delay geometric brownian motion in financial option...

IntroductionThe Delay Geometric Brownian Motion

Delay Effect on OptionsEuler–Maruyama Approximation

Summary

Delay Geometric Brownian Motion in FinancialOption Valuation

Xuerong Mao

Department of Statistics and Modelling ScienceUniversity of Strathclyde

Glasgow, G1 1XH

Xuerong Mao Delay Geometric Brownian Motion



Summary

Outline

1 Introduction

2 The Delay Geometric Brownian Motion

3 Delay Effect on OptionsEuropean OptionsEuropean put optionsLookback OptionsBarrier Options

4 Euler–Maruyama Approximation

5 Summary




Summary

The Black–Scholes World

The price of a risky asset, denoted by S(t) at time t , issupposed to be a geometric Brownian motion

dS(t) = rS(t)dt + σS(t)dW (t),

with initial value S(0) = S0 at time t = 0, where r > 0 is therisk-free interest rate, σ > 0 is the volatility and W (t) is a scalarBrownian motion.




Summary

European call options

At the expiry time T , a European call option with exercise priceE pays S(T )− E if S(T ) exceeds the exercise price, and payszero otherwise, that is the payoff is

S(T )− E)+ := max{S(T )− E , 0}.

The expected payoff at expiry time T is

E(S(T )− E)+.

So the value of the call option at t = 0 is

C = e−rT E(S(T )− E)+.




Summary

The Black–Scholes formula

C = S0Φ(d1)− Ee−rT Φ(d2),

where

Φ(x) =

∫ x

−∞

1√2π

e−y2/2 dy ,

and

d1 =log(S0/E) + (r + 1

2σ2)T

σ√

Tand d2 = d1 − σ

√T .




Summary

Stochastic volatility

In the Black–Scholes model, the volatility σ is assumed to be aconstant. However, it has been shown by many authors that thevolatility is a stochastic process. If the stochastic volatility isdenoted by V (t), then the SDE becomes

dS(t) = rS(t)dt + V (t)S(t)dW (t).




Summary

SDEs for volatility

Hull and White proposed that V (t) obeys

dV (t) = αV (t)dt + βV (t)dW1(t),

where W1(t) is another scalar Brownian motion which maycorrelate with W (t).Heston proposed that V(t) = V 2(t) obeys themean-reverting square root process

dV(t) = α(σ2 − V(t))dt + β√V(t)dW1(t).

Lewis used the stochastic θ-process

dV (t) = αV (t)dt + βV θ(t)dW1(t).




Summary

Other methods for modelling volatility

The implied volatility: the volatility at time t , V (t), isestimated using the option price at previous time, say t − τ ,where τ is a positive constant representing the time lag.

Time series and statistics: Estimate the volatility V (t)(regarded as a parameter of the SDE) using the pastunderlying asset prices S(t − τ1), S(t − τ2), · · · , S(t − τn)by the technique of time series and statistics.




Summary

In the case of implied volatility, the volatility V (t) can berepresented as a function of S(t − τ), say V (S(t − τ)),because the option price at time t − τ is clearly dependenton the underlying asset price S(t − τ). As a result, theSDE becomes

dS(t) = rS(t)dt + V (S(t − τ))S(t)dW (t).

In the other case, the volatility is a function of the paststates S(t − τ1), S(t − τ2), · · · , S(t − τn), whence the assetprice may obey

dS(t) = rS(t)dt + V (S(t − τ1), · · · , S(t − τn))S(t)dW (t).




Summary

Delay geometric Brownian motions

In both ways, stochastic differential delay equations (SDDEs)appear naturally in the modelling of an asset price. As bothSDDEs evolve from the classical geometric Brownian motion,we will call them the delay geometric Brownian motions(DGBMs).




Summary

Key points in modelling a financial quantity

The SDDEs have a unique positive or nonnegative solutionunder relatively weak conditions on the volatility function sothat a wide class of functions may be used to fit a widerange of financial quantities.The solutions have finite probability expectations so thatthe valuations of various associated options may be welldefined.The time-delay effect is not too sensitive in the sense thatshould the time lag τ have a little change, the underlyingasset price S(t) and its related option prices will notchange too much.The valuations of various associated options arecomputable at least numerically if not theoretically.




Summary

The delay geometric Brownian motion (DGBM):

dS(t) = rS(t)dt + V (S(t − τ))S(t)dW (t) (2.1)

on t ≥ 0 with initial data S(u) = ξ(u) on u ∈ [−τ, 0]. Hereτ is a positive constant,r > 0 is the risk-free interest rate,W (t) is a scalar Brownian motion,the initial data ξ := {ξ(u) : u ∈ [−τ, 0]} ∈ C([−τ, 0]; (0,∞)),the volatility function V ∈ C(R+; R+).




Summary

Theorem

The SDDE (2.1) has a unique global positive solution x(t) ont ≥ 0, which can be computed step by step as follows: fork = 0, 1, 2, · · · and t ∈ [kτ, (k + 1)τ ],

S(t) = S(kτ) exp(

r(t − kτ)− 12

∫ t

kτV 2(S(u − τ))du

+

∫ t

kτV (S(u − τ))dW (u)

). (2.2)




Summary

Theorem

For any R large enough for R > ‖ξ‖, define the stopping time

ρR = inf{t ≥ 0 : S(t) > R}.

ThenES(t ∧ ρR) ≤ ξ(0)ert (2.3)

and

P(ρR ≤ t) ≤ ξ(0)ert

R(2.4)

for all t ≥ 0. In particular,

ES(t) ≤ ξ(0)ert , ∀t ≥ 0. (2.5)




Summary

Comments

Assume that one holds a European call option at t = 0 withthe exercise price E at the expiry date T . His mean payoffat the expiry date is E(S(T )− E)+, which is clearlywell-defined by (2.5). Hence the price of the European calloption at t = 0 is

C(ξ, 0) = e−rT E(S(T )− E , 0)+

which is well-defined.For some more complicated options and their associatedmathematical analysis, it is useful for the solution ofequation (2.1) to obey, for example

E(

sup0≤t≤T

S(t))

< ∞ ∀T > 0.




Summary

TheoremAssume that

V (x) ≤ K ∀x ≥ 0. (2.6)

Let p ≥ 1. Then

ESp(t) ≤ ξ(0)ep[r+0.5(p−1)K 2]t (2.7)

for any t ≥ 0 and

E(

sup0≤t≤T

Sp(t))≤ ξp(0)

(2+

9p2K 2

p[r + 0.5(p − 1)K 2]

)ep[r+0.5(p−1)K 2]T

(2.8)for any T ≥ 0.




Summary

European OptionsEuropean put optionsLookback OptionsBarrier Options

We observe that there is a time lag τ when we estimate thevolatility. It is very important to know whether the time lag τ issensitive in the sense that a little change of τ will have asignificant effect on the underlying asset price and itsassociated option price. If this is the case, then the time lagneeds to be controlled tightly in practice; otherwise the delayeffect is robust.




Summary


Outline

1 Introduction




5 Summary




Summary


Assume that one holds a European call option at t = 0 on theunderlying asset price with the exercise price E at the expirydate T . Originally, the holder thinks the underlying asset pricefollows the DGBM (2.1) so the price of the European call optionat t = 0 is

Cτ = e−rT E(S(T )− E)+. (3.1)




Summary


On the second thought, the holder may wonder that if thevolatility at time t is estimated by the corresponding option priceat time t − τ̄ , instead of t − τ , then the underlying asset pricecould follow an alternative DGBM

dS̄(t) = r S̄(t)dt + V (S̄(t − τ̄))S̄(t)dW (t), (3.2)

whence the price of the European call option at t = 0 could be

Cτ̄ = e−rT E(S̄(T )− E)+. (3.3)




Summary


If the difference between Cτ and Cτ̄ is small when thedifference between τ and τ̄ is small, then the holder can simplychoose either (2.1) or (3.2) as the equation for the underlyingasset price; otherwise the holder has to control the time delaytightly.




Summary


Assumption

The volatility function V is locally Lipschitz continuous. That is,for each R > 0, there is a KR > 0 such that

|V (x)− V (x̄)| ≤ KR|x − x̄ | ∀x , x̄ ∈ [0, R].




Summary


Theorem

Let the volatility function V be locally Lipschitz continuous.Then, with the definitions of (3.1) and (3.3), we have

limτ−τ̄→0

|Cτ − Cτ̄ | = 0. (3.4)




Summary


Outline

1 Introduction




5 Summary




Summary


Let us assume that one holds a European put option at t = 0on the underlying asset price with the exercise price E at theexpiry date T . According to equation (2.1) or (3.2) that theunderlying asset price follows, the price of the European putoption at t = 0 is

Pτ = e−rT E(E − S(T ))+ or Pτ̄ = e−rT E(E − S̄(T ))+, (3.5)

respectively.




Summary


Theorem

Let the volatility function V be locally Lipschitz continuous andlet R be any sufficiently large number such that R > ‖ξ‖. Then,with the definitions of (3.5), we have

|Pτ − Pτ̄ | ≤ cRTe(cR−r)T (δ(τ − τ̄) +√

τ − τ̄) +2Eξ(0)

R, (3.6)

where cR is a positive constant independent of T and τ − τ̄ . Inparticular, we have

limτ−τ̄→0

|Pτ − Pτ̄ | = 0. (3.7)




Summary


Outline

1 Introduction




5 Summary




Summary


If the underlying asset price follows the DGBM (2.1), then thepayoff of a lookback option at the expiry date T isS(T )−min0≤t≤T S(t). Hence the price of a lookback option att = 0 is

Lτ = e−rT E(

S(T )− min0≤t≤T

S(t)). (3.8)

Alternatively, if the asset price follows the DGBM (3.2), then theprice of the lookback option is

Lτ̄ = e−rT E(

S̄(T )− min0≤t≤T

S̄(t)). (3.9)




Summary


Theorem


limτ−τ̄→0

|Lτ − Lτ̄ | = 0. (3.10)




Summary


Outline

1 Introduction




5 Summary




Summary


Let us now consider a barrier option under the DGBM (2.1).That is, consider an up-and-out call option, which, at expirytime T , pays the European value with the exercise price E ifS(t) never exceeded a given fixed barrier, B, and pays zerootherwise. Hence, the expected payoff at expiry time T is

E((S(T )− E)+ I{0≤S(t)≤B, 0≤t≤T}

).

Accordingly, the price of the barrier option at t = 0 is

Bτ = e−rT E((S(T )− E)+ I{0≤S(t)≤B, 0≤t≤T}

). (3.11)

Alternatively, if the asset price obeys the DGBM (3.2), theoption price is

Bτ = e−rT E((S̄(T )− E)+ I{0≤S̄(t)≤B, 0≤t≤T}

). (3.12)




Summary


Theorem


limτ−τ̄→0

|Bτ − Bτ̄ | = 0. (3.13)




Summary

Although, in theory, the DGBM S(t) can be computedexplicitly step by step, it is still unclear what the probabilitydistribution the DGBM S(t) is. It is therefore difficult tocompute the expected payoff of a European call optionE(S(T )− E)+, not mentioning more complicatedpath-dependent options e.g. the lookback and barrieroptions.

The recent working paper by Arriojas, Hu, Mohammed andPap provides us with a delay Black-Scholes formula interms of a conditional expectation based on an equivalentmartingale measure. However, it is nontrivial to computethe conditional expectation based on an equivalentmartingale measure.




Summary

The Euler–Maruyama (EM) scheme

To define the EM approximate solution to the DGBM (2.1), letus first extend the definition of the volatility function V from R+

to the whole R by setting V (x) = V (0) for x < 0.

This does not have any effect on the solution of the DGBM(2.1) as the solution is always positive.

Such an extension also preserves the local Lipschitzcontinuity or the boundedness of the volatility functionshould it be imposed.

More importantly, this enable us to define the EMapproximate solution to the DGBM (2.1).




Summary

Definition of the EM scheme

Let the time-step size ∆t ∈ (0, 1) be a fraction of τ , that is∆t = τ/N for some sufficiently large integer N. The discreteEM approximate solution is defined as follows: Set sk = ξ(k∆t)for k = −N,−(N − 1), · · · , 1, 0 and form

sk = sk−1[1 + r∆t + V (sk−1−N)∆Wk ], k = 1, 2, · · · , (4.1)

where ∆Wk = W (k∆t)−W ((k − 1)∆t).




Summary

To define the continuous extension, we introduce the stepprocess

s(t) =∞∑

k=−N

sk I[k∆t ,(k+1)∆t)(t), t ∈ [−τ,∞). (4.2)

The continuous EM approximate solution is then defined bysetting s(t) = ξ(t) for t ∈ [−τ, 0] and forming

s(t) = ξ(0) +

∫ t

0rs(u)du +

∫ t

0V (s(u − τ))s(u)dW (u), t ≥ 0.

(4.3)It is easy to see that s(k∆t) = s(k∆t) = sk for allk = −N,−(N − 1), · · · .




Summary

Assumption

The initial data ξ is Hölder continuous with order γ ∈ (0, 12 ], that

is

sup−τ≤u<v≤0

|ξ(v)− ξ(u)|(v − u)γ

< ∞.




Summary

Theorem

Let the volatility function V be bounded and locally Lipschitzcontinuous. Let the initial data ξ be Hölder continuous withorder γ ∈ (0, 1

2 ]. Then the continuous approximate solution(4.3) will converge to the true solution of the DGBM (2.1) in thesense

lim∆t→0

E(

sup0≤t≤T

|S(t)− s(t)|2)

= 0, ∀T ≥ 0. (4.4)

Moreover, the step process (4.2) and the continuousapproximate solution (4.3) obey

lim∆t→0

(sup

0≤t≤TE|s(t)− s(t)|2

)= 0, ∀T ≥ 0. (4.5)




Summary

Based on the strong convergence properties described in thistheorem, we can show that the expected payoff from thenumerical method converges to the correct expected payoff as∆t → 0 for various options.




Summary

For a European call option, it is straightforward to show

lim∆t→0

|E(S(T )− E)+ − E(s(T )− E)+| = 0.

Note that using the step function s(T ) in the above is equivalentto using the discrete solution (4.1). Hence, for a sufficientlysmall ∆t , e−rT E(s(T )− E)+ gives a nice approximation to theEuropean call option price e−rT E(S(T )− E)+.




Summary

Consider an up-and-out call option, which, at expiry time T ,pays the European value if S(t) never exceeded the fixedbarrier, B, and pays zero otherwise. We suppose that theexpected payoff is computed from a Monte Carlo simulationbased on the method (4.2).




Summary

Theorem

For the DGBM (2.1) and numerical method (4.2), define

Γ := E[(S(T )− E)+1{0≤S(t)≤B, 0≤t≤T}

], (4.6)

Γ̄∆t := E[(s(T )− E)+1{0≤s(t)≤B, 0≤t≤T}

], (4.7)

where the exercise price, E, and barrier, B, are constant. Then

lim∆t→0

|Γ− Γ̄∆t | = 0.




Summary

The DGBM (2.1) has a unique positive solution with a finiteexpected value provided the volatility function V iscontinuous. This enables us to use a wide class of volatilityfunctions to fit a wide range of financial quantities and toprice various associated options.

The time-delay effect is robustness provided the volatilityfunction V is locally Lipschitz continuous.

Although the DGBM can be computed explicitly step bystep, it is still hard to compute its associated option prices.We therefore introduce the Euler–Maruyama numericalscheme and show that this numerical methodapproximates option prices very well.


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