New Tools to Solve Your Option Pricing Problems
For nonlinear PDEs encountered in quantitative finance, advanced probabilistic methods are needed to address dimensionality issues. Written by two leaders in quantitative research—including Risk magazine’s 2013 Quant of the Year—Nonlinear Option Pricing compares various numerical methods for solving high-dimensional nonlinear problems arising in option pricing. Designed for practitioners, it is the first authored book to discuss nonlinear Black-Scholes PDEs and compare the efficiency of many different methods.
Real-World Solutions for Quantitative Analysts
The book helps quants develop both their analytical and numerical expertise. It focuses on general mathematical tools rather than specific financial questions so that readers can easily use the tools to solve their own nonlinear problems. The authors build intuition through numerous real-world examples of numerical implementation. Although the focus is on ideas and numerical examples, the authors introduce relevant mathematical notions and important results and proofs. The book also covers several original approaches, including regression methods and dual methods for pricing chooser options, Monte Carlo approaches for pricing in the uncertain volatility model and the uncertain lapse and mortality model, the Markovian projection method and the particle method for calibrating local stochastic volatility models to market prices of vanilla options with/without stochastic interest rates, the a + bλ technique for building local correlation models that calibrate to market prices of vanilla options on a basket, and a new stochastic representation of nonlinear PDE solutions based on marked branching diffusions.
Table of Contents
Option Pricing in a Nutshell
The super-replication paradigm
Stochastic representation of solutions of linear PDEs
The Monte Carlo method
Euler discretization error
Some Excursions in Option Pricing
Complete market models
Beyond replication and super-replication
Nonlinear PDEs: A Bit of Theory
Nonlinear second order parabolic PDEs: some generalities
Why is a pricing equation a parabolic PDE?
Finite difference schemes
Stochastic control and the Hamilton-Jacobi-Bellman PDE
Examples of Nonlinear Problems in Finance
The uncertain volatility model
Transaction costs: Leland’s model
Super-replication under delta and gamma constraints
The uncertain mortality model for reinsurance deals
Credit valuation adjustment
The passport option
Early Exercise Problems
Super-replication of American options
American options and semilinear PDEs
The dual method for American options
On the ownership of the exercise right
On the finiteness of exercise dates
On the accounting of multiple coupons
Finite difference methods for American options
Monte Carlo methods for American options
Case study: pricing and hedging of a multi-asset convertible bond
Introduction to chooser options
Regression methods for chooser options
The dual algorithm for chooser options
Numerical examples of pricing of chooser options
Backward Stochastic Differential Equations
First order BSDEs
Reflected first order BSDEs
Second order BSDEs
The Uncertain Lapse and Mortality Model
The deterministic lapse and mortality model
The uncertain lapse and mortality model
Pricing the option on the up-and-out barrier
An example of PDE implementation
Monte Carlo pricing
Monte Carlo pricing of the option on the up-and-out barrier
Link with first order BSDEs
Numerical results using PDE
Numerical results using Monte Carlo
The Uncertain Volatility Model
The parametric approach
Solving the UVM with BSDEs
McKean Nonlinear Stochastic Differential Equations
The particle method in a nutshell
Propagation of chaos and convergence of the particle method
Calibration of Local Stochastic Volatility Models to Market Smiles
The calibration condition
Existence of the calibrated local stochastic volatility model
The PDE method
The Markovian projection method
The particle method
Adding stochastic interest rates
The particle method: numerical tests
Calibration of Local Correlation Models to Market Smiles
The FX triangle smile calibration problem
A new representation of admissible correlations
The particle method for local correlation
Some examples of pairs of functions (a, b)
Some links between local correlations
Joint extrapolation of local volatilities
Price impact of correlation
The equity index smile calibration problem
Numerical experiments on the FX triangle problem
Generalization to stochastic volatility, stochastic interest rates, and stochastic dividend yield
Marked Branching Diffusions
Nonlinear Monte Carlo algorithms for some semilinear PDEs
Marked branching diffusions
Application: Credit valuation adjustment algorithm
System of semilinear PDEs
Exercises appear at the end of each chapter.
"... 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.""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."
—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
—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