Variance Reduction Techniques in Pricing Financial Derivatives Emmanuel R

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2008 Variance Reduction Techniques in Pricing Financial Derivatives Emmanuel R. Salta

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COLLEGE OF ARTS AND SCIENCES

VARIANCE REDUCTION TECHNIQUES IN PRICING FINANCIAL DERIVATIVES

EMMANUEL R. SALTA

A Dissertation submitted to the Department of Mathematics in partial fulﬁllment of the requirements for the degree of Doctor of Philosophy

Degree Awarded: Fall Semester, 2008 The members of the Committee approve the Dissertation of Emmanuel R. Salta defended on September 25, 2008.

Giray Okten¨ Professor Directing Dissertation

Ashok Srinivasan Outside Committee Member

Bettye Anne Case Committee Member

Brian Ewald Committee Member

Craig Nolder Committee Member

John R. Quine Committee Member

The Oﬃce of Graduate Studies has veriﬁed and approved the above named committee members.

ii ACKNOWLEDGEMENTS

I would like to thank my advisor, Giray Okten,¨ and my committee members Bettye Anne Case, Brian Ewald, Craig Nolder, John Quine, and Ashok Srinivasan. My sincerest thanks go to my fellow graduate students and graduates who made my graduate student life pleasant: Dervi¸sBayazıt, Ahmet Göncü,Kostas Mavroudis, C. Andres Proaño(M.S. ’06), Partha Srinivasan (Ph.D. ’05), and Ahmet Emin Tatar. Special mention also goes to two former Math Department staff members Grace Godfrey-Brock and Susan Minnerly for their valuable assistance.

iii TABLE OF CONTENTS

List of Tables ...... v

List of Figures ...... viii

Abstract ...... x

1. Introduction and Preliminaries ...... 1 1.1 Introduction ...... 1 1.2 The Monte Carlo Technique ...... 3 1.3 Notations and Assumptions ...... 5 1.4 Examples of Options ...... 6

2. Conditional Expectation and Control Variate Monte Carlo Estimators .... 8 2.1 Conditional Expectation Monte Carlo Estimator ...... 8 2.2 Control Variate Monte Carlo Estimator ...... 16 2.3 Combined Conditional Expectation and Control Variate Monte Carlo Estimator ...... 23

3. Importance Sampling Monte Carlo Estimators ...... 33 3.1 Importance Sampling Monte Carlo Estimator and Heuristics ...... 33 3.2 Importance Sampling Monte Carlo Estimator and Simulated Annealing . 49 3.3 Combined Conditional Expectation and Importance Sampling Monte Carlo Estimator ...... 56

4. Equivalent Estimators ...... 67 4.1 Ross and Shanthikumar’s Estimators ...... 67 4.2 Glasserman and Staum’s Estimators ...... 72 4.3 Properties of the RS Estimators ...... 74 4.4 Applications ...... 77

5. Conclusion ...... 80

REFERENCES ...... 82

BIOGRAPHICAL SKETCH ...... 85

iv LIST OF TABLES

2.1 Relative mean square errors of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for a down-and- in barrier option, and their ratios (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 11

2.2 Relative mean square errors of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for a down-and- in barrier option, and their ratios (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223)...... 11

2.3 Average of ﬁfty price estimates, variance and ratios of variances of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for an arithmetic Asian ASO. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88)...... 15

2.4 Average of ﬁfty price estimates, variance and ratios of variances of crude Monte Carlo (MC) estimates and control variate (ContVar) Monte Carlo estimates for an arithmetic Asian ASO option. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88). 21

2.5 Average of ﬁfty price estimates and variance of crude Monte Carlo (MC) estimates and combined conditional expectation and control variate (CECV) Monte Carlo estimates for an arithmetic Asian ASO option. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88)...... 29

2.6 Ratios of variances of crude Monte Carlo (MC) estimates to variances of various combined conditional expectation and control variate (CECV) estimates from Table 2.5...... 30

3.1 Relative mean square errors of crude Monte Carlo (MC) estimates, conditional expectation (CondExp) Monte Carlo estimates, and importance sampling (ImpSamp-h) Monte Carlo heuristics estimates, and ratios of errors for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 43

v 3.2 Relative mean square errors of crude Monte Carlo (MC) estimates, conditional expectation (CondExp) Monte Carlo estimates, and importance sampling (ImpSamp-h) Monte Carlo heuristics estimates, and ratios of errors for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223)...... 44

3.3 Estimated relative MSE of crude Monte Carlo (MC) estimates and importance sampling (ImpSamp-h) Monte Carlo heuristic estimates, and ratios of errors for an up-and-out barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 150, T =0.2, m = 50, H = 200 and estimated true price = 0.00981841855492). . 49

3.4 Estimated relative MSE of crude Monte Carlo (MC) estimates and importance sampling (ImpSamp-h) Monte Carlo heuristic estimates, and ratios of errors for an up-and-out barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 110, T =0.2, m = 50, H = 160 and estimated true price = 2.46413112486174). . 49 3.5 Relative mean square errors of crude Monte Carlo (MC) estimates, importance sampling (ImpSamp-h) Monte Carlo heuristics, and importance sampling (ImpSamp-SA) Monte Carlo simulated annealing estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 57

3.6 Relative mean square errors of crude Monte Carlo (MC) estimates, importance sampling (ImpSamp-h) Monte Carlo heuristics, and importance sampling (ImpSamp-SA) Monte Carlo simulated annealing estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223)...... 58

3.7 Relative mean square errors of conditional expectation Monte Carlo (Cond- Exp), importance sampling Monte Carlo heuristics (ImpSamp-h), and combined conditional expectation and importance sampling Monte Carlo heuristics (Combined-h) for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784). 62

3.8 Relative mean square errors of conditional expectation Monte Carlo (Cond- Exp), importance sampling Monte Carlo heuristics (ImpSamp-h), and combined conditional expectation and importance sampling Monte Carlo heuristics (Combined-h) for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223). 63

3.9 Relative mean square errors of crude Monte Carlo estimator, combined conditional expectation and importance sampling Monte Carlo estimator using heuristics (Combined-h) and simulated annealing (Combined-SA), and ratios of errors for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 64

vi 3.10 Relative mean square errors of crude Monte Carlo estimator (MC), combined conditional expectation and importance sampling Monte Carlo estimators using heuristics (Combined-h) and simulated annealing (Combined-SA), and ratios of errors for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223)...... 65

3.11 Comparison of computation times of MC estimator, CondExp estimator, ImptSamp-h estimator, and Combined-h estimator for a down-and-in barrier option using 1000 sample price paths (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95)...... 66

4.1 Glasserman and Staum’s estimators for knock-out barrier options...... 73

4.2 Glasserman and Staum’s estimators for knock-in barrier options...... 74

4.3 Relative mean square errors of Ross and Shanthikumar’s raw simulation (Raw) Monte Carlo estimates and weighted (Weighted) Monte Carlo estimates for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 78

4.4 Ratios of variances of crude Monte Carlo (MC) estimates from Chapter 2 to variances of Ross and Shanthikumar’s (Raw and Weighted) estimates (σ =0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784)...... 78

4.5 Relative mean square errors of Ross and Shanthikumar’s raw simulation (Raw) Monte Carlo estimates and weighted (Weighted) Monte Carlo estimates for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223)...... 78

4.6 Ratios of variances of crude Monte Carlo (MC) estimates from Chapter 2 to variances of Ross and Shanthikumar’s (Raw and Weighted) estimates (σ =0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 91, and true price = 0.3670447223)...... 79

vii LIST OF FIGURES

2.1 Convergence of down-and-in barrier option price estimators using crude Monte Carlo (MC) and conditional expectation Monte Carlo (BBG, RS, CondExp)(σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784)...... 12 2.2 Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and conditional expectation Monte Carlo (CondExp). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). .. 15

2.3 Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and conditional expectation Monte Carlo (CondExp). Fifty sample estimates in increments of 5000 simulations. Final estimates: MC = 2.3171, CondExp = 2.3206. Averages: MC = 2.3216, CondExp = 2.3255 (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88)...... 16

2.4 Fifty arithmetic Asian ASO price estimates using control variate (ContVar) Monte Carlo. Each realization uses 5000 simulations (σ = 0.4, r = 0.07, S(0) = 40, T = 88/265, and m = 88)...... 20

2.5 Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and control variate (ContVar) Monte Carlo. Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). .... 22

2.6 Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and control variate (ContVar) Monte Carlo. Fifty sample estimates in increments of 5000 simulations. Final estimates: MC = 2.3171, ContVar = 2.3232. Averages: MC = 2.3216, ContVar = 2.3232 (σ = 0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88)...... 22

1 m 1 − 2.7 Scatter plot of BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) and penultimate − − − 1 m 1 + S t − S t payoff ( m 1) m 1 i=1 ( i) for 5000P sample points. The correlation coefficient for− BSM− − samples and payoff samples is 0.994 (σ = 0.4, r = 0.07, S(0) = 40, T = 88/265,P and m = 88). ...... 27

viii 2.8 Fifty arithmetic Asian ASO price estimates using combined conditional expectation and control variate Monte Carlo (CECV). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). .... 30

2.9 Ten arithmetic Asian ASO price estimates using combined conditional expectation and control variate Monte Carlo with control variates ciX(ti) X(ti) (CECVx) and cie (CECVex). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88)...... P 31 P 2.10 Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and combined conditional expectation and control variate Monte Carlo (CECV). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88)...... 31

2.11 Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and combined conditional expectation and control variate (CECV) Monte Carlo. (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). ... 32

3.1 Fifty up-and-out barrier option price estimates using crude Monte Carlo (MC) and importance sampling (ImpSamp-h) Monte Carlo heuristics. Each estimate uses 10000 simulations (σ = 0.3, r = 0.1, S(0) = 100, K = 150, T =0.2, m = 50, and H = 200)...... 50

3.2 MSE surface plot. Included are the importance sampling heuristic point h and the importance sampling simulated annealing point SA. Each point on the surface is the MSE of ﬁfty price estimates each estimated using 10000 price paths (σ =0.3, r =0.1, S(0) = 100, K = 150, T =0.2, m = 50, H = 95, and true price = 1.4373238784)...... 55

ix ABSTRACT

In this dissertation, we evaluate existing Monte Carlo estimators and develop new Monte Carlo estimators for pricing ﬁnancial options with the goal of improving precision. In Chapter 2, we discuss the conditional expectation Monte Carlo estimator for pricing barrier options, and show that the formulas for this estimator that are used in the literature are incorrect. We provide a correct version of the formula. In Chapter 3, we focus on importance sampling methods in estimating the price of barrier options. We show how a simulated annealing procedure can be used to estimate the parameters required in the importance sampling method. We end this chapter by evaluating the performance of the combined importance sampling and conditional expectation method. In Chapter 4, we analyze the estimators introduced by Ross and Shanthikumar in pricing barrier options and present a numerical example to test their performance.

x CHAPTER 1

Introduction and Preliminaries

1.1 Introduction

The Monte Carlo method has been a popular alternative approach in pricing options whenever price formulas either are too complicated to implement or do not exist at all. For example, the price of an arithmetic Asian option is widely known to have no closed-form solution. As such, a Monte Carlo approach is useful in this problem. By its very nature, the Monte Carlo method is probabilistic; thus, a precise estimate of the price of an option requires a fair amount of observational data generated from random numbers. In some instances, however, one needs an impractical amount of labor in generating information in order to reduce the uncertainty in our price estimates. As we already know, this uncertainty is usually measured by a quantity called variance, and methods that reduce this quantity are what is typically known as variance reduction techniques. This dissertation focuses on variance reduction techniques in pricing financial derivatives. Reducing variance is not a new idea in the simulation literature. However, research on their application to finance is fairly recent, being active only within the last two decades or so. For a brief sampling of the literature, the reader is referred to the following articles. Kemna and Vorst [21] apply control variate techniques in pricing Asian options. Boyle, et al. [3] include a survey of variance reduction techniques in finance. We focus on their heuristic approach to the importance sampling method. Ross and Shanthikumar [27] propose simulation methods for barrier options, Asian options, and lookback options. We supply numerical examples for these methods. Broadie and Glasserman [5] discuss a pathwise method and a likelihood ratio method to price Asian options. Clewlow and Carverhill [8] adapt a hedging approach in antithetic variate and control variate techniques in pricing a standard call option and a lookback call option. L’Ecuyer and Lemieux [22] use randomized

1 lattice rules in reducing variance in Monte Carlo price estimates of Asian options. Duffie and Glynn [12] provide an asymptotically efficient algorithm for pricing security prices. Staum [31] provides an overview of techniques for improving the efficiency of option pricing simulations. Vasquez-Abad and Dufresne [13] combine control variable and the change of measure methods in pricing Asian options. This dissertation is organized as follows. In Chapter 1, we review the Monte Carlo method in the context of option pricing and highlight its properties. We also include in this chapter the notations and examples that will be used throughout the dissertation. In the succeeding chapters that discuss variance reduction methods, we begin each chapter with a proof of the unbiasedness of the estimators followed by an exposition showing the potential reduction in variance in the estimator. Then, we apply the method to pricing different options. Specifically, in Chapter 2, we discuss the conditional expectation estimator and apply the method to pricing barrier options and Asian options. This section also rectifies the conditional expectation estimators for pricing barrier options that are currently accepted in the literature. In the next section, we discuss the control variate estimator and apply the method in pricing arithmetic Asian options. We shall see that correlation between underlyings plays an important role in reducing variance in this method. To end Chapter 2, we combine the conditional expectation and control variate estimators and use these estimators in pricing arithmetic Asian options. In Chapter 3, we simulate price estimates in a slightly more complicated approach. We introduce the importance sampling estimator in pricing barrier options. The parameters that the importance sampling approach yields are estimated by heuristics. For our applications, we estimate first the price of a down-and-in barrier option. Next, we extend the method to pricing an up-and-out barrier option. In the next section, the parameters that the importance sampling approach yields are estimated using a simulated annealing procedure. We shall see that this procedure gives only slightly better improvements over the heuristic approach. In Chapter 4, we shall discuss two barrier option estimators, Ross and Shanthikumar’s [27] raw simulation estimator and Glasserman and Staum’s [16] one-step survival estimator, and show that they are equivalent. We give a numerical example to verify Ross and Shanthikumar’s claim regarding the efficiency of their estimators. Finally, Chapter 5 concludes the dissertation.

2 1.2 The Monte Carlo Technique

In this dissertation, we apply the Monte Carlo technique in estimating an option price expressed as the expected present value of the option payoff, θ = E[X], as efficiently as possible. We define the crude Monte Carlo estimator as follows:

Deﬁnition 1.2.1. Let X be a random variable with mean θ and variance σ2. The crude Monte Carlo estimator is deﬁned as the sample mean estimator

1 N X = X , N i i=1 X where the Xi are i.i.d. copies of X and N is a positive integer. The random variable X has mean θ and variance σ2/N.

Suppose the random variable Z is another estimator of the parameter θ. Let us assume 2 Z has mean θ and variance σZ . Following the above deﬁnition, we can form the Monte Carlo N estimator associated with the estimator Z by taking the sample mean Z = i=1 Zi/N, with 2 mean θ and variance σZ /N. We say that the Monte Carlo estimator Z “estimatesP θ better 2 2 than” or “gives less error than” the crude Monte Carlo estimator X if σZ <σ .

1.2.1 Properties of Estimators

In this section, we review the important properties of Monte Carlo estimators that help us interpret the numerical results in our applications.

Unbiasedness

Deﬁnition 1.2.2. The random variable Z is an unbiased estimator of θ if E[Z]= θ.

Remark 1.2.1. If Z is an unbiased estimator of θ, then the corresponding Monte Carlo estimator Z is also an unbiased estimator since

1 N 1 N 1 E Z = E Z = E [Z ]= Nθ = θ. N i N i N " i=1 # i=1 X X All the estimators that we discuss in this dissertation are unbiased.

3 Error

We measure the error of the various Monte Carlo estimators Z in two ways: First, if the value of θ is known, we use the mean square error

2 MSE(Z)= E Z θ , (1.2.1) − h i or the mean square relative error

2 Z θ MS relative error (Z)= E − , θ " # or their square roots to get the root mean square error

2 RMSE(Z)= E Z θ , r − h i or the root mean square relative error

2 Z θ RMS relative error (Z)= E − . v u " θ # u t We estimate MSE(Z) by 1 N MSE\(Z)= (Z θ)2, N j − j=1 X and RMSE(Z) by 1 N RMSE\ (Z)= (Z θ)2. vN j − u j=1 u X Estimators for the relative errors are similar.t Second, if the value of θ is unknown, the error of the estimator Z is measured by its variance which is estimated by the unbiased sample variance estimator

N 2 1 2 V ar(Z)= sN 1 = Zj E[Z] . − N 1 − − j=1 X d This means, we generate Z1, Z2, ..., ZN Monte Carlo estimates of E[Z], take their average 1 N 2 E[Z]= N j=1 Zj, and compute sN 1. − P

4 Consistency

For issues regarding convergence and large sample size, we resort to the consistency of an estimator.

Deﬁnition 1.2.3. Let θN denote any estimator of the parameter θ that is dependent on the sample size N. We say θN is a consistent estimator of the parameter θ if θN converges in b probability to θ. b b N For now, let us denote the sample mean j=1 Zj/N by ZN . The Weak Law of Large Numbers (WLLN) justiﬁes the Monte Carlo estimatorP ZN being a consistent estimator of the parameter θ, and hence, the convergence of the estimator ZN to the true value θ. Errors associated with a given Monte Carlo computation may be measured by recalling the Central Limit Theorem (CLT). Regardless of the distribution of Z, CLT says that, as the sample size N increases, the distribution of the sample mean ZN approaches the normal distribution 2 with mean θ and variance σZ /N: 1 Z D θ, σ2 . N −→ N N Z 1.3 Notations and Assumptions

All of the option examples in this dissertation are based on the underlying stock S(t), modeled as the geometric Brownian motion GBM(r, σ2) and whose prices are observed at equally spaced, discrete time steps 0 = t < t < < t = T . The exercise or strike 0 1 ··· m price will be denoted by K, the risk-free interest rate by r, and the expiry by T . For barrier options, the barrier level will be denoted by H, and the time the barrier is breached will be denoted by the random variable t , where τ is an integer, 0 τ m + 1. For Asian τ ≤ ≤ m options, the arithmetic mean of the stock price will be denoted by A(tm) = i=1 S(ti)/m, m 1/m and its geometric mean by G(tm) = [ i=1 S(ti)] . The true option prices areP denoted by θ = E[X] or θ = E[h(X)], where the randomQ variable X or the function h(X) of the random variable X represents the discounted payoﬀ of the option.

5 1.4 Examples of Options

1.4.1 Barrier Options

We consider simple forms of barrier options. A down-and-in European call option written on a stock is a type of barrier option that becomes alive (“in”) as a European call option when the stock price path hits the barrier level that is initially set below (“down”) the initial stock price. Its payoﬀ at the expiry is given by the product

1(t

rT + θ = E e− 1(t

1(t >T ) (S(T ) K)+, τ · − and its risk-neutral price is

rT + θ = E e− 1(t >T )(S(T ) K) . (1.4.2) τ − Other combinations of up-down, in-out, and call-put are possible for barrier options.

1.4.2 Asian Options

An Asian option diﬀers from a plain vanilla option in that the average of the stock price over the life of the option replaces the terminal stock price S(T ) or the strike price K in the payoﬀ of the plain vanilla option. An arithmetic Asian option uses the arithmetic mean

A(tm) for the average, while a geometric Asian option uses the geometric mean G(tm). When the average of the stock price replaces the terminal stock price, we get the average rate (or

6 price) option (ARO). The risk-neutral price of an arithmetic call Asian ARO at time 0 is the discounted payoﬀ rT + θ = E e− (A(t ) K) m − while that of a geometric call Asian ARO is

rT + θ = E e− (G(t ) K) . m − When the average of the stock price replaces the strike price, we get the average strike (or ﬂoating average) option (ASO). The risk-neutral price of an arithmetic call Asian ASO at time 0 is the expected discounted payoﬀ

rT + θ = E e− (S(T ) A(t )) − m while that of a geometric call Asian ASO is

rT + θ = E e− (S(T ) G(t )) . − m Arithmetic Asian options do not have closed-form price formulas because the distribution of the sum of lognormal prices has no explicit representation. Thus, solutions for the arithmetic Asian option are more complicated. See pages 77-78 of Clewlow and Strickland [9] for a summary of diﬀerent approaches in pricing arithmetic Asian options in the literature. On the other hand, the product of lognormal prices is itself lognormal, hence the geometric average

G(tm) is also lognormal making the valuation of geometric Asian options more tractable. See page 77 of of Clewlow and Strickland [9] for the derivation of closed-form solutions.

7 CHAPTER 2

Conditional Expectation and Control Variate Monte Carlo Estimators

2.1 Conditional Expectation Monte Carlo Estimator

2.1.1 Principles

Departing from the crude Monte Carlo estimator, we estimate the parameter θ = E[X] alternatively by using the conditional expectation estimator Z = E[X Y ]. This approach is | useful when we can compute E[X Y = y] by other means. | The Law of Total Expectation justiﬁes the unbiasedness of the conditional expectation estimator. For any random variables X and Y , with E[ X ] < , we have E[X] = | | ∞ E[E[X Y ]] = E[Z]. (For proof, see, for example Billingsley [2]). Furthermore, the | variance of our conditional expectation estimator may be reduced since, in general, we have V ar[E[X Y ]] V ar[X]. To see this, rewrite V ar[X] using the Variance Decomposition | ≤ Formula:

V ar[X]= E[V ar[X Y ]] + V ar[E[X Y ]]. (2.1.1) | | The random variable V ar[X Y ] 0, thus E[V ar[X Y ]] 0. Moreover, V ar[E[X Y ]] 0. | ≥ | ≥ | ≥ Thus, we get the potential variance reduction. In [3], Boyle et al. note that when we replace an estimator by its conditional expectation, we reduce variance because we are doing part of the integration analytically and leaving less to be done by Monte Carlo simulation. In [20], Haugh notes that using conditional expectation Monte Carlo is worthwhile if the random variables X and Y are dependent1.

1Otherwise, we get the trivial case. If X and Y are independent, then E[X Y ]= E[X]. That is, simulated runs of Y have no eﬀect on the expected value of X. Consequently, we might| as well use the crude Monte Carlo estimator.

8 2.1.2 Applications

We shall apply the conditional expectation Monte Carlo technique to ﬁnd the price of two types of ﬁnancial derivatives: a down-and-in barrier option and an Asian option.

Down-and-In Barrier Option

Recall the risk-neutral price of the down-and-in European call option from Equation (1.4.1).

rT + θ = E e− 1(t

rtτ by e− to time t0 gives an estimate of the price of the down-and-in barrier option. We can express the above process quantitatively as follows. If the stock price does not hit the barrier before the expiration, assume τ = m+1. Otherwise, condition the discounted payoﬀ on the hitting time tτ

rT + θ = E[e− (S(T ) K) ] − rT + = E e− E (S(T ) K) t , S(t ) − | τ τ rT r(T tτ ) r(T tτ ) + = E e− e − E e− − (S(T ) K) τ, S(t ) − | τ rtτ = E e− BSM(S(tτ ),tτ ,T,K) . (2.1.3) Notice that the inner expectation is replaced by the closed-form expression BSM(S(tτ ),tτ ,T,K) denoting the Black-Scholes-Merton formula for the price of a European call option at time tτ with initial stock price S(tτ ), expiry T , and strike price K:

r(T ti) + BSM(S(t ),t ,T,K) = E e− − (S(T ) K) i i − r(T ti) = S(t )Φ(d ) Ke− − Φ( d ), i 1 − 2 with 2 log S(ti) + r + σ (T t ) K 2 − i d1 = d2 + σ T ti = . − σ(T ti) p − 9 So the conditional expectation Monte Carlo estimator simulates j =1, 2, ..., N stock price paths and averages over the values

e rtτ BSM(S(t ),t ,T ) if t

N j 1 rtτ j j j θ = e− 1(t

2Combining a down-and-in barrier option and a down-and-out barrier option with the same parameter values gives the price of a European call option. See Haug [19] for a discussion on in-out parity.

10 Table 2.1: Relative mean square errors of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for a down-and-in barrier option, and their ratios (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784).

N MC CondExp MC/CondExp 3 4 +1 5K 1.66 10− 1.36 10− 1.22 10 × 4 × 5 × +1 10K 9.73 10− 8.40 10− 1.16 10 × 4 × 5 × +1 50K 2.00 10− 1.65 10− 1.21 10 × × × Cross Barrier 62% 61% n/a Cross Exercise 29% n/a n/a

Table 2.2: Relative mean square errors of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for a down-and-in barrier option, and their ratios (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223).

N MC CondExp MC/CondExp 3 4 0 5K 3.77 10− 4.66 10− 8.09 10 × 3 × 4 × 0 10K 2.60 10− 4.17 10− 6.24 10 × 4 × 5 × 0 50K 5.47 10− 6.98 10− 7.84 10 × × × Cross Barrier 41% 40% n/a Cross Exercise 15% n/a n/a

We infer from Tables 2.1 and 2.2 that as we lower the barrier level from 95 to 91, and thus reducing the barrier crossing from 62% to 41%, the factors of improvement also decrease. On average, the factor of improvement in Table 2.1 is 11.9701 while that in Table 2.2 is 7.3873. Figure 2.1 shows the simulation results of the conditional expectation Monte Carlo estimators using the formulas reported by Boyle et al. [3] (denoted by BBG) and by Ross and Shanthikumar [27] (denoted by RS), and contrasts them with the results of the correct conditional expectation Monte Carlo estimator and the crude Monte Carlo estimator. The ﬁgure displays price estimates of a down-and-in barrier option with parameter values σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50 and H = 95, and true price of 1.4373238784.

11 Figure 2.1: Convergence of down-and-in barrier option price estimators using crude Monte Carlo (MC) and conditional expectation Monte Carlo (BBG, RS, CondExp)(σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784).

Each price estimate uses increments of 10000 simulations. In the figure, we mark the true price of the option by a solid horizontal line passing through the value 1.4373238784. We also include a band around this true price marked by dashed horizontal lines measuring 0.001 above and below the true price. The figure shows more CondExp estimates landing closer within the band than MC estimates suggesting that the conditional expectation Monte Carlo estimator gives more accurate and precise estimates than the crude Monte Carlo estimator. Moreover, the figure highlights the bias of the estimates where the correct discounting factor was inadvertently omitted. The BBG estimator underestimates the true price while the RS estimator overestimates the true price. In addition, an incorrect discounting factor (in BBG estimator) creates a larger bias than the absence of a discounting factor (in RS estimator).

12 Asian Option

In my next example, Ross and Shanthikumar [27] construct a conditional Monte Carlo estimator for the price of an arithmetic Asian call average strike option (ASO). Recall the risk-neutral price of the arithmetic ASO whose strike price is the average end-of-day stock price:

rT + θ = E e− (S(T ) A(t )) − m m + rT 1 = E e− S(T ) S(t ) − m i " i=1 ! # X m 1 + − rT m 1 1 = E e− − S(T ) S(t ) . (2.1.5)  m − m 1 i  i=1 ! − X As before, the goal of the conditional Monte Carlo estimator is to reduce estimation error by rewriting the discounted payoﬀ in Equation (2.1.5) as a conditional expectation and then by replacing this conditional expectation with a closed-form expression. Let us condition the 3 discounted payoﬀ by the vector of stock prices Sm 1 =(S(t0), ..., S(tm 1)) and rewrite the − − resulting inner expectation using the Black-Scholes-Merton formula:

m 1 + − rT m 1 1 θ = E e− − S(T ) S(t )  m − m 1 i  i=1 ! − X  m 1  + − rT m 1 1 = E e− − E S(T ) S(ti) Sm 1  m  − m 1 −  i=1 ! − X   m 1 + m 1 1 − rT X(tm) = E e− − E S(tm 1)e S(ti) Sm 1  m  − − m 1 −  i=1 ! − X   m 1 + m 1 T T 1 − rT r m r m X(tm) = E e− − e E e− S(tm 1)e S(ti) Sm 1  m  − − m 1 −  i=1 ! − X m 1   −  rT m 1 r T 1 = E e− − e m BSM S(tm 1),tm 1,T, S(ti) m − − m 1 " − i=1 !# Xm 1 − r(T T ) m 1 1 = E e− − m − BSM S(tm 1),tm 1,T, S(ti) , (2.1.6) m − − m 1 " i=1 !# − X 3 Ross and Shanthikumar [27] condition instead on the vector Xm−1 = (X(t0),...,X(tm−1)) of random walks that deﬁne the stock prices. We condition on Sm−1 for uniformity across examples. It has the same eﬀect.

13 1 m 1 − where BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) denotes the Black-Scholes-Merton formula − − − for the price of a European call option at time tm 1 with initial stock price S(tm 1), expiry T P − − 1 m 1 − and strike price K = m 1 i=1 S(ti). So the conditional expectation Monte Carlo estimator − simulates j =1, 2, ..., N stockP price paths and averages over the values: m 1 − r(T T ) m 1 1 e− − m − BSM S(tm 1),tm 1,T, S(ti) . m − − m 1 i=1 ! − X In other words, we estimate the price of the arithmetic Asian call ASO by

N m 1 − 1 r(T T ) m 1 1 θ = e− − m − BSM Sj(tm 1),tm 1,T, Sj(ti) . N m − − m 1 j=1 i=1 ! X − X b r T Equation (2.1.6) was erroneously reported without the additional discounting factor e− m in Ross and Shanthikumar [27] (Equation (4), p. 325). For our numerical example, we follow the parameter values in Cox and Rubinstein [11] (p. 216) and Kemna and Vorst [21] (p. 122): σ =0.2, 0.3, and 0.4; r =0.03, 0.05, and 0.07; S(0) = 40; T = 88/265 which is roughly four months; and m = 88. Table 2.3 reports the average and the variance of fifty price estimates using the crude Monte Carlo (MC) estimator and the conditional expectation (CondExp) Monte Carlo estimator. Each price estimate is computed using 5000 stock price paths. From the table, one can immediately observe that there is no clear decrease in variance when this conditional expectation method was applied to our example. Figure 2.2 shows the near-identical graphs of fifty MC and CondExp price estimates for volatility σ = 0.4 and interest rate r = .07. We tried increasing the volatility to a realistic level (e.g. from σ =0.4 to σ =0.9) to allow price paths to cover a wider range in their lifetime. We also tried increasing the length of the time step (e.g. from ∆t = 1/265 to ∆t = 1/20) to give the price path a longer time to vary, notably in the last step where conditional expectation is applied. As before, results from these modifications did not show any improvement over the Monte Carlo method. We believe that, unlike the conditional expectation method for barrier options where conditional expectation was applied at random hitting times, the conditional expectation method of Ross and Shanthikumar [27] for Asian options does not yield errors lower than that of the crude Monte Carlo, contrary to the authors’ claim. However, we shall see in the next section that combining the conditional expectation method with the control variate gives some improvement in pricing Asian options.

14 Table 2.3: Average of ﬁfty price estimates, variance and ratios of variances of crude Monte Carlo (MC) estimates and conditional expectation (CondExp) Monte Carlo estimates for an arithmetic Asian ASO. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88).

r σ MC CondExp MC/CondExp 4 4 0 0.2 1.1489 (6.8844 10− ) 1.1479 (6.4909 10− ) 1.06 10 × 3 × 3 × 0 0.03 0.3 1.6646 (1.5000 10− ) 1.6657 (1.4000 10− ) 1.07 10 × 3 × 3 × 0 0.4 2.1912 (2.6000 10− ) 2.1886 (2.5000 10− ) 1.04 10 × 4 × 4 × 1 0.2 1.2169 (8.2791 10− ) 1.2163 (8.8082 10− ) 9.40 10− × 3 × 3 × 0 0.05 0.3 1.7336 (1.3000 10− ) 1.7338 (1.3000 10− ) 1.00 10 × 3 × 3 × 0 0.4 2.2646 (2.7000 10− ) 2.2645 (1.5000 10− ) 1.80 10 × 4 × 4 × 0 0.2 1.2826 (6.6800 10− ) 1.2829 (4.4842 10− ) 1.49 10 × 3 × 3 × 0 0.07 0.3 1.8020 (1.2000 10− ) 1.8030 (1.1000 10− ) 1.09 10 × 3 × 3 × 0 0.4 2.3147 (2.3000 10− ) 2.3171 (2.3000 10− ) 1.00 10 × × ×

Figure 2.2: Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and conditional expectation Monte Carlo (CondExp). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

15 Figure 2.3: Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and conditional expectation Monte Carlo (CondExp). Fifty sample estimates in increments of 5000 simulations. Final estimates: MC = 2.3171, CondExp = 2.3206. Averages: MC = 2.3216, CondExp = 2.3255 (σ = 0.4, r = 0.07, S(0) = 40, T = 88/265, and m = 88).

2.2 Control Variate Monte Carlo Estimator 2.2.1 Principles

The control variate method uses information about the errors in estimates of known quantities to reduce the error in an estimate of an unknown quantity. We deﬁne the control variate as follows:

Deﬁnition 2.2.1. Let X be an estimator for θ with E[X]= θ. Then Y is a control variate of X if (i) Y is a random variable with known mean E[Y ]; (ii) Y is correlated with X.

Let c R. A control variate estimator of θ comes in the general form ∈ Z = X + c(Y E[Y ]). − The estimator Z is an unbiased estimator of θ since E[Z]= E[X +c(Y E[Y ])] = E[X]= θ. − Moreover, V ar[Z] is minimized at Cov[X, Y ] c∗ = . (2.2.1) − V ar[Y ]

16 To see this (p.139, [26]), note that

V ar [X + c(Y E[Y ])] = V ar [X + cY ] − = V ar [X]+ c2V ar [Y ]+2cCov [X, Y ] .

Calculus shows the above is minimized at c∗ = Cov[X, Y ]/V ar[Y ]. Consequently, the − minimum variance of the estimator Z is Cov2[X, Y ] V ar[Z]= V ar[X + c∗(Y E[Y ])] = V ar[X] . − − V ar[Y ] Note that we can measure the improvement of the control variate estimator Z by taking the ratio of the variances:

V ar[Z] V ar[X + c∗(Y E[Y ])] = − V ar[X] V ar[X] 2 V ar[X] Cov [X,Y ] = − V ar[Y ] V ar[X] = 1 Corr2[X, Y ], − which implies V ar[Z]= 1 Corr2[X, Y ] V ar[X]. − Thus, whenever the random variables X and Y are highly correlated, positively or negatively, V ar[Z] will reduce to a small fraction of V ar[X].

Since Cov[X, Y ] and V ar[Y ] are unknown, we estimate the weight c∗ in Equation (2.2.1) using the unbiased sample covariance and sample variance formulas:

Cov[X, Y ] c∗ = − V ar[Y ] d1 N b N 1 i=1 Xi X Yi Y = −d − − − 1 N 2 PN 1 i=1 Yi Y − − N i=1 XiP X Yi Y = − 2− . (2.2.2) − N Y Y P i=1 i − Observing that c is equal to the ordinaryP least squares estimator b for the slope coeﬃcient − ∗ in the regression model X = a + bY + e with random variable e having mean 0 and variance b b 2 σe . See, for example, Gujarati [17] (page 55).

17 Ross [26] oﬀers a heuristic explanation on how the control variate method works. Consider the case when X and Y are positively correlated, thus implying that the weight c∗ in Equation (2.2.1) is negative. So when simulated runs of Y are large, so are the simulated runs of X. But large runs of Y create large overshoots of Y from its mean E[Y ], these overshoots being expressed as the diﬀerence Y E[Y ]. Since simulated runs of X are also large, the same − overshooting is probably true for X as well. So in order to correct the overshooting of the estimator X, we need to subtract from X the weighted overshooting of Y , hence we get the estimator X + c∗(Y E[Y ]). We get a similar correction when the random variables X and − Y are negatively correlated.

2.2.2 Applications

Estimating the price of an arithmetic Asian option illustrates the effectiveness of the control variate Monte Carlo approach very well. Since the geometric Asian option has a closed- form formula in the lognormal model and its payoff is highly correlated with that of the arithmetic Asian option, we can let the geometric Asian option payoff be the control variate to the arithmetic Asian option payoff. In the following example, we consider arithmetic and geometric Asian ASOs. Re- call the risk-neutral prices of these options expressed as expected discounted payoffs: rT + θ = E e− (S(T ) A(t )) for the price of an arithmetic Asian ASO, and θ = A − m G rT + E e− (S(T ) G(t )) for the price of a geometric Asian ASO. Letting − m

rT + X = e− (S(T ) A(t )) , and − m rT + Y = e− (S(T ) G(t )) , − m the control variate estimator is

Z = X + c(Y E[Y ]) − rT + rT + = e− (S(T ) A(t )) + c(e− (S(T ) G(t )) θ ). − m − m − G

Note that the price θG of the geometric Asian ASO at time t = 0 can be computed analytically as follows4: 4For a generalization of this formula, that is, pricing at any time t, we refer the readers to Clewlow and Strickland [9], pages 77 and 91-93. Note also that the notation in that reference is slightly diﬀerent from the notation that we are using in this dissertation.

18 With the underlying stock S(t) modeled as GBM(r, σ2), the formula for the price of a geometric call ASO at time t = 0 with initial stock price S(0), stock dividend d, expiry T and strike price K is given by

1 2 rT + dT µG+ σ rT E e− (S(T ) G(t ) = S(0)e− Φ(y ) e 2 G− Φ(y ), − m 1 − 2 where

1 2 1 2 ln S(0)+(r d)T µG σ + Σ y = − − − 2 G 2 1 Σ y = y Σ 2 1 − Σ2 = σ2 + σ2T 2ρ σ σ . G − G G T Note that ln G(t )= 1 m ln(S ) and is distributed as (µ ,σ2 ) with m m i=1 ti N G G P 1 h µ = ln S(0) + r d σ2 t + (m 1) G − − 2 1 2 − h(2m 1)(m 1) σ2 = σ2 t + − − . G 1 6m

The covariance for ln G(tm) and ln S(tm) is

h ρ σ σ = σ2 t + (m 1) . G G T 1 2 − Thus, we can write the price of the arithmetic Asian ASO as follows:

rT + θ = E e− (S(T ) A(t )) A − m rT + rT + = E e− (S(T ) A(t )) + c(e− (S(T ) G(t )) θ ) . − m − m − G So the control variate Monte Carlo estimator simulates j =1, 2, ..., N stock price paths and rT + rT + averages over the values e− (S(T ) A(t )) +c(e− (S(T ) G(t )) θ ). In other words, − m − m − G we estimate the price of the arithmetic Asian ASO by

N 1 rT + rT + θ = e− (S (T ) A (t )) + c(e− (S (T ) G (t )) θ ) . A N j − j m j − j m − G j=1 ! X c Table 2.4 reports the average of ﬁfty price estimates for an arithmetic Asian ASO and their variance using the crude Monte Carlo estimator and the control variate Monte Carlo estimator for parameter values in Cox and Rubinstein ([11], page 216) and Kemna and Vorst

19 Figure 2.4: Fifty arithmetic Asian ASO price estimates using control variate (ContVar) Monte Carlo. Each realization uses 5000 simulations (σ = 0.4, r = 0.07, S(0) = 40, T = 88/265, and m = 88).

([21], page 122): σ =0.2, 0.3, and 0.4; r =0.03, 0.05, and 0.07; S(0) = 40; T = 88/265 which is roughly four months; and m = 88. Each price estimate is simulated using 5000 stock price paths. For the choice of the control weight c, we use c = 1 5 and c = b from the regression − − + rT + model (S(T ) A(tm)) = a + b(e− (S(T ) G(tm)) )+ e. − − b Figure 2.4 shows fifty control variate Monte Carlo (ContVar) price estimates whose averages are reported in the last row of Table 2.4 (r = 0.07 and σ = 0.4). Figure 2.5 appends the previous figure with fifty crude Monte Carlo (MC) price estimates. Figure 2.6 shows the convergence of the MC and the ContVar estimators. From the above table, we are able to reduce the crude Monte Carlo variance by at most a factor of 6.6291 103 (r =0.05,σ =0.2) using the control variate Monte Carlo estimator × when the control weight c = b. Control variate Monte Carlo estimates on Table 2.4 clearly − show noticeable improvement over the conditional expectation Monte Carlo estimates on b Table 2.3 in estimating the price of an arithmetic Asian ASO.

5Kemna and Vorst [21] and Clewlow and Strickland [10] estimate the price of an arithmetic Asian ARO in the context of a hedged portfolio. This is equivalent to using a control weight c = 1. −

20 Table 2.4: Average of ﬁfty price estimates, variance and ratios of variances of crude Monte Carlo (MC) estimates and control variate (ContVar) Monte Carlo estimates for an arithmetic Asian ASO option. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88).

r σ MC ContVar1 MC/ContVar1 ContVar2 MC/ContVar2 (c = 1) (c = b) 21 4 − 7 +3 − 7 +3 0.2 1.1489 (6.8844 10− ) 1.1474 (3.6986 10− ) 1.86 10 1.1502 (1.7976 10− ) 3.83 10 × 3 × 6 × +3 × 7 × +3 0.03 0.3 1.6646 (1.5000 10− ) 1.6558 (1.1076 10− ) 1.35 10 1.6731 (4.1899b 10− ) 3.58 10 × 3 × 6 × +2 × 6 × +3 0.4 2.1912 (2.6000 10− ) 2.1867 (5.4674 10− ) 4.76 10 2.1957 (1.9696 10− ) 1.32 10 × 4 × 7 × +3 × 7 × +3 0.2 1.2169 (8.2791 10− ) 1.2156 (4.1470 10− ) 2.00 10 1.2182 (1.2489 10− ) 6.63 10 × 3 × 6 × +2 × 7 × +3 0.05 0.3 1.7336 (1.3000 10− ) 1.7289 (1.5325 10− ) 8.48 10 1.7382 (6.0554 10− ) 2.15 10 × 3 × 6 × +2 × 6 × +3 0.4 2.2646 (2.7000 10− ) 2.2706 (3.5080 10− ) 7.70 10 2.2588 (1.6767 10− ) 1.61 10 × 4 × 7 × +3 × 7 × +3 0.2 1.2826 (6.6800 10− ) 1.2883 (2.2424 10− ) 2.98 10 1.2885 (1.1626 10− ) 5.75 10 × 3 × 6 × +2 × 7 × +3 0.07 0.3 1.8020 (1.2000 10− ) 1.7989 (1.6475 10− ) 7.28 10 1.8051 (6.7274 10− ) 1.78 10 × 3 × 6 × +2 × 6 × +3 0.4 2.3147 (2.3000 10− ) 2.3258 (6.2943 10− ) 3.65 10 2.3233 (2.2262 10− ) 1.03 10 × × × × × Figure 2.5: Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and control variate (ContVar) Monte Carlo. Each realization uses 5000 simulations (σ = 0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

Figure 2.6: Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and control variate (ContVar) Monte Carlo. Fifty sample estimates in increments of 5000 simulations. Final estimates: MC = 2.3171, ContVar = 2.3232. Averages: MC= 2.3216, ContVar = 2.3232 (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

22 2.3 Combined Conditional Expectation and Control Variate Monte Carlo Estimator

2.3.1 Principles

In estimating the parameter θ = E[X], we can combine the conditional expectation estimator from Section 2.1 and the control variate estimator from Section 2.2 to get a new estimator Z = E[X Y ]+ c (W E[W ]) = f(Y )+ c(W E[W ]), where c is a constant. This new | − − estimator gives rise to the combined conditional expectation and control variate Monte Carlo estimator denoted by

1 N Z = E[X Y ]+ c (W E[W ]) = f(Y )+ c (W E[W ]) , (2.3.1) | − N j j − j=1 X where f(Y ) is a closed-form solution. The estimator Z is an unbiased estimator of θ since:

E[Z] = E [E[X Y ]+ c (W E[W ])] | − = E[E[X Y ]] + E[c (W E[W ])] | − = E[X] = θ.

It follows that the combined conditional expectation and control variate Monte Carlo estimator Z is also an unbiased estimator of θ. The variance Z is computed as follows: 1 Cov2[E[X Y ], W ] V ar[Z]= V ar[E[X Y ]+ c (W E[W ])] = V ar[E[X Y ]] | , | ∗ − N | − V ar[W ] where the control weight that minimizes V ar[Z] is given by:

Cov[E[X Y ], W ] c∗ = | . − V ar[W ]

As in Equation (2.2.2), the weight c∗ can be estimated either by using the sample covariance and variance estimators to give us:

N E[X Y ] E[X Y ] W W i=1 | i − | i − c∗ = 2 . (2.3.2) −P N W W i=1 i − b P 23 Another way to see this is by estimating the slope coeﬃcient in the regression model E[X Y ]= a + bW + e to yield c = b. | ∗ − We can measure the improvement of the combined conditional expectation and control b b variate Monte Carlo estimator Z over the crude Monte Carlo estimator X ﬁrst by taking the ratio of the variances V ar[Z] and V ar[E[X Y ]]: |

1 V ar[Z] V ar[E[X Y ]+ c∗(W E[W ])] = N | − V ar[E[X Y ]] 1 V ar[E[X Y ]] | N | Cov2[E[X Y ],W ] V ar[E[X Y ]] | = | − V ar[W ] V ar[E[X Y ]] | = 1 Corr2[E[X Y ], W ], − | implying V ar[Z]=(1 Corr2[E[X Y ], W ])V ar[E[X Y ]]. − | | We then invoke the relationship V ar[E[X Y ]] V ar[X]. Thus, we see that the combined | ≤ conditional expectation and control variate Monte Carlo estimator can reduce the variance of the crude Monte Carlo estimator since

V ar[Z] = (1 Corr2[E[X Y ], W ])V ar[E[X Y ]] − | | (1 Corr2[E[X Y ], W ])V ar[X] ≤ − | V ar[X], ≤ whenever the pairs of random variables X and Y are dependent and E[X Y ] and W are | highly correlated, either positively or negatively.

2.3.2 Applications

We now continue the discussion in Ross and Shanthikumar [27] of estimating the price of an arithmetic Asian call ASO, this time using the combined conditional expectation and control variate Monte Carlo estimator. Recall the pricing Equation (2.1.6) for an arithmetic Asian ASO using conditional

24 expectation:

m 1 − (rT T ) m 1 1 θ = E e− − m − BSM S(tm 1),tm 1,T, S(ti) m − − m 1 " − i=1 !# mX1 − (rT T ) m 1 1 = e− − m − E BSM S(tm 1),tm 1,T, S(ti) , m − − m 1 " i=1 !# − X 1 m 1 − where the quantity BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) serves as the conditional expec- − − − tation estimator E[X Y ]. The random variable X is represented by the discounted payoff | P of the option while the random variable Y is represented by the conditioning vector of stock prices Sm 1. We wish to find a control variate W that is highly correlated with − 1 m 1 − BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) so that we can form the combined conditional − − − expectation and control variateP estimator m 1 1 − ZW = BSM S(tm 1),tm 1,T, S(ti) + c∗ (W E[W ]) , (2.3.3) − − m 1 − i=1 ! − X where c∗ is the weight that minimizes V ar[ZW ]. Once we find a suitable control variate W , we can then rewrite the pricing equation in terms of the above combined conditional expectation and control variate estimator:

(rT T ) m 1 θ = e− − m − E [Z ] . m W The combined conditional expectation and control variate Monte Carlo estimator simulates T (rT m ) m 1 j = 1, 2, ..., N stock price paths and averages over the values e− − m− ZW . In other words, we estimate the price of the arithmetic Asian ASO by

N (rT T ) m 1 1 j θ = e− − m − Z , (2.3.4) m N W j=1 ! X j b where ZW denotes the value of the estimator ZW in Equation (2.3.3) on path j:

m 1 j 1 − ZW = BSM Sj(tm 1),tm 1,T, Sj(ti) + c∗ (Wj E[W ]) . − − m 1 − i=1 ! − X Before suggesting possible candidates for the control variate W , Ross and Shanthikumar [27] ﬁrst observe that

m 1 m 1 + 1 − 1 − BSM S(tm 1),tm 1,T, S(ti) S(tm 1) S(ti) . (2.3.5) − − m 1 ≈ − − m 1 i=1 ! i=1 ! − X − X 25 The right-hand side of the above approximation is equal to the payoff, hence the value at time tm 1, of a European call option with terminal stock price S(tm 1) and strike − − 1 m 1 − price m 1 i=1 S(ti). The left-hand side is the price at time tm 1 of a European call − − option with initial stock price S(tm 1), expiring one time step later at T with payoff P − 1 m 1 + − S(T ) m 1 i=1 S(ti) . − − We may justifyP their observation by arguing loosely as follows. Since we do not expect stock prices that are modeled as geometric Brownian motion to jump big distances after a T small time step m , the terminal stock price S(T ) will not stray far from the penultimate 1 m 1 + − stock price S(tm 1). Thus, the difference between the payoff S(tm 1) m 1 i=1 S(ti) − − − − 1 m 1 + − at time tm 1 and the payoff S(T ) m 1 i=1 S(ti) at time T will not be large.P − − − 1 m 1 − This approximation implies that the quantityP BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) − − − 1 m 1 + − is positively correlated with the quantity S(tm 1) m 1 i=1 S(ti) . Thus,P we can now − − − use the latter quantity as a proxy for the former in determiningP the control variate W in Equation (2.3.3). To illustrate the correlation, we simulate 5000 sample points for both quantities in approximation (2.3.5). The values at each sample point are derived from a stock price path simulated until time tm 1. Figure 2.7 shows the scatter plot of these quantities. − The correlation coefficient is 0.9994. Ross and Shanthikumar [27] then observe that the proxy will be large if the tail end of the series of random walks X(t1),X(t2), ..., X(tm 1) defining the lognormal stock price − X(t1)+X(t2)+...+X(tm−1) 6 S(tm 1)= S(t0)e is also large . The case works in reverse if the tail end − of the random walks is small. This conveys some degree of positive correlation between the proxy quantity and the random walks. So, in choosing a control variate W , the authors first suggest the vector (X(t1),X(t2), ..., X(tm 1)) of random walks. Using these i.i.d. normal − steps X(t ) with mean µ =(r 1 σ2)T/m and variance σ2 = σ2T/m, we get the following i X − 2 X combined conditional expectation and control variate estimator

m 1 m 1 − − 1 1 2 T ZX = BSM S(tm 1),tm 1,T, S(ti) + ci∗ X(ti) r σ , − − m 1 − − 2 m − i=1 ! i=1 X X (2.3.6) where the ci∗ are the negative of the estimated slope coefficients bi in the multiple regression 1 m 1 m 1 − − BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) = a + i=1 biX(ti)+ e. − − − b 6 1 m−1 Of course, the stock price S(tPm−1) will rise faster thanP the average m−1 i=1 S(ti). Nevertheless, the + 1 m−1 proxy S(tm−1) i S(ti) will still rise. P − m−1 =1 P 26 1 m 1 − Figure 2.7: Scatter plot of BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) and penultimate payoff − − − 1 m 1 + S(tm 1) − S(ti) for 5000 sample points. The correlation coefficient for BSM − m 1 i=1 P samples and− payoff− samples is 0.994 (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). P

X(ti) Since the random variables X(ti) and e are correlated, the authors suggest the vector (eX(t1),eX(t2), ..., eX(tm−1)) for the control. The entries in this vector are i.i.d. lognormal with 1 2 mean eµX + 2 σX . This vector yields a second combined conditional expectation and control variate estimator m 1 m 1 1 − − 1 2 X(ti) µX + 2 σX ZeX = BSM S(tm 1),tm 1,T, S(ti) + ci∗ e e , (2.3.7) − − m 1 − i=1 ! i=1 − X X where the ci∗ are the negative of the estimated slope coeﬃcients bi in the multiple regression

1 m 1 m 1 X(ti) BSM S(tm 1),tm 1,T, − S(ti) = a + − bie + e. − − m 1 i=1 i=1 − m 1 b Still another possible controlP variate is the sumP i=1− X(ti) of the random walks giving us a third combined conditional expectation and controlP variate estimator m 1 m 1 1 − − ZΣX = BSM S(tm 1),tm 1,T, S(ti) + c∗ X(ti) (m 1)µX , (2.3.8) − − m 1 − − i=1 ! i=1 ! − X X 1 m 1 m 1 − − where c∗ = b in the regression BSM S(tm 1),tm 1,T, m 1 i=1 S(ti) = a+b i=1 X(ti)+ − − − − e. b P P In estimating the price of the arithmetic Asian ASO using the three diﬀerent combined conditional expectation and control variate estimators ZX , ZeX and ZΣX above, we simulate

27 j j j =1, 2, ..., N stock price paths, evaluate each of these estimators at path j to get ZX , ZeX j j and ZΣX , and replace ZW by these path j values when getting the sample mean in Equation (2.3.4) For our numerical example, we follow the parameter values in Cox and Rubinstein [11] (p. 216) and Kemna and Vorst [21] (p. 122): σ =0.2, 0.3, and 0.4; r =0.03, 0.05, and 0.07; S(0) = 40; T = 88/265 which is roughly four months; and m = 88. Table 2.5 reports the average and variance of fifty price estimates for the combined conditional and control variate estimators ZX , ZeX and ZΣX denoted, respectively, in the table by CECVx, CECVex and CECVsumx. Table 2.6 shows the factors of improvement of the combined estimators on the crude Monte Carlo estimator. These factors are computed as the ratios of the crude Monte Carlo variance to the variances of the combined conditional and control variate. From the table, we observe that variance ratios of the crude Monte Carlo to CECVx and to CECVex trail each other by a factor roughly equal to 1. This is not surprising since the control variates of CECVx and CECVex are highly correlated. Among the three estimators, the table also shows that CECVx and CECVex give higher factors of improvement over MC than CECVsumx. These factors of improvement, however, all less than 5. The numerical results above can also be seen in Figure 2.8 which shows the fifty price estimates of the combined estimators whose averages are reported in the last row of Table 2.5. Each price estimate uses 5000 stock price paths. Figure 2.9 magnifies the first ten price estimates of CECVx and CECVex to show how closely their price estimates trail each other. Figure 2.10 incorporates crude Monte Carlo estimates into Figure 2.8. Finally, Figure 2.11 shows the convergence of the combined estimators relative to the crude Monte Carlo estimator. From these numerical results and figures, we can conclude that Ross and Shanthikumar’s combined conditional expectation and control variate estimator for pricing Asian options is slightly more effective than the crude Monte Carlo estimator. This improvement may be attributed more to the effectiveness of the control variate method than to the conditional expectation method, the latter method as suggested by Ross and Shanthikumar [27].

28 Table 2.5: Average of ﬁfty price estimates and variance of crude Monte Carlo (MC) estimates and combined conditional expectation and control variate (CECV) Monte Carlo estimates for an arithmetic Asian ASO option. Each price estimate uses 5000 simulations. Variances are in parentheses (Various σ and r, S(0) = 40, T = 88/265, m = 88).

r σ MC CECVx CECVex CECVsumx X(ti) control = ciX(ti) control = cie control = c X(ti)

29 4 4 4 4 0.2 1.1489 (6.8844 10− ) 1.1464 (2.0842 10− ) 1.1464 (2.0726 10− ) 1.1480 (3.3793 10− ) × 3 P × 4 P × 4 P× 4 0.03 0.3 1.6646 (1.5000 10− ) 1.6685 (4.0626 10− ) 1.6685 (4.0029 10− ) 1.6681 (6.9874 10− ) × 3 × 4 × 4 × 3 0.4 2.1912 (2.6000 10− ) 2.1939 (8.5683 10− ) 2.1939 (8.4851 10− ) 2.1967 (1.2000 10− ) × 4 × 4 × 4 × 4 0.2 1.2169 (8.2791 10− ) 1.2165 (1.7567 10− ) 1.2166 (1.7590 10− ) 1.2142 (3.0824 10− ) × 3 × 4 × 4 × 4 0.05 0.3 1.7336 (1.3000 10− ) 1.7355 (2.9786 10− ) 1.7356 (2.9399 10− ) 1.7356 (3.7920 10− ) × 3 × 4 × 4 × 4 0.4 2.2646 (2.7000 10− ) 2.2605 (8.6680 10− ) 1.2607 (8.6040 10− ) 2.2574 (1.1855 10− ) × 4 × 4 × 4 × 4 0.2 1.2826 (6.6800 10− ) 1.2881 (1.6213 10− ) 1.2881 (1.6084 10− ) 1.2875 (2.4123 10− ) × 3 × 4 × 4 × 4 0.07 0.3 1.8020 (1.2000 10− ) 1.8005 (4.9153 10− ) 1.8006 (4.9073 10− ) 1.8022 (5.8024 10− ) × 3 × 4 × 4 × 4 0.4 2.3147 (2.3000 10− ) 2.3423 (8.7991 10− ) 2.3243 (8.7933 10− ) 2.3218 (9.3828 10− ) × × × × Table 2.6: Ratios of variances of crude Monte Carlo (MC) estimates to variances of various combined conditional expectation and control variate (CECV) estimates from Table 2.5.

r σ MC/CECVx MC/CECVex MC/CECVsumx X(ti) control = ciX(ti) control = cie control = c X(ti) 0.2 3.3031 3.3216 2.0372 0.03 0.3 3.6922P 3.7473P 2.1467P 0.4 3.0344 3.0642 2.1667 0.2 4.7129 4.7067 2.7035 0.05 0.3 4.3645 4.4219 3.4283 0.4 3.1149 3.1381 2.7752 0.2 4.1202 4.1532 2.7691 0.07 0.3 2.4414 2.4453 2.0681 0.4 2.6139 2.6156 2.4513

Figure 2.8: Fifty arithmetic Asian ASO price estimates using combined conditional expectation and control variate Monte Carlo (CECV). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

30 Figure 2.9: Ten arithmetic Asian ASO price estimates using combined conditional expectation and control variate Monte Carlo with control variates ciX(ti) (CECVx) and X(ti) cie (CECVex). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88). P P

Figure 2.10: Fifty arithmetic Asian ASO price estimates using crude Monte Carlo (MC) and combined conditional expectation and control variate Monte Carlo (CECV). Each realization uses 5000 simulations (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

31 Figure 2.11: Convergence of arithmetic Asian ASO price estimators using crude Monte Carlo (MC) and combined conditional expectation and control variate (CECV) Monte Carlo. (σ =0.4, r =0.07, S(0) = 40, T = 88/265, and m = 88).

32 CHAPTER 3

Importance Sampling Monte Carlo Estimators

3.1 Importance Sampling Monte Carlo Estimator and Heuristics

3.1.1 Principles

Suppose we want to estimate the parameter θ = E [h(X)], where E [ ] indicates expectation f f · using the density f(x) and h(X) is a function of the random variable X. One can estimate the parameter θ directly by applying the crude Monte Carlo technique of drawing i.i.d. samples 1 N X1,X2, ..., XN from the density f(x) and computing the sample mean N j=1 h(Xj). If in using f(x), simulation of the random samples becomes diﬃcult or the estimatorP yields a large variance, we can resort to another variance reduction technique called importance sampling. “The object in importance sampling,” write Hammersley and Handscomb in [18] (p. 58), “is to concentrate the distribution of the sample points in the parts of the interval that are of most ‘importance’ instead of spreading them out evenly.” In order not to bias the result, we compensate for the introduction of the new distribution g(x) that gives more importance on certain parts of the interval by taking h(X)f(X)/g(X) in place of h(X) as our estimator. More formally, the importance sampling technique requires picking another density function g(x), also called the biasing density or the importance sampling distribution, that satisﬁes the condition f(x) = 0 whenever g(x) = 0 and rewriting the parameter θ as an expectation in terms of the biasing density g(x):

f(x) f(X) θ = E [h(X)] = h(x)f(x)dx = h(x) g(x)dx = E h(X) , f g(x) g g(X) Z Z

33 f(X) f(X) where g(X) is called the likelihood ratio. Letting Z = h(X) g(X) and drawing samples

X1,X2..., XN from the biasing density g(x), we call the sample mean

1 N f(X ) Z = h(X ) j (3.1.1) N j g(X ) j=1 j X the importance sampling Monte Carlo estimator associated with the importance sampling estimator Z. The importance sampling Monte Carlo estimator Z is an unbiased estimator of the parameter θ since:

1 N f(X ) E [Z] = E h(X ) j g g N j g(X ) " j=1 j # X 1 N f(X ) = E h(X ) j N g j g(X ) j=1 j X 1 = Nθ N = θ.

Reduction of the crude Monte Carlo estimator variance by importance sampling depends largely on the choice of the biasing density g(x). To see this, let us compare the variance f(X) of h(X) g(X) with respect to the density g(x), with the variance of h(X) with respect to the density f(x). We have:

f(X) f(X) 2 f(X) 2 V ar h(X) = E h(X) E h(X) g g(X) g g(X) − g g(X) " # f 2(X) = E h2(X) θ2 g g2(X) − f 2(x) = h2(x) g(x)dx θ2 g2(x) − Z f(x)f(x) = h2(x) dx θ2 g(x) − Z f(X) = E h2(X) θ2, (3.1.2) f g(X) − and

V ar [h(X)] = E h2(X) (E [h(X)])2 = E h2(X) θ2. (3.1.3) f f − f f − 34 A quick inspection of Equations (3.1.2) and (3.1.3) may lead one to conclude that the only sufficient condition for the importance sampling estimator to reduce the variance of our original estimator is that f(X) 1. Ross ([26], p. 167) cautions the reader about this careless g(X) ≤ conclusion with this warning: V ar h(X) f(X) may become infinite even if f(X) 1. Ross g g(X) g(X) ≤ f(X) h i notes that Eg g(X) = 1 so that on occasion, random draws of X will give large values of f(X) g(X) greater thanh 1.i Thus, we need to find biasing densities that will yield low values of f(X) f(X) h(x) for high values of g(X) in order to prevent V arg h(X) g(X) from exploding. A similar observation is also made by Robert and Casella ([25],h p. 85) andi Glasserman ([15], p. 256). Among the biasing densities that lead to finite variances for the importance sampling Monte Carlo estimator (3.1.1), an optimal density can be found. We quote the result from Rubinstein [28] (p. 122) and Robert and Casella [25] (p. 84).

Theorem 3.1.1. The biasing density g∗(x) that minimizes the variance of the importance sampling Monte Carlo estimator (3.1.1) is

h(x) f(x) g∗(x)= | | . h(z) f(z)dz | | Corollary 3.1.2. If h(x) is nonnegative, thenR the biasing density that minimizes the variance of the importance sampling Monte Carlo estimator is h(x)f(x) g∗(x)= , h(z)f(z)dz

f(XR) ∗ and the minimum variance V arg h(X) g(X) =0. h i In estimating option prices, the function h(x) usually represents the discounted payoff, which is nonnegative. So, the above corollary applies to our price estimations. However, the optimal biasing density g∗(x) requires that we know the value of h(z)f(z)dz, which is precisely the quantity Ef [h(x)] that we wish to estimate in the first place.R This renders the above optimal biasing density useless for practical purposes. One popular choice for the biasing density is exponential tilting (also known as exponential twisting, exponential change of measure, or exponential shift)1. The tilted density is defined as follows: 1Srinivasan ([30], pp. 10–25) notes that three biasing methods are popular in importance sampling. These methods are scaling, translation, and exponential twisting. In this paper, we focus only on exponential twisting.

35 Deﬁnition 3.1.1. Let f(x) be a given density function. The tilted density function of f(x) is etxf(x) f (x)= , t M(t) where the tilting parameter t satisﬁes

We mention some properties of the tilted density in conjunction with a random variable X distributed (µ, σ2) that will be useful in our example later: N

Property 1. Let f(x) be the normal density with mean µ and variance σ2. Then its tilted 2 2 density ft(x) is normal with mean µ + σ t and variance σ . The proof is as follows. A random variable X distributed (µ, σ2) has density N 1 1 x µ 2 f(x)= exp − , σ√2π −2 σ ! and moment generating function 1 M(t) = exp µt + σ2t2 . 2 So, the tilted density of X is given by etxf(x) f (x) = t M(t) tx 1 1 x µ 2 e exp − σ√2π 2 σ = − σ2t2 exp µt + 2 2 1 1 x µ σ2t2 = exp tx − µt + σ√2π − 2 σ − 2 ! 1 1 x µ 2 σ2t2 = exp 2tx + − +2 µt + σ√2π −2 − σ 2 !! 1 1 2txσ2 + x2 2xµ + µ2 +2µσ2t + σ4t2 = exp − − σ√2π −2 σ2 1 1 x2 2x(µ + σ2t)+ µ2 +2µσ2t + σ4t2 = exp − σ√2π −2 σ2 1 1 x (µ + σ2t) 2 = exp − . σ√2π −2 σ ! 36 In other words, intervals around µ + σ2t are given more importance in the tilted density ft(x) than in the original density f(t) where those same intervals may lie to the left or to the right of the mean µ, and possibly at the tail ends, depending on the value of t. Similarly, if X is distributed r 1 σ2 h,σ2h then its tilted density is the normal N − 2 density with mean r 1 σ2 h + σ2ht and variance σ2h. − 2 We also see that for normal densities, exponential twisting is equivalent to translating.

Property 2. We can rewrite the tilted density ft(x) in terms of the logarithm of the moment tX generating function M(t) = Ef [e ], also called the cumulant generating function ψ(t), of f(x): tx ψ(t) ft(x)= e − f(x).

The cumulant generating function of the random variable X distributed (µ, σ2) is N 1 ψ(t)= µt + σ2t2. 2

Property 3. The mean of the random variable X with respect to the tilted density ft(x) is equal to the ﬁrst derivative of the cumulant generating function, ψ′(t). The proof follows from Glasserman [15] (p. 261). Since the cumulant generating function ψ(t) is equal to the logarithm of the moment generating function M(t), we have

tX ψ(t) = log Ef [e ].

Taking the derivative of ψ(t), we get

1 tX ′ ψ (t) = tX Ef [Xe ] Ef [e ] E [XetX ] = f eψ(t) tX ψ(t) = Ef [Xe − ]

= Eft [X].

For the random variable X distributed (µ, σ2), the ﬁrst derivative of the cumulant N generating function is 2 ψ′(t)= µ + σ t,

37 which is the mean of the tilted density in Property 1. Similarly, if the random variable X is distributed ((r 1 σ2)h,σ2h), then the ﬁrst derivative of its cumulant generating function N − 2 and the mean of its tilted density are equal:

1 2 2 ψ′(t)= r σ h + σ ht. (3.1.4) − 2

Property 4. The variance of the random variable with respect to the tilted density ft(x) is equal to the second derivative of the cumulant generating function, ψ′′(t). We prove the statement as follows. From Property 2 above, we know

tX ψ(t) ψ′(t)= Ef [Xe − ].

This implies

tX ψ(t) ψ′′(t) = E Xe − (X ψ′(t)) f − 2 tX ψ(t) tX ψ(t) = E X e − E Xe − ψ′(t) f − f 2 tX ψ(t) tX ψ(t) = E X e − ψ′(t)E Xe − f − f 2 tX ψ(t) tX ψ(t) tX ψ(t) = E X e − E Xe − E Xe − f − f f = E X2 (E [X])2 ft − ft

= V arft [X].

For the random variable X distributed (µ, σ2), the second derivative of its cumulant N generating function is 2 ψ′′(t)= σ , which is the variance of the tilted density in Property 1.

As a ﬁnal remark, the tilted normal density in Property 1 clearly illustrates the importance sampling character of a tilted density. For the eﬀectiveness of the tilted density in reducing variance, we refer the reader to Bucklew [6] (pp. 82, 86-87).

38 3.1.2 Applications Down-and-In Barrier Option

Our first example estimates the price of a down-and-in barrier option. A down-and-in barrier option is a type of barrier option whose barrier level H is below (“down”) the initial stock price S(0) and becomes alive (“in”) as a standard European call option if the stock price path hits the barrier level before expiration. Boyle et al. [3] first applied the concept of importance sampling in estimating the price of a down-and-in barrier option and used heuristic arguments in estimating the tilting parameter. A slightly expanded version of this topic appears in Glasserman [15] (pp. 264– 267). Importance sampling changes the probability distribution of the underlying stock so that the stock price crosses the barrier level and then overshoots the strike price, both events with greater likelihood than in the original distribution. For our down-and-in barrier option example, we again derive the heuristic solution of Boyle et al. [3] and provide a numerical example. Estimating the tilting parameter using a simulated annealing procedure is deferred until Chapter 3.2. We begin by recalling the risk-neutral price of a down-and-in barrier option which is the expected discounted payoff:

rT + θ = E e− 1(t

STEP 1. Choose a model for the underlying stock. We model the stock in the form n

S(tn)= S(0) exp(Ln), Ln = X(ti), i=1 X where X(t1),X(t2), ..., X(tm) are i.i.d. random variables with common density f(x). We assume X(0) = 0 so that L0 = 0. In particular, we model the stock price S(tn) as the 2 geometric Brownian motion GBM(r, σ ) with equally spaced time steps tm = mh. Each increment is distributed 1 X(t ) ∼ N r σ2 h,σ2h i − 2 with cumulant generating function 1 1 ψ(t)= r σ2 ht + σ2ht2 (3.1.6) − 2 2

39 whose derivative is

1 2 2 ψ′(t)= r σ h + σ ht. (3.1.7) − 2

STEP 2. Rewrite the expected discounted payoﬀ (3.1.5) in terms of the likelihood ratio. Note two items here. First, we shall introduce two likelihood ratios that contain biasing densities that increase the likelihood of more “important” events. One likelihood ratio contains the biasing density ft− (x) that increases the likelihood of the event that the stock price hits the barrier level at some random time tτ . The other likelihood ratio contains the biasing density ft+ (x) that increases the likelihood of the event that the stock overshoots the exercise price at time tm. Second, although the payoﬀ is a function of the stock price vector (S(t1), S(t2), ..., S(tm)), we shall invoke importance sampling by changing the distribution f(x) of the i.i.d. random variables X(t1),X(t2), ..., X(tm) that drive our stock prices instead of changing the distribution of the stock price itself. Thus, we get the importance sampling identity

Ef [h(S(t1), S(t2), ..., S(tm))] τ m f(X(ti)) f(X(ti)) = Ef − ,f + h(S(t1), S(t2), ..., S(tm)) , (3.1.8) t t f − (X(t )) f + (X(t )) " i=1 t i i=τ+1 t i # Y Y provided the independence of X(ti) is preserved under the biasing densities ft− (x) and ft+ (x), and the expectation is ﬁnite.

STEP 3. Introduce the following tilted densities as the biasing densities in the likelihood ratios in Equation (3.1.8):

− + et xf(x) et xf(x) − + ft (x)= and ft (x)= + , M(t−) M(t ) where

t−X M(t−) = M(t−)= Ef [e ]

+ + t+X M(t ) = M(t )= Ef [e ]

40 are the moment generating functions of the set of random variables X(t1), ..., X(tτ ) and

X(tτ+1), ..., X(tm), respectively. We can now simplify the likelihood ratio in Equation (3.1.8) to τ m f(X(ti)) f(X(ti)) f − (X(t )) f + (X(t )) i=1 t i i=τ+1 t i Y Y τ + m τ M(t−) M(t ) − = + exp(t−(X(t1)+ X(t2)+ ... + X(tτ ))) exp(t (X(tτ+1)+ X(tτ+2)+ ... + X(tm))) τ M(t−) + m + + = M(t ) exp t t− L t L . (3.1.9) M(t+) − τ − m

+ STEP 4. Determine the values of the tilting parameters t− and t that increase the probabilities of the important events in Step 2. We do so using heuristic arguments of Boyle et al. [3] as follows. Heuristic Condition 1: Our goal in introducing the likelihood ratio is to reduce the variance in our original estimator. In the likelihood ratio (3.1.9) above, the sources of variability are the hitting time index τ, the sum Lτ of the increments X(ti) up to time index

τ, and the sum Lm of the increments X(ti) up to time index m. Among these three random quantities, Boyle et al. [3] argue that the hitting time index τ contributes the most in the variation of the price estimates because one can expect stock price paths not to make big jumps close to the barrier level or the exercise price. That is, unlike the random time index τ, these two quantities are approximately constant: Lτ ≈ log (H/S(0)) and Lm ≈ log (K/S(0)). We can remove the strong dependence of the likelihood ratio on the hitting time index τ by + choosing tilting parameters t− and t that satisfy the ﬁrst heuristic condition

+ M(t−)= M(t ). (3.1.10)

This heuristic condition (3.1.10) simpliﬁes the likelihood ratio (3.1.9) to

τ m f(X(ti)) f(X(ti)) + m + + = M(t ) exp t t− Lτ t Lm . f − (X(t )) f + (X(t )) − − i=1 t i i=τ+1 t i Y Y Heuristic Condition 2. We require the condition that the random walk that pays a positive payoﬀ, on average, completes its path of hitting the barrier level then reaching the exercise price within m steps. That is, if we let c = log (K/S(0)) and b = log (S(0)/H),

41 where c> 0 and b> 0, we require the second heuristic condition b c + b − + + = m. ψ′(t−) ψ′(t ) We derive this condition as follows. Before it reaches the barrier level, we let the random walk travel by downward increments of ψ′(t−) on average. By time index τ, the random walk will have traveled a downward distance of b = ψ′(t−)τ. For the rest of the time indices − + m τ, we let the random walk travel by upward increments of ψ′(t ) on average. By time − + index m, the random walk will have traveled an upward distance of b + c = ψ′(t )(m τ) − from b to c. We use these two equations to justify the second heuristic condition above. − Recall the cumulant generating function ψ(t) in (3.1.6) for a stock price modeled as 2 + GBM(r, σ ) with equally spaced time points. The ﬁrst heuristic condition M(t−)= M(t ) + implies ψ(t−)= ψ(t ) since ψ(t) = log M(t). The symmetry of the graph (parabola) of ψ(t) + + and the condition ψ(t−) = ψ(t ) together imply ψ′(t−) = ψ′(t ). The latter statement − means our random walk, whether it is heading downwards to the barrier level or upwards to the strike price, is traveling at a constant rate. In other words, the combined distance of + 2b + c is traversed within time steps m at the rate ψ′(t )= ψ′(t−) : | |

+ 2b + c = ψ′(t−) m and 2b + c = ψ′(t )m. | |

+ Replacing ψ′(t ) and ψ′(t−) by Equation (3.1.7) and solving for the tilting parameters t− and t+, we get

1 r 2b + c + 1 r 2b + c t− = and t = + . (3.1.11) 2 − σ2 − mσ2h 2 − σ2 mσ2h

STEP 5. Simulate stock price paths and take the average of the discounted payoﬀs that have been rewritten in terms of the likelihood ratio. The importance sampling Monte Carlo estimator simulates j =1, 2, ..., N stock price paths using the tilted density ft− (x) from the initial time until they hit the barrier level and using the tilted density ft+ (x) from the hitting time until expiration, and averages over the values

e rT (S(T ) K)+M(t+)m exp((t+ t )L t+L ) if t

42 Table 3.1: Relative mean square errors of crude Monte Carlo (MC) estimates, conditional expectation (CondExp) Monte Carlo estimates, and importance sampling (ImpSamp-h) Monte Carlo heuristics estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784).

N MC CondExp MC/CondExp ImpSamp-h MC/ImpSamp-h 3 4 +1 4 0 5K 1.66 10− 1.36 10− 1.22 10 3.33 10− 4.98 10 × 4 × 5 × +1 × 4 × 0 10K 9.73 10− 8.40 10− 1.16 10 1.64 10− 5.93 10 × 4 × 5 × +1 × 5 × 0 50K 2.00 10− 1.65 10− 1.21 10 4.29 10− 4.66 10 × × × × × Cross Barrier 62% 61% n/a 84% n/a Cross Exercise 29% n/a n/a 43% n/a

In other words, we estimate the price of the down-and-in barrier option using the heuristics- based importance sampling approach by

N 1 rT j + + m + j + j θ = e− 1(t

In the following numerical examples, we recall the parameter values of the two down- and-in barrier options that were presented in Chapter 2.1 (Conditional Expectation Monte Carlo Estimator). These two options have the following common parameter values: σ =0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2 and m = 50. The first option has a barrier level H = 95 with the true price of 1.4373238784, while the second option has a barrier level H = 91 with the true price of 0.3670447223. Tables 3.1 and 3.2 compare the relative mean square errors of fifty option estimates using the crude Monte Carlo (MC) estimator, the conditional expectation (CondExp) Monte Carlo estimator, and the importance sampling (ImpSamp-h) Monte Carlo estimator using heuristics. The fifth row of the tables, labeled “Cross barrier”, gives the percentage of price paths that cross the barrier, and the next row displays the percentage of paths that cross the exercise price among those that crossed the barrier.

43 Table 3.2: Relative mean square errors of crude Monte Carlo (MC) estimates, conditional expectation (CondExp) Monte Carlo estimates, and importance sampling (ImpSamp-h) Monte Carlo heuristics estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 91, and true price = 0.3670447223).

N MC CondExp MC/CondExp ImpSamp-h MC/ImpSamp-h 3 4 0 4 +1 5K 3.77 10− 4.66 10− 8.09 10 3.56 10− 1.06 10 × 3 × 4 × 0 × 4 × +1 10K 2.60 10− 4.17 10− 6.24 10 1.74 10− 1.49 10 × 4 × 5 × 0 × 5 × +1 50K 5.47 10− 6.98 10− 7.84 10 4.90 10− 1.12 10 × × × × × Cross Barrier 41% 40% n/a 86% n/a Cross Exercise 15% n/a n/a 42% n/a

The ImpSamp-h estimator yields roughly a 10+1 improvement in the relative mean square error over the MC estimator. The ImpSamp-h estimator gives higher relative mean square error than the CondExp estimator in Table 3.1, where the barrier level is 95 and 61% of the price paths reach the barrier. When the barrier is lowered to 91 in Table 3.2, and thus only 40% of the price paths reach the barrier, we observe that the ImpSamp-h estimator becomes better than the CondExp estimator. Note that 84% and 86% of paths reach the barrier when importance sampling is used in Tables 3.1 and 3.2, respectively. ImpSamp-h can be made much better than CondExp by taking the barrier price farther from the initial stock price.

Up-and-Out Barrier Option

Our second example estimates the price of an up-and-out barrier option. An up-and-out barrier option is a barrier option whose barrier level H is above (“up”) its initial stock price S(0), begins as a standard European call option and dies (“out”) if it hits the barrier level before the expiration T . We consider only the case where the exercise price K is below the barrier level H, for otherwise the option dies before getting the chance of yielding a positive payoﬀ. Furthermore, in order to avoid high variability in price estimates from an abundance of zero payoﬀs, resulting from either dead options or failure of the terminal stock price S(T ) to overshoot the exercise price K, we invoke importance sampling. We change the probability

44 distribution of the stock price paths so that the event that the terminal stock price S(T ) lands within the band bounded by the exercise price K and the barrier level H has greater probability than in the original distribution. This event incorporates two important events: the ﬁrst is the event that stock price paths remain below the barrier level H, and the second is the event that stock price paths overshoot the exercise price K at expiration. In order to satisfy both events with the greatest probability, we shall indirectly tilt the distribution of the random variable S(T ) by relocating the mean of S(T ) that maximizes the probability of this band. As in the previous example, we shall derive the importance sampling Monte Carlo estimator using a heuristic. We begin by recalling the risk-neutral price of an up-and-out barrier option:

rT + θ = E [e− 1(t >T )(S(T ) K) ] f τ − = Ef [h(S(t1), S(t2), ..., S(tm))]. (3.1.12)

STEP 1. Choose a model for the underlying stock. As before, we model the stock in the form n

S(tn)= S(0) exp(Ln), Ln = X(ti), i=1 X where X(t1),X(t2), ..., X(tn) are i.i.d. random variables with common density f(x). We assume X(0) = 0 so that L0 = 0. Speciﬁcally, we model the stock price S(tn) as the 2 geometric Brownian motion GBM(r, σ ) with equally spaced time steps tn = nh. Each increment is distributed 1 X(t ) ∼ r σ2 h,σ2h n N − 2 with cumulant generating function 1 1 ψ(t)= r σ2 ht + σ2ht2 − 2 2 whose derivative is 1 2 1 2 ψ′(t)= r σ h + σ ht − 2 2 and moment generating function 1 1 M(t) = exp r σ2 ht + σ2ht2 . − 2 2 45 STEP 2. Rewrite the expected discounted payoﬀ in Equation (3.1.12) in terms of the likelihood ratio. Unlike in the previous example, we shall introduce only one likelihood ratio containing the biasing density ft(x) that increases the likelihood of this important event: that the terminal stock price S(T ) is contained in the band bounded by the strike price K and the barrier level H. As before, we shall invoke importance sampling by changing the distribution f(x) of the i.i.d. random variables X(t1),X(t2), ..., X(tm) that drive our stock prices instead of changing the distribution of the stock price itself. We get the importance sampling identity

Ef [h(S(t1), S(t2), ..., S(tm))] m f(X(t )) = E h(S(t ), S(t ), ..., S(t )) i , ft 1 2 m f (X(t )) " i=1 t i # Y provided the independence of X(ti) is preserved under the biasing density ft(x) and the expectation is ﬁnite.

STEP 3. Introduce the following tilted density as the biasing density in the likelihood ratio in Step 2: etxf(x) f (x)= , t M(t) where M(t) is the moment generating function of the random variable X(ti). Note from Property (1), the tilted density is the normal density with mean (r 1 σ2)h + σ2ht and − 2 variance σ2h. We can now write the likelihood ratio in Step 2 as

m f(X(t )) M(t)m i = f (X(t )) exp(t(X(t )+ X(t )+ ... + X(t ))) i=1 t i 1 2 m Y M(t)m = . exp(tLm)

STEP 4. Determine the value of the tilting parameter t in Step 3 that increases the probability of the important event in Step 2. That is, we want to maximize the probability K S(T ) H P (K < S(T ) < H)= P < < S(0) S(0) S(0) K H = P < exp(L ) < S(0) m S(0) 46 K H = P log < L < log S(0) m S(0) = P (c < Lm < b) , (3.1.13)

where c = log K , b = log H , and L = m X(t ) ∼ (mµ , mσ2 ). Note that S(0) S(0) m i=1 i N X X b > 0 since an up-and-out barrier option requires HP > S(0), while the sign of the quantity c depends on whether K < S(0),K = S(0), or K > S(0). Because the normal distribution is symmetric about its mean, the probability in (3.1.13) is maximized if the mean mµX is located at the midpoint p of the interval [c, b] . That is, c + b mµt = = p, (3.1.14) X 2 t where µX denotes the mean of the tilted step in the random walk that maximizes probability

(3.1.13). The last equation implies that each i.i.d. step X(ti) in the random walk Lm must have an average length p µt = X m in order for the random walk to reach the midpoint p after m steps.

To solve for the tilting parameter t, recall the derivative ψ′(t) is the mean of the tilted density. Now, consider the following cases: Case 1: If 0 < p, then we tilt the random walk upwards.

p p 1 2 2 = ψ′(t)= = r σ h + σ ht m ⇒ m − 2 1 r p = t = + ; (3.1.15) ⇒ 2 − σ2 mσ2h Case 2: If 0= p, then we tilt the random walk to induce zero mean in the steps of the random walk.

1 2 2 0= ψ′(t)= 0= r σ h + σ ht ⇒ − 2 1 r = t = ; (3.1.16) ⇒ 2 − σ2 Case 3: If 0 > p, then we tilt the random walk downwards.

p p 1 2 2 = ψ′(t)= = r σ h + σ ht −m ⇒−m − 2 1 r p = t = . (3.1.17) ⇒ 2 − σ2 − mσ2h

47 STEP 5. Simulate stock price paths and take the average of the discounted payoﬀs. The importance sampling Monte Carlo estimator simulates j =1, 2, ..., N stock price paths using the tilted density ft(x) and averages over the values

rT + M(t)m e− (S(T ) K) if tτ >T − exp(tLm) , ( 0 otherwise where the tilting parameter t is determined by the diﬀerent cases in Step 4 Equations (3.1.15), (3.1.16), and (3.1.17). In other words, we estimate the price of the up-and-out barrier option using the heuristics-based importance sampling approach by

N m 1 rT j + M(t) θ = e− 1(t >T )(S (T ) K) . N τ j − j j=1 exp(tLm) X b

For our numerical examples, we consider two up-and-out barrier options with common parameter values: σ =0.3, r =0.1, S(0) = 100. The first example has barrier level H = 200 and exercise K = 150, while the second example has H = 160 and K = 110. Tables 3.3 and 3.4 compare the estimated variances of fifty option estimates using the crude Monte Carlo (MC) estimator and the importance sampling (ImpSamp-h) Monte Carlo estimator using heuristics at three sample sizes: N = 5000, 10000, and 50000. The last row of the tables, labeled “Positive Payoff”, reports the percentage of stock price paths that yield positive payoffs, that is, paths that land in the band bounded simultaneously below by the exercise price and above by the barrier level at expiration. The widths of the bands are the same in both examples. The higher percentage of paths landing in the band in ImpSamp-h than in MC for both tables indicates that importance sampling indeed gives more importance to the band than the crude Monte Carlo does. Table 3.3 indicates an improvement in the variance estimate by at most a factor of 10+2 when we use importance sampling, while Table 3.4 indicates an improvement by a factor of less than 10. The greater improvement by ImpSamp-h over MC occurs when the band is farther away from the initial stock price. Furthermore, given our assumptions of the model, the tables suggest that the lower the probability of paths landing in the band in the crude Monte Carlo case, the bigger the improvement is in the variance estimates using importance sampling. A 0.16% positive payoff in MC yields a 10+2

48 Table 3.3: Estimated relative MSE of crude Monte Carlo (MC) estimates and importance sampling (ImpSamp-h) Monte Carlo heuristic estimates, and ratios of errors for an up-and- out barrier option (σ =0.3, r =0.1, S(0) = 100, K = 150, T =0.2, m = 50, H = 200 and estimated true price = 0.00981841855492).

N MC ImpSamp-h MC/ImpSamp-h 4 +2 5K 0.2810 7.0672 10− 3.97 10 × 4 × +2 10K 0.1238 5.2031 10− 2.38 10 × 4 × +1 50K 0.0242 4.0396 10− 5.99 10 × × Positive Payoﬀ 0.16% 71% n/a

Table 3.4: Estimated relative MSE of crude Monte Carlo (MC) estimates and importance sampling (ImpSamp-h) Monte Carlo heuristic estimates, and ratios of errors for an up-and- out barrier option (σ =0.3, r =0.1, S(0) = 100, K = 110, T =0.2, m = 50, H = 160 and estimated true price = 2.46413112486174).

N MC ImpSamp-h MC/ImpSamp-h 4 4 0 5K 9.5519 10− 1.7883 10− 5.34 10 × 4 × 4 × 0 10K 5.0906 10− 1.0047 10− 5.06 10 × 4 × 5 × 0 50K 1.0123 10− 2.1188 10− 4.78 10 × × × Positive Payoﬀ 26% 82% n/a

improvement in ImpSamp-h, while a 26% positive payoﬀ in MC yields only an approximately 5 100 improvement in ImpSamp-h. × Figure 3.1 shows the ﬁfty price estimates for both MC and ImpSamp-h whose variances are reported in Table 3.3 for N = 10000. 3.2 Importance Sampling Monte Carlo Estimator and Simulated Annealing

Simulated annealing algorithms try to solve optimization problems, minx S f(x), over a ∈ discrete feasible set by mimicking the “annealing process” in which a metal cools and freezes into a minimum energy crystalline structure. A typical simulated annealing algorithm uses two techniques in ﬁnding the optimal solution. First, along the initial course of the search,

49 Figure 3.1: Fifty up-and-out barrier option price estimates using crude Monte Carlo (MC) and importance sampling (ImpSamp-h) Monte Carlo heuristics. Each estimate uses 10000 simulations (σ =0.3, r =0.1, S(0) = 100, K = 150, T =0.2, m = 50, and H = 200).

the algorithm may accept some non-optimal solutions with some probability in order to explore even more the possible space of solutions. This feature allows the search to escape from local optimal solutions. Second, as in the annealing process, the algorithm lowers the “temperature” to limit non-optimal solutions, eventually capturing the local optimum solution among the good solutions. We introduce the simulated annealing technique as an alternative method in estimating tilting parameters in the importance sampling approach. The materials in this section are taken largely from Alrefaei and Andrad´ottir [1].

3.2.1 Principles Simulated Annealing

For a deterministic objective function f(x) defined over a finite discrete feasible set S and possibly having many local minima, one can use the standard simulated annealing algorithm to minimize f(x) to yield a local minimum. We now state some assumptions and definitions before outlining the algorithm:

Deﬁnition 3.2.1. (Existence of Neighbor System) For each x S, there exists a subset ∈ 50 N(x) of S x, which is called the set of neighbors of x. \

Assumption 1. (Travel Between Feasible Points) For any x, x S, the point x′ is reachable ∈ from the point x, that is, there exists a ﬁnite sequence, n l for some l, such that one can { i}i=0 travel from point x to point x′ by jumping from neighbor to neighbor:

x = x, x = x′, x N(x ), i =0, 1, 2, ...l 1. n0 nl ni+1 ∈ ni − Deﬁnition 3.2.2. (Transition Probability on Neighbor System) Let R : S S [0, 1] be × −→ a nonnegative function that satisﬁes:

(1) R(x, x′) > 0 x′ N(x), and ⇐⇒ ∈ (2) x′ S R(x, x′)=1. ∈ Then RP(x, x′) is called the probability of generating the neighbor x′ from the point x.

Assumption 2. (Symmetry) The neighbor system N(x): x S and the transition { ∈ } probability function R are symmetric, that is,

(1) x′ N(x) x N(x), and ∈ ⇐⇒ ∈ (2) R(x, x′)= R(x′, x).

Now let T be a sequence of positive scalars representing the cooling (temperature) { m} schedule.

Algorithm 1 Select a starting point X S. 0 ∈ 1: Given X = x, choose a candidate Z N(x) with probability distribution m m ∈ P [Z = z X = x]= R(x, z), m | m where N(x) and R (x, z) are deﬁned as above. 2: Given Z = z, generate a uniform random variable U U[0, 1], and set m m ∼ z if U p X = m m m+1 x otherwise,≤ where [f(z) f(x)]+ p = exp − − m T m and [a]+ = max a, 0 . { } 51 3: Let m = m + 1. Go to Step 1.

Observe in Step 2 that the escape probability p is determined by the distance f(z) f(x) m − and the temperature Tm. If the neighbor z is better than the current state x, that is f(z) f(x), then the next state of the Markov chain is chosen as z. If f(z) > f(x), then ≤ the Markov chain moves to state z (aptly called a hill-climbing move) with small escape [f(z) f(x)]+ probability exp − − . Also, the higher the temperature T , the closer p is to 1 Tm m m giving a higher escapeh probability.i Conversely, the cooler the temperature, the less likely the Markov chain changes states. Typically, the last feasible point that is visited is the estimate for the optimization problem.

Modiﬁed Simulated Annealing

In some optimization problems, the objective function may be diﬃcult to evaluate or may include noise so that its values need to be estimated or simulated. For example, say one wants to solve the following problem:

min E[h(x, Yx)], x S ∈ where Yx is a random variable that depends on the parameter x and h is a deterministic function. If i.i.d. observations Y (1), Y (2), ..., Y (n) can be generated for all x S for ﬁnite x x x ∈ feasible discrete set S, one can approximate the above minimization problem by

min hn(x), x S ∈ where h (x)= 1 n h(x, Y (i)) for all x S. n n i=1 x ∈ Alrefaei and AndradóttirP [1] introduce a modified simulated annealing algorithm to solve this minimization problem. Their algorithm differs from the typical simulated annealing algorithm in two respects: (1) the modified algorithm uses a constant cooling temperature schedule instead of a decreasing one, and (2) the estimated optimal solution is the state that yields the smallest sample mean hn(x) instead of the last visited state. In addition to the above definitions and assumptions, they include the following assumption:

Assumption 3. The temperature T is a constant positive real number. In addition, we let K be an unbounded sequence of positive integers. { m}

52 Algorithm 2 Select a starting point X S. 0 ∈ 1: Given X = x, choose a candidate Z N(x) with probability distribution m m ∈ P [Z = z X = x]= R(x, z), m | m where N(x) and R (x, z) are deﬁned as above.

2: Given Zm = z, generate i.i.d. observations Yz(1), Yz(2), ..., Yz(Km) of Yz and Y (1), Y (2), ..., Y (K ) of Y . Evaluate h (x) and h (z) where h (s)= 1 Km h(s, Y (i)) x x x m x m m m Km i=1 s for s = x, z. P

3: Given Z = z, generate U U[0, 1], and set m m ∼ z if U p X = m m , m+1 x otherwise≤ where + [hm(z) hm(x)] pm = exp − − . " T # 4: Let m = m + 1. Go to Step 2.

The state x S that yields the smallest sample mean h (x) is the solution to the ∈ n minimization problem.

Applications

Recall the down-and-in barrier option examples in Section 3.1.2. After applying the importance sampling principle in our price estimating procedure, we arrive at Step 4 of + Section 3.1.2 where the tilting parameters t− and t of the likelihood function are estimated using heuristics. We now introduce a variant of this step which we shall call STEP 4’:

+ STEP 4’. Determine the values of the tilting parameters t−and t that increase the probabilities of the important events in Step 2. We do so using the modiﬁed simulated annealing procedure of Alrefaei and Andrad´ottir as follows.

Our objective is to ﬁnd the values of the tilting parameters that minimize the simulation + error. The parameter x corresponds to the tilting parameters (t−,t ) in importance

53 sampling. As before, we will use the option prices computed by Tse et al. [32] as the true solutions. The function h computes the relative square error of the option price estimate for a fixed sample size. In other words, if Yx(1), ..., Yx(Km) are option price estimates each of which is computed using N stock price paths then we would like to minimize the objective function Y (i) True 2 E[h(x, Y (i))] = E x − x True " # that is estimated by 1 Km Y (i) True 2 h (s)= x − m K True m i=1 X + at the mth iteration. We pick the tilting parameters (t−,t ) that yield the smallest mean hm(s) to be our estimated simulated annealing tilting parameters. One way of evaluating the quality of the tilting parameters obtained from heuristic (3.1.11) and of those derived from simulated annealing is by locating their positions on the stochastic mean square error surface. + We first introduce new variables (d, u) that are related to the tilting parameters (t−,t ) as follows: 1 2 2 1 2 2 + d = r σ h + σ ht− and u = r σ h + σ ht . − 2 − 2 We introduce these variables for their intuitive meaning: d and u are the means of the tilted densities ft− (x) and ft+ (x), respectively. (See Equation (3.1.4). Here, the variable h represents the length of the discretized time step.) The quantity r 1 σ2 h − 2 2 is the mean of the original density f(x). Clearly, one expects σ ht− to be negative and 1 2 2 r σ h + σ ht− < 0 so that the stock price is “pulled down” to the barrier faster than − 2 before. Conversely, one expects σ2ht+ to be positive and r 1 σ2 h + σ2ht+ > 0 so that − 2 the stock price is “pushed up” to the exercise price faster than before. For the parameters of Table 3.1, the heuristic gives u =0.0021 and d = u. We want to − measure the proximity of the heuristic point ( 0.0021, 0.0021) to the simulated annealing − point at which the mean square error (MSE) of the option estimate attains its minimum. This will be done numerically by using Monte Carlo simulation. Consider the grid defined by the points (d, u) [ 0.008, 0] [0, 0.008] with an increment ∈ − × 4 size of 10− . For each point (d, u) on the grid, we run a Monte Carlo simulation using the importance sampling estimator and compute the MSE of fifty option price estimates where

54 Figure 3.2: MSE surface plot. Included are the importance sampling heuristic point h and the importance sampling simulated annealing point SA. Each point on the surface is the MSE of ﬁfty price estimates each estimated using 10000 price paths (σ = 0.3, r = 0.1, S(0) = 100, K = 150, T =0.2, m = 50, H = 95, and true price = 1.4373238784).

each estimate is based on 10000 stock price paths. We use the option parameters of Table 3.1 and use the option price obtained from Tse et al. [32] as the true value in the MSE calculations. Figure 3.2 plots the surface of MSE values over this grid. 4 The point labeled by “h” on the surface has the MSE value 4.67 10− and corresponds to × the heuristic point ( 0.0021, 0.0021). The simulated annealing point, on the other hand, is − 4 located at ( 0.004, 0.005) and has the MSE value 3.18 10− . This point is labeled by “SA” − × 4 on the surface. The minimum MSE on the surface reads 0.5 10− . Simulated annealing × gives better tilting parameters than the heuristic in this problem. We will next present additional numerical results to investigate potential beneﬁts of simulated annealing. Table 3.5 reports the relative mean square errors of the crude Monte Carlo (MC)

55 estimator, importance sampling Monte Carlo estimator using heuristics (ImpSamp-h) from Chapter 3.1 and importance sampling Monte Carlo estimator using simulated annealing (ImpSamp-SA) for the down-and-in barrier option with parameter values σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784. Table 3.6 reports the relative mean square errors of the estimators for the barrier option with parameter values σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 91, and true price = 0.3670447223. Each relative mean square error result is computed using fifty option price estimates for stock price paths of size N = 5000, 10000, and 50000. The simulated annealing-based estimators give less error than the heuristic-based estimators when N = 5K, 10K. The opposite is true for N = 50K. The differences however are very small. Together with the simplicity of computing the heuristic parameters, these results suggest that the heuristic is an effective strategy in computing tilting parameters. 3.3 Combined Conditional Expectation and Importance Sampling Monte Carlo Estimator

3.3.1 Principles

We begin with the importance sampling estimator of the parameter θ : f(X) θ = E [h(X)] = E h(X) . f g g(X) f(X) Consider the conditional expectation Z = E h(X) g(X) Y = H(Y ), where the function h i H(Y ) indicates that the conditional expectation is a function of the random variable Y .

This also means that we compute the conditional expectation above according to the density of Y . We will call the random variable Z the combined conditional expectation and importance sampling estimator. Taking the sample mean of Z, we get the combined conditional expectation and importance sampling Monte Carlo estimator:

f(X) 1 N Z = E h(X) Y = H(Y ). g(X) N j j=1 X

Note that neither the density f nor the density g is directly involved in our sampling procedure for Z. Because the conditional expectation Z is a function of the random variable Y , our sample mean is governed by the distribution of Y . That is, we draw N samples from

56 Table 3.5: Relative mean square errors of crude Monte Carlo (MC) estimates, importance sampling (ImpSamp-h) Monte Carlo heuristics, and importance sampling (ImpSamp-SA) Monte Carlo simulated annealing estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784). 57 N MC ImpSamp-h MC/ImpSamp-h ImpSamp-SA MC/ImpSamp-SA 3 4 0 4 0 5K 1.66 10− 3.33 10− 4.98 10 2.85 10− 5.82 10 × 4 × 4 × 0 × 4 × 0 10K 9.73 10− 1.64 10− 5.95 10 1.55 10− 6.28 10 × 4 × 5 × 0 × 5 × 0 50K 2.00 10− 4.29 10− 4.66 10 9.99 10− 2.00 10 × × × × × Cross Barrier 62% 84% n/a 95% n/a Cross Exercise 29% 43% n/a 79% n/a Table 3.6: Relative mean square errors of crude Monte Carlo (MC) estimates, importance sampling (ImpSamp-h) Monte Carlo heuristics, and importance sampling (ImpSamp-SA) Monte Carlo simulated annealing estimates, and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 91, and true price = 0.3670447223). 58 N MC ImpSamp-h MC/ImpSamp-h ImpSamp-SA MC/ImpSamp-SA 3 4 +1 4 +1 5K 3.77 10− 3.56 10− 1.06 10 2.58 10− 1.46 10 × 3 × 4 × +1 × 4 × +1 10K 2.60 10− 1.74 10− 1.49 10 1.46 10− 1.78 10 × 4 × 5 × +1 × 5 × 0 50K 5.47 10− 4.90 10− 1.12 10 9.70 10− 5.64 10 × × × × × Cross Barrier 41% 86% n/a 97% n/a Cross Exercise 15% 42% n/a 77% n/a the distribution of Y , evaluate H(Yj) at each sample Yj, and take their average to get the combined conditional expectation and importance sampling Monte Carlo estimate. By the law of total expectation, we see that for any random variable Y , f(X) θ = E h(X) g g(X) f(X) = E E h(X) Y g(X)

= E [Z] .

In other words, the estimator Z is an unbiased estimator of the parameter θ. By the unbiasedness of the sample mean estimator, it follows then that the combined conditional expectation and importance sampling Monte Carlo estimator Z is also an unbiased estimator of the parameter θ. Now, from the variance decomposition formula Equation (2.1.1), it follows that

1 f(X) 1 f(X) V ar[Z]= V ar E h(X) Y V ar h(X) , N g(X) ≤ N g g(X) where V ar [ ] indicates that the expectation in computing the variance uses the density g. g · f(X) So by transitivity, if the importance sampling estimator h(X) g(X) decreases the variance of the estimator h(X), then the estimator Z decreases the variance of the estimator h(X). See the variance comparison of Equations (3.1.2) and (3.1.3). Consequently, we get V ar[Z] V ar [h(X)], where h(X) is the crude Monte Carlo estimator. ≤ f 3.3.2 Applications

Ross and Shanthikumar [27] observe that the conditional expectation and importance sampling methods can be combined to estimate the price of a down-and-in barrier option. Importance sampling can be applied to the stock price path until the path crosses the barrier level. When the barrier level is crossed, the conditional expectation can be used to compute the option price, that is, we use the Black-Scholes-Merton formula for the price of a call option to compute the price at the hitting time tτ of the European call option that has just come alive. For our example, we modify slightly the down-and-in barrier option example using

59 importance sampling in Section 3.1 by incorporating the conditional expectation to yield

rT + θ = E e− 1(t

τ f(X(t )) M(t)τ M(t)τ i = = . f (X(t )) exp(t(X(t )+ X(t )+ ... + X(t )) exp(tL ) i=1 t i 1 2 τ τ Y The combined conditional expectation and importance sampling Monte Carlo estimator simulates j = 1, 2, ..., N stock price paths using the tilted density ft(x) from the initial time until they hit the barrier level, stops the simulation to use the Black-Scholes-Merton formula, and averages over the values

τ rtτ M(t) e− BSM(S(tτ ),tτ ,T,K) if tτ

S(0) K 1 2 log H + log S(0) b = r σ2 h + . (3.3.1) − 2 m This formula was erroneously reported as b = rh [2 log (S(0)/H) + log (K/S(0))] /m in − Ross and Shanthikumar [27] (p. 321), where h is replaced by 1/N and m is replaced by

60 n. An alternative estimate for the tilting parameter can be found by using the method of simulated annealing in Section 3.2. We recall the two down-and-in barrier options that we considered in Section 2.1 (Condi- tional Expectation Monte Carlo Estimator), Section 3.1 (Importance Sampling Monte Carlo Estimator and Heuristics), and Section 3.2 (Importance Sampling Monte Carlo Estimator and Simulated Annealing). These options have the following common parameters: σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, and m = 50. The first barrier option has barrier level H = 95 and true price 1.4373238784. The second barrier option has barrier level H = 91 and true price 0.3670447223. We first compare the performance of the conditional expectation Monte Carlo (CondExp) estimator, the importance sampling Monte Carlo heuristic estimator (ImpSamp-h), and the combined conditional expectation and importance sampling Monte Carlo heuristic estimator (Combined-h). Table 3.7 and Table 3.8 report the relative mean square error of fifty option estimates using the three estimators for the first barrier option and second barrier option, respectively. Three sample sizes are considered: N = 5000, 10000, and 50000. The fifth row of the tables, labeled “Cross Barrier” gives the percentage of price paths that cross the barrier level, and the next row labeled “Cross Exercise” displays the percentage of paths that cross the exercise price among those that crossed the barrier. Our results show the Combined-h estimator gives the lowest relative mean square error among all estimators. In the next two tables, we compare the estimators Combined-h and Combined-SA by the relative mean square error of the fifty option estimates that they produce. Table 3.9 and Table 3.10 use the option parameters of Table 3.7 and Table 3.8, respectively. The simulated annealing-based estimator gives less error than the heuristic-based estimator when N = 5000, 10000. The opposite is true for N = 50000. The differences are very small. Thus, these results suggest that the heuristic approach, with its simplicity, is the more effective strategy in computing the tilting parameters. The last table in this chapter, Table 3.11, shows the computation time (in seconds) of different estimators for estimating the price of a down-and-in barrier option using 1000 sample stock price paths. The results are based on codes written in MATLAB R Version

7.0.4.365 (R14) Service Pack 2 running on Intel R CoreT M 2 CPU T7200 at 2.00GHz. We follow the parameter values on Table 3.7. The ratio of the computing time of the slowest estimator to the fastest estimator is 1.08. Therefore, the improvements in the relative mean

61 Table 3.7: Relative mean square errors of conditional expectation Monte Carlo (CondExp), importance sampling Monte Carlo heuristics (ImpSamp-h), and combined conditional expectation and importance sampling Monte Carlo heuristics (Combined-h) for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784). 62 N CondExp MC/CondExp ImpSamp-h MC/ImpSamp-h Combined-h MC/Combined-h Parameters n/a n/a u =0.0021; d = u n/a b =0.0023 n/a 4 +1 4 − 0 5 +1 5K 1.36 10− 1.22 10 3.33 10− 4.98 10 6.66 10− 2.49 10 × 5 × +1 × 4 × 0 × 5 × +1 10K 8.40 10− 1.16 10 1.64 10− 5.93 10 3.48 10− 2.80 10 × 5 × +1 × 5 × 0 × 5 × +1 50K 1.65 10− 1.21 10 4.29 10− 4.66 10 1.25 10− 1.60 10 × × × × × × Cross Barrier 61% n/a 84% n/a 84% n/a Cross Exercise n/a n/a 43% n/a n/a n/a Table 3.8: Relative mean square errors of conditional expectation Monte Carlo (CondExp), importance sampling Monte Carlo heuristics (ImpSamp-h), and combined conditional expectation and importance sampling Monte Carlo heuristics (Combined-h) for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223). 63 N CondExp MC/CondExp ImpSamp-h MC/ImpSamp-h Combined-h MC/Combined-h Parameters n/a n/a u =0.0038; d = u n/a b =0.0040 n/a 4 0 4 − +1 5 +1 5K 4.66 10− 8.09 10 3.56 10− 1.06 10 9.78 10− 2.66 10 × 4 × 0 × 4 × +1 × 5 × +1 10K 4.17 10− 6.24 10 1.74 10− 1.49 10 5.11 10− 5.09 10 × 5 × 0 × 5 × +1 × 5 × +1 50K 6.98 10− 7.84 10 4.90 10− 1.12 10 1.66 10− 3.30 10 × × × × × × Cross Barrier 40% n/a 86% n/a 86% n/a Cross Exercise n/a n/a 42% n/a n/a n/a Table 3.9: Relative mean square errors of crude Monte Carlo estimator, combined conditional expectation and importance sampling Monte Carlo estimator using heuristics (Combined-h) and simulated annealing (Combined-SA), and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 95, and true price = 1.4373238784). 64 N MC Combined-h MC/Combined-h Combined-SA MC/Combined-SA Parameters n/a b =0.0023 n/a b =0.0055 n/a 3 5 +1 5 +1 5K 1.66 10− 6.66 10− 2.49 10 3.34 10− 4.97 10 × 4 × 5 × +1 × 5 × +1 10K 9.73 10− 3.48 10− 2.80 10 2.63 10− 3.70 10 × 4 × 5 × +1 × 5 × +1 50K 2.00 10− 1.25 10− 1.60 10 1.44 10− 1.39 10 × × × × × Cross Barrier 62% 84% n/a 98% n/a Cross Exercise 29% n/a n/a n/a n/a Table 3.10: Relative mean square errors of crude Monte Carlo estimator (MC), combined conditional expectation and importance sampling Monte Carlo estimators using heuristics (Combined-h) and simulated annealing (Combined-SA), and ratios of errors for a down-and-in barrier option (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T = 0.2, m = 50, H = 91, and true price = 0.3670447223). 65 N MC Combined-h MC/Combined-h Combined-SA MC/Combined-SA Parameters n/a b =0.0040 n/a b =0.0065 n/a 3 5 +1 5 +1 5K 3.77 10− 9.78 10− 2.66 10 4.35 10− 8.67 10 × 3 × 5 × +1 × 5 × +1 10K 2.60 10− 5.11 10− 5.09 10 2.83 10− 9.19 10 × 4 × 5 × +1 × 5 × +1 50K 5.47 10− 1.66 10− 3.30 10 1.76 10− 3.11 10 × × × × × Cross Barrier 41% 86% n/a 98% n/a Cross Exercise 15% n/a n/a n/a n/a Table 3.11: Comparison of computation times of MC estimator, CondExp estimator, ImptSamp-h estimator, and Combined-h estimator for a down-and-in barrier option using 1000 sample price paths (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95).

Estimator Time (in secs.) MC 1.2044 CondExp 1.2982 ImpSamp-h 1.2606 Combined-h 1.2873

square errors that we observed in this chapter, which were roughly between factors of 4 and 20, imply the same ratios of improvement in the eﬃciency, deﬁned as the product of error and computing time.

66 CHAPTER 4

Equivalent Estimators

In this chapter, we introduce Ross and Shanthikumar’s (RS) raw simulation estimator for barrier options and its variant from their paper [27]. We shall numerically test the authors’ claim that the variant estimator ε1 is a better estimator than their combined conditional expectation and importance sampling estimator. 4.1 Ross and Shanthikumar’s Estimators

4.1.1 Simulation of the Raw Simulation Estimator

In all of the previous chapters, we simulated a stock price path that follows a lognormal distribution, in particular the geometric Brownian motion GBM(r, σ2), by generating a series of stock prices determined by the following equation 1 S(t )= S(t ) exp r σ2 (t t )+ σ t t Z . i+1 i − 2 i+1 − i i+1 − i i+1 p In this equation, we ﬁnd S(ti+1) by generating a standard normal random variable Zi+1, perhaps by transforming a uniform U(0, 1) random variable using the Box-Muller algorithm or the Moro algorithm. If at any time step ti before the expiration, a stock price S(ti) crosses the barrier level, we continue the simulation until the expiration and compute the discounted payoﬀ to give a price estimate. Otherwise, the price estimate is nil. On the other hand, the RS raw simulation estimator does not depend on the single random variable Zi+1 directly, but rather on the sums of the steps of the random walk. We outline the steps of the simulation as follows.

STEP 1. Choose a model for the underlying stock. We model the stock price as

67 lognormal in the form n

STEP 2. Generate a ﬁnite sequence of good states and bad states that comprise the ﬁnite state space of the Markov chain. We generate the sequence

B(t1), G(t1),B(t2), G(t2), ..., B(tm), G(tm) as follows (See steps 1 and 2 on p. 322, [27].): Assume B(t0) = 0, that is the initial stock price S(0) > H. Now assume at time ti that B(ti)= u. We do the following:

Step 2.1. Generate G(ti+1) by letting its distribution be the conditional distribution of the next state of the Markov chain given that the present state is Li = u and given that this next state is in G, that is L G. i+1 ∈ 68 Step 2.2. Generate B(ti+1) by letting its distribution be the conditional distribution of the next state of the Markov chain given that the present state is Li = u and given that this next state is in B, that is L B. i+1 ∈

In order to perform the above steps, observe that given Li = u, we have

Li+1 = Li + X(ti+1)= u + X(ti+1), where X(t ) r 1 σ2 h,σ2h . Thus, we get i+1 ∼ N − 2 1 L r σ2 h + u, σ2h . (4.1.4) i+1 ∼N − 2 Thus, from condition (4.1.2), simulating a state such that L G given L = u is equivalent i+1 ∈ i to simulating a normal random variable from the distribution (4.1.4) restricted to the interval ( , ln (H/S(0))). Similarly, simulating a state such that L B given L = u is −∞ i+1 ∈ i equivalent to simulating a normal random variable from the distribution (4.1.4) restricted to the interval (ln (H/S(0)) , + ) . In other words, Steps 2.1 and 2.2 simulate states from a ∞ truncated normal distribution. We suggest two methods in simulating a normal random variable from the interval (a, b), where a and b are either ln (H/S(0)) or approach and + , respectively. −∞ ∞ Let f(x) be the density function and F (x) be the cumulative distribution function of r 1 σ2 h + u, σ2h . The ﬁrst method requires the inverse of the cumulative distri- N − 2 bution function. Codes for the cumulative distribution function and its inverse for a normal random variable may be found, for example, in Burkardt [7].

Algorithm 1 Inverse Method for Truncated Distribution. Generate U U(0, 1). Let V = F (a∼) + [F (b) F (a)] U. 1 − · Return X = F − (V ).

The second method is an accept-reject method. We ﬁrst let

f(x) F (b) F (a) , a x b f ∗(x)= − ≤ ≤ . ( 0, otherwise

69 Algorithm 2 Accept-Reject Method for Truncated Distribution. Generate X with pdf f(x). 1 Generate Y U 0, F (a) F (b) f(X) . ∼ − If Y f ∗(X), thenh return X; otherwisei go to the ﬁrst step. ≤

For discussions on simulating from a truncated normal distribution, readers are referred to Bratley [4], Forsythe [14], Marsaglia [23], Robert [24], and Schmeiser [29].

STEP 3. Given the chain L , L , ..., L B with L = u, generate a random variable 0 1 i ∈ i J(ti+1) according to the following distribution 1, with probability P (L G L = u) J(t )= i+1 i . i+1 0, with probability P (L ∈ B|L = u) i+1 ∈ | i These probabilities are computed as follows: H P (L G L = u)= P L < ln L = u i+1 ∈ | i i+1 S(0) i H = P u + X(t ) < ln i+1 S(0) H = P X(t ) < ln u i+1 S(0) − H = F ln u , (4.1.5) S(0) − where F ( ) represents the cumulative distribution function of X(t ) r 1 σ2 h,σ2h ; · i+1 ∼N − 2 and

P (L B L = u)=1 P (L G L = u). (4.1.6) i+1 ∈ | i − i+1 ∈ | i We repeat Step 3 until the Markov chain reaches a good state or the (m + 1)th state.

Algorithm 3 Generating J(ti+1). Generate U U(0, 1). ∼ If U F ln H u then J(t ) = 1 and let L = G(t ). ≤ S(0) − i+1 t+1 i+1 Else J(ti+1) = 0 and letLt+1 = B(ti+1).

STEP 4. Simulate the raw simulation estimator by generating the successive states of the Markov chain, stopping either when the chain reaches the good state G or after m states

70 have been generated. Let

τ = min m +1, min i : J(t )=1 . { { i }

If τ < m then the states are B(t1),B(t2), ..., B(tτ 1), G(tτ ). If τ m, then the states are − ≥ B(t1),B(t2), ..., B(tm 1),B(tm). A realization of the RS raw simulation estimator ε is given − by h(B(t1), ..., B(tτ 1), G(tτ )), if tτ

θ = E[ε]= E [1(tτ

The Monte Carlo estimator associated with the RS raw simulation estimator ε for a down- and-in barrier option is

N 1 j j ε = 1(tτ

4.1.2 A Variant of the Raw Simulation Estimator

Let M = 1, 2, ..., m . We have, { } m

E [X1Y M ] = E [X Y = j] P (Y = j) ∈ | j=1 X = E [X Y = 1] P (Y = 1)+ E [X Y = 2] P (Y = 2)+ ... + | | E [X Y = m] P (Y = m), | 1I am slightly altering their notation for consistency in the notation used in this dissertation. Also, the term I(T = n + 1)G(U1,...,Un) in [27] is ignored since this term represents the event that the underlying stock price hits the barrier level after the expiration. The option is dead at that time, hence the discounted payoﬀ G(U1,...,Un) is equal to zero.

71 and conditioning this expectation with any random variable V , we get m

E [X1Y M V ] = E [X V, Y = j] P (Y = j V ) ∈ | | | j=1 X = E [X V, Y = 1] P (Y =1 V )+ E [X V, Y = 2] P (Y =2 V )+ ... + | | | | E [X V, Y = m] P (Y = m V ). | | Consider realizations of the random vector of good states and bad states V ≡ (B(t1), G(t1), ..., B(tm), G(tm)) of the Markov chain in the RS raw simulation estimator ε. Let us then condition the raw estimator ε with respect to the random vector V and take its expectation to get the new estimator ε = E [ε V ] with the same mean as the mean of ε and 1 | possibly with a lower variance. Using the properties of the conditional expectation above, the estimator ε1 can be expanded as follows:

ε = E [ε V ] 1 | = E [I(τ m)h(B(t1), ..., B(tτ 1), G(tτ )) V ] − m ≤ |

= E [h(B(t1), ..., B(tτ 1), G(tτ )) V, τ = j] − | j=1 X = E[h(G(t )) V )]P (τ =1 V )+ E[h(B(t ), G(t )) V ]P (τ =2 V )+ ...+ 1 | | 1 2 | | E[h(B(t ),B(t ), ..., G(t )) V ]P (τ = m V ) 1 2 m | | = h(G(t1))p(L0)+ h(B(t1), G(t2))q(L0)p(L1)+ ...+

h(B(t1),B(t2), ..., B(tm 1), G(tm))q(L0)q(L1)...q(Lm 2)p(Lm 1), (4.1.8) − − − where p(L )= P (L G L = u) and q(L )=1 p(L ), i i+1 ∈ | i i − i from Equations (4.1.5) and (4.1.6). Observe that we derive the last equality in the above expansion of ε1 by “taking out what is known” in conditional expectation and expanding the probability weights P (τ = j V ). | 4.2 Glasserman and Staum’s Estimators

In this section, we introduce Glasserman and Staum’s (GS) barrier option estimators conditional on one-step survival from their paper [16]. We shall show the GS estimators are similar to the other estimators we have discussed in the previous chapters. This similarity allows us to verify the properties of the RS estimators.

72 Table 4.1: Glasserman and Staum’s estimators for knock-out barrier options.

Estimators Without rebate With rebate m Standard Xm = Amf(S(tm)) Xm = Amf(S(tm)) + i=1(Ai 1 Ai)g(S(ti 1)) m − − − Exact Xm = mf(S(tm)) Xm = mf(S(tm)) + i 1(1 p(Si 1))g(S(ti 1)) L L i=1PL − − − − b b P 4.2.1 Knock-Out Estimators

Glasserman and Staum’s estimators begin with estimators for knock-out barrier options. For example, a down-and-out barrier option is a type of knock-out barrier option whose barrier level is set below the initial stock price, starts as an ordinary European call option, and dies if the stock price hits the barrier level at any time before the expiration. The holder of the down-and-out barrier option may receive a rebate if the option is knocked out. Table 4.1 summarizes the GS knock-out estimators, both the standard estimator and the exact estimator with full importance sampling. The random variable Am is the indicator function for the event that the option remains alive until the expiration time tm, that is, Am = 1 if the stock price has not reached the barrier at expiration. The function f(S(tm)) represents the discounted payoﬀ of the option. The quantity represents the likelihood that the option Li survives at time ti:

i 1 − = p(S(t )), where p(S(t )) = P (S(t ) = ∆ S(t )= s). Li j j j+1 6 | j j=0 Y The event S(t ) = ∆ means that the stock price at time t has not been absorbed by j+1 6 j+1 the absorbing state ∆. For a down-and-out barrier option with barrier level H, this event is equivalent to S(tj+1) > H. If the option writer agrees to compensate the knock-out option holder with a rebate in case the option dies before expiration, we modify the estimators using the rebate function g(S(ti 1)). For example, if the barrier is knocked out at time ti, the function g(S(ti 1)) − − represents the expected present value of the rebate at time ti. The term (1 p(Si 1)) − − represents the probability P (S(ti) = ∆ S(ti 1)= s), that is, the probability that the option | − is knocked out at time ti given a stock price s at the previous time step.

73 Table 4.2: Glasserman and Staum’s estimators for knock-in barrier options.

Estimators With “rebate” m Standard Xm = i=1(Ai 1 Ai)fi(S(ti)) m − − Exact Xm = i 1(1 p(S(ti 1)))fi(S∗(ti)) i=1PL − − − b P 4.2.2 Knock-In Estimators

Since a knock-in barrier option becomes alive as a European call option at the time it hits the barrier level H, a knock-in barrier option may be viewed as a knock-out barrier option with rebate. Table 4.2 summarizes the GS estimators for a knock-in barrier option, both the standard estimator and the exact estimator with full importance sampling. The “rebate” at time ti is represented by the function fi(S(ti)), the discounted value of the European option originating at time ti with payoﬀ f(Sm), when S(ti) < H. The use of the stock price S∗(ti) indicates a price that has hit the barrier level. We note the similarity of these GS knock-in estimators with the other estimators we have already discussed. Taking the expectation of the GS standard estimator yields the conditional expectation estimator in Equation (2.1.3). The RS ε1 estimator in (4.1.8) and the GS exact estimator with importance sampling are similar in form except that the RS

S(ti) estimator “rebate” is a function of Li = ln S(0) while the GS estimator “rebate” is a function of S(ti). Thus, statistical properties that hold true for the GS estimators must also hold true for the RS estimators. 4.3 Properties of the RS Estimators

4.3.1 Properties of the RS Estimator ε

As we have mentioned at the beginning of this chapter, the RS estimator ε and the conditional expectation estimator for pricing a down-and-in barrier option are equivalent. Thus we expect the RS estimator ε to be unbiased:

rT + θ = E e− 1(t

74 The Monte Carlo estimator ε corresponding to the RS estimator ε will also be unbiased:

1 N θ = E[ε]= E ε . N j " j=1 # X

4.3.2 Properties of the RS Estimator ε1

Since the RS estimator ε is deﬁned as the conditional expectation E[ε V ], where V is a 1 | random vector of good states and bad states, we can immediately invoke the properties of the conditional expectation estimator from Section 2.1 and conclude the following properties.

The RS estimator ε1 is an unbiased estimator for the price of the down-and-in barrier option, that is:

θ = E[ε1].

The variance of the RS estimator ε1 is at most equal to the variance of the RS raw simulation estimator ε, that is: V ar [ε ] V ar [ε] . 1 ≤

The Monte Carlo estimator ε1 corresponding to the RS estimator ε1 will also be unbiased and possibly have a lower variance than that of the Monte Carlo estimator ε, that is:

θ = E[ε1] and V ar [ε ] V ar[ε]. 1 ≤

A finer approach in proving the unbiasedness of the RS estimator ε1 is by indirectly proving the unbiasedness of its equivalent GS knock-in exact estimator with full importance sampling Xm in Table 4.2. This approach is justified since transforming the underlying sigma-algebra i in Glasserman and Staum [16] from one that is generated by stock price b F prices (S(t1), ..., S(ti)) to one that is generated by the logarithm of stock prices (L1, ..., Li), where L = log S(ti) , qualifies the lemmas and theorems in their paper not only for their i S(t0) discounted payoff function f(S(t )) but also for other -measurable functions, in particular, m Fi Ross and Shanthikumar’s discounted payoff function h (B(t1), ..., B(tτ ), G(tτ )) in Equation (4.1.7). We need the following information from Glasserman and Staum [16]. First, the authors define the appropriate probability spaces that were used in their proofs. For the standard

75 (crude) Monte Carlo simulation, the authors let P be the probability under which the underlying state vector process (S(t1), ..., S(tm)) has its usual joint distribution on a universe called Ω. The simulated price vectors obey the probability P. The corresponding sigma- algebra is the sigma-algebra generated by the state vector process (S(t ), ..., S(t )). On Fm 1 m the other hand, consider the subset Ω of price vectors for which the indicator function Ai ⊂ Ai = 1, that is, where the option is alive at time ti or the stock price has not crossed the barrier at time t . Let be the sigma-algebra generated by (S(t ), ..., S(t )). The authors i Fi 1 i deﬁne the probability P deﬁned on the set relative to the sigma-algebra through the Ai Fi conditional distribution2 b P [S(t ) Q S(t )] = P [S(t ) Q S(t ),A = 1] . i+1 ∈ | i i+1 ∈ | i i+1 Note that since theb probability P is conditioned on survival at the next step, all stock price paths simulated under P survive until expiry. The authors let E to denote expectation with b respect to P . b b Second, we need a lemma and a theorem. Lemma 1 in Glasserman and Staum (p. 930, b [16]) shows the exact estimator with full importance sampling for a knock-out barrier option without rebate is unbiased,

E[ Y ]= E[A Y ], (4.3.1) Li i for any -measurable function Y . Theb authors also showed in their Theorem 2 (p. 930, Fi [16]) that the estimator for knock-out options with rebates (see Table 4.1) is unbiased:

E mf(S(tm)) + i 1(1 p(Si 1))g(S(ti 1)) L L − − − − " i=1 # X b m = E Amf(S(tm)) + (Ai 1 Ai)g(S(ti 1)) . (4.3.2) − − − " i=1 # X Thus, in order to prove the unbiasedness of the RS estimator ε1, we need only show that the expectations of the Glasserman and Staum estimators in Table 4.2 are equal to each other, that is,

m m

E i 1 (1 p(S(ti 1)) fi(S∗(ti)) = E (Ai 1 Ai)fi(S(ti)) . (4.3.3) L − − − − − " i=1 # " i=1 # X X 2The setb Q was not explicitly deﬁned in Glasserman and Staum [16]. We can assume the set Q to be the set of rational numbers that represents stock price.

76 Here is now a sketch of the proof that uses Theorem 2 above for the unbiasedness of the RS estimators. By the linearity of expectation, Equation (4.3.3) follows if we can show for each i:

E [ (1 p(S(t )) f (S∗(t ))] = E [(A A )f (S(t ))] . Li − i i+1 i+1 i − i+1 i+1 i+1 3 We can claim that f (S∗(t )) is -measurable So, the expression (1 p(S(t )) f (S∗(t )) b i+1 i+1 Fi − i i+1 i+1 is -measurable. By Lemma 1 (4.3.1), we have Fi

E [ (1 p(S(t )) f (S∗(t ))] Li − i i+1 i+1 = E [A f (S∗(t ))] b i i+1 i+1 = E [(A A )f (S∗(t ))] since A =0 at S∗(t ) i − i+1 i+1 i+1 i+1 i+1 = E [(A A )f (S(t ))] . i − i+1 i+1 i+1 By the same argument in the proof of Theorem 2 (4.3.2), we get the desired result in Equation (4.3.3). 4.4 Applications

For our numerical example, we recall the parameter values for the two down-and-in European call options from the conditional expectation Monte Carlo estimator section. The two options have the following common parameter values: σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2 and m = 50. The ﬁrst option has a barrier H = 95 with a true price of 1.4373238784. The second option has a barrier H = 91 with a true price of 0.3670447223. Table 4.3 shows the relative mean square errors of Ross and Shanthikumar’s raw simulation (Raw) Monte Carlo estimates and the weighted (Weighted) Monte Carlo estimates for the ﬁrst option. Observe that the improvement of the Weighted estimator over the Raw estimator is roughly a factor of 2. However, Table 4.4 shows that both the Raw and the Weighted estimators indicate an approximate factor of 10 improvement of the crude MC estimates from Chapter 2. A similar conclusion can be made for the second option as indicated by Tables 4.5 and 4.6. 3 ∗ This claim is true, that is, the option value fi+1(S (ti+1)) is known at time ti. In fact, Glasserman ∗ and Staum write immediately before Equation (25), p. 929 in [16]: “The Si+1 are not part of the path, but are used to estimate the value of the option should it be knocked in at step i +1, since in this case it is not realistic to expect that this value should be known at time i.” Moreover, in this context, we can adopt Ross and Shanthikumar’s treatment of the estimator ε1 conditioned on the Markov chain, which indirectly pre-draws the values of the good states and bad states, hence the stock price values, before estimating the price of the option.

77 Table 4.3: Relative mean square errors of Ross and Shanthikumar’s raw simulation (Raw) Monte Carlo estimates and weighted (Weighted) Monte Carlo estimates for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784).

N Raw Weighted Raw/Weighted 4 5 5K 1.78 10− 8.38 10− 2.12 × 5 × 5 10K 8.99 10− 4.51 10− 1.98 × 5 × 6 50K 1.53 10− 8.46 10− 1.81 × × Cross Barrier 62% n/a n/a

Table 4.4: Ratios of variances of crude Monte Carlo (MC) estimates from Chapter 2 to variances of Ross and Shanthikumar’s (Raw and Weighted) estimates (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 95, and true price = 1.4373238784).

N MC/Raw MC/Weighted 5K 9.33 100 1.98 101 10K 1.08 × 101 2.16 × 101 50K 1.31 × 101 2.36 × 101 × ×

Table 4.5: Relative mean square errors of Ross and Shanthikumar’s raw simulation (Raw) Monte Carlo estimates and weighted (Weighted) Monte Carlo estimates for a down-and-in barrier option (σ =0.3, r =0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223).

N Raw Weighted Raw/Weighted 4 4 5K 4.58 10− 3.30 10− 1.39 × 4 × 4 10K 1.51 10− 2.00 10− 0.75 × 5 × 5 50K 3.80 10− 2.99 10− 1.27 × × Cross Barrier 41% n/a n/a

78 Table 4.6: Ratios of variances of crude Monte Carlo (MC) estimates from Chapter 2 to variances of Ross and Shanthikumar’s (Raw and Weighted) estimates (σ = 0.3, r = 0.1, S(0) = 100, K = 100, T =0.2, m = 50, H = 91, and true price = 0.3670447223).

N MC/Raw MC/Weighted 5K 8.23 100 1.14 101 10K 1.72 × 101 1.30 × 101 50K 1.44 × 101 1.83 × 101 × ×

79 CHAPTER 5

Conclusion

This dissertation evaluated existing unbiased Monte Carlo estimators and offered some new estimators for pricing financial derivatives from the perspective of variance reduction. Estimates of the crude Monte Carlo estimator act as the benchmark in measuring the reduction in variance. We summarize each chapter and make the following conclusions. Chapter 2 discussed conditional expectation and control variate methods. In this chapter, we corrected the conditional expectation estimators for barrier options proposed by Boyle et al. [3] and by Ross and Shanthikumar [27], and found that the conditional expectation estimator for a down-and-in barrier option can reduce variance approximately by a factor of 10. Ross and Shanthikumar [27] also suggested a conditional expectation method for pricing Asian options. Their method, however, did not offer any significant error reduction since the conditional expectation is applied only in the last time step of the stock simulation. Increasing either the length of the time step or the volatility did not improve the errors. Even after their method was combined with the control variate method, an estimator that is known in the literature to perform very well, the result yielded improvements only with factors less than 10. Chapter 3 discussed estimators using an importance sampling method and using a combination of importance sampling and conditional expectation. In each estimator, we used heuristics and a simulated annealing process to determine the tilting parameters. Although the simulated annealing method gave errors that are slightly lower than those yielded by heuristics, the heuristics is the preferred method because of its ease and speed of implementation. We also corrected a part of the estimator that was erroneously reported by Ross and Shanthikumar [27]. The heuristic approach was also extended to price an up-and- out barrier option. We observed that the importance sampling approach performed better

80 than the crude Monte Carlo approach for both types of barrier options when the barrier is farther away from the initial stock price. Chapter 4 evaluated alternative barrier option estimators proposed by Ross and Shan- thikumar [27]: the raw simulation estimator and a variant of this estimator. We noted that the raw simulation estimator was similar to the conditional expectation estimator except that, in generating the stock price, the former simulated sums of steps of random walks while the latter simulated the standard normal distribution. The variant of the estimator that we considered generated a ﬁnite set of paths and priced the option by placing weights on each path. Our numerical results showed that the variant estimator performed only slightly better (a factor of 2) over the raw simulation estimator. However, the raw simulation estimator showed a factor of 10 improvement over the conditional expectation estimator in pricing a down-and-in barrier option. We also showed the equivalence of the Ross and Shanthikumar [27] estimators to a set of estimators that Glasserman and Staum [16] independently proposed.

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84 BIOGRAPHICAL SKETCH

Emmanuel Salta

Emmanuel Salta was born in the Philippines. He received an M.S. (2005) and a Ph.D. (2008) both in Financial Mathematics at Florida State University.