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Monte Carlo Methods in Financial Engineering

verfasst von: Paul Glasserman

Verlag: Springer New York

Buchreihe : Stochastic Modelling and Applied Probability

Enthalten in: Professional Book Archive

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Über dieses Buch

Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management. These applications have, in turn, stimulated research into new Monte Carlo methods and renewed interest in some older techniques.

This book develops the use of Monte Carlo methods in finance and it also uses simulation as a vehicle for presenting models and ideas from financial engineering. It divides roughly into three parts. The first part develops the fundamentals of Monte Carlo methods, the foundations of derivatives pricing, and the implementation of several of the most important models used in financial engineering. The next part describes techniques for improving simulation accuracy and efficiency. The final third of the book addresses special topics: estimating price sensitivities, valuing American options, and measuring market risk and credit risk in financial portfolios.

The most important prerequisite is familiarity with the mathematical tools used to specify and analyze continuous-time models in finance, in particular the key ideas of stochastic calculus. Prior exposure to the basic principles of option pricing is useful but not essential.

The book is aimed at graduate students in financial engineering, researchers in Monte Carlo simulation, and practitioners implementing models in industry.

Mathematical Reviews, 2004: "... this book is very comprehensive, up-to-date and useful tool for those who are interested in implementing Monte Carlo methods in a financial context."

Inhaltsverzeichnis

Frontmatter

1. Foundations

Abstract

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 description and simple examples of Monte Carlo, and then develops a framework for measuring the efficiency of Monte Carlo estimators. Section 1.2 reviews concepts 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.

Paul Glasserman

2. Generating Random Numbers and Random Variables

Abstract

This chapter deals with algorithms at the core of Monte Carlo simulation: methods for generating uniformly distributed random variables and methods for transforming those variables to other distributions. These algorithms may be executed millions of times in the course of a simulation, making efficient implementation especially important.

Paul Glasserman

3. Generating Sample Paths

Abstract

This chapter develops methods for simulating paths of a variety of stochastic processes important in financial engineering. The emphasis in this chapter is on methods for exact simulation of continuous-time processes at a discrete set of dates. The methods are exact in the sense that the joint distribution of the simulated values coincides with the joint distribution of the continuous-time process on the simulation time grid. Exact methods rely on special features of a model and are generally available only for models that offer some tractability. More complex models must ordinarily be simulated through, e.g., discretization of stochastic differential equations, as discussed in Chapter 6.

Paul Glasserman

4. Variance Reduction Techniques

Abstract

This chapter develops methods for increasing the efficiency of Monte Carlo simulation by reducing the variance of simulation estimates. These methods draw on two broad strategies for reducing variance: taking advantage of tractable features of a model to adjust or correct simulation outputs, and reducing the variability in simulation inputs. We discuss control variates, antithetic variates, stratified sampling, Latin hypercube sampling, moment matching methods, and importance sampling, and we illustrate these methods through examples. Two themes run through this chapter:

The greatest gains in efficiency from variance reduction techniques result from exploiting specific features of a problem, rather than from generic applications of generic methods.
Reducing simulation error is often at odds with convenient estimation of the simulation error itself; in order to supplement a reduced-variance estimator with a valid confidence interval, we sometimes need to sacrifice some of the potential variance reduction.

Paul Glasserman

5. Quasi-Monte Carlo

Abstract

This chapter discusses alternatives to Monte Carlo simulation known as quasi-Monte Carlo or low-discrepancy methods. These methods differ from ordinary Monte Carlo in that they make no attempt to mimic randomness. Indeed, they seek to increase accuracy specifically by generating points that are too evenly distributed to be random. Applying these methods to the pricing of derivative securities requires formulating a pricing problem as the calculation of an integral and thus suppressing its stochastic interpretation as an expected value. This contrasts with the variance reduction techniques of Chapter 4, which take advantage of the stochastic formulation to improve precision.

Paul Glasserman

6. Discretization Methods

Abstract

This chapter presents methods for reducing discretization error — the bias in Monte Carlo estimates that results from time-discretization of stochastic differential equations. Chapter 3 gives examples of continuous-time stochastic processes that can be simulated exactly at a finite set of dates, meaning that the joint distribution of the simulated values coincides with that of the continuous-time model at the simulated dates. But these examples are exceptional and most models arising in derivatives pricing can be simulated only approximately. The simplest approximation is the Euler scheme; this method is easy to implement and almost universally applicable, but it is not always sufficiently accurate. This chapter discusses methods for improving the Euler scheme and, as a prerequisite for this, discusses criteria for comparing discretization methods.

Paul Glasserman

7. Estimating Sensitivities

Abstract

Previous chapters have addressed various aspects of estimating expectations with a view toward computing the prices of derivative securities. This chapter develops methods for estimating sensitivities of expectations, in particular the derivatives of derivative prices commonly referred to as “Greeks.” From the discussion in Section 1.2.1, we know that in an idealized setting of continuous trading in a complete market, the payoff of a contingent claim can be manufactured (or hedged) through trading in underlying assets. The risk in a short position in an option, for example, is offset by a delta-hedging strategy of holding delta units of each underlying asset, where delta is simply the partial derivative of the option price with respect to the current price of that underlying asset. Implementation of the strategy requires knowledge of these price sensitivities; sensitivities with respect to other parameters are also widely used to measure and manage risk. Whereas the prices themselves can often be observed in the market, their sensitivites cannot, so accurate calculation of sensitivities is arguably even more important than calculation of prices. We will see, however, that derivative estimation presents both theoretical and practical challenges to Monte Carlo simulation.

Paul Glasserman

8. Pricing American Options

Abstract

Whereas a European option can be exercised only at a fixed date, an American option can be exercised any time up to its expiration. The value of an American option is the value achieved by exercising optimally. Finding this value entails finding the optimal exercise rule — by solving an optimal stopping problem — and computing the expected discounted payoff of the option under this rule. The embedded optimization problem makes this a difficult problem for simulation.

Paul Glasserman

9. Applications in Risk Management

Abstract

This chapter discusses applications of Monte Carlo simulation to risk management. It addresses the problem of measuring the risk in a portfolio of assets, rather than computing the prices of individual securities. Simulation is useful in estimating the profit and loss distribution of a portfolio and thus in computing risk measures that summarize this distribution. We give particular attention to the problem of estimating the probability of large losses, which entails simulation of rare but significant events. We separate the problems of measuring market risk and credit risk because different types of models are used in the two domains.

Paul Glasserman

Backmatter

Titel: Monte Carlo Methods in Financial Engineering
verfasst von: Paul Glasserman
Verlag: Springer New York
Electronic ISBN: 978-0-387-21617-1
Print ISBN: 978-1-4419-1822-2
DOI: https://doi.org/10.1007/978-0-387-21617-1

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Foundations

2. Generating Random Numbers and Random Variables

3. Generating Sample Paths

4. Variance Reduction Techniques

5. Quasi-Monte Carlo

6. Discretization Methods

7. Estimating Sensitivities

8. Pricing American Options

9. Applications in Risk Management

Backmatter