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,

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

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

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

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

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

We are given that ε=[itex]\frac{I}{TS}[/itex] - [itex]\frac{X}{S}[/itex], τ=T-t and V(S, I, t)=[itex]e^{-Dτ}[/itex]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=[itex]α_{1}[/itex](τ)ε + [itex]α_{0}[/itex](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.

 

[blue_raver22]

for sharing your question with us. The equation you have provided is known as the Black Scholes equation, which is a partial differential equation (PDE) commonly used in finance to determine the theoretical value of European-style options. It is a highly important equation in the field of mathematical finance and has many applications in the financial industry.

In order to solve this PDE, you have correctly identified the need to find a general solution for the variables α_{1}(τ) and α_{0}(τ). This can be done by using the method of separation of variables, where you assume that the solution can be expressed as a product of two functions, one depending only on τ and the other only on ε.

You have also provided us with the conditions for ε and τ, which will help in solving the equation. I would recommend consulting a textbook or seeking guidance from a professor or colleague to find the general solution for α_{1}(τ) and α_{0}(τ) using the method of separation of variables. This will involve solving two ordinary differential equations and then combining the solutions to obtain the general solution for the Black Scholes equation.

I hope this helps and good luck with your sheet on Asian Options and the Black Scholes equation. Keep exploring and learning about PDEs and their applications in finance and other areas of science.