  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.
