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.

References

[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.

[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.