Abstract
We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.
Highlights
We present a stable nite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou’s jump-diffusion model
It is widely recognized that the assumption of log-normal stock diffusion with constant volatility in the standard BlackScholes model [1] of option pricing is not ideally consistent with that of the market price movement
The probability distribution of realized asset returns o en exhibits features that are not taken into account by the standard BlackScholes model: heavy tails, volatility clustering, and volatility smile [2]
Summary
It is widely recognized that the assumption of log-normal stock diffusion with constant volatility in the standard BlackScholes model [1] of option pricing is not ideally consistent with that of the market price movement. D’Halluin et al [12, 13] developed a second-order accurate numerical method with a xed-point iteration method and an implicit nite difference scheme along with a penalty method for pricing American options under jump diffusion processes. We will present a stable nite difference method with a second-order convergency with respect to the spatial variable for solving the partial integrodifferential equation de ned on (0, +∞) for arbitrary volatility and arbitrary interest rate. In this paper we present a stable nite difference scheme on a piecewise uniform mesh along with a power penalty method for pricing American put options under Kou’s jump-diffusion model. By adding a penalty term the partial integrodifferential complementarity problem arising from pricing American put options under Kou’s jump-diffusion model is transformed into a nonlinear parabolic integrodifferential equation.
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