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, such as the 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.

Original languageEnglish
Article number127367
JournalPhysics Letters, Section A: General, Atomic and Solid State Physics
Publication statusPublished - 28 Jun 2021


  • A homotopy perturbation method
  • Black-Scholes equations
  • European options
  • Multi-asset options


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