Black-Scholes PDE and finding the general solution

[meghibbert17]

Hello, I have the PDE

$\frac{-∂v}{∂τ}$+$\frac{1}{2}$σ$^{2}$ε$^{2}$$\frac{∂^{2}v}{∂ε^{2}}$+($\frac{1}{T}$+(r-D)ε)$\frac{∂v}{∂ε}$=0

and firstly I need to seek a solution of the form v=α$_{1}$(τ)ε + α$_{0}$(τ) and then determine the general solution for α$_{1}$(τ) and α$_{0}$(τ).

I am given that ε=$\frac{I}{TS}$ - $\frac{X}{S}$ and that τ=T-t.

Can anybody help me with this problem?

Thankyou

 

[chiro]

Hello, I have the PDE

$\frac{-∂v}{∂τ}$+$\frac{1}{2}$σ$^{2}$ε$^{2}$$\frac{∂^{2}v}{∂ε^{2}}$+($\frac{1}{T}$+(r-D)ε)$\frac{∂v}{∂ε}$=0

and firstly I need to seek a solution of the form v=α$_{1}$(τ)ε + α$_{0}$(τ) and then determine the general solution for α$_{1}$(τ) and α$_{0}$(τ).

I am given that ε=$\frac{I}{TS}$ - $\frac{X}{S}$ and that τ=T-t.

Can anybody help me with this problem?

Thankyou

Hey meghibbert17 and welcome to the forums.

Are you familiar with the idea for transforming the B.S. to a standard PDE heat equation?