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    Thermal Diffusivity: Conversion of Black-Scholes Equation

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    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.

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    Solution Summary

    A conversion of the Black-Scholes equation is investigated.