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Paris-Princeton Lectures on Mathematical Finance

Author

Listed:
  • Jose Scheinkman

    (Department of Economics - Princeton University)

  • René Carmona

    (ORFE - Department of Operations Research and Financial Engineering - Princeton University)

  • Erhan Cinlare

    (Departement of probability - Princeton University)

  • Ivar Ekeland

    (Canada Research Chair in Mathematical Economics - UBC - University of British Columbia)

  • Elyès Jouini

    (CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris sciences et lettres - CNRS - Centre National de la Recherche Scientifique)

  • Nizar Touzi

    (CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique - X - École polytechnique - CNRS - Centre National de la Recherche Scientifique)

Abstract

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.

Suggested Citation

  • Jose Scheinkman & René Carmona & Erhan Cinlare & Ivar Ekeland & Elyès Jouini & Nizar Touzi, 2010. "Paris-Princeton Lectures on Mathematical Finance," Post-Print halshs-00706281, HAL.
  • Handle: RePEc:hal:journl:halshs-00706281
    as

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    Citations

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    Cited by:

    1. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2016. "Numerical Analysis On Local Risk-Minimization For Exponential Lévy Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(02), pages 1-27, March.
    2. S. Heise & R. Kühn, 2012. "Derivatives and credit contagion in interconnected networks," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 85(4), pages 1-19, April.
    3. Alexander M. G. Cox & Jiajie Wang, 2011. "Root's barrier: Construction, optimality and applications to variance options," Papers 1104.3583, arXiv.org, revised Mar 2013.
    4. Alexander M. G. Cox & Christoph Hoeggerl, 2013. "Model-independent no-arbitrage conditions on American put options," Papers 1301.5467, arXiv.org.
    5. Régis Chenavaz & Corina Paraschiv & Gabriel Turinici, 2021. "Dynamic Pricing of New Products in Competitive Markets: A Mean-Field Game Approach," Dynamic Games and Applications, Springer, vol. 11(3), pages 463-490, September.
    6. A. Philip Dawid & Steven de Rooij & Peter Grunwald & Wouter M. Koolen & Glenn Shafer & Alexander Shen & Nikolai Vereshchagin & Vladimir Vovk, 2011. "Probability-free pricing of adjusted American lookbacks," Papers 1108.4113, arXiv.org.
    7. Jos'e E. Figueroa-L'opez & Ruoting Gong & Yuchen Han, 2021. "Estimation of Tempered Stable L\'{e}vy Models of Infinite Variation," Papers 2101.00565, arXiv.org, revised Feb 2022.
    8. {L}ukasz Delong & Antoon Pelsser, 2013. "Instantaneous mean-variance hedging and instantaneous Sharpe ratio pricing in a regime-switching financial model, with applications to equity-linked claims," Papers 1303.4082, arXiv.org.
    9. Vladimir Vovk, 2012. "Continuous-time trading and the emergence of probability," Finance and Stochastics, Springer, vol. 16(4), pages 561-609, October.
    10. Tavin, Bertrand, 2015. "Detection of arbitrage in a market with multi-asset derivatives and known risk-neutral marginals," Journal of Banking & Finance, Elsevier, vol. 53(C), pages 158-178.
    11. B. Cooper Boniece & Jos'e E. Figueroa-L'opez & Yuchen Han, 2022. "Efficient Volatility Estimation for L\'evy Processes with Jumps of Unbounded Variation," Papers 2202.00877, arXiv.org.
    12. Cox, Alexander M.G. & Obłój, Jan, 2015. "On joint distributions of the maximum, minimum and terminal value of a continuous uniformly integrable martingale," Stochastic Processes and their Applications, Elsevier, vol. 125(8), pages 3280-3300.
    13. Christoph Belak & Daniel Hoffmann & Frank T. Seifried, 2020. "Continuous-Time Mean Field Games with Finite StateSpace and Common Noise," Working Paper Series 2020-05, University of Trier, Research Group Quantitative Finance and Risk Analysis.
    14. Takuji Arai & Yuto Imai & Ryoichi Suzuki, 2015. "Numerical analysis on local risk-minimization forexponential L\'evy models," Papers 1506.03898, arXiv.org.

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