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By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform the PDE $\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} - rV=0$ into a heat equation.

After I turn this equation into 2D by adding a term $S\frac{\partial V}{\partial J}$, we have $$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} + S\frac{\partial V}{\partial J}- rV=0$$

Is there an algorithm that helps us transform this PDE into a heat equation, or at least eliminate the $S$ in the coefficient? (Since $S$ is a variable while the other coefficients are constant)

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  • $\begingroup$ @Mattos No, we don't' have $V_{JJ}$ in the pde. Any thoughts? $\endgroup$
    – Wenzel
    Jul 1, 2019 at 1:53
  • $\begingroup$ @Mattos Exactly... $\endgroup$
    – Wenzel
    Jul 1, 2019 at 1:56
  • $\begingroup$ @Mattos It’s Black-Scholes PDE for Asian options. I’m sure the form is correct. So you mean that it’s nontrivial to transform it into a heat equation? $\endgroup$
    – Wenzel
    Jul 1, 2019 at 1:58
  • $\begingroup$ The first link on the google search yields this. $\endgroup$ Jul 1, 2019 at 2:01
  • $\begingroup$ @Mattos thanks a lot $\endgroup$
    – Wenzel
    Jul 1, 2019 at 2:08

1 Answer 1

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Hint:

Let $V=e^{rt}W$ ,

Then $\dfrac{\partial V}{\partial t}=e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W$

$\dfrac{\partial V}{\partial S}=e^{rt}\dfrac{\partial W}{\partial S}$

$\dfrac{\partial^2V}{\partial S^2}=e^{rt}\dfrac{\partial^2W}{\partial S^2}$

$\dfrac{\partial V}{\partial J}=e^{rt}\dfrac{\partial W}{\partial J}$

$\therefore e^{rt}\dfrac{\partial W}{\partial t}+re^{rt}W+\dfrac{\sigma^2S^2}{2}e^{rt}\dfrac{\partial^2W}{\partial S^2}+rSe^{rt}\dfrac{\partial W}{\partial S}+Se^{rt}\dfrac{\partial W}{\partial J}-re^{rt}W=0$

$\dfrac{\partial W}{\partial t}+\dfrac{\sigma^2S^2}{2}\dfrac{\partial^2W}{\partial S^2}+rS\dfrac{\partial W}{\partial S}+S\dfrac{\partial W}{\partial J}=0$

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