Nonlinear Option Pricing: 1st Edition (Hardback) book cover

Nonlinear Option Pricing

1st Edition

By Julien Guyon, Pierre Henry-Labordere

Chapman and Hall/CRC

484 pages | 55 B/W Illus.

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pub: 2013-12-19
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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.


"… 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

Table of Contents

Option Pricing in a Nutshell

The super-replication paradigm

Stochastic representation of solutions of linear PDEs

Monte Carlo

The Monte Carlo method

Euler discretization error

Romberg extrapolation

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

Viscosity solutions

Examples of Nonlinear Problems in Finance

American options

The uncertain volatility model

Transaction costs: Leland’s model

Illiquid markets

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

Reinsurance deals

The deterministic lapse and mortality model

The uncertain lapse and mortality model

Path-dependent payoffs

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 model

The parametric approach

Solving the UVM with BSDEs

Numerical experiments

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

Path-dependent volatility

Marked Branching Diffusions

Nonlinear Monte Carlo algorithms for some semilinear PDEs

Branching diffusions

Marked branching diffusions

Application: Credit valuation adjustment algorithm

System of semilinear PDEs

Nonlinear PDEs



Exercises appear at the end of each chapter.

About the Originator

About the Series

Chapman and Hall/CRC Financial Mathematics Series

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Subject Categories

BISAC Subject Codes/Headings:
MATHEMATICS / Probability & Statistics / General