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## Tools for Computational Finance

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### Tools for Computational Finance (Universitext) (2006)

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This sequence is too easily predictable. Various other properties and requirements are discussed in the literature, in particular in [Kn95]. In case the period is M , the numbers Ui are distributed evenly when exactly M numbers are needed. Then each 1 is occupied once. But for serious computations we recommend to rely on the many suggestions in the literature. But which of the many possible generators are recommendable?

The requirements on good number generators can roughly be divided into three groups. In view of Property 2. This leads to select M close to the largest integer machine number. But a period p close to M is only achieved if a and b are chosen properly. Criteria for relations among M, p, a, b have been derived by number-theoretic arguments.

This is outlined in [Kn95], [Ri87]. A second group of requirements are the statistical tests that check whether the numbers are distributed as intended. Another simple test is to check correlations. For example, it would not be desirable if small numbers are likely to be followed by small numbers.

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A slightly more involved test checks how well the probability distribution is approximated. Calculate j samples from a random number generator, and investigate how the samples distribute on the unit interval. This procedure is just the simplest test; for more ambitious tests, consult [Kn95]. The third group of tests is to check how well the random numbers distribute in higher-dimensional spaces. This issue of the lattice structure is discussed next.

## Tools for Computational Finance | Rüdiger Seydel | Springer

We derive a priori analytical results on where the random numbers produced by Algorithm 2. Then the tupels or the corresponding points Ui , But the statement becomes exciting in case it is valid for a family of parallel planes with large distances between neighboring planes.

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Next we attempt to construct such planes. The points calculated by Algorithm 2. Fixing one tupel z0 , z1 , the equation 2. The question is whether there exists a tupel z0 , z1 such that only few of the straight lines cut the square [0, 1 2? In this case wide areas of the square would be free of random points, which violates the requirement of a uniform distribution of the points. The minimum number of parallel straight lines hyperplanes cutting the square, or equivalently the maximum distance between them serve as measures of the equidistributedness.

We now analyze the number of straight lines, searching for the worst case.

By solving 2. For many applications this must be seen as a severe defect. Alternatively we can directly analyze the lattice formed by consecutive points. For illustration consider again Figure 1 2 2. By vectorwise adding an 1 1 appropriate multiple of the next two points are obtained. The reader may verify this with Example 2. A disadvantage of the linear congruential generators of Algorithm 2. The reader may test this on Example 2. For practical advice we refer to [PTVF92]. Example 2.