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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.

##### Solution Summary

A conversion of the Black-Scholes equation is investigated.

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###### Education
• BSc , Wuhan Univ. China
• MA, Shandong Univ.
###### Recent Feedback
• "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
• "excellent work"
• "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
• "Thank you"
• "Thank you very much for your valuable time and assistance!"

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