Why Arithmetic Asian Options Resist a Heat Equation Transformation

# Answer by David Addison (score 1, accepted)


# 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?

## Answer by David Addison (score 1, accepted)

https://quant.stackexchange.com/a/46402

This is an interesting question. On face value, I don't think it will transform into the heat equation because if it could, someone would've already done it, thereby solving the century old problem of the sums/averages of lognormals. But that's not a strong argument.

A few notes:

- Regular Brownian Motion is normally distributed. The sums/averages of normal distributions are normally distributed.

- Geometric Brownian Motion is lognormally distributed. The sums/averages of lognormals are not lognormally distributed.

- Rather, the sums (or equally, the arithmetic averages) of lognormally distributed variables is believed to result in a Bessel process - known as the Integrated Geometric Brownian Motion (wrt time) - that converges to an inverse gamma distribution. This has been studied extensively (Yor and Geman. Bessel Processes, Asian Options, and Perpetuities. 1993) (Daniel Dufresne. Sums of Lognormals. 200)(Daniel Dufresne, Bessel Processes and Asian Options. 2005) However, no known analytic transformation exists for deriving the parameters of this distribution. Rather, they are typically evaluated using numerical methods or approximations.

- Many approximations (e.g., Fenton-Wilkinson moment matching method) rely on the observation that the sums/average of lognormal distributions resemble a lognormal distribution.

- The exception to the above applies when taking the unconditional expectation of the time integral since randomness is non-consequential to the expected value (needed for risk neutral assumption regardless).

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.