Constructions of Low Discrepancy Sequences for Computation of Derivative Pricing

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When we compute a price of option derivative in financial engineering, an integration over very high dimension is required. We usually use Monte Carlo method for such computation problems with ordinal pseudo random numbers. It is known that a kind of balanced point set is better than random numbers with respect to precisions, called (t,m,s)-nets or (t,s)-sequences.We have studied to construct and analyze such point sets. When the strength t=3, 4, we proved a (t,m,s)-net is equivalent to a configuration, called Theta-configuration, in a finite projective space. Also we proved that a (3,m,s)-net is equivalent to an orthogonal array.There are sequences called Hickrnel sequences which are proposed recently. We define the sequences using polynomials over a finite field and proved it is a (t, s)-sequence.The most uniformly distributed sequences are (0,s)-sequences in (t,s)-sequences. We used to choose parameters randomly. We found a method to choose good parameters. This method is called"De-randomization". Also we proposed an approach using i-binomial property. We analyzed properties of de-randomizations.We studied and improved an ordinal pseudo random number generator, called Mersenne twister, which have very long period. And also we studied many basic combinatorial problems which may have a relation to constructions of (t,m,s)-nets or (t,s)-sequences