# Is it possible to transform arithmetic-average strike continuous sampling Asian Black-Scholes equation to a heat equation?

By Transformation from the Black-Scholes differential equation to the diffusion equation - and back, we are able to transform vanilla European option into a heat equation.

And we know that the arithmetic-average strike continuous sampling Asian Black-Scholes equation is $$\frac{\partial V}{\partial t} +\frac{1}{2}\sigma^2S^2\frac{\partial ^2 V}{\partial S^2} +rS\frac{\partial V}{\partial S} + S\frac{\partial V}{\partial J}- rV=0$$ i.e., only one more term $$S\frac{\partial V}{\partial J}$$ compared with original BS equation.

Since this equation is similar to the original BS equation, I assume that we can transform it into a heat equation. Am I correct?

## 1 Answer

This won't transform a version of the heat equation that can be solved analytically. The extra term results in the time-integral of Geometric Brownian Motion, which has no known analytical transform.

The extra term leads to incremental accounting errors when attempting to use straight integration methods. While Ito calculus accounts for this convexity error by transforming the Black-Schole differential equation of an exponential Brownian Motion, no known analytical method can accounts for these error terms when integrating the sums of lognormals themselves.

• The time integral of arithmetic Brownian Motion is still normally distributed. However, due to the error term, the time integral of geometric Brownian Motion is not still lognormally distributed.
• The exception to the above applies when taking the unconditional expectation of the time integral. In this case, the randomness can be shown to inconsequential to the expectations. However, the use case for the SDE shown above is take a conditional expectation for pricing Asian options and the like.

I feel like this problem will be solved within my lifetime through an analog of the Ito chain rule. But until then we are left with approximations, numerical transforms, and numerical methods.