Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation

A high-order compact finite difference scheme for a fully nonlinear parabolic differential equation is analyzed. The equation arises in the modeling of option prices in financial markets with transaction costs. It is shown that the finite difference solution converges locally uniformly to the unique viscosity solution of the continuous equation. The proof is based on a careful study of the discretization matrices and on an abstract convergence result due to Barles and Souganides.High-order compact finite differences, numerical convergence, viscosity solution, financial derivatives

High-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids

We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme

High-order Compact Difference Schemes for the Modified Anomalous Subdiffusion Equation

In this paper, two kinds of high-order compact finite difference schemes for second-order derivative are developed. Then a second-order numerical scheme for Riemann-Liouvile derivative is established based on fractional center difference operator. We apply these methods to fractional anomalous subdiffusion equation to construct two kinds of novel numerical schemes. The solvability, stability and convergence analysis of these difference schemes are studied by Fourier method in details. The convergence orders of these numerical schemes are $\mathcal {O}(\tau^2+h^6)$ and $\mathcal {O}(\tau^2+h^8)$, respectively. Finally, numerical experiments are displayed which are in line with the theoretical analysis.Comment:

High-order compact schemes for parabolic problems with mixed derivatives in multiple space dimensions

We present a high-order compact finite difference approach for a rather general class of parabolic partial differential equations with time and space dependent coefficients as well as with mixed second-order derivative terms in n spatial dimensions. Problems of this type arise frequently in computational fluid dynamics and computational finance. We derive general conditions on the coefficients which allow us to obtain a high-order compact scheme which is fourth-order accurate in space and second-order accurate in time. Moreover, we perform a thorough von Neumann stability analysis of the Cauchy problem in two and three spatial dimensions for vanishing mixed derivative terms, and also give partial results for the general case. The results suggest unconditional stability of the scheme. As an application example we consider the pricing of European Power Put Options in the multidimensional Black-Scholes model for two and three underlying assets. Due to the low regularity of typical initial conditions we employ the smoothing operators of Kreiss et al. to ensure high-order convergence of the approximations of the smoothed problem to the true solution

Efficient hedging in Bates model using high-order compact finite differences

We evaluate the hedging performance of the scheme developed in B. Düring, A. Pitkin, ”High-order compact finite difference scheme for option pricing in stochastic volatility jump models”, 2017. We compare the scheme’s hedging performance to standard finite difference methods in different examples. We observe that the new scheme achieves fourth-order convergence, outperforming a standard, second-order central finite difference approximation in all our experiments