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L´evy Processes in Finance: Pricing Financial Derivatives.
Wim Schoutens
Copyright @ 2003 John Wiley & Sons, Ltd.
Contents
Preface xi Acknowledgements xv 1 Introduction 1 1.1 Financial Assets 1 1.2 Derivative Securities 3 1.2.1 Options 3 1.2.2 Prices of Options on the S&P 500 Index 5 1.3 Modelling Assumptions 7 1.4 Arbitrage 9 2 Financial Mathematics in Continuous Time 11 2.1 Stochastic Processes and Filtrations 11 2.2 Classes of Processes 13 2.2.1 Markov Processes 13 2.2.2 Martingales 14 2.2.3 Finite- and Infinite-Variation Processes 14 2.3 Characteristic Functions 15 2.4 Stochastic Integrals and SDEs 16 2.5 Financial Mathematics in Continuous Time 17 2.5.1 Equivalent Martingale Measure 17 2.5.2 Pricing Formulas for European Options 19 2.6 Dividends 21 3 The Black–Scholes Model 23 3.1 The Normal Distribution 23 3.2 Brownian Motion 24 3.2.1 Definition 25 3.2.2 Properties 26 3.3 Geometric Brownian Motion 27 viii CONTENTS 3.4 The Black–Scholes Option Pricing Model 28 3.4.1 The Black–Scholes Market Model 29 3.4.2 Market Completeness 30 3.4.3 The Risk-Neutral Setting 30 3.4.4 The Pricing of Options under the Black–Scholes Model 30 4 Imperfections of the Black–Scholes Model 33 4.1 The Non-Gaussian Character 33 4.1.1 Asymmetry and Excess Kurtosis 33 4.1.2 Density Estimation 35 4.1.3 Statistical Testing 36 4.2 Stochastic Volatility 38 4.3 Inconsistency with Market Option Prices 39 5 Lévy Processes and OU Processes 43 5.1 Lévy Processes 44 5.1.1 Definition 44 5.1.2 Properties 45 5.2 OU Processes 47 5.2.1 Self-Decomposability 47 5.2.2 OU Processes 48 5.3 Examples of Lévy Processes 50 5.3.1 The Poisson Process 50 5.3.2 The Compound Poisson Process 51 5.3.3 The Gamma Process 52 5.3.4 The Inverse Gaussian Process 53 5.3.5 The Generalized Inverse Gaussian Process 54 5.3.6 The Tempered Stable Process 56 5.3.7 The Variance Gamma Process 57 5.3.8 The Normal Inverse Gaussian Process 59 5.3.9 The CGMY Process 60 5.3.10 The Meixner Process 62 5.3.11 The Generalized Hyperbolic Process 65 5.4 Adding an Additional Drift Term 67 5.5 Examples of OU Processes 67 5.5.1 The Gamma–OU Process 68 5.5.2 The IG–OU Process 69 5.5.3 Other Examples 70 6 Stock Price Models Driven by Lévy Processes 73 6.1 Statistical Testing 73 6.1.1 Parameter Estimation 73 6.1.2 Statistical Testing 74 CONTENTS ix 6.2 The Lévy Market Model 76 6.2.1 Market Incompleteness 77 6.2.2 The Equivalent Martingale Measure 77 6.2.3 Pricing Formulas for European Options 80 6.3 Calibration of Market Option Prices 82 7 Lévy Models with Stochastic Volatility 85 7.1 The BNS Model 85 7.1.1 The BNS Model with Gamma SV 87 7.1.2 The BNS Model with IG SV 88 7.2 The Stochastic Time Change 88 7.2.1 The Integrated CIR Time Change 89 7.2.2 The IntOU Time Change 90 7.3 The Lévy SV Market Model 91 7.4 Calibration of Market Option Prices 97 7.4.1 Calibration of the BNS Models 97 7.4.2 Calibration of the Lévy SV Models 98 7.5 Conclusion 98 8 Simulation Techniques 101 8.1 Simulation of Basic Processes 101 8.1.1 Simulation of Standard Brownian Motion 101 8.1.2 Simulation of a Poisson Process 102 8.2 Simulation of a Lévy Process 102 8.2.1 The Compound Poisson Approximation 103 8.2.2 On the Choice of the Poisson Processes 105 8.3 Simulation of an OU Process 107 8.4 Simulation of Particular Processes 108 8.4.1 The Gamma Process 108 8.4.2 The VG Process 109 8.4.3 The TS Process 111 8.4.4 The IG Process 111 8.4.5 The NIG Process 113 8.4.6 The Gamma–OU Process 114 8.4.7 The IG–OU Process 115 8.4.8 The CIR Process 117 8.4.9 BNS Model 117 9 Exotic Option Pricing 119 9.1 Barrier and Lookback Options 119 9.1.1 Introduction 119 9.1.2 Black–Scholes Barrier and Lookback Option Prices 121 9.1.3 Lookback and Barrier Options in a Lévy Market 123 x CONTENTS 9.2 Other Exotic Options 125 9.2.1 The Perpetual American Call and Put Option 125 9.2.2 The Perpetual Russian Option 126 9.2.3 Touch-and-Out Options 126 9.3 Exotic Option Pricing by Monte Carlo Simulation 127 9.3.1 Introduction 127 9.3.2 Monte Carlo Pricing 127 9.3.3 Variance Reduction by Control Variates 129 9.3.4 Numerical Results 132 9.3.5 Conclusion 134 10 Interest-Rate Models 135 10.1 General Interest-Rate Theory 135 10.2 The Gaussian HJM Model 138 10.3 The Lévy HJM Model 141 10.4 Bond Option Pricing 142 10.5 Multi-Factor Models 144 Appendix A Special Functions 147 A.1 Bessel Functions 147 A.2 Modified Bessel Functions 148 A.3 The Generalized Hypergeometric Series 149 A.4 Orthogonal Polynomials 149 A.4.1 Hermite polynomials with parameter 149 A.4.2 Meixner–Pollaczek Polynomials 150 Appendix B Lévy Processes 151 B.1 Characteristic Functions 151 B.1.1 Distributions on the Nonnegative Integers 151 B.1.2 Distributions on the Positive Half-Line 151 B.1.3 Distributions on the Real Line 152 B.2 Lévy Triplets 153 B.2.1 γ 153 B.2.2 The Lévy Measure ν(dx) 154 Appendix C S&P 500 Call Option Prices 155 References 157 Index 165
Preface
The story of modelling financial markets with stochastic processes began in 1900 with the study of Bachelier (1900). He modelled stocks as a Brownian motion with drift. However, the model had many imperfections, including, for example, negative stock prices. It was 65 years before another, more appropriate, model was suggested by Samuelson (1965): geometric Brownian motion. Eight years later Black and Scholes (1973) and Merton (1973) demonstrated how to price European options based on the geometric Brownian model. This stock-price model is now called the Black–Scholes model, for which Scholes and Merton received the Nobel Prize for Economics in 1997 (Black had already died). It has become clear, however, that this option-pricing model is inconsistent with options data. Implied volatility models can do better, but, fundamentally, these consist of the wrong building blocks. To improve on the performance of the Black–Scholes model, Lévy models were proposed in the late 1980s and early 1990s, since when they have been refined to take account of different stylized features of the markets. This book is concerned with the pricing of derivative securities in market models based on Lévy processes. Financial mathematics has recently enjoyed considerable prestige as a result of its impact on the finance industry. The theory of Lévy processes has also seen exciting developments in recent years. The fusion of these two fields of mathematics has provided new applied modelling perspectives within the context of finance and further stimulus for the study of problems within the context of Lévy processes. This book is aimed at peopleworking in the areas of mathematical finance and Lévy processes, with the intention of convincing the former that the rich theory of Lévy processes can lead to tractable and attractive models that perform significantly better than the standard Black–Scholes model. For those working with Lévy processes, we hope to show how the objects they study can be readily applied in practice. We have taken great care not to use too much esoteric mathematics, nor to get too involved in technicalities, nor to give involved proofs. We focus on the ideas, and the intuition behind the modelling process and its applications. Nevertheless, the processes involved in the modelling are described very accurately and in great detail. These processes lie at the heart of the theory and it is very important to have a clear view of their properties. xii PREFACE This book is organized as follows. In Chapter 1 we introduce the phenomena we want to model: financial assets. Then we look at some of the basic modelling assumptions that are used throughout the book. Special attention is paid to the noarbitrage assumption. In Chapter 2, we recall the basics of mathematical finance in continuous time. We briefly discuss stochastic processes in continuous time together with stochastic integration theory. The main focus is on different pricing methods, arbitrage-free and (in)complete markets. Chapter 3 introduces the famous Black–Scholes model.We give an overview of the model together with its basic properties and then we have a close look at the pricing formulas under this model. Chapter 4 discusses why the Black–Scholes model is not such an appropriate model. First, we argue that the underlying Normal distribution is not suitable for the accurate modelling of stock-price behaviour. Next, we show that the model lacks the important feature of stochastic volatility. These imperfections are shownbased on historical data. Moreover, we show that the model prices do not correspond as they should to market prices. The above discussed imperfections cause this discrepancy between model and market prices. Chapter 5 is devoted to the main ingredients of the more sophisticated models introduced later on. We give an overview of the theory of Lévy processes and the theory of Ornstein–Uhlenbeck processes (OU processes). Lévy processes are based on infinitely divisible distributions.Asubclass of these distributions is self-decomposable and leads toOUprocesses.Awhole group of very popular examples of these processes is looked at in detail. Besides stating the defining equations, we also look at their properties. In Chapter 6, for the first time we use non-Brownian Lévy processes to describe the behaviour of a stock-price process.We discuss the Lévy market model, in which the stock price follows the exponential of a Lévy process. The market models proposed are no longer complete and an equivalent martingale measure has to be chosen. Comparing model prices with market prices demonstrates that Lévy models are a significant improvement on the Black–Scholes model, but that they still fall short. It is in Chapter 7 that we introduce stochastic volatility in our models. This can be done in several ways. We could start from the Black–Scholes model and make the volatility parameter involved itself stochastic.We focus on models where this volatility follows an OU process. Or we could introduce stochastic volatility by making time stochastic. If time goes fast, the market is nervous. If time goes slow, volatility is low. This technique can be applied not only in the Black–Scholes model but also in the Lévy market model. Different choices of stochastic time are considered: the rate of change of time can be described by the classical mean-reverting Cox–Ingersoll–Ross (CIR) stochastic process, but the OU processes are also excellent candidates. Chapter 8 discusses simulation techniques. Attention is paid to the simulation of Lévy processes. Here we use the general method of approximating a Lévy process by compound Poisson processes. Particular examples of Lévy processes can be simulated by making use of some of their properties. When the process is a time change or a PREFACE xiii subordination of a simpler process, this method can be of special advantage. Paths fromOUprocesses can be simulated by using a series representation or by the classical Euler scheme approximation. Chapter 9 gives an overviewof the pricing of exotic options under the different models. Typically, more or less explicit solutions are available under the Black–Scholes model. The situation worsens, however, under the Lévy market model. For barrier and lookback options, some results are available; however, the explicit calculations of prices in these cases are highly complex. Multiple integrals and inversion techniques are needed for numerical evaluation. In the even more advanced stochastic volatility models, prices can only be estimated by Monte Carlo simulations. In contrast with the Black–Scholes model, no closed formulas are available for the pricing of exotic options such as barrier and lookback options. The simulation techniques from Chapter 8 are used intensively here. Finally, Chapter 10 focuses on interest-rate modelling.We have so far assumed that interest rates are constant; in practice, of course, they are not.We follow the Heath– Jarrow–Merton approach and model the entire yield curve. As with the stock-price models, the underlying Brownian motion does not describe the empirical behaviour as it should. In order to make the model more realistic, we replace it again by a more flexible Lévy process.
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