Analysis of a Hybrid Finite Difference Scheme for the Black-Scholes Equation Governing Option Pricing

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1 ISSN (print), (online) International Journal of Nonlinear Science Vol.4(2007) No.3,pp Analysis of a Hybrid Finite Difference Sceme for te Black-Scoles Equation Governing Option Pricing Zongdi Cen, Anbo Le, Lifeng Xi Institute of Matematics, Zejiang Wanli University, Ningbo , Zejiang, P. R. Cina Abstract. In tis paper we present a ybrid finite difference sceme on a piecewise uniform mes for a class of Black-Scoles equations governing option pricing wic is pat-dependent. In spatial discretization a ybrid finite difference sceme combining a central difference metod wit an upwind difference metod on a piecewise uniform mes is used. For te time discretization, we use an implicit difference metod on a uniform mes. Applying te discrete maximum principle and barrier function tecnique we prove tat our sceme is second-order convergent in space for te arbitrary volatility and te arbitrary asset price. Numerical results support te teoretical results. Keywords: Black-Scoles equation, option valuation, central difference sceme, upwind difference sceme, piecewise uniform mes 1 Introduction An option is a financial contract tat gives its owner te rigt to buy or sell a specified amount of a particular asset at a fixed price, called te exercise price, on or before a specified date, called te maturity date. Options tat can be exercised at any time up to te maturity are called American, wile options tat can only be exercised on te maturity data are European. Options wic provide te rigt to buy te underlying asset are known as calls, wereas options conferring te rigt to sell te underlying asset are referred to as puts. It was sown by Black and Scoles [1] tat tese option prices satisfy a second-order partial differential equation wit respect to te time orizon t and te underlying asset price x. Tis equation is now known as te Black-Scoles equation, and can be solved exactly wen te coefficients are constants or space-independent. However, in many practical situations, numerical solutions are normally sougt. Terefore, efficient and accurate numerical algoritms are essential for solving tis problem accurately. Te first accurate numerical approac to te Black-Scoels equations was te lattice tecnique proposed in Cox et al. [4] and improved in Hull and Wite [5]. Tat approac is equivalent to an explicit time-stepping sceme. Oter numerical scemes based on classical finite difference metods applied to constant-coefficient eat equations ave also been developed (cf. Rogers [10]; Scwartz [11]; Courtadon [3]; Wilmott et al. [15]; Cai et al. [2]; Li et al. [7]-[9]). Te reason for tis is tat wen te coefficients of te Black-Scoles equation are constant or space-independent, te equation can be transformed into a diffusion equation. In tis case te problem is said to be pat-independent. However, wen a problem is pat-dependent, tis transformation is impossible, and tus te Black-Scoles equation in te original form need to be solved. Te standard finite difference metod is widely applied to valuating te option pricing problems, see Seydel [12], Tavella and Randall [13] and Wilmott et al. [15]. However, as stated in Seydel [12], te standard finite difference metod encounters some disadvantages. It is well known tat wen using te standard finite difference metod to solve tose problems involving te convection-diffusion operator, suc as te Black-Scoles partial differential operator, numerical difficulty can be caused. Te main reason is tat wen te volatility or te asset price is small, te Black-Scoles partial differential operator becomes a Corresponding autor. Tel. : ; Fax: xilifengningbo@tom.com. Copyrigt c World Academic Press, World Academic Union IJNS /117

2 236 International Journal of Nonlinear Science,Vol.4(2007),No.3,pp convection-dominated operator. Hence, te implicit Euler sceme wit central spatial difference metod will lead to nonpysical oscillations in te computed solution. Tis is due to a loss in stability. Te implicit Euler sceme wit upwinded spatial difference metod do not ave tis disadvantage, but tis difference sceme is only first-order convergent. Wang [14] applied a fitted finite volume sceme to solve te Black-Scoles equation, and sowed tat te fitted volume sceme is also first-order convergent. In tis paper we present a stability ybrid difference sceme on a piecewise uniform mes. Our ybrid difference sceme used central difference metod wenever te local mes size is small enoug to ensure te stability of tat sceme. Oterwise we use an upwind difference sceme. Our sceme is stable for te arbitrary volatility and te arbitrary asset price. Applying te discrete maximum principle and barrier function tecnique we prove tat our sceme is second-order convergent in space for te arbitrary volatility and te arbitrary asset price. Witout loss of generality, we sall discuss te metod using te model for European options in our paper. Naturally, te metod is applicable to American option if it is used togeter wit a tecnique for free boundary problems. Te rest of te paper is organized as follows. In te next section we discuss te continuous model of te Black-Scoles equations. Te discretization metod is described in section 3. In section 4, we present a stability and error analysis for te ybrid finite difference sceme. It is sown tat te finite difference solution converges to te exact solution at te rate of O( 2 +τ). Finally, numerical experiments are provided to support tese teoretical results in section 5. Notation. Trougout te paper, C will denote a generic positive constant (possibly subscripted) tat is independent of te mes. Note tat C is not necessarily te same at eac occurrence. 2 Te continuous problem Let V denote te value of a European call or put option and let x denote te price of te underlying asset. It is well known tat V satisfies te following Black-Scoles equation (see, for example, Wilmott et al. [15]): V t 1 2 σ2 (t)x 2 2 V (r(t)x D(x, t)) V x2 x wit compatibility boundary and final (or payoff) conditions + rv = 0, for (x, t) Ω (2.1) V (0, t) = g 1 (t), V (X, t) = g 2 (t), t [0, T ), (2.2) V (x, T ) = g 3 (x), x Ī, (2.3) were Ω = I (0, T ), I = (0, X) R, σ(t) > 0 denotes te volatility of te asset, T > 0 te expiry date, r(t) 0 te interest rate and D(x, t) te dividend. Wang [14] transform (2.1) wit te non-omogeneous Diriclet boundary conditions in (2.2) and (2.3) into te following self-adjoint form wit te omogeneous boundary condition: Lu u t u [a(t)x2 + b(x, t)xu] + c(x, t)u = f(x, t), for (x, t) Ω, (2.4) x x u(0, t) = u(x, t) = 0, for t [0, T ), u(x, T ) = g(x, T ), for x Ī, (2.5) were a(t) = 1 2 σ2 (t), b(x, t) = r(t) D(x,t) x σ 2, c(x, t) = 2r(t) σ 2 D x and f(x, t) are sufficiently smoot functions. We assume tat a(t) α > 0, β b(x, t) β > 0. For te sake of simplicity we sall also assume tat c(x, t) b(x, t) x b x 0. Tis can always be acieved by a transformation u = ũ exp(χx), wit χ cosen appropriately. We also assume tat te problem satisfies sufficient regularity and compatibility conditions wic guarantee te problem as a unique solution u(x, t) C 2 ( Ω) C 4 (Ω). Our interest lies in constructing iger order numerical metod for te Black-Scoles equation. 3 Discretization In tis section we will describe a piecewise-uniform mes and a ybrid finite difference sceme. IJNS for contribution: editor@nonlinearscience.org.uk

3 Z. Cen, A., L. Xi: Analysis of a Hybrid Finite Difference Sceme for te Black-Scoles Equation 237 Te use of central difference sceme on uniform mes may produces nonpysical oscillations in te computed solution. To overcome tis oscillation we use a piecewise uniform mes Ω N on te space interval [0, X]: x i = { i = 1, [1 + α β (i 1)] i = 2,, N, were = X 1 + α β (N 1). For te time discretization, we use a uniform mes Ω K on [0, T ] wit K mes elements. Ten te piecewise uniform mes Ω N,K on Ω is defined to be te tensor product Ω N,K = Ω N Ω K. It is easy to see tat te mes sizes i = x i and τ j = t j satisfy { for i = 1, i = α β for i = 2,, N, and τ = τ j = T/K for j = 1,, K respectively. We discretize (2.4) using a central difference sceme on te uniform mes [x 2, x N ]. Integrating bot sides of (2.4) over (, x i+1/2 ) we ave /2 = /2 u u xi+1/2 dx [a(t)x2 t x + b(x, t)xu]x i+1/2 + c(x, t)udx f(x, t)dx for i = 2,, N 1. Applying te mid-point quadrature rule and difference discretization we obtain from te above v U j i Uj+1 i τ j+1 U j i 1 (a j x 2 U j i+1 U j i i+1/2 i i+1 a j x 2 U j i U j i 1 i 1/2 ) i 1 (b j i+1/2 x U j i + U j i+1 i+1/2 b j i 2 i 1/2 x U j i 1 + U j i i 1/2 ) + c j i 2 U j i = f j i (3.1) for i = 2,, N 1, were i = ( i + i+1 )/2, s i 1/2 = (s i 1 + s i )/2 for any function s(x). To get a stable second-order discretisation we use te following upwind difference sceme at x 1 : u U j i Uj+1 i τ j+1 U j i a j x 2 i 1 i ( U j i+1 U j i i+1 U j i U j i 1 ) 2a j U j i+1 x U j i i i i+1 bj i+1 x i+1u j i+1 bj i x iu j i i+1 + c j i U j i = f j i for i = 1. (3.2) We set U j i = { u U j i for i = 1, v U j i for i = 2,, N. (3.3) Ten our sceme reads: Find U j i RN+1 R K+1 wit U j i = f j i for i = 1, 2,, N 1, j = K 1,, 1, 0, (3.4) U j 0 = U j N = 0 for j = K,, 1, 0, (3.5) Ui K = gi K for i = 1,, N 1. (3.6) IJNS omepage:ttp://

4 238 International Journal of Nonlinear Science,Vol.4(2007),No.3,pp Analysis of te metod Our analysis is based on discrete maximum principle, truncation error analysis and barrier function tecniques. Lemma 1 (Maximum principle ) Te operator defined by (3.3) on te piecewise uniform mes Ω N,K satisfies a discrete maximum principle, i.e. if {v j i } and {wj i } are mes functions tat satisfy vj 0 wj 0, vj N w j N (j = 0, 1,, K), vk i wi K (i = 0, 1,, N) and v j i w j i (i = 1,, N 1, j = K 1,, 1, 0), ten v j i wj i for all i, j. Proof. By te assumptions of a(t) α > 0, β b(x, t) β > 0 and c(x, t) b(x, t) x b x 0, it is easy to verify tat te matrix associated wit is an M-matrix, as in te proof of [6, Lemma 3.1]. Te next lemma gives us a useful formula for te truncation error. Lemma 2 Let s(x, t) be a smoot function defined on Ω N,K. Ten te following estimate for te truncation error old true: and s j i (Ls)j i C tj+1 2 s t 2 (x i, t) dt + C [x 2 i 4 s x 4 (x, t j) +x i 3 s x 3 (x, t j) ]dx for 2 i < N, 0 < j < K, s j i (Ls)j i C tj+1 2 s t 2 (x i, t) dt + C [x 2 i 3 s x 3 (x, t j) +x i 2 s x 2 (x, t j) ]dx for i = 1, 0 < j < K. Proof. It can be easily obtained by using Taylor s formula wit te integral form of te remainder. Now we can get te main result for our difference sceme. Teorem 1 Let u be te solution of (2.4)-(2.5) and U be te solution of te finite difference sceme (3.4)- (3.6). Ten u(x i, t j ) U j i C(2 + τ) for i = 0, 1,, N, j = K,, 1, 0. Proof. From equation (2.4) we can easily get by using te assumption u C 2 ( Ω) wic implies tat 2 u is bounded. x 2 Applying te definition of iger order derivative we ave u t C for (x, t) Ω (4.1) k u x k Cx2 k for k = 3, 4, (x, t) Ω, (4.2) were we also ave used te assumption u C 2 ( Ω). Now applying Lemma 2 we ave (u j i U j i tj+1 ) = LN,K u j i (Lu)j i C +C 2 u t 2 (x i, t) dt (x 2 i 4 u x 4 (x, t j) + x i 3 u x 3 (x, t j) )dx C(τ + 2 ), for 2 i < N, 0 < j < K, IJNS for contribution: editor@nonlinearscience.org.uk

5 Z. Cen, A., L. Xi: Analysis of a Hybrid Finite Difference Sceme for te Black-Scoles Equation 239 Table 1: Numerical results for Example 1 K N error rate e e e e e-3 - Table 2: Numerical results for Example 2 K N error rate e e e e e-4 - and (u j i U j i tj+1 ) = LN,K u j i (Lu)j i C +C 2 u t 2 (x i, t) dt (x 2 i 3 u x 3 (x, t j) + x i 2 u x 2 (x, t j) )dx C(τ + 2 ), for i = 1, 0 < j < K, were we ave used te estimates (4.1) and (4.2). Hence, using te barrier function w j i = C(τ + 2 )(1 + T t j ) (wit te constant C sufficient large), Lemma 1 implies tat for all i, j, u(x i, t j ) U j i C(2 +τ) wic completes te proof. 5 Numerical experiments In tis section we verify experimentally te teoretical results obtained in te preceding section. Errors and convergence rates for te ybrid difference sceme are presented for two test problems. Example 1. Consider te problem u t u [2x2 + xu] + 2u = f(x, t), for (x, t) (0, 1) (0, 1), x x u(0, t) = u(1, t) = 0, for t [0, 1), u(x, 1) = (1 x 3 )(e x 1) + ex(1 x), for x [0, 1], were f(x, t) is cosen suc tat u(x, t) = (1 x 3 )(e x 1) + e t x(1 x). Example 2. Consider te problem u t x [(1 u + t)x2 2 x + (1 + xt)xu] + 3et u = f(x, t), for (x, t) (0, 1) (0, 1), u(0, t) = u(1, t) = 0, for t [0, 1), u(x, 1) = (1 x 3 )(e x 1) + x(1 x), for x [0, 1], were f(x, t) is cosen suc tat u(x, t) = t(1 x 3 )(e x 1) + t 3 x(1 x). IJNS omepage:ttp://

6 240 International Journal of Nonlinear Science,Vol.4(2007),No.3,pp For our tests we take K = 1024 wic is a sufficiently large coice to bring out second-order convergence in space. We measure te accuracy in te discrete maximum norm u U. Te rates of convergence r N are computed using te following formula: r N = log 2 ( u U N u U 2N ). Te Table 1 and 2 correspond to te above problems respectively. Te numerical results are clear illustrations of te convergence estimate of Teorem 1. Tey indicate tat te teoretical results are fairly sarp. Acknowledgements Te work was supported by State Key Laboratory of Scientific and Engineering Computing (Cinese Academy of Sciences) and National Natural Science Foundation (Grant No ) of Cina, Natural Science Foundation (2007A610048) of Ningbo Municipal of Cina and Researc Project (Jd061052, Jd061053) of Education department of Ningbo. References [1] F. Black, M. S. Scoles: Te pricing of options and corporate liabilities. J. Political Economy. 81: (1973) [2] Guoliang Cai, Juanjuan Huang, Lixin Tian, Qingcao Wang: Adaptive control and slow manifold analysis of a new caotic system. International Journal of Nonlinear Science. 2(1):50-60 (2006) [3] G. Courtadon: A more accurate finite difference approximation for te valuation of options. J. Fin. Econ. Quant. Anal.. 17: (1982) [4] J. C. Cox, S. Ross and M. Rubinstein: Option pricing: a simplified approac. J. Fin. Econ. 7: (1979) [5] J. C. Hull, A. Wite: Te use of control variate tecnique in option pricing. J. Fin. Econ. Quant. Anal. 23: (1988) [6] R. B. Kellogg, A. Tsan: Analysis of some difference approximations for a singular perturbation problem witout turning points. Mat. Comp. 32: (1978) [7] Guangqin Li, Lifeng Xi: Stability Analysis on a Kind of Nonlinear and Unbalanced Cobweb Model. International Journal of Nonlinear Science. 4(2): (2007) [8] Meifeng Dai, Xi Liu: Lipscitz equivalence between two Sierpinski gasketse. International Journal of Nonlinear Science. 2(2),77-82 (2006) [9] Qiuli Guo: Hausdorff dimension of level set related to symbolic system. International Journal of Nonlinear Science. 3(1),63-67 (2007) [10] L. C. G. Rogers, D. Talay: Numercial Metods in Finance. Cambridge: Cambridge University Press (1997) [11] E. Scwartz: Te valuation of warrants: implementing a new approac. J. Fin. Econ.. 13: (1997) [12] R. Seydel: Tools for computational finance. Berlin: Springer (2004) [13] D. Tavella, C. Randall: Pricing financial instruments: Te finite difference metod. New York: Wiley (2000) [14] S. Wang: A novel fitted finite volume metod for te Black-Scoles equation governing option pricing. IAM J. Numer. Anal.. 24, (2004) [15] P. Wilmott. J. Dewynne and S. Howison: Option Pricing: Matematical Models and Computation. Oxford: Oxford Financial Press (1993) IJNS for contribution: editor@nonlinearscience.org.uk