Integration of Piecewise Polynomials against a Gaussian probability density function

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In a number of financial contexts it can be important to calculate the following integral:



For example, this integral is relevant to calculating moments of a fat-tailed distribution, i.e. one whose quantile-quantile plot versus the Normal distribution, , is not unity. The faster  increases as  the greater is the impact of fat-tailed behaviour, i.e. deviations from a density function of the form . Thus risk measures such as expected shortfall (effectively a first moment computation, in which the leading element of  is of order ) are more sensitive to fat-tailed effects than Value-at-Risk risk measures (effectively a zero moment computation, in which the leading element of  is of order ).


This integral can also appear in derivative pricing analysis, if payoff is being approximated by a piecewise polynomial function and the movement of the underlying is of a certain type (but see Valuing polynomial payoffs in a Black Scholes World, which suggests that some modification may be needed to cater for exponentials arising when converting the partial differential equation arising under a Gauss-Weiner process to one with ‘standard’ parabolic form).


Hence, it becomes helpful to be able to calculate the following integral rapidly:



If  is large then we note that:






If  is small then for higher order coefficients the above computation runs into machine rounding problems. It is then better to use a Taylor series expansion, bearing in mind that:



Hence (if  is integral and ) we have:



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