AN INTRODUCTION TO LEVY´ PROCESSES WITH APPLICATIONS IN FINANCE ANTONIS PAPAPANTOLEON Abstract. These lectures notes aim at introducing L´evyprocesses in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L´evyprocesses. We analyze a `toy' example of a L´evyprocess, viz. a L´evyjump-diffusion, which yet offers significant insight into the distributional and path structure of a L´evyprocess. Then, we present several important results about L´evy processes, such as infinite divisibility and the L´evy-Khintchine formula, the L´evy-It^odecomposition, the It^oformula for L´evyprocesses and Gir- sanov's transformation. Some (sketches of) proofs are presented, still the majority of proofs is omitted and the reader is referred to textbooks instead. In the second part, we turn our attention to the applications of L´evyprocesses in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L´evyprocesses and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of L´evy processes, and we also discuss three methods for pricing financial deriva- tives. Finally, some indicative evidence from applications to market data is presented. Contents Part 1. Theory 2 1. Introduction2 2. Definition5 3. `Toy' example: a L´evyjump-diffusion6 4. Infinitely divisible distributions and the L´evy-Khintchine formula8 5. Analysis of jumps and Poisson random measures 11 6. The L´evy-It^odecomposition 12 7. The L´evymeasure, path and moment properties 14 8. Some classes of particular interest 17 8.1. Subordinator 17 arXiv:0804.0482v2 [q-fin.PR] 3 Nov 2008 8.2. Jumps of finite variation 17 8.3. Spectrally one-sided 18 8.4. Finite first moment 18 2000 Mathematics Subject Classification. 60G51,60E07,60G44,91B28. Key words and phrases. L´evyprocesses, jump-diffusion, infinitely divisible laws, L´evy measure, Girsanov's theorem, asset price modeling, option pricing. These lecture notes were prepared for mini-courses taught at the University of Piraeus in April 2005 and March 2008, at the University of Leipzig in November 2005 and at the Technical University of Athens in September 2006 and March 2008. I am grateful for the opportunity of lecturing on these topics to George Skiadopoulos, Thorsten Schmidt, Nikolaos Stavrakakis and Gerassimos Athanassoulis. 1 2 ANTONIS PAPAPANTOLEON 9. Elements from semimartingale theory 18 10. Martingales and L´evyprocesses 21 11. It^o'sformula 22 12. Girsanov's theorem 23 13. Construction of L´evyprocesses 28 14. Simulation of L´evyprocesses 28 14.1. Finite activity 29 14.2. Infinite activity 29 Part 2. Applications in Finance 30 15. Asset price model 30 15.1. Real-world measure 30 15.2. Risk-neutral measure 31 15.3. On market incompleteness 32 16. Popular models 33 16.1. Black{Scholes 33 16.2. Merton 34 16.3. Kou 34 16.4. Generalized Hyperbolic 35 16.5. Normal Inverse Gaussian 36 16.6. CGMY 37 16.7. Meixner 37 17. Pricing European options 38 17.1. Transform methods 38 17.2. PIDE methods 40 17.3. Monte Carlo methods 42 18. Empirical evidence 42 Appendix A. Poisson random variables and processes 44 Appendix B. Compound Poisson random variables 45 Appendix C. Notation 46 Appendix D. Datasets 46 Appendix E. Paul L´evy 46 Acknowledgments 47 References 47 Part 1. Theory 1. Introduction L´evyprocesses play a central role in several fields of science, such as physics, in the study of turbulence, laser cooling and in quantum field theory; in engineering, for the study of networks, queues and dams; in economics, for continuous time-series models; in the actuarial science, for the calculation of insurance and re-insurance risk; and, of course, in mathematical finance. A comprehensive overview of several applications of L´evyprocesses can be found in Prabhu (1998), in Barndorff-Nielsen, Mikosch, and Resnick (2001), in Kyprianou, Schoutens, and Wilmott (2005) and in Kyprianou (2006). INTRODUCTION TO LEVY´ PROCESSES 3 150 USD/JPY 145 140 135 130 125 120 115 110 105 100 Oct 1997 Oct 1998 Oct 1999 Oct 2000 Oct 2001 Oct 2002 Oct 2003 Oct 2004 Figure 1.1. USD/JPY exchange rate, Oct. 1997{Oct. 2004. In mathematical finance, L´evy processes are becoming extremely fashion- able because they can describe the observed reality of financial markets in a more accurate way than models based on Brownian motion. In the `real' world, we observe that asset price processes have jumps or spikes, and risk managers have to take them into consideration; in Figure 1.1 we can observe some big price changes (jumps) even on the very liquid USD/JPY exchange rate. Moreover, the empirical distribution of asset returns exhibits fat tails and skewness, behavior that deviates from normality; see Figure 1.2 for a characteristic picture. Hence, models that accurately fit return distributions are essential for the estimation of profit and loss (P&L) distributions. Simi- larly, in the `risk-neutral' world, we observe that implied volatilities are con- stant neither across strike nor across maturities as stipulated by the Black and Scholes (1973) (actually, Samuelson 1965) model; Figure 1.3 depicts a typical volatility surface. Therefore, traders need models that can capture the behavior of the implied volatility smiles more accurately, in order to handle the risk of trades. L´evyprocesses provide us with the appropriate tools to adequately and consistently describe all these observations, both in the `real' and in the `risk-neutral' world. The main aim of these lecture notes is to provide an accessible overview of the field of L´evyprocesses and their applications in mathematical finance to the non-specialist reader. To serve that purpose, we have avoided most of the proofs and only sketch a number of proofs, especially when they offer some important insight to the reader. Moreover, we have put emphasis on the intuitive understanding of the material, through several pictures and simulations. We begin with the definition of a L´evyprocess and some known exam- ples. Using these as the reference point, we construct and study a L´evy jump-diffusion; despite its simple nature, it offers significant insights and an intuitive understanding of general L´evyprocesses. We then discuss infinitely divisible distributions and present the celebrated L´evy{Khintchine formula, which links processes to distributions. The opposite way, from distributions 4 ANTONIS PAPAPANTOLEON 80 60 40 20 0 −0.02 −0.01 0.0 0.01 0.02 Figure 1.2. Empirical distribution of daily log-returns for the GBP/USD exchange rate and fitted Normal distribution. to processes, is the subject of the L´evy-It^odecomposition of a L´evypro- cess. The L´evymeasure, which is responsible for the richness of the class of L´evyprocesses, is studied in some detail and we use it to draw some conclu- sions about the path and moment properties of a L´evyprocess. In the next section, we look into several subclasses that have attracted special atten- tion and then present some important results from semimartingale theory. A study of martingale properties of L´evyprocesses and the It^oformula for L´evyprocesses follows. The change of probability measure and Girsanov's theorem are studied is some detail and we also give a complete proof in the case of the Esscher transform. Next, we outline three ways for constructing new L´evyprocesses and the first part closes with an account on simulation methods for some L´evyprocesses. The second part of the notes is devoted to the applications of L´evypro- cesses in mathematical finance. We describe the possible approaches in mod- eling the price process of a financial asset using L´evyprocesses under the `real' and the `risk-neutral' world, and give a brief account of market incom- pleteness which links the two worlds. Then, we present a primer of popular L´evymodels in the mathematical finance literature, listing some of their key properties, such as the characteristic function, moments and densities (if known). In the next section, we give an overview of three methods for pric- ing options in L´evy-driven models, viz. transform, partial integro-differential equation (PIDE) and Monte Carlo methods. Finally, we present some em- pirical results from the application of L´evyprocesses to real market financial data. The appendices collect some results about Poisson random variables and processes, explain some notation and provide information and links re- garding the data sets used. Naturally, there is a number of sources that the interested reader should consult in order to deepen his knowledge and understanding of L´evypro- cesses. We mention here the books of Bertoin (1996), Sato (1999), Apple- baum (2004), Kyprianou (2006) on various aspects of L´evyprocesses. Cont and Tankov (2003) and Schoutens (2003) focus on the applications of L´evy INTRODUCTION TO LEVY´ PROCESSES 5 14 13.5 13 12.5 12 implied vol (%) 11.5 11 10.5 10 10 20 30 40 9 10 50 8 60 6 7 70 4 5 80 2 3 delta (%) or strike 90 1 maturity Figure 1.3. Implied volatilities of vanilla options on the EUR/USD exchange rate on November 5, 2001. processes in finance. The books of Jacod and Shiryaev (2003) and Prot- ter (2004) are essential readings for semimartingale theory, while Shiryaev (1999) blends semimartingale theory and applications to finance in an im- pressive manner. Other interesting and inspiring sources are the papers by Eberlein (2001), Cont (2001), Barndorff-Nielsen and Prause (2001), Carr et al. (2002), Eberlein and Ozkan(¨ 2003) and Eberlein (2007). 2. Definition Let (Ω; F; F;P ) be a filtered probability space, where F = FT and the filtration F = (Ft)t2[0;T ] satisfies the usual conditions.