A Multinomial Theorem for Hermite Polynomials and Financial Applications

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.

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Conflicts of Interest

The authors declare no conflicts of interest.

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.

 [1] Madan, D. and Milne, F. (1994) Contingent Claims Valued and Hedged by Pricing and Investing in a Basis. Mathematical Finance, 4, 223-245. http://dx.doi.org/10.1111/j.1467-9965.1994.tb00093.x [2] 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. http://dx.doi.org/10.1080/14697680903193371 [3] 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 [4] Buet-Golfouse, F. and Owen, A. (2015) The Application of Hermite Polynomials to Risk Allocation. Barclays Quantitative Credit, Working Paper. [5] Voropaev, M. (2011) An Analytical Framework for Credit Portfolio Risk Measures. Risk, November, 72-78. [6] Owen, A., Bryers, J. and Buet-Golfouse, F. (2015) Hermite Polynomial Approximations in Credit Risk Modelling with PD-LGD Correlation. Journal of Credit Risk, Accepted Paper. [7] Abramowitz, M. and Stegun, I. (1964) Handbook of Mathematical Functions. Dover Publications, New York. [8] Portait, R. and Poncet, P. (2009) Finance de marche. 2nd Edition, Dalloz, Paris. [9] Brezis, H. (2011) Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York.