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Portfolio Optimization and Option Pricing Under Defaultable Lévy

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Portfolio Optimization and Option Pricing Under Defaultable Lévy Universit`adegli Studi di Padova Dipartimento di Matematica Scuola di Dottorato in Scienze Matematiche Indirizzo Matematica Computazionale XXVI◦ ciclo Portfolio optimization and option pricing under defaultable L´evy driven models Direttore della scuola: Ch.mo Prof. Paolo Dai Pra Coordinatore dell’indirizzo: Ch.ma Prof.ssa. Michela Redivo Zaglia Supervisore: Ch.mo Prof. Tiziano Vargiolu Universit`adegli Studi di Padova Corelatori: Ch.mo Prof. Agostino Capponi Ch.mo Prof. Andrea Pascucci Johns Hopkins University Universit`adegli Studi di Bologna Dottorando: Stefano Pagliarani To Carolina and to my family “An old man, who had spent his life looking for a winning formula (martingale), spent the last days of his life putting it into practice, and his last pennies to see it fail. The martingale is as elusive as the soul.” Alexandre Dumas Mille et un fantˆomes, 1849 Acknowledgements First and foremost, I would like to thank my supervisor Prof. Tiziano Vargiolu and my co-advisors Prof. Agostino Capponi and Prof. Andrea Pascucci, for coauthoring with me all the material appearing in this thesis. In particular, thank you to Tiziano Vargiolu for supporting me scientifically and finan- cially throughout my research and all the activities related to it. Thank you to Agostino Capponi for inviting me twice to work with him at Purdue University, and for his hospital- ity and kindness during my double stay in West Lafayette. Thank you to Andrea Pascucci for wisely leading me throughout my research during the last four years, and for giving me the chance to collaborate with him side by side, thus sharing with me his knowledge and his experience. The material collected in this thesis has been also coauthored by Paolo Foschi, Matthew Lorig and Candia Riga; thank to all of them for this. In particular, thank to Matt for helping me with my job search, and for what became a stable and fruitful collaboration. A special thank goes to Prof. Wolfgang Runggaldier for kindly advising me in the decisions I had to make during the last three years, and for bringing to my attention some issues related to my research, along with his useful suggestions and sincere opinions. Thank you to the anonymous Referee for carefully reading this thesis, and for giving me the chance to improve it by means of his useful comments and suggestions. Thank to all the colleagues of the Department of Mathematics of Padova who helped me throughout my research, systematically or occasionally, by means of useful discussions, comments or suggestions. In particular, thank you to Prof. Paolo Dai Pra for his comments and suggestions regarding Chapter 2 of this thesis. Thank you to Dr. Daniele Fontanari for helping me with the writing of the Mathematica notebooks uploaded on [167]. Thank you to Dr. Martino Garonzi for helping me with the proof of Lemma 3.11, Chapter 2, and to Dr. Giulio Miglietta for the beautiful endless discussions about random topics in Stochastic Analysis. Thank you to Carolina and to my family for their constant support; and again many thanks to Carolina, to Anna, Daniele and all the other friends in Padova, who made the last three years in Padova epic and unforgettable. Last, but not least, a great thank to all the administrative people, the technicians, the cleaning service, and all the people working in the Department of Mathematics of Padova, who make Archimede’s Tower an amazing place to work, study and socialize. In particular, thank you to Angela Puca for assigning me to the best office of the building (n. 732) in terms of location and atmosphere. v Abstract In this thesis we study some portfolio optimization and option pricing problems in market models where the dynamics of one or more risky assets are driven by L´evy processes, and it is divided in four independent parts. In the first part we study the portfolio optimization problem, for the logarithmic terminal utility and the logarithmic consumption utility, in a multi-defaultable L´evy driven model. In the second part we introduce a novel technique to price European defaultable claims when the pre-defaultable dynamics of the underlying asset follows an expo- nential L´evy process. In the third part we develop a novel methodology to obtain analytical expansions for the prices of European derivatives, under stochastic and/or local volatility models driven by L´evy processes, by analytically expanding the integro-differential operator associated to the pricing problem. In the fourth part we present an extension of the latter technique which allows for obtaining analytical expansion in option pricing when dealing with path-dependent Asian-style derivatives. KEYWORDS: portfolio optimization, stochastic control, dynamic programming, HJB equation, jump-diffusion, multi-default, direct contagion, information-induced contagion, L´evy, exponential, default, equity-credit, default intensity, change of measure, Girsanov theorem, Esscher transform, characteristic function, abstract Cauchy problem, eigenvec- tors expansion, Fourier inversion, local volatility, analytical approximation, partial integro- differential equation, Fourier methods, local-stochastic volatility, asymptotic expansion, pseudo-differential calculus, implied volatility, CEV, Heston, SABR, Asian options, arith- metic average process, hypoelliptic operators, ultra-parabolic operators, Black and Scholes, option pricing, Greeks. vii Riassunto In questa tesi studiamo alcuni problemi di portfolio optimization e di option pricing in modelli di mercato dove le dinamiche di uno o pi`utitoli rischiosi sono guidate da processi di L´evy. La tesi `edivisa in quattro parti indipendenti. Nella prima parte studiamo il problema di ottimizzare un portafoglio, inteso come massimizzazione di un’utilit`alogaritmica della ricchezza finale e di un’utilit`aloga- ritmica del consumo, in un modello guidato da processi di L´evy e in presenza di fallimenti simultanei. Nella seconda parte introduciamo una nuova tecnica per il prezzaggio di opzioni europee soggette a fallimento, i cui titoli sottostanti seguono dinamiche che prima del fallimento sono rappresentate da processi di L´evy esponenziali. Nella terza parte sviluppiamo un nuovo metodo per ottenere espansioni analitiche per i prezzi di derivati europei, sotto modelli a volatilit`astocastica e locale guidati da processi di L´evy, espandendo analiticamente l’operatore integro-differenziale associato al problema di prezzaggio. Nella quarta, e ultima parte, presentiamo un estensione della tecnica precedente che consente di ottenere espansioni analitiche per i prezzi di opzioni asiatiche, ovvero particolari tipi di opzioni il cui payoff dipende da tutta la traiettoria del titolo sot- tostante. KEYWORDS: portfolio optimization, stochastic control, dynamic programming, HJB equation, jump-diffusion, multi-default, direct contagion, information-induced contagion, L´evy, exponential, default, equity-credit, default intensity, change of measure, Girsanov theorem, Esscher transform, characteristic function, abstract Cauchy problem, eigenvec- tors expansion, Fourier inversion, local volatility, analytical approximation, partial integro- differential equation, Fourier methods, local-stochastic volatility, asymptotic expansion, pseudo-differential calculus, implied volatility, CEV, Heston, SABR, Asian options, arith- metic average process, hypoelliptic operators, ultra-parabolic operators, Black and Scholes, option pricing, Greeks. viii Table of contents Introduction 1 I Stochastic optimization in multi-defaultable market models 7 1 Portfolio optimization in a defaultable L`evy-driven market model 9 1.1 Introduction.................................... 10 1.2 Asimplifiedmodel ................................ 12 1.3 Thegeneralsetting............................... 13 1.4 The portfolio optimization problem . ...... 15 1.5 Dynamicprogrammingsolution . 17 1.6 Examples ..................................... 24 1.6.1 Diffusiondynamicswithdefault. 25 1.6.2 One default-free stock and one defaultable bond . ...... 28 1.6.3 Two stocks, one of which defaultable . 29 1.6.4 Severaldefaultablebonds . 31 1.6.5 Two defaultable bonds with direct contagion . ..... 32 1.7 GOPandGOP-denominatedprices. 34 1.7.1 Strict supermartingale inverse-GOP . 35 II Pricing european claims in defaultable exponential L´evy models 37 2 Pricing vulnerable claims in a L´evy driven model 39 2.1 Introduction.................................... 40 2.2 Pricingofadefaultableclaim . 42 2.3 A characteristic function approach in the negative power intensity case . 45 2.3.1 A useful characterization of the characteristic function . 47 2.3.2 Eigenvaluesexpansion . 50 2.4 Numericalanalysis ............................... 57 2.4.1 Characteristicfunction. 57 2.4.2 Fourierpricingformulas . 58 2.4.3 Option implied volatilities . 59 2.4.4 Bondprices................................ 60 2.5 Conclusions .................................... 61 ix 2.6 Appendix ..................................... 62 2.6.1 Proofs of Lemmas 3.6, 3.11, 3.13 and 3.14 . 62 2.7 Figuresandtables ................................ 64 III Analytical expansions for parabolic PIDE’s and application to op- tion pricing 73 3 Adjoint expansions in local L´evy models 75 3.1 Introduction.................................... 76 3.2 Generalframework ................................ 78 3.3 LVmodelswithGaussianjumps . 81 3.3.1 Simplified Fourier approach for LV models . 87 3.4 LocalL´evymodels ................................ 89 3.4.1 Highorderapproximations . 95 3.5 Numericaltests.................................. 96 3.5.1 Tests under CEV-Merton
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