# Thermal Diffusivity: Conversion of Black-Scholes Equation

Show that the form of the Black-Scholes equation given can be converted into Ut=((sigma^2)/2)*Uxx

Please see the attached PDF file.

Define u(X, τ) V(X, τ)

e (α ⋅X+β ⋅τ)

= , where α and β are constants yet to be specified.

Then: V(X, τ) u(X, τ) e = ⋅(α ⋅X+β ⋅τ).

Now, starting with the PDE for V(X, τ):

Now, by divine inspiration, introduce the ratio γ 2⋅r

Show that your choice of α and β, when expressed in terms of this γ, reads:

Finally, show that the I.C. for V(X, τ) is equivalent to the following I.C. for u(X, τ):

u(X,0) [0____________________For⋅(X ≤ 0)]

Note that the system describing u(X, τ) is formally identical to the heat equation of

an infinite rod with thermal diffusivity k

Thus, the volitility of the commodity

which underlies our call option is generating a mathematical diffusion process.

https://brainmass.com/physics/heat-transfer/thermal-diffusivity-conversion-black-scholes-equation-17675

#### Solution Summary

A conversion of the Black-Scholes equation is investigated.