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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, &#964;) V(X, &#964;)
e (&#945; &#8901;X+&#946; &#8901;&#964;)
= , where &#945; and &#946; are constants yet to be specified.
Then: V(X, &#964;) u(X, &#964;) e = &#8901;(&#945; &#8901;X+&#946; &#8901;&#964;).
Now, starting with the PDE for V(X, &#964;):

Now, by divine inspiration, introduce the ratio &#947; 2&#8901;r

Show that your choice of &#945; and &#946;, when expressed in terms of this &#947;, reads:

Finally, show that the I.C. for V(X, &#964;) is equivalent to the following I.C. for u(X, &#964;):
u(X,0) [0____________________For&#8901;(X &#8804; 0)]

Note that the system describing u(X, &#964;) 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.