Second order accurate IMEX methods for option pricing under Merton and Kou jump-diffusion models. (English) Zbl 1331.91191
Summary: In this paper three implicit-explicit (IMEX) time semi-discrete methods, namely IMEX-BDF1, IMEX-BDF2 and CN-LF, are developed for solving parabolic partial integro-differential equations which arise in option pricing theory when the underlying asset follows a jump diffusion process. It is shown that IMEX-BDF2 and CN-LF are stable and second order accurate, whereas IMEX-BDF1 is stable but only first order accurate. After time semi-discretization, the resulting linear differential equations are solved by using a cubic B-spline collocation method. The methods so developed have computational complexity of \(O(MN\log_2(M))\) for Merton model and of \(O(MN)\) for Kou model, where \(N\) denotes the number of time steps and \(M\) the number of collocation points. Some numerical examples, for pricing European options under Merton and Kou jump-diffusion models with constant as well as variable volatility, are presented to demonstrate the stability, convergence and computational complexity of the methods.
MSC:
| 91G60 | Numerical methods (including Monte Carlo methods) |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M70 | Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs |
| 91G20 | Derivative securities (option pricing, hedging, etc.) |
Keywords:
option pricing; jump-diffusion model; partial integro-differential equation; finite differences; spline collocationReferences:
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