Variance Reduction Techniques of Importance Sampling Monte Carlo Methods for Pricing Options

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.


Introduction
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.The advantage of Monte Carlo method is that its convergence rate is independent on the number of state variables.Monte Carlo simulation is often the only way available for the pricing of complex path-dependent options if the number of relevant underlying assets is greater than three.However, Monte Carlo simulation is constantly criticized for its slow convergence.Let be a random variable and we want to calculate . We can generate independently and identically distributed samples of V .Law of Large Numbers guarantees that and Central Limit Theorem guarantees that  asymptotically falls in the confidence interval 2 2 , hence, in order to reduce the err by a factor of 10 one has to generate 100 times as mu as samples as well as computation time.For this reason, Monte Carlo simu-or ch lation needs to be run on large parallel computers with a high financial cost in terms of hardware and software developments.The computational demands of simulation have motivated substantial interest in the financial industry in demands for increased efficiency.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 riables, importance sampling and stratification(see Boyle, Broadie and Glasserman [1], and Glasserman [2].These techniques aim to reduce the variance per Monte Carlo observation so that a given level of accuracy can be obtained with a smaller number of simulations.Control variates and antithetic variables are the most widely used variance reduction techniques, mainly because of the simplicity of their implementations, and the fact that they can be accommodated in an existing Monte Carlo calculator with a small effort.Examples of successful implementations of control variates for pricing the derivatives include Hull and White [3], Kemna and Vorst [4], Turnbull and Wakeman [5], Ma and Xu [6]. Importance sampling has the capacity to exploit detailed knowledge about a model (often in the form of asymptotic approximations) to produce potential variance re o tantially by increasing the drift in simulation fo ampling combined with stratified sampling to dr ampling attempts to reduce variance by hich samples , consider the duction.Unfortunately, importance sampling technique has not been widely used as other variance reduction techniques in pricing financial derivatives until recently.This is mainly because there is no general way to implement importance sampling.If the transformation of probability measure is chosen improperly, this method does not work.Importance sampling attempts to reduce variance by changing the probability measure from which paths are generated.Our goal is to obtain a more convenient representation of the expected value.The idea behind the importance sampling is to reduce the statistical uncertainty of Monte Carlo calculation by focusing on the most important region of the space from which the random samples are drawn.Such regions depend both on the random process simulated, and the structure of the security priced.Just as mentioned by Glasserman [2], an effective importance sampling density should weight more points to the region where the product of their probability and their payoff is large.For example, for a deep out-of-the-money call option, most of the time the payoff from simulation is 0 , so simulating more paths with positive payoffs should reduce the variance in the estimation.
An early example of imp rtance sampling applied to security pricing is Reider [7], where the variance was reduced subs r deep out-of-the-money European call options.Glasserman, Heidelberger and Shahabuddin [8] applied importance sampling to reduce substantial variance by combining stratification in the stochastic volatility model.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 different methods proposed in the literature mainly differ in the way where such a change of drift is found, and can be divided into two families based on the strategy adopted.The first one is proposed by Glasserman, Heidelberge and Shahabuddin in a remarkable paper [14] (GHS for short), relies on a deterministic optimization procedure which can be applied for a specific class of payoffs.Xu and Zhang [13] improve the optimization algorithm of the importance sampling by Newton Raphson algorithm based on direct simulation.The second one is the so-called adaptive Monte Carlo method, such as Vázquez-Abad and Dufresne [9], Su and Fu [10], Arouna [11], that aims to determine the optimal drift through stochastic optimization techniques that typically involve an iterative algorithm.
Most closely related to our work is Glasserman, Heidelberge and Shahabuddin [14], who applied importance s amatically reduce variance in derivative pricing.In this paper, we propose a new importance sampling method by modifying the drift term and the quadratic term of the simulated process simultaneously.In the previous literature, the variance for the linear part is eliminated by importance sampling and those for the quadratic part is reduced by stratification.However, we eliminate both kinds of variances just by importance sampling.The corresponding space for the eigenvalues of the Hessian matrix of the log function of payoff is enlarged.Illustrations of the use of the method with European options and Asian options are given, which show the high efficiency of the method.The method proposed in the paper can be extended to the pricing of other financial derivaives directly.

Importance Sampling Method
Importance s changing the probability measure from w are generated.To make this idea concrete problem of estimating where X is a random vector of with probability density , , , M X X X  with independent draws from .Let f g be any other prob n  satisfying ability density on , can be interpreted as an expectation with respect t density , , , M X X X  are now independent draws from g , the im pling estimator associated with portance sam The weight is the likelihood ratio evaluated at i X .It follows from ( 1) and (4) that ˆˆ, V and ˆg V are un .To compare variances with and without im bi ed estimators of portance sampling it suffices to compare their second moments.

as V
With importance sampling we have Depending on the choice of g , this could be larger or smaller than the second mo   w importance sampling.Succe pling lies on t ment ssful denotes the riance va under the measure with density function f .Similarly, from (5) and (6), the variance of ˆg V is he optimal density function.This choice of density is precluded unless the estimated quantity is known from the outset.N observation provides a useful insight: An effective im-V evertheless, this portance sampling density should weight points according to the product of their probability and payoff.Generally, the key of importance sampling lies to find g satisfying (3) to solve the following optimal problem, Unfortunately, there is no general way to find the (8) optimal g for an ordinary function f .So, we find the relatively optimal turn to g to m small as possible.

Im i
a dea of GHS.Consider the ake the variance as

New portance Sampling S mulation
In this section, we propose new importance sampling simulation following the i problem of estimating the expectation of the payoff where X is a n -dimensional vector of standard normal variable with probability density when choosing the new probability functio 2π e , , , , .
n, the previous literature featuring GHS has only taken drift into consideration.They choose drift  satisfying where Based term into on GHS's idea, we take both drift and variance consideration.If the covariance matrix  is positive definite, we can choose thus, we obtain by changing measure where the likelihood ratio is Note that the covariance ma there exists a nonsingular trix is positive definite, ensional matrix and is twice continuously differentiable on , a second er Taylor expansion formula shows By (12) and ( 13), we have where we assume that all the eigenvalues of the Hessian ix Rogers and Talay are less than .(Following the analysis [15], th ondition most of options including Asian option 1 e c is satisfied for , barrier option and various path-dependent options with stochastic volatility.)(15) can be solved using iteration method in GHS.Then,  is easily obtained by direct substitution of  .So the linear and quadratic part of the function F is may be removed.Our choice of drift vector and variance matrix viewed as eliminating the variance contribution due to the linear and the quadratic part of F .We conside the new density function as a proper measure transformation.Finally, the new estimator with smaller variance can be obtained b In GHS, the variance caused by y tic part of (5). the quadra F is reduced by use of stratification whi relatively complicated numerical calculation.Meanwhile, ch needs the efficiency of the stratification demands that all the eigenvalues of the Hessian matrix of F at the value  are less than 1 2 .Obviously, the new method can be applied to more types of financial derivatives.

Numerical Simulation
In For simplicity, we set 1 , 1, 2, , Experiment 1: Consider European call option under the Black-Scholes model.The discounted pa ff function yo is We set and use 1, 000, 000 M  paths to estimate the variance reduction ratio between      1 show est ated prices and variance ratios, relative to ordinary Monte Carlo m g importance sampling with GH and the new m f the proced r this problem, eseduction ra im ethod, usin S method ethod respectively, which confirm the effectiveness o ure fo pecially for in-the-money options.Variance r tio is the variance per replication using standard Monte Carlo method divided by the variance per replication using the above two methods.The large ratio, the great the improvement. In and use 1000000 M  paths to estimate the variance reduction ratio between     [8] P. Glasserman, P. Heidelberger and P. Shahabuddin, "Im-his use th most s with p m pled oney option makes it ineffective just by changing the drift.When the volatility  gets smaller, the underlying asset changes mor slow so that most payoffs are positive.This leads to the GHS method in vain and makes the new method effective.
Thirdly, for an out-of-the-money option, the GHS method performs better than the new method especially when the volatility e  is small.In this case, by changing the drift, most paths with zero payoff are replaced by nonzero-payoff paths, leaves no room for more variance re portance Sampling in the Heath-Jarrow-Morton Framework," Journal of Derivatives, Vol. 7, No. duction by changing the form of the sampling density function.

Figure 1 , 2 :
the optimal density denotes the density represented by (7).The GHS density figure can be obtained through translating the original density figure right by  .After modifying the quadratic term, we can change the shape of the GHS density figure and get the new density figure.Obviously, the new density figure is closer e to the optimal one than the GHS density figure, thus achieving greater variance reduction.Experiment As a typical test case treated in recent papers, w will consider arithmetic Asian call option.The discounted payoff function is


Var g G X rical results are illustrated in Tables2-6These tables above show the simulation results for different volatilities and strike prices.Firstly, GHS m es larger variance ratios when th deeper out f money, which coincides with results in Reider[7].For a f-mone t paths with zero payoff are sampled in simulation anging the drift of sampling density, a large part of zero-payoff paths are replaced by positive-payoff paths.Hence, simulating more samples with positive payoff reduces the variance.The effect of variance reduction by changing the drift will be strengthened or weakened by changing the form of the sampling density figure.
this section, we illustrate the results developed in the previous sections.We use two examples in compare our method with GHS method to