Nonlinear Option Pricing

Julien Guyon, Pierre Henry-Labordere

"... provides a wide overview of the advanced modern techniques applied in financial modeling. It gives an optimal combination of analytical and numerical tools in quantitative finance. It could provide guidance on the development of nonlinear methods of option pricing for practitioners as well as for analysts."—Nikita Y. Ratanov, from Mathematical Reviews Clippings, January 2015"… anyone with interest in quantitative finance and partial differential equations/continuous time stochastic analysis will not only greatly enjoy this book, but he or she will find both many numerical ideas of real practical interest as well as material for academic research, perhaps for years to come."—Peter Friz, The Bachelier Finance Society"This textbook provides a comprehensive treatment of numerical methods for nonlinear option pricing problems."—Zentralblatt MATH 1285"It is the only book of its kind. … The contribution of this book is threefold: (a) a practical, intuitive, and self-contained derivation of various of the latest derivative pricing models driven by diffusion processes; (b) an exposition of various advanced Monte Carlo simulation schemes for solving challenging nonlinear problems arising in financial engineering; (c) a clear and accessible survey of the theory of nonlinear PDEs. The authors have done a brilliant job providing just the right amount of rigorous theory required to understand the advanced methodologies they present. … Julien Guyon and Pierre Henry-Labordère, as befitting their reputations as star quants, have done an excellent job presenting the latest theory of nonlinear PDEs and their applications to finance. Much of the material in the book consists of the authors’ own original results. I highly recommend this book to seasoned mathematicians and experienced quants in the industry … Mathematicians will be able to see how practitioners argue heuristically to arrive at solutions of the toughest problems in financial engineering; practitioners of quantitative finance will find the book perfectly balanced between mathematical theory, financial modelling, and schemes for numerical implementation."—Quantitative Finance, 2014"Ever since Black and Scholes solved their eponymous linear PDE in 1969, the complexity of problems plaguing financial practitioners has exploded (non-linearly!). How fitting it is that nonlinear PDEs are now routinely used to extend the original framework. Written by two leading quants at two leading financial houses, this book is a tour de force on the use of nonlinear PDEs in financial valuation."—Peter Carr, PhD, Global Head of Market Modeling, Morgan Stanley, New York, and Executive Director of Masters in Mathematical Finance, Courant Institute of Mathematical Sciences, New York University "Finance used to be simple; you could go a long way with just linearity and positivity but this is not the case anymore. This superb book gives a wide array of modern methods for modern problems."—Bruno Dupire, Head of Quantitative Research, Bloomberg L.P."In this unique and impressive book, the authors apply sophisticated modern tools of pure and applied mathematics, such as BSDEs and particle methods, to solve challenging nonlinear problems of real practical interest, such as the valuation of guaranteed equity-linked annuity contracts and the calibration of local stochastic volatility models. Not only that, but sketches of proofs and implementation details are included. No serious student of mathematical finance, whether practitioner or academic, can afford to be without it."—Jim Gatheral, Presidential Professor, Baruch College, CUNY, and author of The Volatility Surface"Guyon and Henry-Labordère have produced an impressive textbook, which covers options and derivatives pricing from the point of view of nonlinear PDEs. This book is a comprehensive survey of nonlinear techniques, ranging from American options, uncertain volatility, and uncertain correlation models. It is aimed at graduate students or quantitative analysts with a strong mathematical background. They will find the book reasonably self-contained, i.e., discussing both the mathematical theory and the applications, in a very balanced approach. A must-read for the serious quantitative analyst."—Marco Avellaneda, Courant Institute of Mathematical Sciences, New York University

• Some Excursions in Option Pricing. Nonlinear PDEs: A Bit of Theory. Examples of Nonlinear Problems in Finance. Early Exercise Problems. Backward Stochastic Differential Equations. The Uncertain Lapse and Mortality Model. The Uncertain Volatility Model. McKean Nonlinear Stochastic Differential Equations. Calibration of Local Stochastic Volatility Models to Market Smiles. Calibration of Local Correlation Models to Market Smiles. Marked Branching Diffusions. References. Index.