# Reducing to 2D heat equation

• April 20th 2010, 12:06 PM
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Reducing to 2D heat equation
How can I use the transformations:

$x_1^* = \log(\frac{S_1}{X_1})$ and $x_2^* = \log(\frac{S_2}{X_2})$

and using other transformations analogous to those used in the one-dimensional case.

To show that the two-dimensional Black-Scholes equation can be reduced to the two-dimensional heat equation i.e.

$
\frac{\partial V}{\partial t} + \frac{1}{2}\sum_{i=1}^{2}\sum_{j=1}^{2}\sigma_i\si gma_jS_iS_j\rho_{ij}\frac{\partial^2 V}{\partial^2 S_iS_j} + \sum_{i=1}^{2}(r - D_i) S_i \frac{\partial V}{\partial S_i} - rV = 0
$

can be reduced to the two-dimensional heat equation i.e.

$
\frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x_1^2} + \frac{\partial^2 u}{\partial x_2^2}
$

I am really stuck with it, and don't know how to begin, any help is appreciated.