ToCamilla Whatgoodisitforaman togainthewholeworld, yetforfeithislife? Mc.8,36 Andrea Pascucci PDE and Martingale Methods in Option Pricing AndreaPascucci DepartmentofMathematics UniversityofBologna [email protected] B&SS–Bocconi&SpringerSeries ISSNprintedition:2039-1471 ISSNelectronicedition:2039-148X ISBN978-88-470-1780-1 e-ISBN978-88-470-1781-8 DOI10.1007/978-88-470-1781-8 LibraryofCongressControlNumber:2010929483 SpringerMilanDordrechtHeidelbergLondonNewYork ©Springer-VerlagItalia2011 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerial isconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmsorinotherways,andstorageindatabanks.Duplicationofthis publicationorpartsthereofispermittedonlyundertheprovisionsoftheItalianCopyrightLawinits currentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violationsareliableto prosecutionundertheItalianCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply,even in the absence of a specific statement,that such names are exempt from the relevant protectivelawsandregulationsandthereforefreeforgeneraluse. 9 8 7 6 5 4 3 2 1 Cover-Design:KDesign,Heidelberg TypesettingwithLATEX:PTP-Berlin,ProtagoTEX-ProductionGmbH,Germany(www.ptp-berlin.eu) PrintingandBinding:GrafichePorpora,Segrate(Mi) PrintedinItaly Springer-VerlagItaliasrl–ViaDecembrio28–20137Milano SpringerisapartofSpringerScience+BusinessMedia(www.springer.com) Preface This book gives an introduction to the mathematical, probabilistic and nu- merical methods used in the modern theory of option pricing. It is intended as a textbook for graduate and advanced undergraduate students, but I hope itwillbeusefulalsoforresearchersandprofessionalsinthefinancialindustry. Stochasticcalculusanditsapplicationstothearbitragepricingoffinancial derivatives form the main theme. In presenting these, by now classic, topics, the emphasis is put on the more quantitative rather than economic aspects. Being aware that the literature in this field is huge, I mention the following incomplete list of monographs whose contents overlap with those of this text: in alphabetic order, Avellaneda and Laurence [14], Benth [43], Bjo¨rk [47], DanaandJeanblanc[84],Dewynne,HowisonandWilmott[340],Dothan[100], Duffie [102], Elliott and Kopp [120], Epps [121], Follmer and Schied [134], Glasserman [158], Huang and Litzenberger [171], Ingersoll [178], Karatzas [200; 202], Lamberton and Lapeyre [226], Lipton [239], Merton [252], Musiela and Rutkowski [261], Neftci [264], Shreve [310; 311], Steele [315], Zhu, Wu and Chern [349]. What distinguishes this book from others is the attempt to present the matterby giving equal weighttothe probabilistic point of view,basedon the martingale theory, and the analytical one, based on partial differential equa- tions. Thepresentbook doesnotclaimtodescribe the latestdevelopmentsin mathematical finance: that target would indeed be very ambitious, given the speed of progress of research in the field. Instead, I have chosen to develop some of the essential ideas of the classical pricing theory to devote space to the fundamental mathematical and numerical tools when they arise. Thus I hope to provide a sound background of basic knowledge which may facilitate the independent study of newer problems and more advanced models. The theory of stochastic calculus, for continuous and discontinuous pro- cesses,constitutes thebulk ofthebook: Chapters3onstochasticprocesses,4 on Brownian integration and 9 on stochastic differential equations may form the material for an introductory course on stochastic calculus. In these chap- ters, I have constantly sought to combine the theoretical concepts to the in- VI Preface sightonthefinancialmeaning,inordertomakethepresentationlessabstract and more motivated: in fact many theoretical concepts naturally lend them- selves to an intuitive and meaningful economic interpretation. The origin of this book can be traced to courses on option pricing which I taught at the master program in Quantitative Finance of the University of Bologna, which I have directed with Sergio Polidoro since its beginning, in 2004. I wrote the first version as lecture notes for my courses. During these years, I substantially improved and extended the text with the inclusion of sections on numerical methods and the addition of completely new chapters on stochastic calculus for jump processes and Fourier methods. Nevertheless, during these years the original structure of the book remained essentially unchanged. I am grateful to many people for the suggestions and helpful comments with which supported and encouraged the writing of the book: in particular I would like to thank several colleagues and PhD students for many valuable suggestions on the manuscript, including David Applebaum, Francesco Car- avenna, Alessandra Cretarola, Marco Di Francesco, Piero Foscari, Paolo Fos- chi, Ermanno Lanconelli, Antonio Mura, Cornelis Oosterlee, Sergio Polidoro, Valentina Prezioso, Enrico Priola, Wolfgang Runggaldier, Tiziano Vargiolu, Valeria Volpe. I also express my thanks to Rossella Agliardi, co-author of Chapter 13, and to Matteo Camaggi for helping me in the translation of the book. It is greatly appreciated if readers could forward any errors, misprints or suggested improvements to: [email protected] Corrections received after publication will be posted on the website: http://www.dm.unibo.it/∼pascucci/ Bologna, November 2010 Andrea Pascucci Contents Preface ....................................................... V General notations............................................. XV 1 Derivatives and arbitrage pricing ......................... 1 1.1 Options ............................................... 1 1.1.1 Main purposes ................................... 3 1.1.2 Main problems................................... 4 1.1.3 Rules of compounding ............................ 4 1.1.4 Arbitrage opportunities and Put-Call parity formula .. 5 1.2 Risk-neutral price and arbitrage pricing ................... 7 1.2.1 Risk-neutral price ................................ 7 1.2.2 Risk-neutral probability ........................... 8 1.2.3 Arbitrage price................................... 8 1.2.4 A generalization of the Put-Call parity.............. 10 1.2.5 Incomplete markets............................... 11 2 Discrete market models................................... 15 2.1 Discrete markets and arbitrage strategies .................. 15 2.1.1 Self-financing and predictable strategies ............. 16 2.1.2 Normalized market ............................... 19 2.1.3 Arbitrage opportunities and admissible strategies..... 20 2.1.4 Equivalent martingale measure..................... 21 2.1.5 Change of numeraire.............................. 24 2.2 European derivatives.................................... 26 2.2.1 Pricing in an arbitrage-free market ................. 27 2.2.2 Completeness .................................... 30 2.2.3 Fundamental theorems of asset pricing .............. 31 2.2.4 Markov property ................................. 34 2.3 Binomial model ........................................ 35 2.3.1 Martingale measure and arbitrage price ............. 38 2.3.2 Hedging strategies................................ 40 VIII Contents 2.3.3 Binomial algorithm ............................... 45 2.3.4 Calibration ...................................... 50 2.3.5 Binomial model and Black-Scholes formula .......... 53 2.3.6 Black-Scholes differential equation .................. 60 2.4 Trinomial model........................................ 62 2.4.1 Pricing and hedging in an incomplete market ........ 66 2.5 American derivatives.................................... 72 2.5.1 Arbitrage price................................... 74 2.5.2 Optimal exercise strategies ........................ 80 2.5.3 Pricing and hedging algorithms .................... 83 2.5.4 Relations with European options ................... 88 2.5.5 Free-boundary problem for American options ........ 90 2.5.6 American and European options in the binomial model 93 3 Continuous-time stochastic processes ..................... 97 3.1 Stochastic processes and real Brownian motion............. 97 3.1.1 Markov property ................................. 100 3.1.2 Brownian motion and the heat equation............. 102 3.2 Uniqueness ............................................ 103 3.2.1 Law of a continuous process ....................... 103 3.2.2 Equivalence of processes........................... 105 3.2.3 Modifications and indistinguishable processes ........ 107 3.2.4 Adapted and progressively measurable processes...... 110 3.3 Martingales............................................ 111 3.3.1 Doob’s inequality................................. 113 3.3.2 Martingale spaces: M2 and M2.................... 114 c 3.3.3 The usual hypotheses ............................. 117 3.3.4 Stopping times and martingales .................... 120 3.4 Riemann-Stieltjes integral ............................... 125 3.4.1 Bounded-variation functions ....................... 127 3.4.2 Riemann-Stieltjes integral and Itoˆ formula........... 131 3.4.3 Regularity of the paths of a Brownian motion........ 134 4 Brownian integration ..................................... 139 4.1 Stochastic integral of deterministic functions ............... 140 4.2 Stochastic integral of simple processes..................... 141 4.3 Integral of L2-processes ................................. 145 4.3.1 Itoˆ and Riemann-Stieltjes integral .................. 149 4.3.2 Itoˆ integral and stopping times..................... 151 4.3.3 Quadratic variation process........................ 153 4.3.4 Martingales with bounded variation ................ 156 4.3.5 Co-variation process .............................. 157 4.4 Integral of L2 -processes ................................ 159 loc 4.4.1 Local martingales ................................ 161 4.4.2 Localization and quadratic variation ................ 163 Contents IX 5 Itoˆ calculus ............................................... 167 5.1 Itoˆ processes........................................... 168 5.1.1 Itoˆ formula for Brownian motion ................... 169 5.1.2 General formulation .............................. 174 5.1.3 Martingales+and parabolic equations ............... 176 5.1.4 Geometric Brownian motion ....................... 176 5.2 Multi-dimensional Itoˆ processes .......................... 179 5.2.1 Multi-dimensional Itoˆ formula...................... 183 5.2.2 Correlated Brownian motion+and martingales ....... 188 5.3 Generalized Itoˆ formulas ................................ 191 5.3.1 Itoˆ formula and+weak derivatives .................. 191 5.3.2 Tanaka formula+and local times ................... 194 5.3.3 Tanaka+formula for Itoˆ processes .................. 197 5.3.4 Local+time and Black-Scholes formula .............. 198 6 Parabolic PDEs with variable coefficients: uniqueness..... 203 6.1 Maximum principle and Cauchy-Dirichlet problem.......... 206 6.2 Maximum principle and Cauchy problem .................. 208 6.3 Non-negative solutions of the Cauchy problem ............. 213 7 Black-Scholes model ...................................... 219 7.1 Self-financing strategies ................................. 220 7.2 Markovian strategies and Black-Scholes equation ........... 222 7.3 Pricing................................................ 225 7.3.1 Dividends and time-dependent parameters........... 228 7.3.2 Admissibility and absence of arbitrage .............. 229 7.3.3 Black-Scholes analysis: heuristic approaches.......... 231 7.3.4 Market price of risk............................... 233 7.4 Hedging............................................... 236 7.4.1 The Greeks...................................... 236 7.4.2 Robustness of the model .......................... 245 7.4.3 Gamma and Vega-hedging......................... 246 7.5 Implied volatility ....................................... 248 7.6 Asian options .......................................... 252 7.6.1 Arithmetic average ............................... 253 7.6.2 Geometric average................................ 255 8 Parabolic PDEs with variable coefficients: existence ...... 257 8.1 Cauchy problem and fundamental solution................. 258 8.1.1 Levi’s parametrix method ......................... 260 8.1.2 Gaussian estimates and adjoint operator ............ 261 8.2 Obstacle problem....................................... 263 8.2.1 Strong solutions.................................. 265 8.2.2 Penalization method.............................. 268