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 th e 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 i1 as.. developments. The computational demands of simulation 1 n n have motivated substantial interest in the financial VVi1 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, nn22antithetic va riables, 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 e xpectation 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 i1 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

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fx fX  2 fX VG x gxxEGXdg    . (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 i1 gX i simulation following the idea of GHS. Consider the The weight problem of estimating the expectation of the payoff

fXii 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 ,,,. xn and thus both Vˆ and Vˆ are unbias ed estimators of f g when choosing the new probability function, the previous V . To compare variances with and without importance literature featuring GHS has only taken drift into sampling it suffices to compare their second moments. consideration. They choose drift  satisfying With importance sampling we have

2  2 fX fX fX min  EGXf    , (9)  2  gX EGXgf  EGX . (6)    gX gX where n 1 T Depending on the choice of g , this could be larger or  xx   2 2 2  gx  2π  e . smaller than the second moment EGXf  without importance sampling. Successful importance sampling Based on GHS’s idea, we take both drift and variance lies on the art of selecting an effective importance term into consideration. If the covariance matrix  is sampling density g . positive definite, we can choose From (2), the variance of Vˆ is f n 1 1 T 1   xx   11 gx 2π 2 2 e,2 (10) VarVGˆ  VarXEGXV22 , ff f f M M thus, we obtain by changing measure where Var denotes the va riance under the measure f   fx  with density function f . Similarly, from (5) and (6), the VGxgxx d Rn   variance of Vˆ is gx g (11) 1 2 fX Var Vˆ EGXwXV2 . EGX EGXwX, gg  g   gg  M gX In particularly, if V 0 , a zero-variance estimate is  where the likelihood ratio is thus obtained by choosing 11 1 xxT  x T 1  x 1 22 g xGxfx  . (7) wx2 e. V Note that the covariance matrix  is positive definite, We call this function as the optimal density function. there exists a nonsingular n -dimensional matrix A This choice of density is precluded unless the estimated satisfying   AAT . quantity V is known from the outset. Nevertheless, this If observation provides a useful insight: An effective im- portance sampling density should weight points ac- Z  NI0, n , cording to the product of their probability and payoff. and Generally, the key of importance sampling lies to find g satisfying (3) to solve the following optimal problem, X    AZ ,

Open Access JMF 434 Q. ZHAO ET AL. then may be viewed as eliminating the variance contribution XN , . due to the linear and the quadratic part of F . We consider the new density function Denote 1 1 n  xxˆ T ˆ 1  ˆ  ˆ 2 2 F xG log  x , gx 2π 2 e (17) where as a proper measure transformation. Finally, the new estimator with smaller variance can be obtained by (5). xDDxRGx,:n 0.  In GHS, the variance caused by the quadratic part of It follows that F is reduced by use of stratification which needs relatively complicated numerical calculation. Meanwhile, VE G XwX g  the efficiency of the stratification demands that all the EGf  AZw AZ eigenvalues of the Hessian matrix of F at the value ˆ 1 1 EF2 exp  AZ are less than . Obviously, the new method can be f  2

11TT1 applied to more types of financial derivatives. exp AZAZAZAZ    22 1 4. Numerical Simulation  T 2 Ef  exp F  AZ AZ In this section, we illustrate the results developed in the  previous sections. We use two examples in which we 11TT T compare our method with GHS method to reveal the ZAAIZn . 22 efficiency of the new method. (12) In the Black-Scholes model, the price of unde rlying Suppose that F is twice continuously differentiable asset St  obeys on D , a second-ord er Taylor expansion formula shows ddd,St   rStt  St Wt (18) T FAZF  F AZ where r is the risk free rate,  is the volatility and (13) Wt is a standard Brownian motion. 1 T 2 2   AZFAZoZ   Suppose that 0, T is the valid period of th e option 2    and n is the number of observations, t n are the By (12) and (13), we have i i1

1 observati on dates and Sti is the price of the   1 T T 2 VEf  exp  F  F AZ underlying asset at the i th observation date.   2 The underlying asse t can be simulated by 1 T 212  1 2 AZ FI  n AZ o Z . 2  Stii St 11 exp r tiiiii tt t1 X  2 (14) in 1, 2, , , We can choose ˆ and ˆ satisfying (19) F ˆˆ (15) where T and X  XX12,,, Xn N 0, I. 1 ˆ IF 2 ˆ , (16) For simplicity, we set ttii 1  tTni ,  1, 2, , n. where we assume that all the eigenvalues of the Hessian matrix F ˆ are less than 1. (Following the analysis Experiment 1: Consider European call option under of Rogers and Talay [15], the condition is satisfied for the Black-Scholes model. The discounted payo ff function most of options including Asian option, barrier option is and various path-dependent options with stochastic vola- GX emaxrT  ST K ,0. tility.) (15) can be solved using iteration method in GHS. We set SrTn0  50, 0.05, 1.0, 16 and use Then, ˆ is easily obtained by direct substitution of ˆ . M  1, 000, 000 paths to estimate the variance reduction So the linear and quadratic part of the function F is ratio between Varf GX  and Varg GXwX  . removed. Our choice of drift vector and variance matrix The numerical results are illustrated in Table 1.

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Numerical results in Table 1 show estim ated prices and variance ratios, relative to ordinary Monte Carlo method, usin g importance sampling with GHS method and the new method respectively, which confirm the effectiveness of the procedure fo r this problem, es- pecially for in-the-money options. Variance reduction ratio 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 Figure 1, 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 to the optimal one than the GHS density figure, Figure 1. Sampling probability density function for Euro- thus achieving greater variance reduction. pean call option with   0.1, K  50 . Experiment 2: As a typical test case treated in recent papers, we will consider arithmetic Asian call option. Table 2. Estimated variance reduction ratios for arithme- The discounted payoff function is tic Asian call option with   0.05 . 1 n Parameters IS (GHS method) IS (new method) GX emaxrT St K ,0.   i K Price VarRatio Price VarRatio n i1 We set SrTn0 50, 0.05, 1.0, 16 and use 45 6.042 39.8 6.042 931.2 M  1000000 paths to estimate the variance reduction 50 1.437 6.4 1.438 11.5 55 0.007 13 8.1 0.007 35.9 ratio between Varf GX and Varg GXwX . The numerical results are illustrated in Tables 2-6. These tables above show the simulation results for Table 3. Estimated variance reduction ratios f or arithme - different volatilities and strike prices. Firstly, GHS tic Asian call option with   0.1 . method achieves larger variance ratios when the option is deeper out of money, which coincides with results in Parameter s IS (GHS method) IS (new method) Reider [7]. For a deep out-o f-mone y Call option, most K Price VarRatio Price VarRati o paths with zero payoff are sampled in simulation. By 45 6.055 10.8 6.055 24.5 changing the drift of sampling density, a large part of 50 1.919 7.0 1.919 11.9 zero-payoff paths are replaced by positive-payoff paths. Hence, simulating more samples with positive payoff 55 0.202 21.2 0.202 13.3 reduces the variance. The effect of variance reduction by changing the drift will be strengthened or weakened by Table 4. Estimated variance reduction ratios f or arithme - changing the form of the sampling density figure. tic Asian call option with   0.2 . Parameter s IS (GHS method) IS (new method) Table 1. Estimated variance reduction ratios for European call option. K Price VarRatio Price VarRati o 45 6.418 7.7 6.419 14.5 Parameters IS (GHS method) IS (new method) 50 3.028 8.2 3.028 13.4  K Price VarRatio Price VarRatio 55 1.106 13.0 1.106 14.0 0.1 30 21.463 103.3 21.463 931.2

45 7.317 8.2 7.316 15.9 Table 5. Estimated varianc e reduction ratios f or arithme - 50 3.404 7.2 3.405 12.1 tic Asian call option with   0.3 . 55 1.087 11.2 1.088 12.5 Parameter s IS (GHS method) IS (new method) 0.3 30 21.602 14.9 21.601 30.0 K Price VarRati o Price VarRati o 45 9.869 9.5 9.855 15.9 45 7.150 8.3 7.151 15.0 50 7.118 10.3 7.121 15.8 50 4.169 9.2 4.169 14.9 55 5.010 11.8 5.014 5.9 55 2.211 12.2 2.210 15.4

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Table 6. Estimated variance reduction ratios for arithme- http://dx.doi.org/10.2307/2331065 tic Asian call option with   0.5 . [4] A. G. Kemna and A. C. F. Vorst, “A Pricing Method for Options Based on Average Asset Values,” Journal of Parameter s IS (GHS method) IS (new method) Banking & Finance, Vol. 14, No. 1, 1990, pp. 113-129. K Price VarRati o Price VarRati o http://dx.doi.org/10.1016/0378-4266(90)90039-5 45 8.996 10.6 8.996 18.6 [5] S. Turnbull and L. Wakeman, “A Quick Algorithm for 50 6.459 11.6 6.457 18.7 Pricing European Average Options,” Journal of Financial and Quantitative Analysis, Vol. 26, No. 3, 1991, pp. 377- 55 4.455 13.5 4.543 18.9 389. http://dx.doi.org/10.2307/2331213 [6] J. Ma and C. Xu, “An Efficient Control Variate Method Secondly , for an in-the-m oney optio n, the n ew method for Pricing Variance Derivatives under Stochastic Vola- achieves larger variance r atios tha n the GH S metho d. tility and Jump Diffusion Models,” Journal of Computa- With the d ecrease of  , the supe riority be comes o b- tional and Applied Mathematics, Vol. 235, No. 1, 2010, vious. This is because th e fact that most paths with pp. 108-119. http://dx.doi.org/10.1016/j.cam.2010.05.017 positive payoff are sampled when simulating an in-the- [7] R. Reider, “An Efficient Monte Carlo Technique for money option makes it ineffective just by changing the Pricing Options,” Working Paper, Wharton School, Uni- versity of Pennsylvania, Philadelphia, 1993. drift. When the volatility  gets smaller, the under- lying asset changes more slow so that most payoffs are [8] P. Glasserman, P. Heidelberger and P. Shahabuddin, “Im- positive. This leads to the GHS method in vain and portance Sampling in the Heath-Jarrow-Morton Frame- work,” Journal of Derivatives, Vol. 7, No. 1, 1999, pp. makes the new method effective. 32-50. http://dx.doi.org/10.3905/jod.1999.319109 Thirdly, for an out-of-the-money option, the GHS [9] F. J. Vázquez-Abad and D. Dufresne, “Accelerated Simu- method performs better than the new method especially lation for Pricing Asian Options,” Proceedings of 1998 when the volatility  is small. In this case, by chang- Winter Simulation Conference, Washington DC, 13-16 ing the drift, most paths with zero payoff are replaced by December 1998, pp. 1493-1500. nonzero-payoff paths, leaves no room for more variance http://dx.doi.org/10.1109/WSC.1998.746020 reduction by changing the form of the sampling density [10] Y. Su and M. C. Fu, “Simulation in Financial Engineering: function. Importance Sampling in Derivative Securities Pricing,” Proceedings of the 32nd Conference on Winter Simula- 5. Concluding Remarks tion, Society for Computer Simulation International, 2000, pp. 587-596. In this paper, a new importance sampling Monte Carlo [11] B. Arouna, “Robbins-Monro Algorithms and Variance method for pricing options is proposed. Unlike the Reduction in Finance,” Journal of Computational Fi- classical importance sampling procedure, both kinds of nance, Vol. 7, No. 2, 2004, pp. 35-62. variance caused by the linear part and the quadratic part [12] L. Capriotti, “Least-squares Importance Sampling for of the logarithmic function of payoff are eliminated. The Monte Carlo Security Pricing,” Quantitative Finance, Vol. corresponding space for the eigenvalues of the Hessian 8, No. 5, 2008, pp. 485-497. matrix of the logarithmic function of payoff is enlarged. http://dx.doi.org/10.1080/14697680701762435 Computational Simulation shows the high efficiency of [13] C. Xu and L. Zhang, “On Optimal Drift Vector and Im- the new method. portance Sampling for Pricing Options Based on Direct Simulations,” Working Paper, 2011. [14] P. Glasserman, P. Heidelberger and P. Shahabuddin, REFERENCES “Asymptotically Optimal Importance Sampling and Stra- tification for Pricing Path-Dependent Options,” Mathe- [1] P. Boyle, M. Broadie and P. Glasserman, “Monte Carlo matical Finance, Vol. 9, No. 2, 1999, pp. 117-152. Methods for Security Pricing,” Journal of Economic Dy- http://dx.doi.org/10.1111/1467-9965.00065 namic and Control, Vol. 21, No. 8, 1997, pp.1267-1321. http://dx.doi.org/10.1016/S0165-1889(97)00028-6 [15] L. C. G. Rogers and D. Talay, “Numerical Methods in Finance,” Cambridge University Press, Cambridge, 1997. [2] P. Glasserman, “Monte Carlo Methods in Financial En- http://dx.doi.org/10.1017/CBO9781139173056 gineering,” Springer-Verlag, New York, 2004.

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