Different aspects of mathematical finance benefit from the use Hermite polynomials, and this is particularly the case where risk drivers have a Gaussian distribution. They support quick analytical methods which are computationally less cumbersome than a full-fledged Monte Carlo framework, both for pricing and risk management purposes. In this paper, we review key properties of Hermite polynomials before moving on to a multinomial expansion formula for Hermite polynomials, which is proved using basic methods and corrects a formulation that appeared before in the financial literature. We then use it to give a trivial proof of the Mehler formula. Finally, we apply it to no arbitrage pricing in a multi-factor model and determine the empirical futures price law of any linear combination of the underlying factors.
Tanaka, K., Yamada, T. and Watanabe, T. (2010) Applications of Gram-Charlier Expansion and Bond Moments for Pricing of Interest Rates and Credit Risk. Quantitative Finance, 10, 645-662.
Schloegl, E. (2013) Option Pricing Where the Underlying Assets Follow a Gram/Charlier Density of Arbitrary Order. Journal of Economic Dynamics and Control, 37, 611-632. http://dx.doi.org/10.1016/j.jedc.2012.10.001