Removing non-smoothness in solving Black-Scholes equation using a perturbation method
Abstract
Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, including homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by {first} performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.
Cite
@article{arxiv.2104.07839,
title = {Removing non-smoothness in solving Black-Scholes equation using a perturbation method},
author = {Endah R. M. Putri and Lutfi Mardianto and Amirul Hakam and Chairul Imron and Hadi Susanto},
journal= {arXiv preprint arXiv:2104.07839},
year = {2021}
}