# 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?

• What is $S\frac{\partial V}{\partial J}$ doing here? What is the partial derivate of V wrt to J? Moreover, where is J coming from? Commented Apr 4, 2023 at 18:54
• @DavidAddison I think what the OP means with J is $$J = \int_0^t S_u du$$ where $t$ is current valuation date and the contract inception is at time $0$. Commented Apr 4, 2023 at 21:17

Intuitively, the extra term $$S\frac{\partial V}{\partial J}$$ is responsible for convexity errors terms which are analogous to Jensen's inequality. Ito's lemma is able to account for these errors by adding the term $$\frac{\sigma^2}{2}t$$ to GBM. The issue, though, is finding a transformation of variables a known distribution. This will probably be solved sometime in my lifetime.