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.
Highlights
Hermite polynomials are widely used in finance, for various purposes including option pricing and risk management
This paper proposes a simple way of expanding the Hermite polynomial of a linear combination of factors into simpler elements
This method allows us to prove the celebrated Mehler formula in a very simple way, and enables us to derive the empirical prices of functions of linear combination of factors in a market with no arbitrage and facilitates credit risk modelling
Summary
Hermite polynomials are widely used in finance, for various purposes including option pricing and risk management. Madan and Milne [1] have built a framework applying functional analysis results to the particular case of Hermite polynomials and inferred pricing formulas for general payoffs expressed as linear combinations of Hermite polynomials. They applied their framework to the simple case of calls to determine the implicit basis prices in the market data and imply an empirical futures price law. The second main result is a multinomial expansion theorem for Hermite polynomials (and its extensions) Both provide a solid foundation to derive the no-arbitrage price of a contingent claim stemming from a linear combination of factors. Empirical applications of the described methodology can be found in [4] and [6]
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