Paper: Oct 07,2024
q-fin.CP
ID:2410.04745
Numerical analysis of American option pricing in a two-asset jump-diffusion model
This paper addresses a significant gap in rigorous numerical treatments for
pricing American options under correlated two-asset jump-diffusion models using
the viscosity solution approach, with a particular focus on the Merton model.
The pricing of these options is governed by complex two-dimensional (2-D)
variational inequalities that incorporate cross-derivative terms and nonlocal
integro-differential terms due to the presence of jumps. Existing numerical
methods, primarily based on finite differences, often struggle with preserving
monotonicity in the approximation of cross-derivatives-a key requirement for
ensuring convergence to the viscosity solution. In addition, these methods face
challenges in accurately discretizing 2-D jump integrals. We introduce a novel
approach to effectively tackle the aforementioned variational inequalities,
seamlessly managing cross-derivative terms and nonlocal integro-differential
terms through an efficient and straightforward-to-implement monotone
integration scheme. Within each timestep, our approach explicitly tackles the
variational inequality constraint, resulting in a 2-D Partial
Integro-Differential Equation (PIDE) to solve. Its solution is then expressed
as a 2-D convolution integral involving the Green's function of the PIDE. We
derive an infinite series representation of this Green's function, where each
term is strictly positive and computable. This series facilitates the numerical
approximation of the PIDE solution through a monotone integration method, such
as the composite quadrature rule. The proposed method is demonstrated to be
both $\ell_{\infty} $-stable and consistent in the viscosity sense, ensuring
its convergence to the viscosity solution of the variational inequality.
Extensive numerical results validate the effectiveness and robustness of our
approach, highlighting its practical applicability and theoretical soundness.
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Paper Author: Hao Zhou,Duy-Minh Dang
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