In this paper, we propose an unconditionally stable numerical technique for a multi-dimensional Black-Scholes equation to price an option with high accuracy. The proposed scheme uses the operator splitting method to reduce the multi-dimensional partial differential equation to a set of one-dimensional sub-problems. The computational domain is discretized with a uniform space and time step size to approximate the option values and asset correlation-related terms by piecewise quadratic polynomials. In order to obtain the numerical solutions of the sub-problems, we analytically integrate the polynomials to estimate the expectation of the Feynman-Kac formula. We compare our method with the implicit operator splitting method (OSM), a widely used finite difference method in the industry. Numerical experiments show that the proposed method outperforms OSM in terms of convergence in space and time directions. We also provide analysis to guarantee the unconditional stability of our method by exploiting the Feynman-Kac recursively.
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© 2023 Elsevier Ltd
All Science Journal Classification (ASJC) codes
- Modelling and Simulation
- Computational Theory and Mathematics
- Computational Mathematics