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PricingMulti-dimensional options is a challenging problem in financial mathematics. In this paper, we price these options with importance sampling for moment reduction. That is, instead of minimizing the second moment or relative variance of an estimator, the optimal parameters of a candidate measure are obtained by minimizing the relative centered moment of order p and the relative origin moment of order p respectivly. We investigate the use of different importance sampling for moment reduction techniques to improve the efficiency of the Monte Carlo estimators. Some numerical experiments on multi-dimensional options are used to investigate the performance of these approaches.

How to allocate the weights of stocks is an interesting technology in stock index optimized replicate. This paper proposed a hybrid algorithm of adaptive genetic algorithm and pattern search (AGA-PS) to find the optimal portfolio weights. In AGA-PS, the crossover probability and mutation probability are adjusted adaptively. The weight from adaptive genetic algorithm is as the search start point of pattern search. The experiment result of AGA-PS, which has smaller tracking error compared with GA and AGA, has shown that AGA-PS model is feasible and effective to the stock portfolio. Keywords-adaptive genetic algorithm; pattern search; index optimized replicate; stock; portfolioI.

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