# B-S derivative with another boundary condition

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}

$$\Downarrow$$ $$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}

How was look equation 5.8 in "The mathematics of financial derivative" p.88

• Maybe you could specify what equation 5.8 is? – Sanjay Jul 30 '19 at 16:07