# PDE and finding a general solution

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

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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 ?

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

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.