Good Path Generation Methods in Quasi-Monte Carlo for Pricing Financial Derivatives

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Quasi-Monte Carlo (QMC) method is a powerful numerical tool for pricing complex derivative securities, whose accuracy is affected by the smoothness of the integrands. The payoff functions of many financial derivatives involve two types of non-smooth factors — an indicator function (called jump structure) and a positive part of a smooth function (called kink structure). This paper develops a good path generation method (PGM) for recovering the superiority of QMC method on problems involving multiple such structures. This is achieved by realigning these structures such that the associated non-smooth surfaces are parallel to as many coordinate axes as possible. The proposed method has the advantage of addressing different structures according to their importance. We also offer a systematic measurement of different structures for quantifying and then ranking their importance. Numerical experiments demonstrate that the proposed method is more efficient than traditional PGMs for pricing exotic options, such as straddle Asian options, digital options and barrier options. The numerical results confirm that both the jumps and kinks have tremendous impacts on the performance of QMC method.

SIAM Journal on Scientific Computing, 36(2), B171-B197