Paris-Princeton Lectures on Mathematical Finance

This is the fourth volume of the Paris-Princeton Lectures in Mathematical Finance. The goal of this series is to publish cutting edge research in self contained articles prepared by established academics or promising young researchers invited by the editors. Contributions are refereed and particular attention is paid to the quality of the exposition, the goal being to publish articles that can serve as introductory references for research. The series is a result of frequent exchanges between researchers in finance and financial mathematics in Paris and Princeton. Many of us felt that the field would benefit from timely exposés of topics in which there is important progress. René Carmona, Erhan Cinlar, Ivar Ekeland, Elyes Jouini, José Scheinkman and Nizar Touzi serve in the first editorial board of the Paris-Princeton Lectures in Financial Mathematics. Although many of the chapters involve lectures given in Paris orPrinceton, we also invite other contributions. Springer Verlag kindly offered to hostthe initiative under the umbrella of the Lecture Notes in Mathematics series, and weare thankful to Catriona Byrne for her encouragement and her help. This fourth volume contains five chapters. In the first chapter, Areski Cousin, Monique Jeanblanc, and Jean -Paul Laurent discuss risk management and hedging of credit derivatives. The latter are over-the-counter (OTC) financial instruments designed to transfer credit risk associated to are ference entity from one counter party to another. The agreement involves a seller and a buyer of protection, the sellerbeing committed to cover the losses induced by the default. The popularity of theseinstruments lead a runaway market of complex derivatives whose risk management did not developas fast. This first chapter fills the gap by providing rigorous tools for quantifying and hedging counterparty risk in some of these markets. In the second chapter, Stéphane Crépey reviews the general theory of for-ward backward stochastic differential equations and their associated systems of partial integro-differential obstacle problems and applies it to pricing and hedging financial derivatives. Motivated by the optimal stopping and optimal stopping game formulations of American option and convertible bond pricing, he discussesthe well-posedness and sensitivities of reflected and doubly reflected Markovian Backward Stochastic Differential Equations. The third part of the paper is devotedto the variational inequality formulation of these problems and to a detailed discussion of viscosity solutions. Finally he also considers discrete path-dependenceissues such as dividend payments. The third chapter written by Olivier Guéant Jean-Michel Lasry and Pierre-Louis Lions presents an original and unified account of the theory and the applications of the mean field games as introduced and developed by Lasry and Lions in a seriesof lectures and scattered papers. This chapter provides systematic studies illustrating the application of the theory to domains as diverse as population behavior (theso-called Mexican wave), or economics (management of exhaustible resources). Some of the applications concern optimization of individual behavior when inter-acting with a large population of individuals with similar and possibly competing objectives. The analysis is also shown to apply to growth models and for example, to their application to salary distributions. The fourth chapter is contributed by David Hobson. It is concerned with the applications of the famous Skorohod embedding theorem to the proofs of model in dependent bounds on the prices of options. Beyond the obvious importance of thefinancial application, the value of this chapter lies in the insightful and extremely pedagogical presentation of the Skorohodem bedding problem and its application to the analysis of martingales with given one-dimensional marginals, providing a one-to-one correspondence between candidate price processes which are consistent with observed call option prices and solutions of the Skorokhod embedding problem, extremal solutions leading to robust model in dependent prices and hedges for exoticoptions. The final chapter is concerned with pricing and hedging in exponential Lévy models. Peter Tankov discusses three aspects of exponential Lévy models: absenceof arbitrage, including more recent results on the absence of arbitrage in multi dimensional models, properties of implied volatility, and modern approaches tohedging in these models. It is a self contained introduction surveying all the results and techniques that need to be known to be able to handle exponential Lévy models in finance.