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Removing non-smoothness in solving Black-Scholes equation using a perturbation method

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Listed:
  • Endah R. M. Putri
  • Lutfi Mardianto
  • Amirul Hakam
  • Chairul Imron
  • Hadi Susanto

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.

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  • Endah R. M. Putri & Lutfi Mardianto & Amirul Hakam & Chairul Imron & Hadi Susanto, 2021. "Removing non-smoothness in solving Black-Scholes equation using a perturbation method," Papers 2104.07839, arXiv.org, revised Apr 2021.
    • Handle: RePEc:arx:papers:2104.07839
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    References listed on IDEAS

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