High order finite difference schemes on non-uniform meshes for the time-fractional Black-Scholes equation

We construct a three-point compact finite difference scheme on a non-uniform mesh for the time-fractional Black-Scholes equation. We show that for special graded meshes used in finance, the Tavella-Randall and the quadratic meshes the numerical solution has a fourth-order accuracy in space. Numerical experiments are discussed. Introduction The Black-Scholes-Merton model for option prices is an important model in financial mathematics. Since its discovery in the early seventies, it has been widely used in practice and has been studied rigorously using analytical and computational methods. The value of an option, denoted by V , depends on the current market value of the underlying asset s, and the remaining time t until the option expires: V = V (s, t). The Black-Scholes equation (BS) is a backward-in-time parabolic equation [1] LV := ∂V ∂t + 1 2 σs ∂V ∂s2 + (r − d)s ∂V ∂s − rV = 0, (1) where σ is the annual volatility of the asset price, r is the risk-free interest rate, d is the dividend yield and T is the expiry date (t = 0 means "today"). Due to the complexity of the financial markets, a number of improvements and modifications to the model have been proposed in order to improve the its accuracy depending on the state of the market. The change in the option price with time in the fractional model for option prices is a fractional transmission system. This assumption implies that the total flux rate of the option price Y (s, t) per unit time from the current time t to the expiry date T and the option price V (s, t) satisfy ∫ T t Y (s, t)dt = sf ∫ T t H(t − t)[V (s, t)− V (s, T )]dt, (2) where H(t) is the transmission functional and df is the Hausdorff dimension of the fractional transmission system. As pointed in [13], the essence of (2) is a conservation equation containing an explicit reference to the history of the diffusion process of the option price on a fractal structure. We further assume that the diffusion sets are underlying fractals and the transmission function H(t) = Aα Γ(1−α)tα , where Aα and α are constants and α is the transmission exponent. Now, by differentiating (2) with respect to t, we obtain Y (s, t) = sf d dt ∫ T t H(t − t)[V (s, t)− V (s, T )]dt. (3) On the other hand, from the BS equation, we have Y (s, t) = 1 2 σs ∂V ∂s2 + (r − d)s ∂V ∂s − rV, which combined with (3), yields [13] Aαs df−1 ∂V ∂tα + 1 2 σs ∂V ∂s2 + (r − d)s ∂V ∂s − rV = 0, (4) where ∂ V ∂tα is the modified Riemann-Liouville derivative defined as ∂V ∂tα = 1 Γ(n− α) ∂ ∂tn ∫ T t V (s, t)− V (s, T ) (t − t)1+α−n dt for n− 1 ≤ α < n. When α = 1 and under natural conditions for the function V (s, t) the modified Riemann-Liouville derivative ∂V ∂tα is equal to the partial derivative ∂V ∂t and ∂V ∂tα is equal to the Caputo derivative, when 0 < α < 1 [8] ∂V ∂tα = 1 Γ(1− α) ∂ ∂t ∫ T t V (s, t)− V (s, T ) (t − t)α dt = 1 Γ(1− α) ∫ T t Vt(s, t ) (t − t)α dt. Therefore equation (4) transforms to (1) when Aα = df = 1 and α = 1. For consistency with the benchmark Black-Scholes model, following [13], we assume that Aα = df = 1. In fact, the compact difference approximation (9) described below, can be easily extended to other values of Aα and df . In the last decade, a great deal of effort has been devoted to developing high-order compact schemes, which utilize the grid nodes directly adjacent to the central nodes. Three-point compact finite-difference schemes on uniform spacial meshes for the time-fractional advection-diffusion equation are constructed in [10]. The non-uniform meshes improve the efficiency of the numerical solutions of equation (4), which has a second order degeneration at s = 0 [5, 7, 6, 2]. The goal of the present paper is to construct a highorder three-point compact finite-difference scheme for the time-fractional Black-Scholes (TFBS) equation (6) and the time-fractional Black-Scholes equation (7) in diffusion form (TFBSD) on a spacial non-uniform mesh. The outline of the paper is as follows. In section 2, we introduce and analyze a fourth-order compact approximation (5) for the second derivative on a non-uniform mesh. In section 3 we use approximation (5) to construct a compact finite-difference scheme for the TFBSD equation on special non-uniform meshes used in finance and we present the results of the numerical experiments for test examples. Compact approximation on a non-uniform mesh Non-uniform grids are frequently used for numerical solution of differential equations, especially for equations with singular solutions, in order to improve the accuracy of the numerical method. The most commonly used grid in finance is the Tavella-Randall grid, which resolves the effect of the singularity of the initial condition of the BS equation at the striking price s = K. Let φ(x) be an increasing function on the interval [0, 1] with values φ(0) = s and φ(1) = s. Denote by M = {xn = nh} N n=0 the uniform net on the interval [0, 1], where h = 1/N and N is a positive integer. We use the function φ to define non-uniform meshes Mφ on the interval [s, s] by Mφ = {sn = φ(xn)|n = 0, 1, · · · , N} . The mesh Mφ has non-uniform mesh steps hn = sn+1−sn. When the function φ is a differentiable function with a bounded first derivative we determine a bound on the mesh steps, using the mean value theorem hn = sn+1 − sn = φ(xn+1)− φ(xn) = hφ ′(yn), where yn ∈ (xn, xn+1). The maximal length of the subintervals of the mesh M N φ is bounded by the maximal value of the first derivative of the function φ hn ≤ (