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Black-Scholes equation as one of the most celebrated mathematical models hasan explicit analytical solution known as the Black-Scholes formula. Latervariations of the equation, such as fractional or nonlinear Black-Scholesequations, do not have a closed form expression for the corresponding formula.In that case, one will need asymptotic expansions, including homotopyperturbation method, to give an approximate analytical solution. However, thesolution is non-smooth at a special point. We modify the method by {first}performing variable transformations that push the point to infinity. As a testbed, we apply the method to the solvable Black-Scholes equation, whereexcellent agreement with the exact solution is obtained. We also extend ourstudy to multi-asset basket and quanto options by reducing the cases tosingle-asset ones. Additionally we provide a novel analytical solution of thesingle-asset quanto option that is simple and different from the existingexpression.

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