TY - JOUR

T1 - Removing non-smoothness in solving Black-Scholes equation using a perturbation method

AU - Putri, Endah R.M.

AU - Mardianto, Lutfi

AU - Hakam, Amirul

AU - Imron, Chairul

AU - Susanto, Hadi

N1 - Publisher Copyright:
© 2021 Elsevier B.V.

PY - 2021/6/28

Y1 - 2021/6/28

N2 - 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.

AB - 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.

KW - A homotopy perturbation method

KW - Black-Scholes equations

KW - European options

KW - Multi-asset options

UR - http://www.scopus.com/inward/record.url?scp=85104771599&partnerID=8YFLogxK

U2 - 10.1016/j.physleta.2021.127367

DO - 10.1016/j.physleta.2021.127367

M3 - Article

AN - SCOPUS:85104771599

SN - 0375-9601

VL - 402

JO - Physics Letters, Section A: General, Atomic and Solid State Physics

JF - Physics Letters, Section A: General, Atomic and Solid State Physics

M1 - 127367

ER -