Abstract
In this paper we discuss the importance sampling Monte Carlo methods for pricing options. The classical importance sampling method is used to eliminate the variance caused by the linear part of the logarithmic function of payoff. The variance caused by the quadratic part is reduced by stratified sampling. We eliminate both kinds of variances just by importance sampling. The corresponding space for the eigenvalues of the Hessian matrix of the logarithmic function of payoff is enlarged. Computational Simulation shows the high efficiency of the new method.
Highlights
Monte Carlo simulation is a numerical method based on the probability theory
Several techniques to reduce the variance of the Monte Carlo simulation have been proposed, such as control variates, antithetic variables, importance sampling and stratification(see Boyle, Broadie and Glasserman [1], and Glasserman [2]
Other recent work on importance sampling methods in finance has been done for Monte Carlo simulations driven by high-dimensional Gaussian vectors, such as Boyle, Broadie and Glasserman [1], Vázquez-Abad and Dufresne [9], Su and Fu [10], Arouna [11], Capriotti [12], Xu and Zhang [13]
Summary
Monte Carlo simulation is a numerical method based on the probability theory. Its application in finance becomes more and more popular as the demand for pricing and hedging of various complex financial derivatives, which play an important role in the field of investment, risk management and corporate governance. Another way to improve the accuracy is to reduce the standard deviation Motivated by this thought, several techniques to reduce the variance of the Monte Carlo simulation have been proposed, such as control variates, antithetic variables, importance sampling and stratification(see Boyle, Broadie and Glasserman [1], and Glasserman [2]. Other recent work on importance sampling methods in finance has been done for Monte Carlo simulations driven by high-dimensional Gaussian vectors, such as Boyle, Broadie and Glasserman [1], Vázquez-Abad and Dufresne [9], Su and Fu [10], Arouna [11], Capriotti [12], Xu and Zhang [13] In this framework, Importance Sampling is applied by modifying the drift term of the simulated process to construct a new measure in which more weight is given to important outcomes thereby increasing sampling efficiency. The method proposed in the paper can be extended to the pricing of other financial derivaives directly
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