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:
Where
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:
