PDE and finding a general solution

[meghibbert17]

Hi everyone,

I am doing a sheet on Asian Options and The Black Scholes equation.

I have the PDE,

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

I have to seek a solultion of the form v=α_{1}(τ)ε + α_{0}(τ) and determine the general solution for α_{1}(τ) and α_{0}(τ).

We are given that ε=\frac{I}{TS} - \frac{X}{S}, τ=T-t and V(S, I, t)=e^{-Dτ}Sv(ε, τ)

Can anybody help me with this problem?
Thanks

 

[JJacquelin]

Hi everyone,

I am doing a sheet on Asian Options and The Black Scholes equation.

I have the PDE,

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

I have to seek a solultion of the form v=α_{1}(τ)ε + α_{0}(τ) and determine the general solution for α_{1}(τ) and α_{0}(τ).

We are given that ε=\frac{I}{TS} - \frac{X}{S}, τ=T-t and V(S, I, t)=e^{-Dτ}Sv(ε, τ)

Can anybody help me with this problem?
Thanks

Hi meghibbert17 !

several notations are not clear enough. For exemple three different typographies for "v".
What exactly is the list of variables and the list of constants ?
In the first equation, are you sure that dv/dr = 0 ?

 

[meghibbert17]

Hello,

Sorry, I am new to this and it does look rather messy!

in the first equation it is not dv/dr but dv/d(tau).

All the v's in the equation are the same as the v=α_{1}(τ)ε + α_{0}(τ) which we are seeking a solution for and then V(S, I, t) = e−DτSv(ε, τ).

Incase it is also not clear, its tau = T-t
Is that any clearer? Thankyou

 

[JJacquelin]

Sorry, I cannot understand what are the constants and what are the variables and the functions.
Moreover, I see that dV/d(tau)=0 in the first equation. And V is a function of (tau) in the given relationship V(S,I,t)=exp(-D*tau)*S*V(epsilon,tau). This is in contradiction.
All this is too messy for me. I hope that someone else could help you.