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.
A conversion of the Black-Scholes equation is investigated.