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.

作为最著名的数学模型之一，Black-Scholes方程具有一个明确的解析解决方案，称为Black-Scholes公式。方程的后续变式（例如分数或非线性Black-Scholes方程）对于相应的公式没有封闭形式的表达式。在那种情况下，将需要渐近展开，例如同伦摄动法，以给出近似的解析解。但是，该解决方案在特定点上是不平滑的。我们通过首先执行将点推向无穷大的变量转换来修改方法。作为测试平台，我们将该方法应用于可求解的Black-Scholes方程，在该方程中可获得与精确解的极好的一致性。通过将案例减少到单资产案例，我们还将研究扩展到多资产篮子和量化选项。