Koksma Inequality

For the calculation of the European Put in the Boness formula including the special case of Black-Scholes, we have a one dimensional integration. Niederreiter [#!kn:Niederreiter1992!#] Chapter 3 p23-24 discusses numerical integration in this case. We are integrating u from 0 to 1 to calculate the expected payoff of the put. After that we can discount its payoff at a constant rate in the Boness formula or its Black-Scholes subcase.

If we have N points, the minimum Star Discrepancy is $\frac{1}{2N}$. See Theorem 2.6 of Niederreiter. The minimum is obtained for the choice of points

$$x_i = \frac{2i-1}{N}$$ (2.19)

for i=1,...,N.

Theorem 7 (Koksma Inequality European Put)   For a European Put in the Boness framework, the numerical integration over the interval of u from 0 to 1 of the Put Payoff Function is given by

$$\vert\epsilon\vert \le x_i = \frac{K}{2N}$$ (2.20)

where K is the strike price. If B is the price of a zero discount bond, the inequality on the price is

$$\vert\epsilon\vert \le x_i = \frac{K B}{2N}$$ (2.21)

These inequalities are for the case of direct integration of the Put Payoff Function or Put Price. It is somewhat more efficient to first transform as Boness (and Sprenkle [#!kn:SprenkleYale1960!#] ) realized to the form we are familiar with for these integrals in terms of the cumulative normal distribution function.