore that the locus of its singularity has no relationwith the directions of the sparse grid (or sparse tensor product): therefore, theerror of the sparse approximation will increase (blow up) near maturity.

For basket options with a payoff depending on the weighted sum∑d

i=1 αiSi, thechange of variable (2.17) proposed by Reisinger [95] may be used; for a Cartesiangrid in the new variables (yi)1≤i≤d or for a sparse grid obtained by removingnodes from the last Cartesian grid, the locus of the singularity is an hyperplaneperpendicular to one of the grid’s directions. This enables grid refinement in thelast direction, which decreases the error while keeping the size of the discreteproblem reasonable. The resulting grid is sparse in the directions parallel tothe last hyperplane and nonuniformly refined in the remaining direction. Theprice to pay is a more complicated partial differential equation. Of course, thistrick is not possible with other options such as best-of options; more involvedrefinement strategies have then to be used, see e.g. [51] and the examples below.

To compensate the loss of regularity at maturity, Schwab et al [109] haveproposed to use a time stepping with a very nonuniform time-grid suitablyrefined near maturity.An even more difficult case is that of American options, see § 2.3, because thepricing function exhibits a singularity at the exercise boundary which is anunknown and cannot be related to the grid’s directions.As an illustration, we plot the pricing function for a put on an average of twoassets computed with a sparse grid, see Figure 2.7. Here, the singularity ofthe payoff is not aligned with the grid. In Figure 2.8, we show an adaptedsparse grid for the same baskets on two assets. The sparse grid is computed byprogressively enriching an initial coarse grid. The mesh is refined near a node ifthe corresponding coefficient in the multilevel expansion of the discrete solutionis larger than a threshold.

We shall see later that sparse methods prove more useful for option pricing on asingle asset but with a multifactor stochastic volatility, see § 2.4.4. In this case,the payoff function depends on the price variable only. Hence, the singularityis located on an hyperplane in the price/volatilities space, and sparse grids canbe used in an easy way.


0 0.5 1 1.5 2 2.5 3 3.5 4

0 0.5 1 1.5 2 2.5 3 3.5 4

-0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

"Bsk_Put_SparseG"

Figure 2.7: The pricing function of a European option on a basket of two assets,computed on a sparse grid (many thanks to David Pommier who wrote thesoftware and gave us this figure).


0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4

"Basket2D-adapt"

Figure 2.8: Adapted sparse grid for an option on a basket of two assets, (thanksto David Pommier).

2.3. AMERICAN BASKET OPTIONS 67

2.3 American Basket Options

We consider an American option on the d risky assets whose prices Si,t are theprocesses described at the beginning of §2.1. The maturity of the option is Tand its payoff function is P : R+

d 7→ R+.The Black-Scholes model leads to the following formula for the price of theAmerican option at time t: under the risk-neutral probability,

Pt = supτ∈Tt,T

E∗(e−

∫τtr(s)dsP(S1,τ , . . . , Sd,τ)

∣∣∣Ft), (2.91)

where Tt,T denotes the set of stopping times in [t, T ], (see [68, 67]). It is clearthat Pt ≥ P(S1,t, . . . , Sd,t).Under suitable assumptions on the payoff P and on the volatilities, it can beproven that Pt is a function of S1,t, . . . , Sd,t and t, i.e. Pt = P (S1,t, . . . , Sd,t, t),and that the pricing function P is the solution of a variational inequality, see(2.96) below, which is the variational form of the following linear complemen-tarity problem:

∂P

∂t+ LP − rP ≤ 0, in Rd+ × [0, T ),

P ≥ P in Rd+ × [0, T ),(

∂P∂t + LP − rP

)(P − P) = 0, in Rd+ × [0, T ),P |t=T = P in Rd+,

(2.92)

where L is given by (2.3). The proof of this result goes beyond the scope of thisparagraph: it can be found in Bensoussan et al [15] or in Jaillet et al [67].

2.3.1 The Variational Inequality

Here, we assume that P ∈ L2(Rd+). It is possible to deal with other payofffunctions by using Sobolev spaces with different weights at infinity, see [15] or[67]. Calling t the time to maturity, (2.92) becomes

∂P

∂t− LP + rP ≥ 0, in Rd+ × (0, T ],

P ≥ P in Rd+ × (0, T ],(∂P∂t − LP + rP

)(P − P) = 0, in Rd+ × (0, T ],

P |t=0 = P in Rd+,

(2.93)

where L is given by (2.3). To write the variational formulation of (2.93), we usethe same Sobolev space V as for the European option, see (2.31). We call K theclosed and convex subset of V :

K = v ∈ V, v ≥ P a.e. in Rd+. (2.94)

and we introduce K:

K = v ∈ L2(0, T ;V ), s.t. v(t) ∈ K for a.a. t ∈ (0, T ) (2.95)

Following Lions, [75], a variational formulation of (2.93) is to


find P ∈ K such that dPdt ∈ L2(0, T ;V ′) and P (t = 0) = P and satisfying

∀v ∈ K,∫ T

0

⟨dP

dt(t), v(t) − P (t)

⟩dt+

∫ T

0

at(P (t), v(t)−P (t))dt ≥ 0, (2.96)

where 〈, 〉 is the duality pairing between V ′ and V and at is the bilinear formintroduced in (2.30).

By adapting the results in [75], see also [70], one can prove the following theorem:

Theorem 2.7 Under the assumptions (2.33) (2.34) (2.35) and if r is a boundedfunction defined on (0, T ), the variational inequality (2.96) has a unique solution

P . Furthermore, P ∈ C0([0, T ];L2(Rd+)),∂P∂t ∈ L2(Rd+ × (0, T )), SiSj

∂2P∂Si∂Sj

∈L2(Rd+ × (0, T )), i, j = 1, . . . , d, and P satisfies the linear complementarityproblem (2.93).

More properties can be proved under stronger assumptions on the payoff func-tion and on the coefficients: for example, if the volatilities are constant and ifthe payoff function is piecewise linear and continuous with compact support,then P is continuous on Rd+ × [0, T ]. If the coefficients are constant and un-der some special assumptions on the payoff , it can be proved that P (S, t) isnondecreasing with respect to t (t is the time to maturity).

2.3.2 The Exercise Region

The exercise region at time t is the set

S ∈ R

d+, s.t. P (S, t) = P(S)

.

The theoretical results concerning the exercise region for American options onbaskets strongly depend on the payoff. S. Villeneuve [108] has proved that ifthe coefficients are constant with r > 0 and if P is bounded and continuous,then the exercise region is nonempty. The shape of the exercise region and itsbehavior near maturity is studied as well in [108] for a particular class of payofffunctions.Examples of exercise regions for American best-of options computed by finiteelement methods will be given below, see Figure 2.10. It will be seen that theexercise boundary, i.e., the boundary of the exercise region may exhibit ratherstrong singularities.

2.3.3 Finite Element Methods

Assuming that P has compact support, we truncate the domain as for theEuropean option, i.e., we consider Ω = (0, S)d (where S is large enough sothat the support of P is strictly contained in Ω) and Γ0 given in (2.42). Wechoose to impose homogeneous Dirichlet artificial boundary conditions on Γ0.


The Sobolev space V to work with is given in (2.45) and the new definition of Kis K = v ∈ V, v ≥ P a.e in Ω. Changing K accordingly (see (2.95)), the newvariational inequality is (2.96) where at is given by (2.46). We are now readyto propose a finite element discretization.We introduce a partition of the interval [0, T ] into subintervals [tm−1, tm], 1 ≤m ≤M , with δti = ti − ti−1, δt = maxi δti. We choose a triangulation Th of Ω,and we define Vh by the following:

Vh =vh ∈ V, ∀ω ∈ Th, vh|ω ∈ P1

. (2.97)

where P1 is the space of affine functions. For simplicity, we assume that P ∈ Vh.We define the closed and convex subset Kh of Vh by

Kh = v ∈ Vh, vh ≥ P in Ω. (2.98)

The discrete problem arising from the implicit Euler scheme is:

find (Pm)0≤m≤M ∈ Kh satisfying

P 0 = P, (2.99)

and for all m, 1 ≤ m ≤M ,

∀v ∈ Kh,(Pm − Pm−1, v − Pm

)+ δtmatm(Pm, v − Pm) ≥ 0. (2.100)

Expressing Pm, 0 ≤ m ≤M , and v in the nodal basis of Vh, (2.100) is equivalentto the finite dimensional linear complementary system

M (Um −Um−1) + δtmAmUm ≥ 0,

Um ≥ U0,

(Um −U0)T (M(Um −Um−1) + δtmAmUm) = 0,

(2.101)

where for two vectors U and V , U ≥ V means that all the coordinates of U−V

are nonnegative.Assuming that the coefficients satisfy the assumptions (2.33) (2.34) (2.35), itcan be proved that for δt small enough (2.101) has a unique solution for allm = 1, . . . ,M . Stability and convergence in the natural energy norm can beproved as well.Mesh adaptivity based on a posteriori error estimates is possible too. The de-scription of the error estimators for parabolic varitional inequalities goes beyondthe scope of the present article. We refer to [26, 106, 85, 86] for a posteriori errorestimates and mesh refinment strategies for elliptic variational inequalities. Inthe parabolic case, a strategy similar to the one in [16] is studied in [4].

2.3.4 Algorithms

The Projected SOR Algorithm

Let us write (2.101) in the simpler form

Bu ≥ f ,u ≥ g,

(u− g)T (Bu− f) = 0,(2.102)


The projected SOR algorithm is an iterative method for solving (2.102). Let ωbe a positive real number. The idea is to approximate u by using a one steprecursion formula u(k+1) = ψ(u(k)) (starting from an initial guess u(0)), whereψ is the nonlinear mapping in RN :

ψ : v 7→ w = ψ(v) :∀i = 1, . . . , N, wi = max(gi, yi), and yi is given by1

ωBiiyi +

∑

j<i

Bijwj = fi + (1

ω− 1)Biivi −

∑

j>i

Bijvj .(2.103)

This construction is a modification of the so-called successive over relaxationmethod (SOR) used for systems of linear equations, see [12], [49], [100] foriterative methods: for solving approximately the system Bv = f , the SORalgorithm constructs the sequence (v(k))k (starting from an initial guess v(0))by the recursion:

∀i = 1, . . . , N,1

ωBiiv

(k+1)i +

∑

j<i

Bijv(k+1)j = fi+(

1

ω−1)Biiv

(k)i −

∑

j>i

Bijv(k)j .

Proposition 2.3 If B is a diagonal dominant matrix and if 0 < ω ≤ 1, thenthe mapping ψ defined in (2.103) is a contraction in RN for the norm ‖.‖∞(‖v‖∞ = max1≤i≤N |vi|). The fixed point of ψ is u.

Under the assumptions of Proposition 2.3, the sequence constructed by thePSOR algorithm converges to u. The speed convergence of convergence de-pends on the matrix B and on the relaxation parameter ω. The convergence isgenerally slow for ill-conditioned matrices.

Primal-dual Methods

Following [65], we first go back to the semi-discrete problem: find Pm ∈ K suchthat

∀v ∈ K,(Pm − Pm−1, v − Pm

)+ δtmatm(Pm, v − Pm) ≥ 0.

For any positive constant c, this is equivalent to finding Pm ∈ V and a Lagrangemultiplier µ ∈ V ′ such that

∀v ∈ V,1

δtm

(Pm − Pm−1, v

)+ atm(Pm, v)− 〈µ, v〉 = 0,

µ = max(0, µ− c(Pm − P 0)).

(2.104)

When using an iterative method for solving (2.104), i.e., when constructing asequence (Pm,j , µj) for approximating (Pm, µ), the Lagrange multiplier µj maynot be a function if the gradient of Pm,j jumps, whereas µ may be a function.Therefore, a dual method (i.e. an iterative method for computing µ) may be


difficult to use. As a remedy, K.Ito and K.Kunisch [65] consider a one parameterfamily of regularized problems based on smoothing the equation for µ as follows:

µ = αmax(0, µ− c(Pm − P 0)), (2.105)

for 0 < α < 1, which is equivalent to

µ = max(0,−χ(Pm − P 0)), (2.106)

for χ = cα/(1−α) ∈ (0,+∞). We may consider a generalized version of (2.106):

µ = max(0, µ− χ(Pm − P 0)), (2.107)

where µ is a fixed function. This turns out to be useful when the complemen-tarity condition is not strict.It is now possible to study the fully regularized problem

∀v ∈ V,1

δtm

(Pm − Pm−1, v

)+ atm(Pm, v)− 〈µ, v〉 = 0,

µ = max(0, µ− χ(Pm − P 0)),

(2.108)

and prove that it has a unique solution, with µ a square integrable function. Aprimal-dual active set algorithm for solving (2.108) is

Primal-Dual Active Set Algorithm• Choose Pm,0, set k = 0• Loop

1. Set

A−,k+1 = S : µk(S)− χ(Pm,k(S)− P 0(S)) > 0,A+,k+1 = (0, S)\A−,k+1.

2. Solve for Pm,k+1 ∈ V : ∀v ∈ V,

0 =1

δtm

(Pm,k+1 − Pm−1, v

)+ atm(Pm,k+1, v)

− (µ− χ(Pm,k+1 − P 0), 1A−,k+1v).

(2.109)

3. Set

µk+1 =

0 on A+,k+1,

µ− χ(Pm,k+1 − P 0) on A−,k+1 (2.110)

4. Set k = k + 1.

Calling Am the operator from V to V ′: 〈Amv, w〉 = 1δtm

(v, w) + atm(v, w) and

F : V × L2(Rd+) → V ′ × L2(Rd+)

F (v, µ) =

(Amv + µ− Pm−1

δtmµ−max(0, µ− χ(v − P 0))

),


it is proved in [65] that G(v, µ) : V × L2(Rd+) → V ′ × L2(Rd+),

G(v, µ)h =

(Amh1 + h2h2 − χ1µ−χ(v−P 0)>0h1

)

is a generalized derivative of F in the sense that

lim‖h‖→0

‖F (v + h1, µ+ h2)− F (v, µ)−G(v + h1, µ+ h2)h‖‖h‖ = 0.

Note that

G(Pm,k, µk)h =

(Amh1 + h2h2 − χ1A−,k+1h1

).

Thus the primal-dual active set algorithm above can be seen as a semi-smoothNewton method applied to F , i.e.,

(Pm,k+1, µk+1) = (Pm,k, µk)−G−1(Pm,k, µk)F (Pm,k, µk). (2.111)

Indeed, calling (δPm, δµ) = (Pm,k+1 − Pm,k, µk+1 − µk), it is straightforwardto see that in the primal-dual active set algorithm we have

AmδPm + δµ = −AmPm,k − µk +

Pm−1

δtm,

δµ = −µk on A+,k+1,

δµ− χδPm = −µk + µ− χ(Pm,k − P 0) on A−,k+1,

which is precisely (2.111).In [65], by using the results proved in [59], Ito and Kunish establish that theprimal-dual active set algorithm converges from any initial guess and that if theinitial guess is sufficiently close to the solution of (2.108) then the convergenceis superlinear.To solve (2.104), it is possible to successively compute the solutions (Pm(χℓ), µ(χℓ))of (2.108) for a sequence of parameters (χℓ) converging to +∞, using (Pm(χℓ), µ(χℓ))as an initial guess for the primal-dual active set algorithm for (Pm(χℓ+1), µ(χℓ+1)).Of course, it is possible to use the same algorithm for the fully discrete problem.Convergence results hold in the discrete case if there is a discrete maximumprinciple. The algorithm amounts to solving a sequence of systems of linearequations and the matrix of the system varies at each iteration.

Examples The discrete method discussed above has been applied to computethe pricing function of an American best-of put option on a two assets basket,P(S1, S2) = (K−max(S1, S2))

+. The artificial boundary Γ0 is max(S1, S2) =S = 200. Homogeneous Dirichlet conditions are imposed on Γ0. We havechosen two examples:

1. In the first example, the parameters are

σ1 = 0.2, σ2 = 0.1, r = 0.05, ρ = −0.6 K = 100.


Figure 2.9: The adapted mesh and the contours of P one year to maturity.σ1 = 0.2, σ2 = 0.1, ρ = −0.6

2. In the second example, the parameters are

σ1 = σ2 = 0.2, r = 0.05, ρ = 0 K = 50.

The implicit Euler scheme has been used with a uniform time step of 1/250year. For solving the linear complementarity problems (2.101), we have used theregularized active set method with the regularization parameters χ = 107 andµ = 0, see (2.107). Mesh adaption in the (S1, S2) variable has been performedevery 1/10 year. The adaptive strategy is close to the one used in FreeFem, andthe mesh is refined in the contact set. This may be unnecessary if the obstaclebelongs to the finite element space: we refer to [4] for a better adaptive strategybased on local error indicators where the mesh is not refined in the so-calledstrong contact region, see also [26, 106, 85, 86] for elliptic contact problems.In Figure 2.9, we have plotted the adapted mesh (left) and the contours of thepricing function (right) one year to maturity for the first example. Note thatthe contours exhibit right angles in the exercise region. In Figure 2.10, we plotthe exercise region one year to maturity for the first example (top) and for thesecond example (bottom). One sees that the exercise boundary has singularities.It is also visible that the mesh has been adapted near the exercise boundary.Finally, we consider an American binary option whose payoff is

P(S1, S2) = K1max(S1,S2)<K,

with the parameters of the second example. The exercise region does not evolve.The contours of the pricing function one year to maturity are plotted in Figure2.11.


0 10 20 30 40 50 60 70 80 90 100 0

10

20

30

40

50

60

70

80

90

100

"exercise_250"

0 5 10 15 20 25 30 35 40 45 50 0

5

10

15

20

25

30

35

40

45

50

"exercise_250"

Figure 2.10: The exercise region one year to maturity. Top: k = 100, σ1 = 0.2,σ2 = 0.1, ρ = −0.6. Bottom: K = 50, σ1 = σ2 = 0.2, ρ = 0


Figure 2.11: The contours of P for an American binary option one year tomaturity. P(S1, S2) = 50.1max(S1,S2)<50, σ1 = σ2 = 0.2, ρ = 0

Penalty Methods

Assume that P is bounded and that P ∈ V where V is given in (2.31). Apenalized version of (2.93) is to look for Qǫ such that

∂Qǫ∂t

− LQǫ + rQǫ − Vǫ(Qǫ) = LP − rP, in Rd+ × (0, T ],

Qǫ|t=0 = 0 in Rd+,(2.112)

where L is given by (2.3), ǫ is a small positive parameter and Vǫ : R → R+ is asequence of C2 nonincreasing functions such that

• Vǫ(z) = 0 if z > 0.

• Vǫ(z) = − zǫ if z < −ǫ.

• V ′ǫ and V ′′

ǫ are bounded by Cǫ .

It is reasonable to think that Qǫ is a good approximation of Q = P − P as ǫtends to zero.The theory of the weak solutions of parabolic partial differential with monotoneoperators, see [75], can be used because Vǫ is nonincreasing. One can prove that(2.112) has a unique weak solution and that Qǫ tends to Q in the natural normsassociated with the variational problem (2.96). In some cases, error estimatescan be found.


The initial value problem (2.112) can in turn be approximated by the finite ele-ment method: using e.g. an implicit Euler scheme and piecewise affine functionson a triangulation of Ω, each time step is of the form

M(Qm −Qm−1) + δtm (AmQm + Vǫ(Qm)) = −δtmAmU0, (2.113)

where M and A are the matrices in (2.57) and where Vǫ : RN → RN is anonlinear operator such that

(Vǫ(Qm))i ∼

∫

Ω

wiVǫ

N∑

j=1

qmj wj

.

Here the sign ∼ means that a suitable elementary quadrature rule has been used

to approximate∫KVǫ(∑N

j=1 qmj wj

)wi, K ∈ Th. Newton iterations can be used

for solving (2.113), and Qm−1 can be used as an initial guess.A simpler nonlinear problem is

M(Qm −Qm−1) + δtm (AmQm + Vǫ(Qm)) = −δtmAmU0, (2.114)

where Vǫ(Qm) is the vector defined by (Vǫ(Qm))i = Vǫ(qmi ), i = 1, . . . , N .Convergence may be proved for (2.113) or (2.114).A similar method has been carefully tested for American options in the slightlydifferent context of a stochastic volatility model, see R. Zvan et al.,[114].The computing times of the penalty methods and of the primal-dual active setmethods are comparable.

Projection Schemes

Let us go back to the sequence of fully discrete linear complementarity problems(2.101) arising from the implicit Euler scheme. It is possible to write them inan equivalent manner using a Lagrange multiplier Λm ∈ RN ,

M (Um −Um−1) + δtmAmUm = δtmΛm,

Um ≥ U0,Λm ≥ 0,

(Um −U0)TΛm = 0.

(2.115)

Following Lions-Mercier [77], see also [48] and [63], we modify (2.115) in orderto obtain a new operator splitting time scheme where each time step is madeof two substeps: the first substep is that of the implicit Euler scheme for thecorresponding parabolic equation (European option) and the second step is aprojection step. Operator splitting schemes are very popular in computationalfluid dynamics for incompressible fluids and they are often called projectionschemes, see e.g [56, 55, 3]. More precisely, the first substep reads:

M(Um −Um−1) + δtmAmU

m= δtmΛm−1, (2.116)


where the unknown is Um

( (2.116) is a system of linear equations) whereas thesecond substep is

M(Um − Um) = δtm(Λm − Λm−1),

Um ≥ U0,Λm ≥ 0,

(Um −U0)TΛm = 0.

(2.117)

The unknowns of the linear complementarity problem (2.117) are Um and Λm

and the matrix of the problem is M instead of M + δtmAm in (2.115). Since

1. M is symmetric and positive definite,

2. generally, M has a much smaller condition number than M + δtmAm,

the linear complementarity problem (2.117) is easier to solve than (2.115). For(2.117), either the PSOR algorithm or projected gradient algorithms are effi-cient. Furthermore, these algorithms can be easily modified in order to usediagonal preconditioning for M when the finite element mesh is highly nonuni-form.If the operator splitting scheme is used either with mass lumping (see Remark2.6) or with finite difference schemes instead of finite elements as in [63], then(2.117) becomes easier because M is replaced with a diagonal matrix and theprojection substep consists of solving as many decoupled one dimensional linearcomplementarity problems as there are nodes in the grid.Other schemes can be modified to produce operator splitting schemes: for ex-ample, the Crank-Nicolson scheme

M(Um −Um−1) + 12δtm(AmUm +Am−1Um−1) = δtmΛm,

Um ≥ U0,Λm ≥ 0,

(Um −U0)TΛm = 0,

(2.118)

becomes:

• Linear substep

M(Um −Um−1) +

1

2δtm(AmU

m+Am−1Um−1) = δtmΛm−1, (2.119)

• Projection substep: (2.117).

The second order accurate backward Euler scheme

M(Um − 43U

m−1 + 13U

m−2) + 23δtmAmUm = 2

3δtmΛm,Um ≥ U0,Λm ≥ 0,

(Um −U0)TΛm = 0,

(2.120)

becomes:


• Linear substep

M(Um − 4

3Um−1 +

1

3Um−2) +

2

3δtmAmU

m=

2

3δtmΛm−1, (2.121)

• Projection substep

M(Um − Um) = 2

3δtm(Λm − Λm−1),

Um ≥ U0,Λm ≥ 0,

(Um −U0)TΛm = 0.

(2.122)

It is possible to estimate the error produced by using the projection schemesabove. This was done in [63] in the context of finite differences: for the Crank-Nicolson scheme with constant time steps and time-invariant coefficients, it wasshown that the error between the solution of (2.118) and that of (2.119,2.117)is of the order of δt2, if some stability assumptions are satisfied (A is diagonallydominant with positive diagonal entries). Therefore both the Crank-Nicolsonand the projected Crank-Nicolson schemes are second order in time. It is alsopossible to obtain error estimates in Sobolev norms for the three schemes aboveby using the techniques in [56, 55].In terms of computing time, we found that the projection scheme is clearlyfaster than the primal-dual algorithm above for a comparable accuracy. In ourexperience, the only difficulty posed by the projection schemes is that they arenot fully compatible with a strategy of dynamic mesh adaption (i.e., when themesh is adapted between two time steps).

Remark 2.7 Operator splitting can also be applied to the penalized problems(2.113) or (2.114).

Example We use the projected Euler scheme (2.116,2.117) for computing thepricing in the first example in § 2.3.4. In Figure 2.12, we plot the exercise regionone year to maturity. This figure must be compared with Figure 2.10(top). Thedifference is not visible.

Multigrid Methods

Multigrid methods can also be used for linear complementarity problems: onepossibility is to modify the primal-dual algorithm described above: recall thateach iteration of such algorithms requires the solution of a linear boundaryvalue problem in a varying subdomain. The idea is to use a multigrid methodfor these linear substeps. The main difficulty lies in the fact that one cannotobtain a hierarchy of meshes of the subdomains by simply taking subsets of thehierarchical meshes for the whole domain. We refer to [61] for such algorithmsfor finite difference schemes. In the same context but for finite elements, additivemultilevel preconditioners were proposed in [60].Another possible way is to design a multigrid cycle for the full nonlinear problem.This has been done in [87, 96, 64].

2.4. STOCHASTIC VOLATILITY 79

0 10 20 30 40 50 60 70 80 90 100 0

10

20

30

40

50

60

70

80

90

100

"exercise_250"

Figure 2.12: The exercise region one year to maturity. σ1 = 0.2, σ2 = 0.1,ρ = −0.6. This has to be compared with Figure 2.10

Conclusion

We have presented different algorithms relevant in the context of American op-tions pricing. The PSOR and primal-dual methods are algorithms for solvingthe linear complementarity problem at each time step. Multigrid methods canbe used either as a component of the primal-dual methods or for directly solvingthe linear complementarity problem. The penalty method replaces the linearcomplementarity problem by a penalized nonlinear problem. The projectionschemes consist of dividing each time step into two substeps, the first one fordiffusion and the other one to enforce the constraints.In our experience, the projection schemes are fast (faster than the primal-dualmethods not combined with multigrid) and their implementation is very sim-ple and general. Multigrid methods may be faster but their implementationcertainly needs more care.

2.4 Stochastic volatility

2.4.1 Volatility Models with One Stochastic Process

We consider a financial asset whose price is given by the stochastic differentialequation

dSt = µStdt+ σtStdWt, (2.123)

where µStdt is a drift term, (Wt) is a Brownian motion, and (σt) is the volatil-ity. As we have seen before, the simplest models take a constant deterministic


volatility, but these models are generally too coarse to match real market prices.Here, we assume that (σt) is a stochastic process taking nonnegative values, sat-isfying a stochastic differential equation driven by a second Brownian motionZt, not perfectly correlated to Wt. More precisely, using the notations of [44],we assume that σt = f(Yt), where f is some positive function and (Yt) is thedriving process; the most common choices for (Yt) are

• lognormal process:

dYt = c1Ytdt+ c2YtdZt, (2.124)

where c1 and c2 are positive constants.

• mean reverting Orstein-Uhlenbeck (OU) process:

dYt = α(m− Yt)dt+ βdZt, (2.125)

where α and β are positive constants.

• Cox-Ingersoll-Ross (CIR) process:

dYt = κ(m− Yt)dt+ λ√YtdZt, (2.126)

where κ, m and λ are positive constants.

One of the important feature of the second and third processes is their mean-reversion: the drift term in the stochastic differential equation for Yt pulls theprocess back to the long-run mean level m. For example, if (Yt) is a meanreverting OU process satisfying (2.125), then the law of Yt knowing Y0 is

N(m+ (Y0 −m)e−αt,

β2

2α(1− e−2αt)

).

Therefore, m is the limit of the mean value of Yt as t → +∞, and 1α is the

characteristic time of mean reversion. The parameter α is called the rate of

mean reversion. The ratio β2

2α is the limit of the variance of Yt as t→ +∞. The

long-run distribution of the OU process is N(m, β

2

2α

).

The Brownian motion Zt may be correlated to Wt: it can be written as a linearcombination of Wt and an independent Brownian motion Zt:

Zt = ρWt +√1− ρ2Zt, (2.127)

where the correlation factor ρ lies in [−1, 1].The table 2.1 summarizes the most popular choices of stochastic volatility mod-els with one stochastic process


Table 2.1: Frequently used models of stochastic volatilities

Authors ρ f(y) Yt process

Hull-White [62] ρ = 0 f(y) =√y lognormal

Scott ρ = 0 f(y) = ey mean reverting OUStein-Stein [103] ρ = 0 f(y) = |y| mean reverting OU

Heston [58] ρ 6= 0 f(y) =√y CIR

2.4.2 European Options with Stochastic Volatility

Consider a European option on the previously mentioned asset, with expirationdate T and payoff function P(ST ). Its price at time t < T will depend on t, onthe price of the underlying asset St, and on Yt. We denote by P (St, Yt, t) theprice of the option, and by r(t) the interest rate.The option is priced using a no arbitrage principle and the two dimensional Ito’sformula. However, since there are two risk factors, it is not possible to constructa hedged portfolio containing simply one option and shares of the underlyingasset. One says that the market is incomplete. Instead, one can try to build ahedged portfolio containing options with two different maturities and shares ofthe underlying assets; only for simplicity, we restrict the discussion to the casewhen (Yt) is a OU process satisfying (2.125), with (Zt) given by (2.127), butwhat follows can be generalized to any Markovian Ito driving process:

dYt = µY (t, Yt)dt+ σY (t, Yt)dZt.

Let us try to build a self-financing hedged portfolio containing at shares of theunderlying asset, one option with expiration date T1 whose price is

P(1)t = P (1)(St, Yt, t)

and bt options with a larger expiration date T2 > T1, whose price is

P(2)t = P (2)(St, Yt, t).

The value of the portfolio is ct. The no arbitrage principle yields that for t < T1,

dct = atdSt + dP(1)t + btdP

(2)t = rtctdt = rt(atSt + P

(1)t + btP

(2)t )dt (2.128)

The two-dimensional Ito formula permits dP(1)t and dP

(2)t to be expressed as

combinations of dt, dWt and dZt. The right hand side of (2.128) does not containdZt, thus

bt = −∂P (2)

∂y

∂P (1)

∂y

.

From the last equation and since the right hand side of (2.128) does not containdWt, we deduce

at +∂P (1)

∂S+ bt

∂P (2)

∂S= 0.


Comparing the dt terms in (2.128) and substituting the values of at and bt, weobtain

1∂P (1)

∂y

(∂P (1)

∂t + 12f(y)

2S2 ∂2P (1)

∂S2 + ρβSf(y)∂2P (1)

∂S∂y + 12β

2 ∂2P (1)

∂y2 + r(t)(S ∂P

(1)

∂S − P (1)))

=

1∂P (2)

∂y

(∂P (2)

∂t + 12f(y)

2S2 ∂2P (2)

∂S2 + ρβSf(y)∂2P (2)

∂S∂y + 12β

2 ∂2P (2)

∂y2 + r(t)(S ∂P

(2)

∂S − P (2)))

.

In the equation above, the left hand side does not depend on T2 and the righthand side does not depend on T1. Therefore, there exists a function g(S, y, t)such that

1∂P∂y

(∂P

∂t+

1

2f(y)2S2 ∂

2P

∂S2+ ρβSf(y)

∂2P

∂S∂y+

1

2β2 ∂

2P

∂y2+ r(t)

(S∂P

∂S− P

))

=g(S, y, t).

Writing g(S, y, t) = α(y −m) + βΛ(S, y, t) makes the infinitesimal generator ofthe OU process explicit in the last equation. We obtain

∂P

∂t+

1

2f(y)2S2 ∂

2P

∂S2+ ρβSf(y)

∂2P

∂S∂y+

1

2β2 ∂

2P

∂y2

+r(t)

(S∂P

∂S− P

)+(α(m− y)− βΛ(S, y, t)

) ∂P∂y

= 0,

0 ≤ t < T,S > 0, y ∈ R,

(2.129)where

Λ(S, y, t) = ρµ− r(t)

f(y)+√1− ρ2γ(S, y, t), (2.130)

with the terminal condition P (S, y, T ) = P(S).The function γ(S, y, t) (return on the volatility risk ) can be chosen arbitrarily.As explained in [44], we can group the differential operator in (2.129) as follows:

∂P

∂t+

1

2f(y)2S2 ∂

2P

∂S2+ r(t)

(S∂P

∂S− P

)

︸︷︷︸BSf(y)

+ ρβSf(y)∂2P

∂S∂y︸︷︷︸correlation

+1

2β2 ∂

2P

∂y2+ α(m− y)

∂P

∂y︸︷︷︸Orstein Uhlenbeck

− βΛ(S, y, t)∂P

∂y︸︷︷︸premium

.

(2.131)

The term βΛ(S, y, t)∂P∂y is a premium on the volatility risk: the reason to decom-

pose Λ as in (2.130) is that in the perfectly correlated case (|ρ| = 1, completemarket), it is possible to find the equation satisfied by P by a simpler no arbi-trage argument with a hedged portfolio containing only the option and sharesof the underlying assets. In this case, the equation found for P is:

∂P

∂t+

1

2f(y)2S2 ∂

2P

∂S2+ ρβSf(y)

∂2P

∂S∂y+

1

2β2 ∂

2P

∂y2

+r(t)

(S∂P

∂S− P

)+

(α(m − y)− βρ

µ− r(t)

f(y)

)∂P

∂y= 0,

0 ≤ t < T,S > 0, y ∈ R.

(2.132)


The term µ−r(t)f(y) is called the excess return to risk ratio.

Finally, with (2.129), the Ito formula and (2.130)

dP (St, Yt, t) = (Sf(Yt)∂P

∂S+βρ

∂P

∂y)(µ− r

f(Yt)dt+dWt)+β

√1− ρ2

∂P

∂y(γdt+dZt)

from which we see that the function γ is the contribution of the second sourceof randomness dZt to the risk premium. The function γ is called the marketprice of the volatility risk or the risk premium factor.

Similarly, assuming that (Yt) is a CIR process satisfying (2.126), one obtains

∂P

∂t+

1

2f(y)2S2 ∂

2P

∂S2+ r(t)

(S∂P

∂S− P

)

︸︷︷︸BSf(y)

+ ρλS√yf(y)

∂2P

∂S∂y︸︷︷︸correlation

+1

2λ2y

∂2P

∂y2+ κ(m− y)

∂P

∂y︸︷︷︸CIR

−λ√yΛ(S, y, t)

∂P

∂y︸︷︷︸premium

= 0,

(2.133)

where Λ(S, y, t) is given by (2.130).

Remark 2.8 It is possible to obtain (2.129) and (2.133) by using a more math-ematically sound risk-neutral theory, and the market price of the volatility riskappears from Girsanov’s theorem, see [44],§2.5.

Remark 2.9 For the Heston model, a closed form solution in terms of integralsis available, see [58].

The Initial Value Problem for Stein-Stein’s Model We discuss themathematical analysis of the initial value problem with (2.129), in the case whenρ = 0 and f(y) = |y| (Stein-Stein’s model). The goal is to study variationalformulations of (2.129), and obtain global energy estimates. These estimates areuseful for studying discrete approximations by e.g. finite element methods. Vari-ational formulations are also particularly useful for the linear complementarityproblems obtained when pricing American options. This paragraph summarizesthe results contained in [8] and [2].For simplicity, we assume that the market price of risk γ is bounded indepen-dently of t, S and y. The variance of the invariant distribution of the OU

process, i.e. ν2 = β2

2α , will play an important role in what follows.In order to obtain a forward parabolic equation, we work with the time to ma-turity, i.e. T − t → t. With the aim of deriving a variational formulation, wemake the change of unknown

u(S, y, t) = P (S, y, T − t)e−(1−η) (y−m)2

2ν2 , (2.134)

where η is a parameter such that 0 < η < 1; we are going to impose that utends to 0 as y tends to ∞. Indeed, if Λ = 0, then one can find a solution of


(2.129) of the form g(t)e(y−m)2

2ν2 ; imposing that u(S, y, t) tends to 0 as y tends to∞ prevents such a behavior for large values of y. The parameter η will not beimportant for practical computations, because in any case, we have to truncatethe domain and suppress large values of y.With the notations r(t) = r(T − t), γ(t) = γ(T − t) and Λ(t) = Λ(T − t) thenew unknown u satisfies the degenerate parabolic partial differential equation

∂u

∂t− 1

2y2S2 ∂

2u

∂S2− r(t)

(S∂u

∂S− u

)− 1

2β2 ∂

2u

∂y2

+(−α(y −m) + βΛ(S, y, t))∂u

∂y+

(2α

βΛ(S, y, t)(y −m)− α

)u

+η

(2α(y −m)

∂u

∂y+ 2

α2

β2(1− η)(y −m)2u− 2

α

βΛ(y −m)u+ αu

)= 0.

(2.135)The equation is degenerate near the axis y = 0 because the coefficient in front

of S2 ∂2P∂S2 vanishes on y = 0.

Expanding Λ and denoting by Lt the linear partial differential operator

Ltv= −1

2y2S2 ∂

2v

∂S2− 1

2β2 ∂

2v

∂y2− r(t)S

∂v

∂S+ (−(1− 2η)α(y −m) + βγ(S, y, t))

∂v

∂y

+

(r(t) + 2

α2

β2η(1− η)(y −m)2 + 2(1− η)

α

β(y −m)(γ(S, y, t))− α(1 − η)

)v,

(2.136)we obtain

∂u

∂t+ Ltu = 0. (2.137)

We denote by Q the open half plane Q = R+ ×R. Let us consider the weightedSobolev space V :

V =

v :

(√1 + y2v,

∂v

∂y, S|y| ∂v

∂S

)∈ (L2(Q))3

. (2.138)

This space with the norm

|||v|||V =

(∫

Q

(1 + y2)v2 + (∂v

∂y)2 + S2y2(

∂v

∂S)2) 1

2

(2.139)

is a Hilbert space, and it has the following properties:

1. V is separable.

2. Calling D(Q) the space of smooth functions with compact support in Q,D(Q) ⊂ V and D(Q) is dense in V

3. V is dense in L2(Q).

The crucial point is point 2 which can be proved by an argument due toFriedrichs (Theorem 4.2 in [46]). We also have


Lemma 2.4 For any function v in V ,

∫

Q

y2v2 ≤ 4

∫

Q

y2S2(∂v

∂S)2. (2.140)

The semi-norm

‖v‖V =

(∫

Q

v2 + (∂v

∂y)2 + S2y2(

∂v

∂S)2) 1

2

(2.141)

is in fact a norm in V , equivalent to ‖|.|‖V .

We call V ′ the dual of V . In order to use the general theory of Lions andMagenes[76] on parabolic equations, we first need to prove the following lemma:

Lemma 2.5 The operator v → S ∂v∂S is continuous from V into V ′.

Proof. Call X and Y the differential operators

X(v) = Sy∂v

∂S+ β

∂v

∂y, Y (v) = Sy

∂v

∂S− β

∂v

∂y, (2.142)

The operators X and Y are continuous operators from V into L2(Q) and theiradjoints are

XT (v) = −Sy ∂v∂S

− β∂v

∂y− yv = −X(v)− yv,

Y T v = −Sy ∂v∂S

+ β∂v

∂y− yv = −Y (v)− yv.

(2.143)

Consider the commutator [X,Y ] = XY − Y X : it can be checked that

[X,Y ](v) = 2βS∂v

∂S. (2.144)

Therefore, for v ∈ V and w ∈ D(Q),

(2βS∂v

∂S,w) = −

∫

Q

Y (v)(X(w) + yw) +

∫

Q

X(v)(Y (w) + yw), (2.145)

and from (2.140), there exists a constant C such that

|(2βS ∂v∂S

,w)| ≤ C‖v‖V ‖w‖V .

To conclude, we use the density of D(Q) into V .

Lemma 2.5 implies that the operator Lt is continuous from V to its dual V ′.


Calling at the bilinear form defined on V × V by at(u, v) = 〈Ltu, v〉, we have

at(u, v) =1

2

∫

Q

y2S2 ∂u

∂S

∂v

∂S+

∫

Q

y2S∂u

∂Sv +

β2

2

∫

Q

∂u

∂y

∂v

∂y

+r(t)

2β

(∫

Q

Y (u)(X(v) + yv)−∫

Q

X(u)(Y (v) + yv)

)

+

∫

Q

((2η − 1)α(y −m) + βγ(S, y, t))∂u

∂yv

+

∫

Q

(r(t) + (1− η)

(2α2

β2η(y −m)2 + 2

α

β(y −m)γ(S, y, t)− α

))uv.

(2.146)

Proposition 2.4 Assume that r is a bounded function of time and that γ isbounded by a constant. The bilinear form at is continuous on V × V , with acontinuity constant independent of t.

We also need a Garding inequality:

Proposition 2.5 Assume that r is a bounded function of time and that γ isbounded by a constant Γ. If α > β, then there exist two positive constantsC and c independent of t and two constants 0 < η1 < η2 < 1 such that, forη1 < η < η2 and for any v ∈ V ,

at(v, v) ≥ C‖v‖2V − c‖v‖2L2(Q). (2.147)

From Propositions 2.4 and 2.5, we can prove the existence and uniqueness ofweak solutions to the initial value problem with (2.137).

Theorem 2.8 Assume that α > β and that η has been chosen as in Proposi-tion 2.5. Then, for any u ∈ L2(Q), there exists a unique u in L2(0, T ;V ) ∩C0([0, T ];L2(Q)), with ∂u

∂t ∈ L2(0, T ;V ′) such that, for any smooth functionφ ∈ D(0, T ), for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Q

u(t)v

)dt+

∫ T

0

φ(t)at(u, v)dt = 0 (2.148)

and

u(t = 0) = u. (2.149)

The mapping u 7→ u is continuous from L2(Q) to L2(0, T ;V )∩C0([0, T ];L2(Q)).

Remark 2.10 The ratio 2α2

β2 is exactly the ratio between the rate of mean rever-sion and the asymptotic variance of the volatility. The assumption in Theorem2.8 says that the rate of mean reversion should not be too small compared withthe asymptotic variance of the volatility. This condition is usually satisfied in

practice, since α is often much larger than the asymptotic variance β2

2α .


It is possible to prove a maximum principle, see[8]: as a consequence, in thecase of a vanilla put, we see that the weak solution given by Theorem 2.8 has afinancially correct behavior:

Proposition 2.6 Assume that the coefficients are smooth and bounded, andthat α > β. If P(S, y) = (K − S)+, then the function

P (t, S, y) = u(T − t, S, y)e(1−η)(y−m)2

2ν2 ,

where u is the solution to (2.148), (2.149) with u = e(1−η)(y−m)2

2ν2 P, satisfies

(S −Ke−r(T−t))− ≤ P (t, S, y) ≤ Ke−r(T−t), (2.150)

and we have the put-call parity C(t, S, y) − P (t, S, y) = S −Ke−r(T−t), if C isthe pricing function of the corresponding call option.

Consider now Lt as an unbounded operator defined on L2(Q) and call Dt thedomain of Lt, i.e. v ∈ V s.t. Ltv ∈ L2(Q). In [2], it is shown that Dt doesnot depend on t:

Theorem 2.9 If for all t, r(t) > 0, then Dt does not depend on t: Dt = D.Moreover, if there exists a constant r0 > 0 such that r(t) > r0 a.e., and ifα2

β2 > 2, then for well chosen values of η (in particular such that 2α2

β2 η(1−η) > 1),

D =

v ∈ V ; y2S2 ∂

2v

∂S2,∂2v

∂y2, yS

∂2v

∂S∂y, S

∂v

∂S, y∂v

∂y, y2v ∈ L2(Q)

. (2.151)

Then, from general results of Kato, (see [88] Theorem 5.6.8.), regularity resultson the solution to (2.148) (2.149) can be obtained:

Theorem 2.10 Assume that there exists ζ, 0 < ζ ≤ 1 such that γ belongs toCζ([0, T ], L∞(Q)) and r is a Holder function of time with exponent ζ. Assume

also that r(t) > r0 for a positive constant r0 and that α2

β2 > 2. Then for η chosen

as in Proposition 2.5 and Theorem 2.9, if u belongs to D defined by (2.151),then the solution u of (2.148) (2.149) belongs to C1((0, T );L2(Q))∩C0([0, T ];D)and the functional equation in L2(Q)

u′(t) + Ltu(t) = 0 (2.152)

is satisfied pointwise in [0, T ].Furthermore, for u ∈ L2(Q), the solution of (2.148) (2.149) also belongs toC1((τ, T );L2(Q)) ∩ C0([τ, T ];D), for all τ > 0 and satisfies

‖u′(t)‖L2(Q) + ‖Ltu(t)‖L2(Q) ≤C

t, for t > 0.

Remark 2.11 The same kind of analysis is possible for an extended Stein-Stein’s model with a nonzero correlation factor, but in this case, (still assuming


that γ is bounded) one has to cope with the term ρµ−r(t)|y|∂P∂y , which becomes

singular on the axis y = 0: therefore, one may need to impose a Dirichletcondition on the axis y = 0, of the form

P (S, 0, t) = g(S, t), 0 ≤ t < T, S > 0, (2.153)

where∂g∂t + r(t)

(S ∂g∂S − g

)= 0 0 ≤ t < T, S > 0,

g(S, t = T ) = P(S, 0).(2.154)

The Initial Value Problem for Heston’s Model We aim at making ananalysis for Heston’s model, in the same spirit as the one proposed above for theStein-Stein’s model. We consider the partial differential equation (2.133), in thesimple case when ρ = 0. We also assume that γ is bounded independently of t,S and y and we define γ(S, y, t) = γ(S, y, T − t). We need to set the variationalformulation in a suitable weighted Sobolev space compatible with the operatordegeneracy on the axis y = 0. For that, we introduce a smooth positive functionψ defined on R+ such that ψ(y) =

√y on (0,m) and ψ(y) = e−dy on (2m,+∞),

for some positive parameter d. We will fix d later on.We introduce the new unknown function

u(S, y, t) = ψ(y)P (S, y, T − t). (2.155)

Denoting by Lt the linear partial differential operator

Ltv = −1

2yS2 ∂

2v

∂S2− r(t)

(S∂v

∂S− v

)

−λ2

2y

(∂2v

∂y2− 2

ψ′

ψ

∂v

∂y− ψ(

ψ′

ψ2)′v

)− κ(m− y)

(∂v

∂y− ψ′

ψv

)

+λ√yγ(S, y, t)

(∂v

∂y− ψ′

ψv

),

(2.156)we obtain

∂u

∂t+ Ltu = 0. (2.157)

The equation clearly becomes degenerate on the axis y = 0, because the coeffi-

cients in front of the two operators ∂2u∂y2 and S2 ∂2u

∂S2 vanish. We denote by Q theopen sector Q = R+ × R+. Let us consider the weighted Sobolev space V :

V =

v :

(v,

v√y,√y∂v

∂y,√yS

∂v

∂S

)∈ (L2(Q))4

. (2.158)

This space with the norm

‖v‖V =

(∫

Q

(1 +1

y)v2 + y(

∂v

∂y)2 + yS2(

∂v

∂S)2) 1

2

(2.159)

is a Hilbert space, and it has the following properties:


1. V is separable.

2. Calling D(Q) the space of smooth functions with compact support in Q,D(Q) ⊂ V and D(Q) is dense in V

3. V is dense in L2(Q).

4. For any function v in V ,

∫

Q

yv2 ≤ 4

∫

Q

yS2(∂v

∂S)2. (2.160)

Remark 2.12 The reason for imposing that v√y be square integrable will appear

in Lemma 2.6 below. Note that the functions v(y) = logσ(y), with 0 < σ < 12 ,

are such that v and√yv′ are square integrable near y = 0, but v√

y is not square

integrable.

Lemma 2.6 The operator v → S ∂v∂S is continuous from V into V ′.

Proof. Call X and Y the differential operators

X(v) =√yS

∂v

∂S, Y (v) =

√y∂v

∂y, (2.161)

The operators X and Y are continuous operators from V into L2(Q) and theiradjoints are

XT (v) = −X(v)−√yv, Y T v = −Y (v)− 1

2√yv. (2.162)

It can be checked that

[X,Y ](v) = −1

2S∂v

∂S. (2.163)

Therefore, for v ∈ V and w ∈ D(Q),

(S∂v

∂S,w

)= 2

∫

Q

Y (v)(X(w) +√yw) − 2

∫

Q

X(v)(Y (w) +1

2√yw), (2.164)

and from (2.160), there exists a constant C such that

∣∣∣∣(S∂v

∂S,w

)∣∣∣∣ ≤ C‖v‖V ‖w‖V .

To conclude, we use the density of D(Q) into V .

Lemma 2.6 and the assumption made on ψ imply that the operator Lt is contin-uous from V to its dual V ′ because, in particular, the functions ψ′

ψ , 1yddy (y

ψ′

ψ )


and ψ ddy (

ψ′

ψ2 ) are bounded on [2m,+∞).

Calling at the bilinear form defined on V × V by at(u, v) = 〈Ltu, v〉, we have

at(u, v) =1

2

∫

Q

yS2 ∂u

∂S

∂v

∂S+

∫

Q

yS∂u

∂Sv

+λ2

2

(∫

Q

y∂u

∂y

∂v

∂y+

∫

Q

∂u

∂yv + 2

∫

Q

yψ′

ψ

∂u

∂yv +

∫

Q

yψ

(ψ′

ψ2

)′uv

)

−2r(t)

(∫

Q

Y (u)(X(v) +√yv)−

∫

Q

X(u)(Y (v) +1

2√yv)

)+ r(t)

∫

Q

uv

−κ∫

Q

(m− y)

(∂u

∂y− ψ′

ψu

)v + λ

∫

Q

√yγ(t, S, y)

(∂u

∂y− ψ′

ψu

)v.

(2.165)Assume that r is a bounded function of time and that γ is bounded by a con-stant. The bilinear form at is continuous on V × V , with a continuity constantindependent of t. We also need a Garding type inequality:

Proposition 2.7 Assume that r is a bounded function of time and that γ isbounded by a constant Γ. If

κmin(m,κ) >3

4λ2, (2.166)

then one can choose d (see the definition of ψ) such that there exist two positiveconstants C and c independent of t,

at(v, v) ≥ C‖v‖2V − c‖v‖2L2(Q), ∀v ∈ V (2.167)

Proof. It is enough to prove (2.167) for v ∈ D(Q). Several integrations by partlead to

at(v, v) =1

2

∫

Q

yS2

(∂v

∂S

)2

− 1

2

∫

Q

yv2 +3

2r(t)

∫

Q

v2 +λ2

2

∫

Q

y

(∂v

∂y

)2

−∫

Q

(λ2

2

(yψ′

ψ

)′+ κy

ψ′

ψ

)v2 + λ

∫

Q

√yγ(t, S, y)

(∂v

∂y− ψ′

ψv

)v

+

∫

Q

(κm

ψ′

ψ+λ2

2

(yψ

(ψ′

ψ2

)′)− κ

2

)v2

For brevity we skip many details, and we only focus on the main two steps ofthe proof:

First step Note that, if y < m then κmψ′

ψ + λ2

2

(yψ(ψ′

ψ2

)′)= 1

2y (κm− 34λ

2).

From this and (2.166), we see that the quantity∫Q∩y<m

v2

y is bounded by a

positive factor times∫Q∩y<m

(κmψ′

ψ + λ2

2

(yψ(ψ′

ψ2

)′))v2. On the other


hand, the quantity

∫

Q∩y<2m

∣∣∣∣∣−1

2y −

(λ2

2

(yψ′

ψ

)′+ κy

ψ′

ψ

)∣∣∣∣∣ v2

+

∫

Q∩m<y<2m

∣∣∣∣∣κmψ′

ψ+λ2

2

(yψ

(ψ′

ψ2

)′)∣∣∣∣∣ v2

is bounded by a constant times∫Q∩y<2m v

2.

Second step Calling

I =

∫

Q∩y>2m

(−1

2y −

(λ2

2

(yψ′

ψ

)′+ κy

ψ′

ψ

)+

(κm

ψ′

ψ+λ2

2

(yψ

(ψ′

ψ2

)′)))v2

we choose ψ such that there is an estimate of the form:

I ≥ −α∫

Q

yS2

(∂v

∂S

)2

− β

∫

Q

v2,

for some α < 12 and β ≥ 0. In view of (2.160), this will be a consequence of the

estimate

I ≥ −α4

∫

Q

yv2 − β

∫

Q

v2.

Therefore, let us look for ψ such that

∫

Q∩y>2m

((α

4− 1

2)y −

(λ2

2

(yψ′

ψ

)′+ κy

ψ′

ψ

)+

(κm

ψ′

ψ+λ2

2

(yψ

(ψ′

ψ2

)′)))v2

≥ −β∫

Q

v2.

Since ψ(y) = e−dy, y > 2m, the integrand above is

(α

4− 1

2)y − 1

2λ2(d2y − d) + κ(y −m)d,

so the estimate will be true if one can choose d such that − 12λ

2d2 +κd− 38 > 0.

This is the case if κ2 > 34λ

2.

The continuity of at and the Garding’s inequality yield the existence and unique-ness of a weak solution to the initial value problem with equation (2.157).

Theorem 2.11 Under the assumptions of Proposition 2.7, for any u ∈ L2(Q),there exists a unique u in L2(0, T ;V )∩C0([0, T ];L2(Q)), with ∂u

∂t ∈ L2(0, T ;V ′)such that, for any smooth function φ ∈ D(0, T ), for any v ∈ V ,

−∫ T

0

φ′(t)

(∫

Q

u(t)v

)dt+

∫ T

0

φ(t)at(u, v)dt = 0 (2.168)


andu(t = 0) = u. (2.169)

The mapping u 7→ u is continuous from L2(Q) to L2(0, T ;V )∩C0([0, T ];L2(Q)).

Remark 2.13 The assumption (2.166) plays the same role as the assumptionα > β in the discussion of the Stein-Stein’s model.

Remark 2.14 Note that no boundary condition have been imposed on ∂Q.

For example, if P(S) = (K−S)+ then u(S, y) = ψ(y)(K−S)+ is a squareintegrable function, and Theorem 2.11 can be applied to the case of a Europeanput.

A Numerical Method for Pricing Options with Stein-Stein’s ModelWe consider the partial differential equation (2.131) with f(y) = |y|. We assumethat the interest rate r is constant, and we take ρ = 0, γ = 0. The partialdifferential equation is rewritten in divergence form, with the new variable T −t→ t:

0 =∂P

∂t− 1

2

(∂

∂S

(y2S2 ∂P

∂S

)+

∂

∂y

(β2 ∂P

∂y

))

+(y2S − rS

) ∂P∂S

+ α(y −m)∂P

∂y+ rP

(2.170)

A first order implicit Euler scheme is used for time discretization:

0 =Pm − Pm−1

δt− 1

2

(∂

∂S

(y2S2 ∂P

m

∂S

)+

∂

∂y

(β2 ∂P

m

∂y

))

+(y2S − rS

) ∂Pm∂S

+ α(y −m)∂Pm

∂y+ rPm

(2.171)

Aiming at obtaining a discrete version of this equation, we first truncate thedomain, i.e. we introduce the rectangle Ω = (0, S)× (−y, y), with S and y largeenough. We are looking for a numerical approximation of P in Ω. For that, wefirst need to supply reasonable artificial boundary conditions on boundaries of∂Ω, in agreement with the payoff function. Let us discuss the artificial boundarycondition for a vanilla put option, i.e. P(S) = (K − S)+.

• On ∂Ω ∩ S = S, we impose ∂P∂S (S, y, t) = 0. Such a condition is reason-

able if S is large enough compared to K.

• On ∂Ω∩y = ±y, finding an accurate artificial boundary condition is noteasy. However, if y is chosen such that α(y ±m) ≫ β2, then for y ∼ ±y,the coefficient of the advection term in the y direction, i.e. α(y − m),is much larger in absolute value than the coefficient of the diffusion in

the y direction, i.e. β2

2 . Furthermore, near y = ±y, the vertical velocityα(y−m) is directed outward Ω. Therefore, the error caused by an artificialcondition on y = ±y will be damped away from the boundaries, and


localized in boundary layers whose width is of the order of β2

αy . Therefore,

Dirichlet boundary conditions, e.g. P (S,±y, t) = 0, will not cause a largeerror for |y| small enough compared to y, for example |y| ≤ y

2 , even thoughthese conditions are not satisfied at all by the exact solution.

• No boundary condition is needed on S = 0, because of the degeneracy ofthe equation.

Let Vh be the space of continuous piecewise linear functions on a triangulation ofΩ, which are equal to zero on y = ±y. We consider the following finite elementdiscretization: ∀vh ∈ Vh,∫

Ω

(1

δt+ r)Pmv +

1

2

∫

Ω

y2S2 ∂Pm

∂S

∂v

∂S+β2

2

∫

Ω

∂Pm

∂y

∂v

∂y

+

∫

Ω

(y2S − rS)∂Pm

∂Sv + α

∫

Ω

(y −m)∂Pm

∂yv =

1

δt

∫

Ω

Pm−1v

(2.172)To illustrate this, let us take for the parameters

r = 0.05, α = 1, ν = 0.5, m = 0.2, K = 100. (2.173)

The goal is to approximate P in the domain (0, S) × (−1.5, 1.5) for t smallerthan 1. We choose S = 800. For computing the solution in (0, S)× (−1.5, 1.5),we choose the larger domain Ω = (0, S)× (−3, 3), i.e y = 3.We compute the pricing function of the the put option one year to maturity. Thetime step has been set to 6 days. The artificial boundary conditions u(S,±y, t) =Ke−rt have been used. In Figure 2.13, we display the contours of the solutionin Ω = (0, 800)× (−3, 3). On this figure, it is very well seen that the artificialboundary conditions on y = ±3 do not affect the solution in the region |y| < 1.5.

Remark 2.15 The choice of α = 1 is not quite realistic from a financial view-point, if the asset is linked to stocks, because the mean reversion rate is generallylarger. When the asset corresponds to interest rates, such values of α are rea-sonable. When the mean reversion rate is large, it is possible to carry out anasymptotic expansion of the solution as in [44] and we believe that the varia-tional setting introduced above justifies these expansions.

The analysis of convergence of the finite element method for the ellipticpart of operator appearing in (2.170) was performed in [2]. In this paper, itwas shown that the finite element error analysis theory as in [28, 27] couldbe performed, as soon as the family of meshes under consideration satisfy aregularity assumption with respect to an intrinsic metric associated with thedegenerate operator: more precisely, defining X and Y the smooth vector fieldsin R

2:X = Sy∂S , Y = ∂y, (2.174)

we say that an absolutely continuous curve γ : [0, T ] −→ R2 is a sub-unit curvewith respect to X if for any ξ ∈ R2

〈γ(t), ξ〉2 ≤ 〈X(γ(t)), ξ〉2 + 〈Y (γ(t)), ξ〉2


Figure 2.13: The contours of the price computed in Ω = (0, 800)× (−3, 3): notethe boundary layers due to artificial boundary conditions on y = ±3.


for a.e. t ∈ [0, T ]. Following e.g. [83]), we define the intrinsic or Carnot-Caratheodory metric d: ∀P1, P2 ∈ R2,

d(P1, P2) = inf T > 0 : there exists a sub-unit curve γ,

γ : [0, T ] −→ R2, γ(0) = P1, γ(T ) = P2,

If the above set of curves is empty, we take d(P1, P2) = ∞.In [2], it was essentially required that the family of meshes be regular withrespect to the Carnot-Caratheodory metric: there exists a parameter σ suchthat, for any triangle T , one can find two Carnot-Caratheodory balls, the firstone containing T and the other one contained in T , such that, calling r1 andr2 their Carnot-Caratheodory radii, (r1 > r2), we have r1

r2≤ σ. Under such

an assumption, it is possible to construct a local regularization operator similarto that of Clement, [29], and then to obtain optimal error estimates. In [2],examples of meshes satisfying the regularity assumption were given.

A Numerical Method for Pricing Options with Heston’s Model WithHeston’s model, in contrast with the last example, the advection in the y vari-able does not dominate the diffusion as y → ∞. Therefore, inexact boundaryconditions on an artificial boundary y = y may produce large errors. For thisreason, another strategy has been chosen: instead of truncating the domain inthe variable y, we have used a suitable change of variable in order to map they-domain, i.e., R+ onto the interval (0, 1).We want to approximate P given by (2.133). The idea is to make the changeof variables z = y/(y + 1), which maps R+ onto (0, 1). The inverse map isy = z/(1− z).In the variables (T − t, S, z), the partial differential equation becomes

∂P

∂t+ LtP = 0. (2.175)

for t ∈ (0, T ], S ∈ R+ and z ∈ (0, 1), where

Ltv = −1

2

z

1− zS2 ∂

2v

∂S2− r(t)

(S∂v

∂S− v

)− ρλS(1− z)z

∂2v

∂S∂z

−λ2

2z(1− z)2

((1 − z)

∂2v

∂z2− 2

∂v

∂z

)− κ(m− z

1− z)(1 − z)2

∂v

∂z,

(2.176)assuming that γ = 0.No boundary condition is needed on the axis z = 0 because the partial differen-tial operator becomes degenerate there: indeed, all the coefficients of the secondderivatives vanish; moreover, the first order derivatives correspond to an advec-tion with an outgoing velocity, i.e., the coefficient in front of ∂v

∂z , is negativenear z = 0 (its value is −κm). Similarly, no boundary condition is needed onthe axis z = 1 because L becomes degenerate near z = 1.On the other hand, one can truncate the domain in the S variable. The newproblem is posed in the rectangle (0, S) × (0, 1), which allows for the use of a


finite element method. We have refined the mesh near the strike.We have made the following choice of parameters

r = 0, ρ = −0.5, κ = 2.5, λ = 0.5, m = 0.06, K = 1.

We have taken S = 4. An approximation of P solution of (2.133) is obtainedfrom the finite element approximation of P (or possibly e−γyP ) by performingthe inverse change of variable z 7→ y = z/(1−z). The pricing functions of the putand the call half-year to maturity are displayed in Figure 2.14. The computedprices are in good agreement with the closed form obtained by Heston, see [58].

2.4.3 American Options with Stochastic Volatility

In this paragraph, we discuss the pricing of an American put option with theStein-Stein stochastic volatility model. The parameters of the model are givenby (2.173). The domain truncation is also the same as in the example of theEuropean option. Piecewise linear finite element are used for the discretization.We have chosen to use the first order operator splitting or projection schemedescribed in § 2.3.4. A similar method has been studied in [63] for Heston’smodel. In order to capture the exercise boundary, we have adapted the meshin the variables S and y. In Figure 2.15, we have plotted the contours of thepricing function one year to maturity: the exercise zone clearly appears: indeed,it corresponds to the zone where the pricing function matches the functionS 7→ K − S, i.e. where the contours are vertical straight lines. Figure 2.15 hasto be compared with Figure 2.13 for the European option. In Figure 2.16, wehave plotted the exercise region one year to maturity. The mesh is visible too.It is refined near the exercise boundary.

In [64], Ikonen and Toivanen have proposed specific finite difference meth-ods and solution procedures for American options with Heston’s model. Specialdiscretization and grid are designed in such a way that the resulting matrix bean M-matrix. The scheme is a seven points scheme and upwinding is used whennecessary. A specific alternating direction splitting scheme is proposed. Eachsub-step consists of solving a one dimensional linear complementarity problemby the Brennan-Schwartz algorithm, see [22]. Three directions are used: thetwo axes and the first diagonal. For these algorithms to work, one needs thatthe tridiagonal matrices used in the substeps are M-matrices. This is not true ingeneral but the scheme and the grid have been designed so that this conditionholds. It is also necessary that the exercise boundary intersects the directionsonce at most. This condition is not proved but no counterexample has beenfound in the computations. Ikonen and Toivanen have compared this solutionprocedure with four other methods, in particular the previously described pro-jection scheme and a multigrid algorithm; they have shown that the alternatingdirections method performs best. On the other hand, the last solution procedurehas been tailored for this problem and its robustness has to be assessed.


0.5 1 1.5 2 2.5 3 0

0.5

1

1.5

2

2.5

3

0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05

0.8 0.9 1 1.1 1.2 1.3 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

Figure 2.14: The contours of the pricing function of a put(top)/call(bottom)option with Heston model half-year to maturity


Figure 2.15: The pricing function of the American option one year to maturity.The exercise zone is clearly visible.


0 20 40 60 80 100-1

-0.5

0

0.5

1

"exercise"

Figure 2.16: The exercise zone one year to maturity. One clearly sees that themesh has been refined near the exercise boundary.


2.4.4 Volatility Models with Several Stochastic Variables

We give the example of a generalized multifactor Scott model: we consider dfully correlated Orstein-Uhlenbeck processes

dY(i)t = −λiY (i)

t dt+ βidBt, i = 1, . . . d, (2.177)

where Bt is one dimensional Brownian motion. We consider an asset whoseprice is a lognormal process:

dSt = r(t)Stdt+ σtStdWt,

whereWt is a Brownian motion independent of Bt and the volatility σt is of theform

σt = σ

(d∑

i=1

Y(i)t , t

).

The price of the option is P (St, Y(i)t , . . . , Y

(d)t , t) where P satisfies the d + 2

dimensional PDE:

∂P

∂t+

1

2σ2(z1, t)S

2 ∂2P

∂S2+ rS

∂P

∂S

+1

2

∑

i,j

βiβj∂2P

∂yi∂yj+ ρβiσ(z1, t)S

∂2P

∂S∂yi−

d∑

i=1

λiyi∂P

∂yi− rP = 0,

where z1 =∑di=1 yi. One can make the change of variables z = Qy where

Q =

1 1 . . . . . . 1−β2/β1 1 0 . . . 0

−β3/β1 0 1. . .

......

.... . .

. . . 0−βd/β1 0 . . . 0 1

and get the PDE

∂P

∂t+1

2σ2(z1, t)S

2 ∂2P

∂S2+rS

∂P

∂S+1

2β2 ∂

2P

∂z21+ρβσ(z1, t)S

∂2P

∂S∂z1−zTLT∇zP−rP = 0

where β =∑d

i=1 βi and L = QDiag(λ1, . . . , λd) Q−1. This linear PDE is

parabolic with respect to the variables S and z1 and hyperbolic with respect tozi, 1 < i ≤ d.Sparse grid methods can be used for approximating the five variables functionP . We give an example provided: the parameters of the three factors modelsare

• interest rate: r = 5%,


• spot price-volatility correlation: ρ = −0.5,

• mean value of the volatility: σ = 0.2.

• parameters of the OU processes: λ ≈ (29.27, 2.45, 0.108), β = (1.26, 0.42, 0.42).

With these parameters, one may truncate the domain of computation becausethe velocity in the advection terms in the PDE are directed outward near theartificial boundaries, and the error produced by the inexact boundary conditionsare small sufficiently far from the artificial boundaries.In Table 2.2, we compute the price of a European call option one year to maturityfor several spot values, (the strike is 1), for yi = 0, i = 1, 2, 3. The payofffunction depends on the variable S only. Hence, the singularity is located on anhyperplane in the spot/volatilities space, and sparse grids are well adapted. ACrank-Nicolson scheme with a time step of 0.01 year has been used.We compare the results of the sparse grid method with refinement levels of 7, 8and 9 with a Monte-Carlo simulation. Note that the sparse grid approximationis sharper for spots larger than or equal to the strike, because the sparse grid isrelatively coarser for small spots.

Spot Level = 7 Level = 8 Level = 9 Monte Carlo0.80 1.51 1.48 1.45 1.410.85 2.86 2.81 2.78 2.750.90 4.78 4.72 4.71 4.670.95 7.29 7.23 7.24 7.221.00 10.32 10.31 10.31 10.321.05 13.82 13.84 13.85 13.881.10 17.72 17.75 17.77 17.811.15 21.92 21.94 21.96 22.011.20 26.31 26.35 26.38 26.43

Table 2.2: Price of a European call one year to maturity. These results havebeen obtained by D. Pommier.

The tests have been done on a 2.66GHz Intel Xeon processors with 1.5GbRAM. The computing time is approximately 4 minutes for n = 7, 15 minutesfor n = 8 and 1 hour for n = 9.

Chapter 3

Sensitivity and Calibration

3.1 Sensitivity

It is important to compute the sensitivity of options’ prices to parameters suchas the spot price or the volatility. The partial derivatives with respect to the rel-evant parameters are called the Greeks: let C be the price of a vanilla Europeancall:

• the δ (delta) is its derivative with respect to the stock price S : ∂C∂S .

• the Θ or time-decay is its derivative with respect to time: ∂C∂t .

• the vega κ is its derivative with respect to the volatility σ,

• the rho ρ is its derivative with respect to the interest rate, ∂C∂r ,

• η is its derivative with respect to the strike K

• finally, the gamma is the rate of change of its delta : ∂2C∂S2 .

Equations can be derived for these by directly differentiating the partial dif-ferential equation and the boundary conditions which define C. Automaticdifferentiation (AD for brevity) of a computer code for option pricing providesa way to do that efficiently and automatically.

3.1.1 Automatic Differentiation

Automatic differentiation of computer programs is based on the idea that everyline of a program can be differentiated analytically. Consider, for instance, theC-program which computes J = (u2 − u2d) for some values of u and ud

int main()

double J, aux, u=2.5, u_d=1;

aux = u-u_d;

J=aux*(u+u_d);

cout<<"J="<<J<<endl;

103

104 CHAPTER 3. SENSITIVITY AND CALIBRATION

It can be made to compute the derivative of J with respect to u as well byadding above each instruction its derivative with respect to u:

int main()

double dJdu,J, dauxdu,aux, u=2, u_d=0.1;

dauxdu=1;

aux = u-u_d;

dJdu=dauxdu*(u+u_d) + aux;

J=aux*(u+u_d);

cout<<"J="<<J<<" dJdu="<<dJdu<<endl;

However it is more systematic to compute differentials and consider thatevery double or float variable has an infinitesimal variation, potentially andinitially set to zero.

Let J(u) = (u− ud)(u + ud), then its differential is

δJ = (u− ud)(δu+ δud) + (u+ ud)(δu− δud) (3.1)

Obviously the derivative of J with respect to u is obtained by setting δu = 1and δud = 0. Now suppose that J is programmed in C/C++ by

double J(double u, double u_d)

double z = u-u_d;

z = z*(u+u_d);

return z;

int main() double u=2, u_d = 0.1;

cout << J(u,u_d) << endl;

A program which computes J and its differential can be obtained by writingabove each differentiable line its differentiated form:

double JandDJ(double u, double u_d, double du, double du_d, double *pdz)

double dz = du - du_d, z = u-u_d;

double dJ = dz*(u+u_d) + z*(du + du_d);

z = z*(u+u_d);

*pdz = dz;

return z;

int main()

double dJ, u=2,u_d = 0.1;

cout << J(u,u_d,1,0,&dJ) << endl;

Except for the embarrassing problem of returning both z,dz instead of z,the procedure can be automatized by introducing a structured type holding thevalue of the variable and of its derivative:

struct double val[2]; dreal;

dreal JandDJ(dreal u, dreal u_d)

dreal z;

z.val[1] = u.val[1]-u_d.val[1];

3.1. SENSITIVITY 105

z.val[0] = u.val[0]-u_d.val[0];

z.val[1] = z.val[1]*(uval[0]+u_d.val[0])+z.val[0]*(uval[1]+u_d.val[1]);

z.val[0] = z.val[0]*(uval[0]+u_d.val[0]);

return z;

int main()

dreal u, dJ;

u.val[0]=2;u_d.val[0]=0.1;u.val[1]=1;u_d.val[1]=0;

cout <<J(u,u_d).val[0]<<J(u,u_d,1,0).val[1];

The class ddouble

In C++ the program can be simplified further by redefining the operators =,-,+ and *. Then a class has to be used instead of a struct!

class ddouble

public: double val[2];

ddouble(double a, double b=0) v[0] = a; v[1]=b;

ddouble operator=(const ddouble& a)

val[1] = a.val[1]; val[0]=a.val[0];

return *this;

friend ddouble operator - (const ddouble& a, const ddouble& b)

ddouble c;

c.v[1] = a.v[1] - b.v[1]; // (a-b)’=a’-b’

c.v[0] = a.v[0] - b.v[0];

return c;

friend ddouble operator + (const ddouble& a, const ddouble& b)

ddouble c;

c.v[1] = a.v[1] + b.v[1]; // (a-b)’=a’-b’

c.v[0] = a.v[0] + b.v[0];

return c;

friend ddouble operator * (const ddouble& a, const ddouble& b)

ddouble c;

c.v[1] = a.v[1]*b.v[0] + a.v[0]* b.v[1];

c.v[0] = a.v[0] * b.v[0];

return c;

;

For more complex programs all the operators and all the functions of algebrahave to to be redefined in the class ddouble. For instance

ddouble ddouble::sqrt(const ddouble& x)

ddouble c;

c.v[0] = sqrt(x.v[0]);

c.v[1] = 0.5*x.v[1]/sqrt(x.v[0]);

return c;

Obviously it creates a problem at x = 0 but there the function is non-differentiable anyway.


The library ddouble.hpp implements correctly all these; it can be downloadedfrom http://www.ann.jussieu.fr/pironeau.

3.1.2 Sensitivity by Automatic Differentiation

The most obvious application of Automatic Differentiation is for the computa-tion of the sensitivity with respect to the parameters of the models (the Greeks).For this there is very little to change in the computer program; for example,consider a local volatility function of S and t:

σ = σ0(1 + a√|S −K|)(1 + bt). (3.2)

The following changes are made to computes also the sensitivity with respectto b at b = 1 for a = 0.01 and σ0 = 0.3.

1. In the file optionhb.hpp we make now an explicit use of ddouble.hpp

#include "ddouble.hpp" // for automatic differentiation

// typedef double ddouble; // for compatibility with AD

2. and in the main function the definition of b has value 1 in the secondargument to indicate that derivatives are with respect to this variable.

int main()

VarMesh m(50,100,0.5,300.);// uniform nb x, nb t, T, xmax

const ddouble b(1.,1.); // for derivatives with respect to b at b=1

const double K=100, a=0.01;

Option p(1,&m,K,0.05,0.3); // B&S,mesh,strike,r,sigma


for(int j = 1; j < m.nX[i]; j++) // local volatility

p.sigma[i][j] *= (1 + a*sqr(fabs(m.x[i][j]-K)))*(1+b*m.t[i]);

p.calc();

Results are in p.u[i][j].val[0] for the price of the put and in p.u[i][j].val[1] for itsderivative with respect to b. Both are shown on Figure 3.1

3.2 Calibration

3.2.1 Limitation of the Black-Scholes Model: the Need for

Calibration

Consider a European option on a given stock with a maturity T and a payofffunction P0, and assume that this option is on the market. Call p its presentprice. Also, assume the risk-free interest rate is the constant r. One may asso-ciate with p the so-called implied volatility, i.e. the volatility σimp such that theprice given by formula (1.6) at time t = 0 with σ = σimp coincides with p. If theBlack-Scholes model was sharp, then the implied volatility would not depend onthe payoff function P0. Unfortunately, for e.g. vanilla European puts or calls,

3.2. CALIBRATION 107

0 50

100 150

200 250

300 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5

0 10 20 30 40 50 60 70 80 90

100

"u.txt" using 1:2:3

0 50

100 150

200 250

300 0 0.05

0.1 0.15

0.2 0.25

0.3 0.35

0.4 0.45

0.5

0

1

2

3

4

5

6

7

"u.txt" using 1:2:3

Figure 3.1: The pricing function of the put option (left) and its derivative withrespect to the parameter b when σ is given by (3.2)

it is often observed that the implied volatility is far from constant. Rather, itis often a convex function of the strike price. This phenomenon is known asthe volatility smile. A possible explanation for the volatility smile is that thedeeply out-of-the-money options are less liquid, thus relatively more expensivethan the options in-the-money.This shows that the critical parameter in the Black-Scholes model is the volatil-ity σ. Assuming σ constant and using (1.3) often leads to poor predictions of theoptions’prices. The volatility smile is the price paid for the too great simplicityof Black-Scholes’assumptions. It is thus necessary to use more involved modelswhich must be calibrated.Let us first explain what the term calibration means: consider an arbitrage-freemarket described by a probability measure P on a scenario space (Ω,A). Thereis a risk-free asset whose price at time τ is erτ , r ≥ 0 and a risky asset whoseprice at time τ is Sτ . Specifying an arbitrage-free option pricing model necessi-tates the choice of a risk-neutral measure, i.e. a probability P∗ equivalent to P

such that the discounted price (e−rτSτ )τ∈[0,T ] is a martingale under P∗. Such

a probability measure P∗ allows for the pricing of European options; consider aEuropean option with payoff P at maturity t ≤ T : its price at time τ ≤ t isPτ = e−r(t−τ)EP

∗(P(St)|Fτ ), where (Fτ )τ∈[0,T ] is the natural filtration.

The pricing model P∗ must be compatible with the prices of the options ob-served on the market, whose number may be large. Model calibration consistsof finding P∗ such that the discounted price (e−rτSτ )τ∈[0,T ] is a martingale, andsuch that the option prices computed with the model coincide with the observedoption prices. This is an inverse problem.Popular extensions to the Black-Scholes model are:

• local volatility models: the volatility is a function of time and of the spotprice, i.e. σt = σ(St, t). With suitable assumptions on the regularity andthe behavior at infinity of the function σ, (1.6) holds, and Pt = p(St, t),where p satisfies the final value problem (1.3), in which σ varies with t andS. Calibration of local volatility has been much studied, see for example


[37, 9, 66, 5] for volatility calibration with European options and [1, 7]with American options;

• stochastic volatility models: one assumes that σt = f(yt), where yt is acontinuous time stochastic process, correlated or not to the process drivingSt, see § 2.4. Stochastic volatility calibration has been performed in [84].

• Levy driven spot price: one may generalize the Black-Scholes model byassuming that the spot price is driven by a more general stochastic process,e.g. a Levy process [30, 80]. Levy processes are processes with stationaryand independent increments which are continuous in probability. For aLevy process Xτ on a filtered probability space with probability P, theLevy-Khintchine formula says that there exists a function χ : R → C suchthat

E(eiuXτ ) = eτχ(u),

χ(u) = −σ2u2

2+ iβu+

∫

|z|<1

(eiuz − 1− iuz)ν(dz) +

∫

|z|>1

(eiuz − 1)ν(dz),

for σ ≥ 0, β ∈ R and a positive measure ν onR\0 such that∫Rmin(1, z2)ν(dz) <

+∞. The measure ν is called the Levy measure of X . We focus on Levymeasure with a density, ν(dz) = k(z)dz. Assume that the discounted priceof the risky asset is a square integrable martingale under P and that it isrepresented as the exponential of a Levy process:

e−rτSτ = S0eXτ .

The martingale property is that E(eXτ ) = 1, i.e.

∫

|z|>1

ezν(dz) <∞, and β = −σ2

2−∫

R

(ez − 1− z1|z|≤1)k(z)(dz).

and the square integrability comes from the condition∫|z|>1 e

2zk(z)dz <∞.With such models, the pricing function for a European option is obtainedby solving a partial integrodifferential equation (PIDE), with a nonlocalterm, see[89, 30] for the analysis of this equation and [79, 78, 33, 6] fornumerical methods based on the PIDE. Calibration of Levy models withEuropean options has been discussed in [31, 32].

In this paragraph, we assume that the model is characterized by parameters θin a suitable class Θ.The last two classes of models (stochastic volatility and Levy driven assets) de-scribe incomplete markets, see [31]: the knowledge of the historical price processalone does not allow to compute the option prices in a unique manner. Whenthe option prices do not determine the model completely, additional informationmay be introduced by specifying a prior model. If the historical price processhas been estimated statistically from the time series of the underlying asset,


this knowledge has to be injected in the inverse problem; calling P0 the priorprobability measure obtained as an estimation of P, the inverse problem maybe cast in a least-square formulation of the type: find θ ∈ Θ which minimizes

∑

i∈Iωi(P θ(0, S, ti, xi)− pi

)2+ ρJ2(P

θ,P0), (3.3)

where• ωi are suitable positive weights,• S is the price of the underlying asset today,• P θ(0, S, ti, xi) is the price of the option with maturity ti strike xi, computedwith the pricing model associated with θ,• ρJ2(Pθ,P0) is a regularization term which measures the closeness of the modelPθ to the prior. The number ρ > 0 is called the regularization parameter. This

functional has two roles: 1) it stabilizes the inverse problem; for that, ρ shouldbe large enough and J2 should be convex or at least convex in a large enoughregion; 2) it guarantees that Pθ remains close to P0 in some sense. The choiceof J2 is very important: J2(P

θ,P0) is often chosen as the relative entropy ofthe pricing measure Pθ with respect to the prior model P0, see [10], becausethe relative entropy becomes infinite if Pθ is not equivalent to P0. Some au-thors have argued that such a choice may be too conservative in some cases, fortwo reasons: a) the historical data which determine the prior may be missingor partially available -b) in the context of e.g. volatility calibration, once thevolatility is specified under P0, then the volatility under Pθ must be the samefor the relative entropy to be finite. A different approach was considered whichallowed for volatility calibration, see [11].Note that local volatility models describe complete markets: however, an addi-tional regularization cost functional is necessary too, as explained in the para-graph below.

3.2.2 The Calibration of Local Volatility

Consider a family of I call options on the same asset with maturities (Ti)1≤i≤Iand strike prices (Ki)1≤i≤I . Assume that the options are available on the mar-ket, so one can observe their prices today. Call (Ci)1≤i≤I their prices. Assumethat the spot price today is S0. Calibrating the local volatility amounts to findinga local volatility surface S, t → σ(S, t) such that if Ci(S, t,Ki, Ti) is computedby solving the boundary value problem

∂Ci∂t

+σ(S, t)2

2S2 ∂

2Ci∂S2

+ rS∂Ci∂S

+ rCi = 0, Ci(S, Ti) = (S −Ki)+, (3.4)

then Ci(S0, 0) coincides with the observed price Ci, for 1 ≤ i ≤ I.

A natural idea for this is to use least squares, i.e. to minimize a functional

JLS : σ 7→∑

i∈Iωi|Ci − Ci(S0, 0)|2


for σ in a suitable function set Σ, where ωi are positive weights and Ci is com-puted by solving (3.4). The evaluation of JLS requires the solution of I initialvalue problems. The set Σ where the volatility is to be found must be chosenin order to ensure that from a minimizing sequence one can extract at least asubsequence that converges in Σ, and that its limit is indeed a solution of theleast square problem. For example, Σ may be a compact subset of a Hilbertspace W such that the mapping JLS is continuous in W (lower semi-continuouswould be enough). In practice, W has a finite dimension and is compactly em-bedded in the space of bounded and continuous functions σ such that S∂Sσ isbounded. Thus, the existence of a solution to the minimization problem is mostoften guaranteed. What is more difficult to guarantee is uniqueness and stabil-ity: is there a unique solution to the least square problem? If yes, is the solutioninsensitive to small variations of the data? the answer to these questions is noin general, and we say that the problem is ill-posed.As a possible cure to ill-posedness, one usually modifies the problem by mini-mizing the functional

J : σ 7→ JLS(σ) + JR(σ)

instead of JLS , where JR is a sufficiently large strongly convex functional definedon W and containing some financially relevant information. For example, onemay choose

JR(σ) = ω‖σ − σ‖2,

where ω is some positive weight, ‖.‖ is a norm in W and σ is a prior localvolatility, which may come from a historical knowledge. The difficulty is that ωmust not be too large not to perturb the inverse problem too much, but not toosmall to guarantee some stability. The art of the practitioner lies in the choiceof JR.

3.2.3 Gradient Methods

Once the least square problem is chosen, we are left with proposing a strategy forthe construction of minimizing sequences. If JLS and JR are C1 functional, thengradient methods may be used. The principle of gradient methods is as follows:let δσ be an admissible variation and assume that J is Frechet differentiablewith respect to σ; then by definition

δJ = J(σ + δσ)− J(σ) = gradJ(σ) · δσ + o(|δσ|). (3.5)

So δσ = −ρgradJ(σ) causes a decrease of J , at least when ρ is small.

Gradient Algorithm with Armijo Rule

1. Choose an initial σ0 and 2 numbers 0 < α < 1, ρ0 > 0.

2. Set Hm = −gradJ(σm)


3. Compute by dichotomy a signed integer k such that, with ρ = ρ02k

J(σm + ρHm)− J(σm) ≤ ρα gradJ(σm) ·Hm

J(σm + 2ρHm)− J(σm) ≥ 2ρα gradJ(σm) ·Hm (3.6)

4. Set σm+1 = σm + ρHm and proceed to the next iteration.

Conjugate Gradient Method In the Polak-Ribiere [90] version one changesthe definition of Hm, m > 1 to

Hm = −gradJ(σm)+γHm−1 with γ =gradJ(σm) · (gradJ(σm)− gradJ(σm−1))

‖gradJ(σm−1)‖2

Under strong hypothesis such as local convexity and twice differentiability of J ,this algorithm converges to a local minimum superlinearly, i.e. faster than anygeometric progression.The drawbacks and advantages of gradient methods are well known: on the onehand, they do not guarantee convergence to the global minimum if the functionalis not convex, because the iterates can be trapped near a local minimum. Onthe other hand, they are fast and accurate when the initial guess is close enoughto the minimum. For these reasons, gradient methods are often combined withtechniques that permit to localize the global minimum but that are slow, likesimulated annealing or evolutionary algorithms.

3.2.4 A Finite Dimensional Example

To reduce the size of the problem we parametrize the volatility surface by alinear-quadratic spline (LQS). The idea is that we want the local volatility tobe close to linear in the money and to have a convex shape out of the money;this is desired at each time t but the bounds may depend on t linearly. Hencethe surface is given by

σ(S, t) =

a+ (2σ1 − a

S1− σ2 − σ1S2 − S1

)S + (σ2 − σ1S2 − S1

− σ1 − a

S1)S2

S1, if S < S1,

σ2S − S1

S2 − S1+ σ1

S2 − S

S2 − S1, if S1 ≤ S ≤ S2,

σ2 + (S − S2)σ2 − σ1S2 − S1

+

((S − S2)

σ2 − σ1S2 − S1

)2

, if S > S2,

(3.7)where Si, σi, i = 1, 2 are linear with respect to t:

Si = Si1(1−t

T) + Si2

t

T, σi = σi1(1−

t

T) + σi2

t

T. (3.8)

The local volatility is thus C1 regular, linear w.r.t. S for S1 ≤ S ≤ S2; it takesthe values σ1 at S = S1, σ2 at S = S2 and a at S = 0. For a, T given, the local


volatility depends on eight parameters: Sij , σij . However, the representation(3.7) (3.8) does not make sense unless 0 < S1(t) < S2(t), so it is better to set

z0 =√S11, z1 =

√S21, z2 =

√S21 − S11, z3 =

√S22 − S12,

z2∗i+j+1 =

√σij

1 +√σij

, i, j = 1, 2,(3.9)

so as to work with the set of parameters zi70; the last formula gives σij =z22∗i+j+1/(1 + z22∗i+j+1) ∈ (0, 1) for all possible values of z2∗i+j+1. This is asimple way to force σ in (0, 1) for all values of the parameters z.

A Test Case Table 3.1 contains the observed prices of several European callson the same asset. The parameter r is constant: r = 0.03. The spot price is1418.3.The least square problem is

minσ∈Σ

I∑

i=1

|Ci(S0, 0)− Ci|2, subject to (3.4). (3.10)

The conjugate gradient algorithm was used to solve (3.10) with the initial guessfor the parameter vector

z0 =√1500, z1 =

√1700, z2 = z3 =

√150, z4+i =

√0.4

1 + i− 0.4i = 0..3.

The cost function J has I = 57 terms. The other parameters are a = 4, T = 3(years). The results are shown on figure 3.2. The strange shape of the volatilitysurface is due to the parametrization. If in (3.7) we change the definition forS > S2 to be

σ2 + (S − S2)σ2 − σ1S2 − S1

+

((S − S2)

σ2(0)− σ1(0)

S2(0)− S1(0)

)2

, if S > S2 (3.11)

then the shape is more natural, see figure 3.3, and the same precision is obtained.

3.2.5 Dupire’s Equation

Solving (3.10) by a conjugate gradient method is time consuming because theBlack-Scholes PDE has to be solved I times at each iterations and for eachρ-trial in the dichotomy (3.6). Dupire[36] noticed that the function (K, τ) 7→C(S, t,K, τ) (now t and S are fixed) satisfies a forward parabolic PDE. One canobtain this either by reasoning directly on (1.6) or by PDE arguments usingadjoint operators. We present the second approach:


Strike 1 Month 2 Months 6 Months 12 Months 24 Months 36 Months700 733800 650.6900 569.81000 467.81100 385.31150 345.41175 265.21200 242 266.1 306.61215 253.41225 219 2451250 196.6 224.2 269.21275 174.5 203.9 2511300 152.9 184.1 233.21325 131.9 164.9 215.81350 111.7 146.3 198.91365 1001375 50.6 60 92.5 139 182.61380 46.1 55.81385 41.8 51.81390 37.5 47.91395 33.4 441400 29.4 40.3 74.5 128.4 166.7 215.91405 25.6 36.71410 21.9 33.21415 18.7 29.81420 15.4 26.61425 12.7 23.8 58 111.4 151.51430 10 20.71435 8 18.21440 6.3 15.71445 4.4 13.41450 3.1 11.3 43.3 95.2 136.9 187.51455 2.05 9.61460 1.45 7.91475 30.6 80.21500 20.3 54 109.6 160.81525 12.6 42.71550 7.5 331575 24.71600 1.95 18.2 64.5 113,91700 32.7 75,71800 15.51900 5.2

Table 3.1: The prices of a family of calls on the same asset


600 800

1000 1200

1400 1600

1800 0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600 700 800

’od.txt’’ud.txt’

0 500

1000 1500

2000 2500

3000 3500

4000 0.5

1

1.5

2

2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

’s.txt’

Figure 3.2: Difference between observations and model predictions (top) ob-tained with the local volatility (bottom) produced by 100 iterations of Conju-gate Gradient on (3.10). The cost function is reduced from 500000 to 9000 andthe L2 − norm of the gradient from 1011 to 107.


Figure 3.3: Same as Figure 3.2 but with (3.11). Convergence is also not perfect:after 100 iterations J is reduced from 500000 to 12000, giving an average errorat each observed point larger than one percent.

Proposition 3.1 (Dupire) Let v be solution in R+ × (0, T ) of

∂v

∂t− 1

2σ2(S, t)S2 ∂

2v

∂S2+ rS

∂v

∂S= 0, v(S, 0) = (S0 − S)+, (3.12)

thenC(S0, 0,K, τ) = v(K, τ) (3.13)

and p ≡ ∂2v∂S2 satisfies the boundary value problem

∂p

∂t− ∂2

∂S2

(σ2(S, t)S2

2p

)+

∂

∂S(rSp) + rp = 0, p(S, 0) = δS0(S), (3.14)

where δS0 is the Dirac mass at S0.

Proof. Call Q the rectangular domain Q = (Sm, SM ) × (0, τ), with 0 ≤ Sm <SM and consider the boundary value problem in Q

∂u

∂t+ η(S, t)

∂2u

∂S2+ µ

∂u

∂S− ru = 0, u(S, τ) = uτ (S), (3.15)

where η(S, t) = 12σ

2(S, t)S2 and µ = rS. Multiplying (3.15) by the solution p of(3.14), integrating on Q and using an integration by part in time and Green’sformula in the variable S yield

u(S0, 0) =

∫ SM

Sm

uτ (S)p(S, τ)dS +

∫ τ

0

[pη∂u

∂S− u

∂

∂S(ηp) + pµu

]SM

Sm

. (3.16)


We take u(S, t) = C(S, t,K, τ), Sm = 0 and SM = +∞. The second integralin (3.16) vanishes because u ∼ S and p tends to 0 faster than S−1 as S tends

to infinity. Let v be a double primitive of p, i.e., a function such that ∂2v∂S2 = p

then, integrating (3.14) twice yields

∂v

∂t− 1

2σ2(S, t)S2 ∂

2v

∂S2+ rS

∂v

∂S= aS+ b, v(S, 0) = c+dS+(S−S0)

+, (3.17)

where a, b, c, d are integration constants. But uτ = (S − K)+, so ∂2uτ

∂S2 (S) =δK(S), where δK is the Dirac mass at K. A double integration by part appliedto (3.16) yields

u(S0, 0) =

∫ ∞

0

uτ (S)∂2v

∂S2(S)(S, τ)dS =

∫ ∞

0

v(S, τ)∂2uτ∂S2

(S)dS+

[uτ∂v

∂S− v

∂uτ∂S

]∞

0

.

(3.18)Therefore, if v vanishes at ∞, we obtain

u(S0, 0) = v(K, τ),

because uτ (0) = ∂Suτ (0) = 0.By choosing a = b = 0 and c = S0, d = −1, we obtain (3.13) because (S −S0)

+− (S−S0) = (S−S0)− = (S0−S)+ so v vanishes at infinity and the initial

condition in (3.12) is the desired one.

Remark 3.1 If the underlying asset yields a distributed dividend, qStdt, thenthe pricing function satisfies

∂C

∂t+σ2(S, t)S2

2

∂2C

∂S2+ (r − q)S

∂C

∂S− rC = 0 in R+ × [0, τ),

C(S, τ) = (S −K)+ in R+,(3.19)

and the related Dupire’s equation is

∂v

∂t− 1

2σ2(S, t)S2 ∂

2v

∂S2+ (r − q)S

∂v

∂S+ qv = 0, (3.20)

for τ ≥ t > 0 and S ∈ R+.

Numerical Results A finite difference method implicit in time of order one(Euler’s scheme) is used for (3.4) and (3.12); the parameters are K = 100,r = 0.06, σ = 0.4, and 200 time steps and 250 mesh points for S. We havecompared the numerical results for (3.4) with (3.13) by solving (3.12) for all thevalues of S0 used to display S0 → C(S0, 0). Very good accuracy is found, seefigure 3.4.


0

10

20

30

40

50

60

70

80

90

100

0 50 100 150 200 250

"u.txt" using 1:2"u.txt" using 1:3"u.txt" using 1:4

Figure 3.4: The pricing function of European call option one year to maturitycomputed by 3 different methods: a) by solving (3.4), b) by the Dupire formula(3.12, 3.13), c) by the Black-Scholes analytic formula.

Extension to more complex option is possible as long as the PDE is linear.The previous argument does not work with American options (modeled by avariational inequality). It is not always possible to find a double primitive ofthe adjoint equation but the discrete equivalent of (3.16) can be found when avariational numerical method such as the Finite Element Method is used.For simplicity assume that the coefficients in (3.15) do not depend of time andconsider a numerical discretization of (3.15) with an Euler implicit scheme intime with time step δt and piecewise linear finite elements in the variable S.The scheme can be written in matrix form

(B +A)un −Bun+1 = 0, (3.21)

where un is the vector of the nodal values of the piecewise linear function unhwhich approximates u(·, nδt). Given a vector p0, introduce the sequence ofvectors pn obtained by iterating

(A+B)Tpn+1 −BTpn = 0. (3.22)

Now notice that (3.21) multiplied by (pn+1)T gives

0 = (pn+1)T (A+B)un − (pn+1)TBun+1 = (pn)TBun − (pn+1)TBun+1,(3.23)

where the last equality has used (3.22). Summing up over all n gives

(p0)TBu0 = (pN )TBuN . (3.24)


Choosing p0j = δij , j = 0, . . . ,M gives the discrete equivalent of (3.16)

(

M∑

j=0

Bij)u0i ≈ (Bu0)i = (pN )TBuN . (3.25)

3.2.6 A New Least Square Problem

A natural idea is to somehow interpolate the observed prices by a sufficientlysmooth function v : R+ × [0,maxi∈I Ti] → R+, then use (3.20) with v = v inorder to obtain an approximation σ. For examples, bicubic splines may be used.This approach has several serious drawbacks:

• it is difficult to design an interpolation process such that ∂2v∂S2 does not

take the value 0, and such that the obtained approximation of the squaredvolatility is non negative.

• There is an infinity of possible interpolations of Ci at (Ki, Ti), 1 ≤ i ≤ I,and for two possible choices, the volatility obtained by (3.20) may differconsiderably.

We see that financially relevant additional information have to be added to theinterpolation process.Another natural idea is to consider a new least square problem where the statefunction satisfies Dupire’s equation:

minσ∈Σ

J(σ) =

I∑

1

ωi|v(Ki, Ti)− Ci|2 + JR(σ) subject to (3.20).(3.26)

The evaluation of J requires solving only one boundary value problem insteadof I in the first least square problem.

Numerical Results

The same test problem as in § 3.2.4 was solved: the local volatility is repre-sented by a spline and a gradient method with automatic differentiation is used.Naturally the computing time is more than 50 times faster: on a 2Gig Hz dualcore Intel, 50 iterations on a 100 × 50 grid takes 0.5 seconds. The results areshown on figures 3.5, 3.6 and 3.7.

Adjoint equation

Gradient methods require the differentiation of J with respect to σ. The gradi-ent of J can be computed with automatic differentiation of computer programsif the chosen representation of σ is done with a small number of parameters.Alternatively, when σ has a large number of degrees of freedom, automatic dif-ferentiation becomes too expensive, and an analytic procedure is needed.Since JR explicitly depends on σ, its gradient is easily computed. The gradient


0 500 1000 1500 2000 2500 3000 3500 4000 0 0.5

1 1.5

2 2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

"usigma.txt"using 1:2:4

0 500 1000 1500 2000 2500 3000 3500 4000 0 0.5

1 1.5

2 2.5

3

-200 0

200 400 600 800

1000 1200 1400


Figure 3.5: Local volatility surface generated by the optimization algorithmfrom the parametrization (3.7) and using Dupire’s equation. Below is the pricesurface as a function of strike and maturity.


10

20

30

40

50

60

70

0 5 10 15 20 25 30 35 40 45 50

"converge.h"using 1:2"converge.h"using 1:3

600 800

1000 1200

1400 1600

1800 0.5

1

1.5

2

2.5

3

0 100 200 300 400 500 600 700 800

"uduo.txt"using 1:2:3"uduo.txt"using 1:2:4

Figure 3.6: Convergence of J/1000 to its local minimum and of the logarithmof the squared norm of the gradient to −∞ as a function of iteration num-ber. Below is a comparison between all observed prices (target) and the pricesgenerated by the model.


0 500 1000 1500 2000 2500 3000 3500 4000 0 0.5

1 1.5

2 2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1


0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0.5

1 1.5

2 2.5

3

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

"s.txt"

Figure 3.7: Same as Figure 3.5 but with (3.11). 100 iterations of conjugategradient decreases J from 500000 to 1200, giving an average error smaller thana percent

of JLS =∑I

1 |v(Ki, Ti) − Ci|2 is more difficult to evaluate, because the pricesv(Ki, Ti) depend of σ in an indirect way: one needs to evaluate the variations ofv(Ki, Ti) caused by a small variation of σ; calling δσ the variation of σ and δvthe induced variation of v, one sees by differentiating (3.20) that δv(S, t = 0) = 0and

∂tδv −σ2(S, t)S2

2∂2SSδv + (r − q)S∂Sδv + qδv = σδσS2∂2SSv. (3.27)

To express δJ in terms of δσ, an adjoint state function P is introduced, as thesolution to the adjoint problem: find the function P such that P (T , ·) = 0 andfor t < T ,

∂tP+∂2SS(σ2S2

2P)−∂S

(P (r−q)S

)−qP = 2

∑

i∈Iωi(v(Ki, Ti)−Ci)δKi,Ti , (3.28)

where T is an arbitrary time greater than maxi∈I Ti and in the right hand side,δKi,Ti denote Dirac functions at (Ki, Ti). The meaning of (3.28) is the following:

−∫

Q

(∂tw − σ2S2

2∂2SSw + (r − q)S∂Sw + qw

)P = 2

∑

i∈Iωi(v(Ki, Ti)−Ci)w(Ki, Ti).

(3.29)where Q = R+ × (0, T ), and w is any function such that w ∈ L2((0, T ), V ) with∂tw ∈ L2(Q) and S2∂2SSw ∈ L2(Q). Taking v = δv in (3.29) and using (3.27),


one finds

2∑

i∈Iωi(v(Ki, Ti)− Ci)δv(Ki, Ti) = 2

∑

i∈Iωi(v(Ki, Ti)− Ci)〈δKi,Ti , δv〉

= −∫

Q

(∂tδv −

σS2

2∂2SSδv + (r − q)S∂Sδv + qδv

)P

= −∫

Q

σδσS2P∂2SSv.

We have worked in a formal way, but all the integrations above can be justified.This leads to the estimate

∣∣∣∣δJLS +

∫

Q

σδσS2P∂2SSv

∣∣∣∣ ≤ c‖δσ‖2L∞(Q),

which implies that JLS is differentiable, and that its differential at point σ isgiven by

DJLS(σ) : η 7→ −∫

Q

σηS2P (σ)∂2SSv(σ),

where P (σ) satisfies (3.28), and v(σ) satisfies (3.20). We see that the gradientof JLS can be evaluated. When (3.20) is discretized with e.g. finite elements,all what has been done can be repeated (with a discrete adjoint problem), andthe gradient of the functional can be evaluated in the same way. Let us stressthat the gradient DJLS(σ) is computed exactly, which would not be the casewith e.g. a finite difference method.Local volatility can also be calibrated with American options, but it is not pos-sible to find the analogue of Dupire’s equation. Thus, in the context of a leastsquare approach, the evaluation of the cost function requires the solution of Ivariational inequalities, which is computationally expensive, see [1, 7]. Never-theless, it is also possible to find necessary optimality conditions involving anadjoint state.

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partialdifferentialequationsfor …...in order to price the option before maturity, some assumptions...

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