I want to use the derivation of BS for another type of derivative, not an option. Known the derivation of the Black-Scholes differential equation, is it possible to use in the same equation when my boundary condition changes and now it's a free boundary?
Here is the equation of BS from the book: Transformation from the Black-Scholes differential equation to the diffusion equation - and back
My question: If the derivative is:
$$G \left(S,t\right)=\begin{cases} s_{b}-S_{t} & 0\le S_{t}\le s_{b}\\ S_{t}-s_{b} & s_{b}<S_{t},\,S_{t}\ngeq s_{a}\\ s_{a}-S_{t} & S_{t}<s_{a},\,S_{t}\nleq s_{b}\\ S_{t}-s_{a} & s_{a}<S_{t}<\infty \end{cases}$$
$$\Downarrow$$ $$G\left(S,t\right)=\begin{cases} \text{PT}\left(S,s_{b}\right) & 0\le S_{t}\le s_{b}\\ S_{t}-s_{b} & s_{b}<S_{t},\,S_{t}\ngeq s_{a}\\ s_{a}-S_{t} & S_{t}<s_{a},\,S_{t}\nleq s_{b}\\ \text{CL}\left(S,s_{a}\right) & s_{a}<S_{t}<\infty \end{cases}$$
How was look equation 5.8 in "The mathematics of financial derivative" p.88
