Please consider the following modified European Call Option
where $ 0 < a \leq 1$. When $a = 1$ the modified European call option is reduced to the standard European call option.
Transforming the Black-Scholes equation in the standard heat equation and using Fourier transform, I am obtaining the following analytical solution for such modified European call option (please do right click on the image to enlarge it)
When $a=1$ the standard Black-Scholes formula for the usual European call option is recovered, namely
My questions are:
Do you know another method to derive the solution for the modified European Cal option.
Do you know real life cases on which a modified European Call options are applied.