Application to the Pricing of Complex Financial Securities

The risk-neutral valuation paradigm shows that the fair value at time of a financial derivative maturing at time is expressed as an integral. For example, for a European option with pay-off function on an underlying with initial price and volatility

where is the riskless interest rate over the period , the function the density

and is the inverse cummulative normal distribution function (entering from the assumption that the price process of the underlying satisfies a geometric Brownian motion).

In case of a European call option with strike the function (depending on , , , , and ) becomes

Let us extract a one-dimensional sequence of quasi-random numbers from the previously calculated Niederreiter sequence in base 3 and calculate the sample mean:

If we compare this to the analytic solution obtained from the Black-Scholes formula we see a high accuracy of quasi-Monte Carlo integration:

We visualize how good the first (transformed) sequence elements of a Niederreiter sequence can approximate the Gauss distribution. To this end, we throw away the first sequence element (as it is 0) and invert with the inverse cummulative distribution function.

bars = BarChart[buckets,PlotRange->All,BarLabels->None,BarEdges->False];

Finally we superimpose the Gaussian density function with mean and variance the sample mean respectively the sample variance of the transformed samples.

Converted by Mathematica November 18, 1998