Efficient smoothing of Crank-Nicolson method for pricing barrier optionsunder stochastic volatility

M. Yousuf1,∗

1 Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

Most of the option pricing problems have nonsmooth payoff. In barrier options certain aspects of the option are triggered ifthe asset price becomes too high or too low. Standard smoothing schemes used to solve problems with nonsmooth payoff donot work well for the barrier option because a discontinuity is introduced in the time domain each time a barrier is applied. Animproved smoothing strategy is introduced for smoothing the A-stable Cranck-Nicolson scheme at each time when a barrieris applied. A partial differential equation (PDE) approach is utilized for the evaluation of complex option pricing modelsunder stochastic volatility which brings major mathematical and computational challenges for estimation and stability of theestimates.

1 Introduction

It is generally recognized that some assumptions underlying the Black-Scholes pricing formula are imperfect. Realized volatil-ity of traded assets show significant variability. Generally, the implied volatility depend on the strike price in contrast to theunderlying assumption of Black-Scholes model where volatility is considered as constant. This phenomenon stands in empir-ical contradiction to the consistent use of classical Black-Scholes (constant volatility) approach to pricing options and similarsecurities. However, it is clearly desirable to maintain as many of the features as possible that have contributed to this model’spopularity and longevity.

The natural extension pursued both in the literature and in practice has been to modify the specification of volatility in thestochastic dynamics of the underlying asset. Because of the existence of non-flat implied volatility surface, stochastic volatilitymodels have become popular for derivative pricing and hedging. A stochastic volatility model is an incomplete market model,and as such there is whole family of risk-neutral pricing measure, unlike the constant volatility case where there is only onevolatility value.

Wade et al. [5] developed smoothing criteria for the Cranck-Nicolson scheme by combining (0, 1)–Pade scheme with themain (1, 1)–Pade scheme. The (0, 1)–Pade scheme has convergence order equal to 1 whereas it is 2 for the main (1, 1)–Padescheme. We observed that amplification symbols of the (0, 2)–Pade scheme is better approximation to the exponential functione−x than that of (0, 1)–Pade scheme. Also (0, 2)–Pade scheme and (1, 1)–Pade scheme have same convergence order. Thesmoothing criteria of combining (0, 2)–Pade schemes as damping devices for the Cracnk-Nicolson scheme instead of (0, 1)–Pade schemes is introduced in this work. This smoothing scheme is found more robust than the well-known Rannacherschemes [2, 3] and newly developed smoothing schemes by Wade et al [4, 5]. A strategy for barrier options, developed in ourearlier paper [5], is also utilized which is unique in achieving optimal order convergence. Partial fraction technique is appliedto construct parallel version of these schemes.

2 Heston’s Stochastic Volatility Model

In the standard Black-Scholes model volatility is assumed to be constant. This assumption is incorrect because in realityvolatility is not only non-constant it is also not predictable for more than a few months. This fact led to the developmentof stochastic volatility models, in which volatility itself is assumed to be a stochastic process. For a stochastic volatilitymodel, the constant volatility σ is replaced with a function ν(t), that models the variance of S(t). This variance function isalso modeled as Brownian motion, and the form of ν(t) depends on the particular SV model under study. Assume that S(t)satisfies the stochastic differential equation

dS(t) = νS(t) dt +√

νS(t) dW1(t) (1)

and volatility ν(t) follows the stochastic process

dν(t) = k(θ − ν(t)) dt + ξ√

ν(t) dW2(t) (2)

where dW1 and dW2 are correlated with a constant correlation value ρ, that is Cov[dW1(t), dW2(t)] = ρdt.

∗ Corresponding author E-mail: myousuf@kfupm.edu.sa, Phone: +00 966 3860 7196, Fax: +00 966 3860 2340

PAMM · Proc. Appl. Math. Mech. 7, 1081101–1081102 (2007) / DOI 10.1002/pamm.200700249

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Fig. 1 Cranck-Nicolson scheme. Fig. 2 Cranck-Nicolson smoothing.

Now the value of option price V is not only a function of stock price S and time t but also a function of variance ν, that isV = V (S, ν, t). For the Heston’s Model, the governing PDE arises for pricing options values under stochastic volatility is:

∂V

∂t− 1

2νS2

∂2V

∂S2− 1

2σ2ν

∂2V

∂ν2− ρσνS

∂2V

∂S∂ν− rS

∂V

∂S− [κ(θ − ν) − λν]

∂V

∂ν+ rV = 0 (3)

where λ(S, ν, t) = λν is the market price of volatility risk [1].The Method of Lines, MOL, is an approach used for the numerical solution of a wide range of PDEs that involve spatial

variables as well as a time variable. The spatial discretization results in an approximating system of ODEs. For an inho-mogeneous linear PDE, this method leads to an inhomogeneous linear system of ODEs in time whose solution satisfies atwo-term recurrence relation involving the matrix exponential, where the matrix is determined by the form of spatial dis-cretization applied, such as finite element, finite difference, or spectral approach. Central difference formulas are used in ourexperimentation for approximating the spatial derivatives.

2.1 Smoothing Schemes for Barrier Options

We shall use the Heston’s Stochastic Volatility model based PDE for double barrier option pricing problems under stochasticvolatility. The variable, r is the risk free interest rate, θ is the mean level of the variance, κ is the rate of reversion on the meanlevel, and σ is the volatility of the variance. The market price of the risk is denoted by λ. The correlation between the price ofthe underlying asset and its variance is ρ. The payoff with strike price E is:

g(x, y, 0) = max(x − E, 0).

We consider a pricing call option, where the barriers applied are defined as follows:

u(x, y, t) =

{u(x, y, t) x ≤ B1 or x ≥ B2 and tn ∈ B

0 otherwise(4)

Here, B1 and B2 are the lower and upper barriers, respectively and the set B = {τi|τi ∈ [0, T ]} consists of all the times whenbarriers are applied. We assume that barrier times are distributed uniformly in the set B and barriers are applied discretely.These time dependent discontinuities cause a serious computational difficulty.

Table 1 Convergence results for the Crank-Nicolson Smoothing scheme.

k Error B1 Error B2 Order B1 Order B2

0.02500 7.3505e-3 6.8866e-3 0.000 0.0000.01250 1.8219e-3 1.6981e-3 2.012 2.0200.00625 4.5901e-4 4.2843e-4 1.989 1.9870.003125 1.1684e-4 1.0916e-4 1.974 1.973

0.0015625 3.1001e-5 2.8922e-5 1.914 1.916

Acknowledgements This work is supported by King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia.

References

[1] Heston, S, A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options, TheReview of Financial Studies, 6, 327- 343 (1993.

[2] R. Rannacher, Finite Element Solution of Diffusion Problems with Irregular Data, Numer. Math. 43, 309–327 (1984).[3] R. Rannacher, Discretization of the Heat Equation with Singular Initial Data, Zeit. Ang. Math. Meth. (ZAMM) 62, 346– 348 (1982).[4] A.Q.M. Khaliq, B.A. Wade, M.Yousuf, and J. Vigo-Aguiar, Higher Order Smoothing Schemes for Inhomogeneous Parabolic Problems

with Applications to Nonsmooth Payoff in Option Pricing, Numer. Meth. for Partial Differential Equations, 23, 1249-1276 (2007).[5] B.A. Wade, A.Q.M. Khaliq, M.Yousuf, J. Vigo-Aguiar, and R. Deininger, On smoothing of the CrankNicolson scheme and higher

order schemes for pricing barrier options, Journal of Computational and Applied Mathematics, 204 144–158 (2007).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

ICIAM07 Minisymposia – 08 Probability and Statistics 1081102