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this book is the concept of a change of measure. This idea is so central both to derivatives pricing and to Monte Carlo methods that there is simply no avoiding it. The prerequisites to understanding the statement of the Girsanov theorem should suffice for reading this book. Whereas the language of mathematical finance is essential to our topic, its technical subtleties are less so for purposes of computational work. My use of mathematical tools is often informal: I may assume that a local martingale is a martingale or that a stochastic differential equation has a solution, for example, without calling attention to these assumptions. Where convenient, I take derivatives without first assuming differentiability and I take expecta- tions without verifying integrability. My intent is to focus on the issues most important to Monte Carlo methods and to avoid diverting the discussion to spell out technical conditions. Where these conditions are not evident and where they are essential to understanding the scope of a technique, I discuss them explicitly. In addition, an appendix gives precise statements of the most important tools from stochastic calculus. This book divides roughly into three parts. The first part, Chapters 1–3, develops fundamentals of Monte Carlo methods. Chapter 1 summarizes the theoretical foundations of derivatives pricing and Monte Carlo. It explains the principles by which a pricing problem can be formulated as an integra- tion problem to which Monte Carlo is then applicable. Chapter 2 discusses random number generation and methods for sampling from nonuniform dis- tributions, tools fundamental to every application of Monte Carlo. Chapter 3 provides an overview of some of the most important models used in financial engineering and discusses their implementation by simulation. I have included more discussion of the models in Chapter 3 and the financial underpinnings in Chapter 1 than is strictly necessary to run a simulation. Students often come to a course in Monte Carlo with limited exposure to this material, and the implementation of a simulation becomes more meaningful if accompanied by an understanding of a model and its context. Moreover, it is precisely in model details that many of the most interesting simulation issues arise. If the first three chapters deal with running a simulation, the next three deal with ways of running it better. Chapter 4 presents methods for increas- ing precision by reducing the variance of Monte Carlo estimates. Chapter 5 discusses the application of deterministic quasi-Monte Carlo methods for nu- merical integration. Chapter 6 addresses the problem of discretization error that results from simulating discrete-time approximations to continuous-time models. The last three chapters address topics specific to the application of Monte Carlo methods in finance. Chapter 7 covers methods for estimating price sen- sitivities or “Greeks.” Chapter 8 deals with the pricing of American options, which entails solving an optimal stopping problem within a simulation. Chap- ter 9 is an introduction to the use of Monte Carlo methods in risk management. It discusses the measurement of market risk and credit risk in financial port- folios. The models and methods of this final chapter are rather different from

vii

those in the other chapters, which deal primarily with the pricing of derivative securities. Several people have influenced this book in various ways and it is my pleasure to express my thanks to them here. I owe a particular debt to my frequent collaborators and co-authors Mark Broadie, Phil Heidelberger, and Perwez Shahabuddin. Working with them has influenced my thinking as well as the book’s contents. With Mark Broadie I have had several occasions to collaborate on teaching as well as research, and I have benefited from our many discussions on most of the topics in this book. Mark, Phil Heidelberger, Steve Kou, Pierre L’Ecuyer, Barry Nelson, Art Owen, Philip Protter, and Jeremy Staum each commented on one or more draft chapters; I thank them for their comments and apologize for the many good suggestions I was unable to incorporate fully. I have also benefited from working with current and former Columbia students Jingyi Li, Nicolas Merener, Jeremy Staum, Hui Wang, Bin Yu, and Xiaoliang Zhao on some of the topics in this book. Several classes of students helped uncover errors in the lecture notes from which this book evolved.

Paul Glasserman New York, 2003

• 1 Foundations
  • 1.1 Principles of Monte Carlo
    • 1.1.1 Introduction
    • 1.1.2 First Examples
    • 1.1.3 Efficiency of Simulation Estimators
  • 1.2 Principles of Derivatives Pricing
    • 1.2.1 Pricing and Replication
    • 1.2.2 Arbitrage and Risk-Neutral Pricing
    • 1.2.3 Change of Numeraire
    • 1.2.4 The Market Price of Risk
• 2 Generating Random Numbers and Random Variables
  • 2.1 Random Number Generation
    • 2.1.1 General Considerations
    • 2.1.2 Linear Congruential Generators
    • 2.1.3 Implementation of Linear Congruential Generators
    • 2.1.4 Lattice Structure
    • 2.1.5 Combined Generators and Other Methods
  • 2.2 General Sampling Methods
    • 2.2.1 Inverse Transform Method
    • 2.2.2 Acceptance-Rejection Method
  • 2.3 Normal Random Variables and Vectors
    • 2.3.1 Basic Properties
    • 2.3.2 Generating Univariate Normals
    • 2.3.3 Generating Multivariate Normals
• 3 Generating Sample Paths
  • 3.1 Brownian Motion
    • 3.1.1 One Dimension
    • 3.1.2 Multiple Dimensions
  • 3.2 Geometric Brownian Motion
    • 3.2.1 Basic Properties x Contents
    • 3.2.2 Path-Dependent Options
    • 3.2.3 Multiple Dimensions
  • 3.3 Gaussian Short Rate Models
    • 3.3.1 Basic Models and Simulation
    • 3.3.2 Bond Prices
    • 3.3.3 Multifactor Models
  • 3.4 Square-Root Diffusions
    • 3.4.1 Transition Density
    • 3.4.2 Sampling Gamma and Poisson
    • 3.4.3 Bond Prices
    • 3.4.4 Extensions
  • 3.5 Processes with Jumps
    • 3.5.1 A Jump-Diffusion Model
    • 3.5.2 Pure-Jump Processes
  • 3.6 Forward Rate Models: Continuous Rates
    • 3.6.1 The HJM Framework
    • 3.6.2 The Discrete Drift
    • 3.6.3 Implementation
  • 3.7 Forward Rate Models: Simple Rates
    • 3.7.1 LIBOR Market Model Dynamics
    • 3.7.2 Pricing Derivatives
    • 3.7.3 Simulation
    • 3.7.4 Volatility Structure and Calibration
• 4 Variance Reduction Techniques
  • 4.1 Control Variates
    • 4.1.1 Method and Examples
    • 4.1.2 Multiple Controls
    • 4.1.3 Small-Sample Issues
    • 4.1.4 Nonlinear Controls
  • 4.2 Antithetic Variates
  • 4.3 Stratified Sampling
    • 4.3.1 Method and Examples
    • 4.3.2 Applications
    • 4.3.3 Poststratification
  • 4.4 Latin Hypercube Sampling
  • 4.5 Matching Underlying Assets
    • 4.5.1 Moment Matching Through Path Adjustments
    • 4.5.2 Weighted Monte Carlo
  • 4.6 Importance Sampling
    • 4.6.1 Principles and First Examples
    • 4.6.2 Path-Dependent Options
  • 4.7 Concluding Remarks
• 5 Quasi-Monte Carlo Contents xi
  • 5.1 General Principles
    • 5.1.1 Discrepancy
    • 5.1.2 Van der Corput Sequences
    • 5.1.3 The Koksma-Hlawka Bound
    • 5.1.4 Nets and Sequences
  • 5.2 Low-Discrepancy Sequences
    • 5.2.1 Halton and Hammersley
    • 5.2.2 Faure
    • 5.2.3 Sobol’
    • 5.2.4 Further Constructions
  • 5.3 Lattice Rules
  • 5.4 Randomized QMC
  • 5.5 The Finance Setting
    • 5.5.1 Numerical Examples
    • 5.5.2 Strategic Implementation
  • 5.6 Concluding Remarks
• 6 Discretization Methods
  • 6.1 Introduction
    • 6.1.1 The Euler Scheme and a First Refinement
    • 6.1.2 Convergence Order
  • 6.2 Second-Order Methods
    • 6.2.1 The Scalar Case
    • 6.2.2 The Vector Case
    • 6.2.3 Incorporating Path-Dependence
    • 6.2.4 Extrapolation
  • 6.3 Extensions
    • 6.3.1 General Expansions
    • 6.3.2 Jump-Diffusion Processes
    • 6.3.3 Convergence of Mean Square Error
  • 6.4 Extremes and Barrier Crossings: Brownian Interpolation
  • 6.5 Changing Variables
  • 6.6 Concluding Remarks
• 7 Estimating Sensitivities
  • 7.1 Finite-Difference Approximations
    • 7.1.1 Bias and Variance
    • 7.1.2 Optimal Mean Square Error
  • 7.2 Pathwise Derivative Estimates
    • 7.2.1 Method and Examples
    • 7.2.2 Conditions for Unbiasedness
    • 7.2.3 Approximations and Related Methods
  • 7.3 The Likelihood Ratio Method
    • 7.3.1 Method and Examples
    • 7.3.2 Bias and Variance Properties xii Contents
    • 7.3.3 Gamma
    • 7.3.4 Approximations and Related Methods
  • 7.4 Concluding Remarks
• 8 Pricing American Options
  • 8.1 Problem Formulation
  • 8.2 Parametric Approximations
  • 8.3 Random Tree Methods
    • 8.3.1 High Estimator
    • 8.3.2 Low Estimator
    • 8.3.3 Implementation
  • 8.4 State-Space Partitioning
  • 8.5 Stochastic Mesh Methods
    • 8.5.1 General Framework
    • 8.5.2 Likelihood Ratio Weights
  • 8.6 Regression-Based Methods and Weights
    • 8.6.1 Approximate Continuation Values
    • 8.6.2 Regression and Mesh Weights
  • 8.7 Duality
  • 8.8 Concluding Remarks
• 9 Applications in Risk Management
  • 9.1 Loss Probabilities and Value-at-Risk
    • 9.1.1 Background
    • 9.1.2 Calculating VAR
  • 9.2 Variance Reduction Using the Delta-Gamma Approximation
    • 9.2.1 Control Variate
    • 9.2.2 Importance Sampling
    • 9.2.3 Stratified Sampling
  • 9.3 A Heavy-Tailed Setting
    • 9.3.1 Modeling Heavy Tails
    • 9.3.2 Delta-Gamma Approximation
    • 9.3.3 Variance Reduction
  • 9.4 Credit Risk
    • 9.4.1 Default Times and Valuation
    • 9.4.2 Dependent Defaults
    • 9.4.3 Portfolio Credit Risk
  • 9.5 Concluding Remarks
• A Appendix: Convergence and Confidence Intervals
  • A.1 Convergence Concepts
  • A.2 Central Limit Theorem and Confidence Intervals
• B Appendix: Results from Stochastic Calculus Contents xiii
  • B.1 Itˆo’s Formula
  • B.2 Stochastic Differential Equations
  • B.3 Martingales
  • B.4 Change of Measure
• C Appendix: The Term Structure of Interest Rates
  • C.1 Term Structure Terminology
  • C.2 Interest Rate Derivatives
• References
• Index

Foundations

This chapter’s two parts develop key ideas from two fields, the intersection of which is the topic of this book. Section 1.1 develops principles underlying the use and analysis of Monte Carlo methods. It begins with a general descrip- tion and simple examples of Monte Carlo, and then develops a framework for measuring the efficiency of Monte Carlo estimators. Section 1.2 reviews con- cepts from the theory of derivatives pricing, including pricing by replication, the absence of arbitrage, risk-neutral probabilities, and market completeness. The most important idea for our purposes is the representation of derivative prices as expectations, because this representation underlies the application of Monte Carlo.

1.1 Principles of Monte Carlo

1.1.1 Introduction

Monte Carlo methods are based on the analogy between probability and vol- ume. The mathematics of measure formalizes the intuitive notion of probabil- ity, associating an event with a set of outcomes and defining the probability of the event to be its volume or measure relative to that of a universe of possible outcomes. Monte Carlo uses this identity in reverse, calculating the volume of a set by interpreting the volume as a probability. In the simplest case, this means sampling randomly from a universe of possible outcomes and taking the fraction of random draws that fall in a given set as an estimate of the set’s volume. The law of large numbers ensures that this estimate converges to the correct value as the number of draws increases. The central limit theorem provides information about the likely magnitude of the error in the estimate after a finite number of draws. A small step takes us from volumes to integrals. Consider, for example, the problem of estimating the integral of a function f over the unit interval. We may represent the integral

1.1 Principles of Monte Carlo 3

Indeed, the steps outlined above extend to estimating an integral over [0, 1]d (and even d) for all dimensions d. Of course, when we change dimensions we change f and when we change f we change σf , but the standard error will still have the form σf /

n for a Monte Carlo estimate based on n draws from the domain [0, 1]d. In particular, the O(n−^1 /^2 ) convergence rate holds for all d. In contrast, the error in a product trapezoidal rule in d dimensions is O(n−^2 /d) for twice continuously differentiable integrands; this degradation in convergence rate with increasing dimension is characteristic of all deterministic integration methods. Thus, Monte Carlo methods are attractive in evaluating integrals in high dimensions. What does this have to do with financial engineering? A fundamental im- plication of asset pricing theory is that under certain circumstances (reviewed in Section 1.2.1), the price of a derivative security can be usefully represented as an expected value. Valuing derivatives thus reduces to computing expecta- tions. In many cases, if we were to write the relevant expectation as an integral, we would find that its dimension is large or even infinite. This is precisely the sort of setting in which Monte Carlo methods become attractive. Valuing a derivative security by Monte Carlo typically involves simulating paths of stochastic processes used to describe the evolution of underlying asset prices, interest rates, model parameters, and other factors relevant to the security in question. Rather than simply drawing points randomly from [0, 1] or [0, 1]d, we seek to sample from a space of paths. Depending on how the problem and model are formulated, the dimension of the relevant space may be large or even infinite. The dimension will ordinarily be at least as large as the number of time steps in the simulation, and this could easily be large enough to make the square-root convergence rate for Monte Carlo competitive with alternative methods. For the most part, there is nothing we can do to overcome the rather slow rate of convergence characteristic of Monte Carlo. (The quasi-Monte Carlo methods discussed in Chapter 5 are an exception — under appropriate con- ditions they provide a faster convergence rate.) We can, however, look for superior sampling methods that reduce the implicit constant in the conver- gence rate. Much of this book is devoted to examples and general principles for doing this. The rest of this section further develops some essential ideas underly- ing Monte Carlo methods and their application to financial engineering. Sec- tion 1.1.2 illustrates the use of Monte Carlo with two simple types of option contracts. Section 1.1.3 develops a framework for evaluating the efficiency of simulation estimators.

1.1.2 First Examples

In discussing general principles of Monte Carlo, it is useful to have some simple specific examples to which to refer. As a first illustration of a Monte Carlo method, we consider the calculation of the expected present value of the payoff

4 1 Foundations

of a call option on a stock. We do not yet refer to this as the option price; the connection between a price and an expected discounted payoff is developed in Section 1.2.1. Let S(t) denote the price of the stock at time t. Consider a call option granting the holder the right to buy the stock at a fixed price K at a fixed time T in the future; the current time is t = 0. If at time T the stock price S(T ) exceeds the strike price K, the holder exercises the option for a profit of S(T ) − K; if, on the other hand, S(T ) ≤ K, the option expires worthless. (This is a European option, meaning that it can be exercised only at the fixed date T ; an American option allows the holder to choose the time of exercise.) The payoff to the option holder at time T is thus

(S(T ) − K)+^ = max{ 0 , S(T ) − K}.

To get the present value of this payoff we multiply by a discount factor e−rT^ , with r a continuously compounded interest rate. We denote the expected present value by E[e−rT^ (S(T ) − K)+]. For this expectation to be meaningful, we need to specify the distribution of the random variable S(T ), the terminal stock price. In fact, rather than simply specifying the distribution at a fixed time, we introduce a model for the dynamics of the stock price. The Black-Scholes model describes the evolution of the stock price through the stochastic differential equation (SDE)

dS(t) S(t)

= r dt + σ dW (t), (1.1)

with W a standard Brownian motion. (For a brief review of stochastic cal- culus, see Appendix B.) This equation may be interpreted as modeling the percentage changes dS/S in the stock price as the increments of a Brownian motion. The parameter σ is the volatility of the stock price and the coefficient on dt in (1.1) is the mean rate of return. In taking the rate of return to be the same as the interest rate r, we are implicitly describing the risk-neutral dynamics of the stock price, an idea reviewed in Section 1.2.1. The solution of the stochastic differential equation (1.1) is

S(T ) = S(0) exp

[r − 12 σ^2 ]T + σW (T )

As S(0) is the current price of the stock, we may assume it is known. The random variable W (T ) is normally distributed with mean 0 and variance T ; this is also the distribution of

T Z if Z is a standard normal random variable (mean 0, variance 1). We may therefore represent the terminal stock price as

S(T ) = S(0) exp

[r − 12 σ^2 ]T + σ

T Z

The logarithm of the stock price is thus normally distributed, and the stock price itself has a lognormal distribution.

6 1 Foundations

is an asymptotically (as n → ∞) valid 1 − δ confidence interval for C. (For a 95% confidence interval, δ = .05 and zδ/ 2 ≈ 1 .96.) Alternatively, because the standard deviation is estimated rather than known, we may prefer to replace zδ/ 2 with the corresponding quantile from the t distribution with n− 1 degrees of freedom, which results in a slightly wider interval. In either case, the probability that the interval covers C approaches 1 − δ as n → ∞. (These ideas are reviewed in Appendix A.) The problem of estimating E[e−rT^ (S(T ) − K)+] by Monte Carlo is simple enough to be illustrated in a spreadsheet. Commercial spreadsheet software typically includes a method for sampling from the normal distribution and the mathematical functions needed to transform normal samples to terminal stock prices and then to discounted option payoffs. Figure 1.1 gives a schematic illustration. The Zi are samples from the normal distribution; the comments in the spreadsheet illustrate the formulas used to transform these to arrive at the estimate Cˆn. The spreadsheet layout in Figure 1.1 makes the method transparent but has the drawback that it requires storing all n replication in n rows of cells. It is usually possible to use additional spreadsheet commands to recalculate cell values n times without storing intermediate values.

Replication Normals Stock Price Option Payoff 1 Z_1 S_1 C_ 2 Z_2 S_2 C_ 3 Z_3 S_3 C_ 4 Z_4 S_4 C_ 5 Z_5 S_5 C_ 6 Z_6 S_6 C_ 7 Z_7 S_7 C_ 8 Z_8 S_8 C_ 9 Z_9 S_9 C_ 10 Z_10 S_10 C_ 11 Z_11 S_11 C_

n Z_n^ S_n^ C_n AVERAGE(C_1,…,C_n) s_C = STDEV(C_1,…,C_n)

S_1=S(0)exp((r-0.5σ^2)T+σsqrt(T)*Z_1)

C_8=exp(-rT)*max(0,S_8-K)

Cˆn =

M M M M

Fig. 1.1. A spreadsheet for estimating the expected present value of the payoff of a call option.

This simple example illustrates a general feature of Monte Carlo methods for valuing derivatives, which is that the simulation is built up in layers: each of the transformations Zi −→ Si(T ) −→ Ci

exemplifies a typical layer. The first transformation constructs a path of under- lying assets from random variables with simpler distributions and the second calculates a discounted payoff from each path. In fact, we often have additional

1.1 Principles of Monte Carlo 7

layers above and below these. At the lowest level, we typically start from in- dependent random variables Ui uniformly distributed between 0 and 1, so we need a transformation taking the Ui to Zi. The transformation taking the Ci to the sample mean Cˆn and sample standard deviation sC may be viewed as another layer. We include another still higher level in, for example, valuing a portfolio of instruments, each of which is valued by Monte Carlo. Randomness (or apparent randomness) typically enters only at the lowest layer; the sub- sequent transformations producing asset paths, payoffs, and estimators are usually deterministic.

Path-Dependent Example

The payoff of a standard European call option is determined by the terminal stock price S(T ) and does not otherwise depend on the evolution of S(t) between times 0 and T. In estimating E[e−rT^ (S(T ) − K)+], we were able to jump directly from time 0 to time T using (1.3) to sample values of S(T ). Each simulated “path” of the underlying asset thus consists of just the two points S(0) and S(T ). In valuing more complicated derivative securities using more complicated models of the dynamics of the underlying assets, it is often necessary to sim- ulate paths over multiple intermediate dates and not just at the initial and terminal dates. Two considerations may make this necessary:

◦ the payoff of a derivative security may depend explicitly on the values of underlying assets at multiple dates; ◦ we may not know how to sample transitions of the underlying assets exactly and thus need to divide a time interval [0, T ] into smaller subintervals to obtain a more accurate approximation to sampling from the distribution at time T.

In many cases, both considerations apply. Before turning to a detailed example of the first case, we briefly illustrate the second. Consider a generalization of the basic model (1.1) in which the dynamics of the underlying asset S(t) are given by

dS(t) = rS(t) dt + σ(S(t))S(t) dW (t). (1.7)

In other words, we now let the volatility σ depend on the current level of S. Except in very special cases, this equation does not admit an explicit solution of the type in (1.2) and we do not have an exact mechanism for sampling from the distribution of S(T ). In this setting, we might instead partition [0, T ] into m subintervals of length ∆t = T /m and over each subinterval [t, t + ∆t] simulate a transition using a discrete (Euler) approximation to (1.7) of the form S(t + ∆t) = S(t) + rS(t)∆t + σ(S(t))S(t)

∆tZ,

1.1 Principles of Monte Carlo 9

Figure 1.2 gives a schematic illustration of a spreadsheet implementation of this method. The spreadsheet has n rows of standard normal random vari- ables Zij with m variables in each row. These are mapped to n paths of the underlying asset, each path consisting of m steps. From each path, the spread- sheet calculates a value of the time average S¯i and a value of the discounted payoff Ci. The Ci are averaged to produce the final estimate Cˆn.

Path \ Step (^1 2 3) m 1 Z_ 11 Z_ 12 Z_ 13 Z_ 1 m 2 Z_ 21 Z_ 22 Z_ 23 Z_ 2 m 3 Z_ 31 Z_ 32 Z_ 33 Z_ 3 m

n Z_n 1 Z_n 2 Z_n 3 Z_nm 1 S_ 11 S_ 12 S_ 13 S_ 1 m 2 S_ 21 S_ 22 S_ 23 S_ 2 m 3 S_ 31 S_ 32 S_ 33 S_ 3 m

n S_n 1 S_n 2 S_n 3 S_nm AVERAGE(C_1,…,C_n)

S 1 S 2 S 3

S n

C 1 C 2 C 3

C n Cˆn =

S_13=S12exp((r-0.5σ^2)(t_3-t_2)+σsqrt(t_3-t_2)*Z_13)

=AVERAGE(S_31,S_32,…,S_3m)

=exp(-rT)*max(0, (^) S 2 -K)

M

M

M M M

M M M

L L L L L L L

C 2 S 3

Fig. 1.2. A spreadsheet for estimating the expected present value of the payoff of an Asian call option.

1.1.3 Efficiency of Simulation Estimators

Much of this book is devoted to ways of improving Monte Carlo estimators. To discuss improvements, we first need to explain our criteria for compar- ing alternative estimators. Three considerations are particularly important: computing time, bias, and variance. We begin by considering unbiased estimates. The two cases considered in Section 1.1.2 (the standard call and the Asian call) produced unbiased esti- mates in the sense that in both cases E[ Cˆn] = C, with Cˆn the corresponding estimator and C the quantity being estimated. Also, in both cases the esti- mator Cˆn was the mean of n independent and identically distributed samples. We proceed by continuing to consider estimators of this form because this setting is both simple and practically relevant. Suppose, then, that

Cˆn =^1 n

∑^ n

i=

Ci,

with Ci i.i.d., E[Ci] = C and Var[Ci] = σ^2 C < ∞. The central limit theorem asserts that as the number of replications n increases, the standardized esti- mator ( Cˆn − C)/(σC /

n) converges in distribution to the standard normal, a statement often abbreviated as

10 1 Foundations

C^ ˆn − C σC /

n

⇒ N (0, 1)

or, equivalently, as (^) √ n[ Cˆn − C] ⇒ N (0, σ^2 C ). (1.10)

Here, ⇒ denotes convergence in distribution and N (a, b^2 ) denotes the normal distribution with mean a and variance b^2. The stated convergence in distrib- ution means that

lim n→∞

P

Cˆn − C σC /

n

≤ x

= Φ(x)

for all x, with Φ the cumulative normal distribution. The same limit holds if σC is replaced with the sample standard devation sC (as in (1.5)); this is important because σC is rarely known in practice but sC is easily calculated from the simulation output. The fact that we can replace σC with sC without changing the limit in distribution follows from the fact that sC /σC → 1 as n → ∞ and general results on convergence in distribution (cf. Appendix A). The central limit theorem justifies the confidence interval (1.6): as n → ∞, the probability that this interval straddles the true value C approaches 1 − δ. Put differently, the central limit theorem tells us something about the distribution of the error in our simulation estimate:

Cˆn − C ≈ N (0, σ^2 C /n),

meaning that the error on the left has approximately the distribution on the right. This makes precise the intuitively obvious notion that, other things being equal, in comparing two estimators of the same quantity we should prefer the one with lower variance. But what if other things are not equal? In particular, suppose we have a choice between two unbiased estimators and that the one with smaller vari- ance takes longer to compute. How should we balance variance reduction and computational effort? An informal answer was suggested by Hammersley and Handscomb [169]; Fox and Glynn [128] and Glynn and Whitt [160] develop a general framework for analyzing this issue and we now review some of its main conclusions. Suppose that generating a replication Ci takes a fixed amount of comput- ing time τ. Our objective is to compare estimators based on relative compu- tational effort, so the units in which we measure computing time are unim- portant. Let s denote our computational budget, measured in the same units as τ. Then the number of replications we can complete given the available budget is s/τ , the integer part of s/τ , and the resulting estimator is Cˆs/τ . Directly from (1.10), we get

√ s/τ [ Cˆs/τ  − C] ⇒ N (0, σ^2 C )