A Multinomial Theorem for Hermite Polynomials and Financial Applications

Author(s)
Francois Buet-Golfouse

Abstract

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.

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.

Cite this paper

Buet-Golfouse, F. (2015) A Multinomial Theorem for Hermite Polynomials and Financial Applications.*Applied Mathematics*, **6**, 1017-1030. doi: 10.4236/am.2015.66094.

Buet-Golfouse, F. (2015) A Multinomial Theorem for Hermite Polynomials and Financial Applications.

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