These lectures notes aim at introducing L\'{e}vy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L\'{e}vy processes. We analyze a `toy' example of a L\'{e}vy process, viz. a L\'{e}vy jump-diffusion, which yet offers significant insight into the distributional and path structure of a L\'{e}vy process. Then, we present several important results about L\'{e}vy processes, such as infinite divisibility and the L\'{e}vy-Khintchine formula, the L\'{e}vy-It\^{o} decomposition, the It\^{o} formula for L\'{e}vy processes and Girsanov'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\'{e}vy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L\'{e}vy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of L\'{e}vy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.
Lévy processes 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 Lévy processes can be found in Prabhu (1998), in Barndorff-Nielsen, Mikosch, and Resnick (2001), in Kyprianou, Schoutens, and Wilmott (2005) and in Kyprianou (2006). In mathematical finance, Lévy processes are becoming extremely fashionable 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. Similarly, in the 'risk-neutral' world, we observe that implied volatilities are constant 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. Lévy processes 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 Lévy processes 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 Lévy process and some known examples. Using these as the reference point, we construct and study a Lévy jump-diffusion; despite its simple nature, it offers significant insights and an intuitive understanding of general Lévy processes. We then discuss infinitely divisible distributions and present the celebrated Lévy-Khintchine formula, which links processes to distributions. The opposite way, from distributions to processes, is the subject of the Lévy-Itô decomposition of a Lévy process. The Lévy measure, which is responsible for the richness of the class of Lévy processes, is studied in some detail and we use it to draw some conclusions about the path and moment properties of a Lévy process. In the next section, we look into several subclasses that have attracted special attention and then present some important results from semimartingale theory.
A study of martingale properties of Lévy processes and the Itô formula for Lévy processes 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 Lévy processes and the first part closes with an account on simulation methods for some Lévy processes.
The second part of the notes is devoted to the applications of Lévy processes in mathematical finance. We describe the possible approaches in modeling the price process of a financial asset using Lévy processes under the ‘real’ and the ‘risk-neutral’ world, and give a brief account of market incompleteness which links the two worlds. Then, we present a primer of popular Lévy models 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 pricing options in Lévy-driven models, viz. transform, partial integro-differential equation (PIDE) and Monte Carlo methods. Finally, we present some empirical results from the application of Lévy processes to real market financial data. The appendices collect some results about Poisson random variables and processes, explain some notation and provide information and links regarding 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 Lévy processes. We mention here the books of Bertoin (1996), Sato (1999), Applebaum (2004), Kyprianou (2006) on various aspects of Lévy processes. Cont and Tan
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