Journal of Mathematical Finance, 2013, 3, 431-436 Published Online November 2013 (http://www.scirp.org/journal/jmf) http://dx.doi.org/10.4236/jmf.2013.34045 Variance Reduction Techniques of Importance Sampling * Monte Carlo Methods for Pricing Options Qiang Zhao1, Guo Liu1, Guiding Gu2 1School of Finance, Shanghai University of Finance and Economics, Shanghai, China 2Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, China Email: [email protected] Received July 10, 2013; revised September 7, 2013; accepted September 18, 2013 Copyright © 2013 Qiang Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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. Keywords: Monte Carlo Method; Importance Sampling; Variance Reduction; Option Pricing 1. Introduction with probability 1 , where is the standard devia- Monte Carlo simulation is a numerical method based on tion of V , n is the number of samples, is the the probability theory. Its application in finance becomes significance le vel and Z is the quantile of standard more and more popular as the demand for pricing and 2 hedging of various complex financial derivatives, which normal distribution under . play an important role in the field of investment, risk 2 management and corporate governance. The advantage It is clear that the convergence rate of Monte Carlo of Monte Carlo method is that its convergence rate is method is independent on the number of state variables. Monte 1 Carlo simulation is often the only way available for the 2 On. pricing of complex path-dependent options if the number of relevant underlying assets is greater than three. However, Monte Carlo simulation is constantly criticized hence, in order to reduce the error by a factor of 10 one for its slow convergence. Let V be a random variable has to generate 100 times as much as samples as well as and we want to calculate EV . We can generate computation time. For this reason, Monte Carlo simu- independently and identically distributed samples lation needs to be run on large parallel computers with a V n of V . Law of Large Numbers guarantees that high financial cost in terms of hardware and software i i1 as.. developments. The computational demands of simulation 1 n n have motivated substantial interest in the financial VVi1 i n industry in demands for increased efficiency. Another and Central Limit Theorem guarantees that asymp- way to improve the accuracy is to reduce the standard totically falls in the confidence interval deviation . Motivated by this thought, several tech- niques to reduce the variance of the Monte Carlo simu- VZVZnn, lation have been proposed, such as control variates, nn22antithetic variables, importance sampling and stratifi- *This work is supported by Research Innovation Foundation of Shang- cation(see Boyle, Broadie and Glasserman [1], and Gla- hai University of Finance and Economics under Grant No. CXJJ-2013- sserman [2]. These techniques aim to reduce the variance 323; Basic Academic Discipline Program, the 11th five year plan of 211 per Monte Carlo observation so that a given level of Project for Shanghai University of Finance and Economics. Open Access JMF 432 Q. ZHAO ET AL. accuracy can be obtained with a smaller number of two families based on the strategy adopted. The first one simulations. Control variates and antithetic variables are is proposed by Glasserman, Heidelberge and Shaha- the most widely used variance reduction techniques, buddin in a remarkable paper [14] (GHS for short), relies mainly because of the simplicity of their implementations, on a deterministic optimization procedure which can be and the fact that they can be accommodated in an exist- applied for a specific class of payoffs. Xu and Zhang [13] ing Monte Carlo calculator with a small effort. Examples improve the optimization algorithm of the importance of successful implementations of control variates for sampling by Newton Raphson algorithm based on direct pricing the derivatives include Hull and White [3], simulation. The second one is the so-called adaptive Kemna and Vorst [4], Turnbull and Wakeman [5], Ma Monte Carlo method, such as Vázquez-Abad and Du- and Xu [6]. fresne [9], Su and Fu [10], Arouna [11], that aims to Importance sampling has the capacity to exploit determine the optimal drift through stochastic optimi- detailed knowledge about a model (often in the form of zation techniques that typically involve an iterative algo- asymptotic approximations) to produce potential variance rithm. reduction. Unfortunately, importance sampling technique Most closely related to our work is Glasserman, has not been widely used as other variance reduction Heidelberge and Shahabuddin [14], who applied impor- techniques in pricing financial derivatives until recently. tance sampling combined with stratified sampling to This is mainly because there is no general way to dramatically reduce variance in derivative pricing. In this implement importance sampling. If the transformation of paper, we propose a new importance sampling method by probability measure is chosen improperly, this method modifying the drift term and the quadratic term of the does not work. Importance sampling attempts to reduce simulated process simultaneously. In the previous variance by changing the probability measure from literature, the variance for the linear part is eliminated by which paths are generated. Our goal is to obtain a more importance sampling and those for the quadratic part is convenient representation of the expected value. The idea reduced by stratification. However, we eliminate both behind the importance sampling is to reduce the statis- kinds of variances just by importance sampling. The tical uncertainty of Monte Carlo calculation by focusing corresponding space for the eigenvalues of the Hessian on the most important region of the space from which the matrix of the log function of payoff is enlarged. Illu- random samples are drawn. Such regions depend both on strations of the use of the method with European options the random process simulated, and the structure of the and Asian options are given, which show the high effi- security priced. Just as mentioned by Glasserman [2], an ciency of the method. The method proposed in the paper effective importance sampling density should weight more points to the region where the product of their can be extended to the pricing of other financial deriva- probability and their payoff is large. For example, for a ives directly. deep out-of-the-money call option, most of the time the payoff from simulation is 0 , so simulating more paths 2. Importance Sampling Method with positive payoffs should reduce the variance in the Importance sampling attempts to reduce variance by estimation. changing the probability measure from which samples An early example of impo rtance sampling applied to are generated. To make this idea concrete, consider the security pricing is Reider [7], where the variance was problem of estimating reduced substantially by increasing the drift in simulation VEGX Gxfxxd, (1) for deep out-of-the-money European call options. Gla- f sserman, Heidelberger and Shahabuddin [8] applied where X is a random vector of n with probability importance sampling to reduce substantial variance by density f , and E f denotes the expectation under combining stratification in the stochastic volatility model. the original measure with density function f , G is a Other recent work on importance sampling methods in function from n to , genera lly denoting the finance has been done for Monte Carlo simulations payoffs of financial derivatives. The ordinary Monte driven by high-dimensional Gaussian vectors, such as Carlo estimator is Boyle, Broadie and Glasserman [1], Vázquez-Abad and M ˆ 1 Dufresne [9], Su and Fu [10], Arouna [11], Capriotti [12], VGXfi , (2) M i1 Xu and Zhang [13]. In this framework, Importance Sampling is applied by modifying the drift term of the with X12,,,XX M independent draws from f . Let n simulated process to construct a new measure in which g be any other probability density on satisfying more weight is given to important outcomes thereby f x>0 gx >0, (3) increasing sampling efficiency. The different methods proposed in the literature mainly differ in the way where for all x n . Then we can obtain by c hanging mea- such a change of drift is found, and can be divided into sure Open Access JMF Q. ZHAO ET AL. 433 fx fX 2 fX VG x gxxEGXdg . (4) min gfEGX . (8) gx gX gX Thus, V can be interpreted as an expectation with Unfortunately, there is no general way to find the respect to the density g . optimal g for an ordinary function f . So, we turn to If X12,,,XX M are now independent draws from find the relatively optimal g to make the variance as g , the importance sampling estimator associated with small as possible. g is 3. New Im portance Sampling Si mulation M fX ˆ 1 i VGXgi . (5) In this section, we propose a new importance sampling M i1 gX i simulation following the idea of GHS. Consider the The weight problem of estimating the expectation of the payoff fXii gX wX i VEGX Gxfxxd, f Rn is the likelihood ratio evaluated at X i . where X is a n -dimensional vector of standard It follows from (1) and (4) that normal variable with probability density ˆˆ n 1 T EVff EV gg V, xx T 2 2 fx 2π e,x xx12 ,,,.