numerical methods for american options - mimuwapalczew/cfp_lecture9.pdfpenalty method there are many...

Lecture 9

Numerical methods forAmerican options

Lecture Notesby Andrzej Palczewski

Computational Finance – p. 1

American options

The holder of an American option has the right to exercise it at any momentup to maturity. If exercised at t

an American call option has the payoff (St − K)+,

an American put option has the payoff (K − St)+.

Theorem. If the underlying does not pay dividends, the price of anAmerican call option with maturity T and exercise price K is equal tothe price of a European call option with exercise price K expiring at T .

Important! It is only true if there are no dividends. Above theorem does notapply for options on the foreign exchange market and on stock indices.


American instruments

Definition. American type instrument with maturity T and payoff function f isa contingent claim that can be exercised at any moment up to T . Its payoff att equals

f(St, t).

Theorem. The price of an American claim at time t can be written as

V (St, t)

for some function V : (0,∞) × [0, T ] → R.


Properties

V (St, t) is the price of the instrument at t, so it values future payoffs fromthe instrument.

f(St, t) is the payoff of the instrument if it is exercised at t.

At the terminal time T

V (s, T ) = f(s, T ), s > 0.

To preclude arbitrage

V (s, t) ≥ f(s, t), (s, t) ∈ (0,∞) × [0, T ].


If the inequality is strict, i.e. V (s, t) > f(s, t), the holder should not exercisethe option. Because it is more profitable to sell the option for V (s, t) than toexercise it for f(s, t).

If V (s, t) = f(s, t), it is optimal to exercise the option immediately. By holdingit, the holder risks loosing money.

General rule. The holder should exercise the option as soon as

V (St, t) = f(St, t).


American option PDE

Up to the exercise of the option, its price forms a value process of thereplicating strategy. It is therefore a martingale with respect to the risk-neutralmeasure. Applying Itô’s formula gives

dVt = d(

V (St, t))

=

(

σSt

∂V (St, t)

∂s

)

dWt+

+

(

rSt

∂V (St, t)

∂s+

1

2σ2S2

t

∂2V (St, t)

∂s2− rV (St, t) +

∂V (St, t)

∂t

)

dt.

This process is a martingale if and only if the term by dt is zero.


Until the optimal exercise we have

rs∂V (s, t)

∂s+

1

2σ2s2 ∂2V (s, t)

∂s2− rV (s, t) +

∂V (s, t)

∂t= 0.

And of course, we have V (s, t) > f(s, t).

At the optimal exercise moment, we have

V (s, t) = f(s, t),

and

rs∂V (s, t)

∂s+

1

2σ2s2 ∂2V (s, t)

∂s2− rV (s, t) +

∂V (s, t)

∂t≤ 0.

The inequality in the second formula comes from the fact that V (s, t) may notrepresent the value process of some strategy after the optimal exercisemoment.


Free-boundary problem

We summarize our findings in the following system of equations andinequalities called the free-boundary problem

V (s, t) ≥ f(s, t)

12σ2s2 ∂2V (s,t)

∂s2 + rs∂V (s,t)

∂s− rV (s, t) + ∂V (s,t)

∂t≤ 0

V (s, t) = f(s, t) or 12σ2s2 ∂2V (s,t)

∂s2 + rs∂V (s,t)

∂s− rV (s, t) + ∂V (s,t)

∂t= 0

boundary conditions...

The “free-boundary” is the set of points where V (s, t) = f(s, t). In thesepoints the system is not governed by the equation with partial derivatives.


Boundary conditions

Boundary conditions for the American put option :

Terminal condition

V (s, T ) = (K − s)+.

Left-boundary condition

lims→0

V (s, t) = K.

Right-boundary condition

lims→∞

V (s, t) = 0.


Complete free-boundary problem

For s > 0, t ∈ [0, T ]:

V (s, t) ≥ (K − s)+

12σ2s2 ∂2V (s,t)

∂s2 + rs∂V (s,t)

∂s− rV (s, t) + ∂V (s,t)

∂t≤ 0

V (s, t) = (K − s)+ or 12σ2s2 ∂2V (s,t)

∂s2 + rs∂V (s,t)

∂s− rV (s, t) + ∂V (s,t)

∂t= 0

V (s, T ) = (K − s)+

lims→0 V (s, t) = K

lims→∞ V (s, t) = 0


Simple example

We compute the price of an American put option by a simple numericalmethod.

First we make the following change of variables (the same as for theBlack-Scholes PDE)

x := log s + (r −1

2σ2)(T − t),

τ :=σ2

2(T − t),

y(x, τ) := e−r(T− 2τ

σ2 )V

(

ex−( 2r

σ2 −1)τ , T −2τ

σ2

)

,

where x ∈ R and τ ∈[

0, σ2

2 T]

.


Transformation of the problem

y(x, τ) ≥ e−r(T− 2τ

σ2 )

(

K − ex−( 2r

σ2 −1)τ

)+

,

∂y∂τ

(x, τ) − ∂2y∂x2 (x, τ) ≥ 0,

y(x, τ) = e−r(T− 2τ

σ2 )

(

K − ex−( 2r

σ2 −1)τ

)+

or ∂y∂t

(x, τ) − ∂2y∂x2 (x, τ) = 0,

y(x, 0) = e−rT (K − ex)+,

limx→−∞ y(x, τ) = Ke−r(T− 2τ

σ2 ),

limx→∞ y(x, τ) = 0.


Numerical method

Grid

x ∈ (−∞,∞), τ ∈ [0, 12σ2T ]

discretization of time τ : δτ =12 σ2T

N

τν := ν · δτ for ν = 0, 1, . . . , N

discretization of space x

xmin, xmax

δx =xmax − xmin

M

xi := xmin + i · δx for i = 0, 1, . . . , M

wi,ν denotes the approximation of y(xi, τν)


Finite differences

We use explicit method (this is important !).

The consequence is a small time step for the algorithm to be stable.


In the node (xi, τν+1) the expression

u1 = λwi−1,ν + (1 − 2λ)wi,ν + λwi−1,ν ,

where λ = δτ(δx)2 , approximates the value of y(xi, τν+1) in the case of no

exercise.

If this value is smaller than the payoff from the exercise

u2 = e−r(T−2τν+1

σ2 )(

K − exi−( 2r

σ2 −1)τν+1

)+

then it is optimal to exercise immediately and wi,ν+1 = u2.

This is written concisely as

wi,ν+1 = max

(

λwi+1,ν + (1 − 2λ)wi,ν + λwi−1,ν ,

e−r(T−2τν+1

σ2 )(

K − exi−( 2r

σ2 −1)τν+1

)+)

.


Algorithm for an American put option'

&

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%

Input: xmin, xmax, M , N , K, T and the parameters of the model

δτ = σ2T2N

, δx = xmax−xmin

M

Calculate τν , ν = 0, 1, . . . , N , and xi, i = 0, 1, . . . , M

For i = 0, 1, . . . , M

wi,0 = e−rT (K − exi)+

For ν = 0, 1, . . . , N − 1

w0,ν+1 = Ke−r(T−

2τν+1

σ2 )

wM,ν+1 = 0

For i = 1, 2, . . . , M − 1

u1 = λwi+1,ν + (1 − 2λ)wi,ν + λwi−1,ν

u2 = e−r(T−

2τν+1

σ2 )“

K − exi−( 2r

σ2 −1)τν+1”+

wi,ν+1 = max(u1, u2)

Output: wi,ν for i = 0, 1, . . . , M , ν = 0, 1, . . . , N


General American instrument

For a general American instrument we return to original variables onlymaking the change of time t 7→ τ = T − t.

A general American instrument is characterized by a pay-off function g(s, τ).

The free-boundary problem for this instrument is given by

(

∂V (s,τ)∂τ

− rs∂V (s,τ)

∂s− 1

2σ2s2 ∂2V (s,τ)∂s2 + rV (s, τ)

)

(

V (s, τ) − g(s, τ))

= 0,

V (s, τ) − g(s, τ) ≥ 0,∂V (s,τ)

∂τ− rs

∂V (s,τ)∂s

− 12σ2s2 ∂2V (s,τ)

∂s2 + rV (s, τ) ≥ 0,

V (s, 0) = g(s, 0),

lims→+∞ V (s, τ) = lims→+∞ g(s, τ).

lims→0 V (s, τ) = lims→0 g(s, τ).

This problem is also called the linear complementarity problem for theAmerican instrument defined by a pay-off function g(s, τ).


Penalty method

There are many numerical methods which solve the linear complementarityproblem (LCP). We present here the method called the penalty method .This method is simple and efficient (this is particularly visible for morecomplicated instruments like barrier options). On the other hand the methodis only first order (slow convergence).

The basic idea of the penalty method is simple. We replace the linearcomplementarity problem by the nonlinear PDE

∂V (s, τ)

∂τ= rs

∂V (s, τ)

∂s+

1

2σ2s2 ∂2V (s, τ)

∂s2− rV (s, τ)+ρ max

(

g(s, τ)−V (s, τ))

where, in the limit as the positive penalty parameter ρ → ∞, the solutionsatisfies V − g ≥ 0.


Finite difference approximation

We use the same grid as for the Black-Scholes equation with V ni denoting an

approximation to V (si, τn) and gni an approximation to g(si, τn).

The nonlinear PDE for the penalty method becomes in the discrete version

V n+1i − V n

i =

(1 − θ)

(

∆τ∑

j=i±1

(γij + βij)(

V n+1j − V n+1

i

)

− r∆τV n+1i

)

+ θ

(

∆τ∑

j=i±1

(γij + βij)(

V nj − V n

i

)

− r∆τV ni

)

+ P n+1i

(

gn+1i − V n+1

i

)

,

where the choice of θ gives the implicit (θ = 0) and the Crank-Nicolson(θ = 1

2 ) scheme.


Finite difference approximation – cont.

Coefficients from the previous slide are as follows:

P n+1i =

ρ, for V n+1i < gn+1

i ,

0, otherwise,

γij =σ2s2

i

|sj − si| (si+1 − si−1),

βij =

rsi(j−i)si+1−si−1

, for σ2si + r(j − i)|sj − si| > 0,(

2rsi(j−i)si+1−si−1

)+

, otherwise,

where j = i ± 1 and ρ is a penalty factor (a large positive number).



The numerical algorithm can be written in the concise form

(

I + (1 − θ)∆τM + P (V n+1))

V n+1 = (I − θ∆τM)V n + P (V n+1)gn+1,

where V n is a vector with entries V ni and gn a vector with entries gn

i ,

[MV n]i = −

(

∑

j=i±1

(γij + βij)(

V nj − V n

i

)

− rV ni

)

and P (V n) is a diagonal matrix with entries

[P (V n)]i,i =

ρ, for V ni < gn

i ,

0, otherwise.



Matrix M has the property of strict diagonal dominance. It has positivediagonal and non-positive off-diagonals with diagonal entries strictlydominating sum of absolute values of off-diagonal entries. This property of M

is essential for the convergence of the method and is visible from thestructure of the upper left corner of the matrix

M =

0

B

B

@

r + γ12 + β12 −γ12 − β12 0 . . .

−γ21 − β21 r + γ21 + β21 + γ23 + β23 −γ23 − β23 . . .

. . . . . . . . . . . .

1

C

C

A

,

Note that in the vector on the right hand side (I − θ∆τM)V n the first and thelast elements have to be modified to take into account the boundaryconditions.


Convergence

Theorem .Let us assume that

γij + βij ≥ 0,

2 − θ

(

∆τ∑

j=i±1

(γij + βij) + r∆τ

)

≥ 0,

∆τ

∆s< const.

∆τ, ∆s −→ 0,

where ∆s = mini (si+1 − si).


Convergence – cont.

Then the numerical scheme for the LCP from the previous slidessolves

∂V (s, τ)

∂τ− rs

∂V (s, τ)

∂s−

1

2σ2s2 ∂2V (s, τ)

∂s2+ rV (s, τ) ≥ 0,

V n+1i − gn+1

i ≥ −C

ρ, C > 0,

(

∂V (s, τ)

∂τ− rs

∂V (s, τ)

∂s−

1

2σ2s2 ∂2V (s, τ)

∂s2+ rV (s, τ) = 0

)

∨

∨

(

|V n+1i − gn+1

i | ≤C

ρ

)

,

where C is independent of ρ, ∆τ and ∆S.


Iterative solution

Since we get a nonlinear equation for V n+1 it has to be solved by iterations.We shall use here the simple iteration method.

Let (V n+1)(k) be the k-th estimate for V n+1. For notational convenience, wewrite

V (k) ≡ (V n+1)(k) and P (k) ≡ P(

(V n+1)(k))

.

If V (0) = V n, then we have the following algorithm of Penalty AmericanConstraint Iteration.


Algorithm

'

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%

Input: V n, tolerance tol.

V (0) = V n

For k = 0, . . . until convergencesolve

`

I + (1 − θ)∆τM + P (k))´

V (k+1) = (I − θ∆τM)V n + P (k)gn+1.

If

maxi|V

(k+1)i

−V(k)i

|

max(1,|V(k+1)i

|)< tol

orP (k+1) = P (k)

quit.V n+1 = V (k+1)

Output: V n+1


Convergence of iterations

Theorem .Let

γij + βij ≥ 0,

thenthe nonlinear iteration converges to the unique solution to thenumerical algorithm of the penalized problem for any initial iterateV (0);

the iterates converge monotonically, i.e., V (k+1) ≥ V (k) for k ≥ 1;

the iteration has finite termination; i.e. for an iterate sufficientlyclose to the solution of the penalized problem, convergence is ob-tained in one step


Size ofρ

In theory, if we are taking the limit as ∆s, ∆τ → 0, then we should have

ρ = O

(

1

min(

(∆s)2, (∆τ)2)

)

.

This means that any error in the penalized formulation tends to zero at thesame rate as the discretization error. However, in practice it seems easier tospecify the value of ρ in terms of the required accuracy. Then we should take

ρ ≈1

tol.


Speed up iterates convergence

Although the simple iterates converge to the solution of the nonlinear problemits speed of convergence is rather slow.

To make the convergence more rapid we can use the Newton iterates . Thisrequires to write the nonlinear equation in the form F (x) = 0 and solve theiterative procedure

xk+1 = xk − F ′(xk)−1F (xk),

where F ′(x) is the Jacobian of F .

In the penalty method algorithm the only nonlinear term which requiresdifferentiation in order to obtain F ′ is P n

i . Unfortunately, this term isdiscontinuous. A good convergence can be obtained when we define thederivative of the penalty term as

∂P n+1i (gn+1

i − V n+1i )

∂V n+1i

=

−ρ, for V n+1i < gn+1

i ,

0, otherwise.


American call option

Solving linear complementarity problem for an American call option we haveto impose a boundary condition on artificial boundary Smax. Let us recall thatfor an Europen call option the suggested boundary condition is

(

Smax − Ke−r(T−t))+

or for a stock paying dividend of constant rate d

(

Smaxe−d(T−t) − Ke−r(T−t))+

.

Due to the early exercise possibility for an American call option the boundarycondition on artificial boundary Smax should be

(

Smax − K)+

.


American call option – cont.

For large Smax and r > 0 we have

(

Smax − K)+

<(

Smax − Ke−r(T−t))+

.

Hence for large S, the price of an American call option is smaller than theprice of an European call option, which contradicts the theorem that theseprices are equal.

The problem comes from the replacement of an original boundary conditionfor S → ∞ by a condition at artificial boundary Smax. To avoid the problem weshould impose the following boundary condition at Smax

max

(

(

Smax − K)+

,(

Smaxe−d(T−t) − Ke−r(T−t))+

)

.


American barrier options

Contrary to European barrier options for American barrier options we do nothave in-out parity. In general, the sum of prices of knock-in and knock-outoptions is not equal to the price of the corresponding American vanilla option.

Let

τ(D)X = inf{t ≤ T : St ≤ X} τ

(U)X = inf{t ≤ T : St ≥ X}

be the first time the price of the underlying asset falls below (rise above) thebarrier X . Then for options with payoff g(St, t) we have

supτ1∈T0,T

E

[

e−rτ1g(Sτ1 , τ1)11{τ1<τ(·)X

}

]

+ supτ2∈T0,T

E

[

e−rτ2g(Sτ2 , τ2)11{τ2>τ(·)X

}

]

≥

supτ3∈T0,T

E

[

e−rτ3g(Sτ3 , τ3)]

and the equality holds only when all stopping times τ1, τ2 and τ3 are equal.


American knock-in optionsDown-and-in option.

The option condition is: when the asset price St falls below X the holder ofthe option receives an American vanilla option, with maturity date T andstrike price K.

To price that option we consider only the interval [X, +∞). In this intervalthere is no early exercise possibility and we gave an European option whichfulfils the Black-Scholes equation

∂V (S, t)

∂t+ rS

∂V (S, t)

∂S+

1

2σ2S2 ∂2V (S, t)

∂S2− rV (S, t) = 0.

with terminal condition

V (S, T ) = 0 for S > X.

and boundary conditions

V (S, t) → 0 as S → ∞, V (X, t) = C(X, t),

where C(S, t) is the price of the corresponding American vanila option.Computational Finance – p. 33

American knock-in options – cont.

Up-and-in option.

The option condition is: when the asset price St rises above X the holder ofthe option receives an American vanilla option, with maturity date T andstrike price K.

We price the option in the interval [0, X ]. In this interval there is no earlyexercise possibility and we gave an European option which fulfils theBlack-Scholes equation with terminal condition

V (S, T ) = 0 for S < X.

and boundary conditions

V (S, t) → 0 as S → 0,

V (X, t) = C(X, t),

where C(S, t) is the price of the corresponding American vanila option.Computational Finance – p. 34

American knock-out optionsThese options are options with early exercise provision. Terminal conditionsare as for American vanila options modified by limitations coming fromboundary conditions.

Boundary conditions:

Down-and-out option

V (S, t) = h(X, t), for S = X,

V (S, t) = boundary value for vanila option, for S > X.

Up-and-out option

V (S, t) = boundary value for vanila option, for S < X,

V (S, t) = h(X, t), for S = X.

Here h(X, t) is a payoff function on barrier X (see the next slide).


American knock-out options – cont.From both financial and mathematical point of view, payoff function h(X, t) isequal to zero.

But before the stock price crosses the trigger X , the American option isactive and its price must be no less than the option’s payoff function g(St, t)

((St − K)+ for call option and (K − St)+ for put option). Hence, up to the

barrier the price of the option must be bounded from below by the payofffunction g(St, t).

Assuming h(X, t) = 0 will create a jump on the barrier since before the stockprice crosses the trigger X the price V (S, t) ≥ g(S, t). If the option price iscontinuous up to the barrier then we shall get V (X, t) ≥ g(X, t). Thatsuggests to take as the payoff function on the barrier h(X, t) = g(X, t).

This choice does not change the resulting price V (S, t) but has obviousnumerical adventage: we do not solve a problem with jumps on a boundary.As a result we obtain a numerical algorithm which is more stable (jumps on aboundary can create oscillations in numerical solutions).


numerical methods for american options - mimuwapalczew/cfp_lecture9.pdfpenalty method there are many...

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