Black-Scholes PDE and finding the general solution

  • #1
Hello, I have the PDE

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

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

I am given that ε=[itex]\frac{I}{TS}[/itex] - [itex]\frac{X}{S}[/itex] and that τ=T-t.

Can anybody help me with this problem?

Thankyou
 

Answers and Replies

  • #2
chiro
Science Advisor
4,790
132
Hello, I have the PDE

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

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

I am given that ε=[itex]\frac{I}{TS}[/itex] - [itex]\frac{X}{S}[/itex] 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?
 

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