Analysis, Geometry, and Modeling in Finance : Advanced Methods in Option Pricing book cover
1st Edition

Analysis, Geometry, and Modeling in Finance
Advanced Methods in Option Pricing

ISBN 9781420086997
Published September 22, 2008 by Chapman and Hall/CRC
391 Pages - 30 B/W Illustrations

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Book Description

Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing is the first book that applies advanced analytical and geometrical methods used in physics and mathematics to the financial field. It even obtains new results when only approximate and partial solutions were previously available.

Through the problem of option pricing, the author introduces powerful tools and methods, including differential geometry, spectral decomposition, and supersymmetry, and applies these methods to practical problems in finance. He mainly focuses on the calibration and dynamics of implied volatility, which is commonly called smile. The book covers the Black–Scholes, local volatility, and stochastic volatility models, along with the Kolmogorov, Schrödinger, and Bellman–Hamilton–Jacobi equations.

Providing both theoretical and numerical results throughout, this book offers new ways of solving financial problems using techniques found in physics and mathematics.

Table of Contents


A Brief Course in Financial Mathematics

Derivative products

Back to basics

Stochastic processes

Itô process

Market models

Pricing and no-arbitrage

Feynman–Kac’s theorem

Change of numéraire

Hedging portfolio

Building market models in practice

Smile Dynamics and Pricing of Exotic Options

Implied volatility

Static replication and pricing of European option

Forward starting options and dynamics of the implied volatility

Interest rate instruments

Differential Geometry and Heat Kernel Expansion

Multidimensional Kolmogorov equation

Notions in differential geometry

Heat kernel on a Riemannian manifold

Abelian connection and Stratonovich’s calculus

Gauge transformation

Heat kernel expansion

Hypo-elliptic operator and Hörmander’s theorem

Local Volatility Models and Geometry of Real Curves

Separable local volatility model

Local volatility model

Implied volatility from local volatility

Stochastic Volatility Models and Geometry of Complex Curves

Stochastic volatility models and Riemann surfaces

Put-Call duality

λ-SABR model and hyperbolic geometry

Analytical solution for the normal and log-normal SABR model

Heston model: a toy black hole

Multi-Asset European Option and Flat Geometry

Local volatility models and flat geometry

Basket option

Collaterized commodity obligation

Stochastic Volatility Libor Market Models and Hyperbolic Geometry


Libor market models

Markovian realization and Frobenius theorem

A generic SABR-LMM model

Asymptotic swaption smile


Solvable Local and Stochastic Volatility Models


Reduction method

Crash course in functional analysis

1D time-homogeneous diffusion models

Gauge-free stochastic volatility models

Laplacian heat kernel and Schrödinger equations

Schrödinger Semigroups Estimates and Implied Volatility Wings


Wings asymptotics

Local volatility model and Schrödinger equation

Gaussian estimates of Schrödinger semigroups

Implied volatility at extreme strikes

Gauge-free stochastic volatility models

Analysis on Wiener Space with Applications


Functional integration

Functional-Malliavin derivative

Skorohod integral and Wick product

Fock space and Wiener chaos expansion


Portfolio Optimization and Bellman–Hamilton–Jacobi Equation


Hedging in an incomplete market

The feedback effect of hedging on price

Nonlinear Black–Scholes PDE

Optimized portfolio of a large trader

Appendix A: Saddle-Point Method

Appendix B: Monte Carlo Methods and Hopf Algebra



Problems appear at the end of each chapter.

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Featured Author Profiles


… this book is very compact and succinctly written, yet very rich in examples, exercise problems and proofs. There are many figures which support theories, both pure mathematics and mathematical finance. Among numerous tables, the comparison tables of financial models are especially helpful. … it is a pure joy to read the current edition … as a textbook and a quick reference guide to financial engineering. …
Mathematical Reviews, Issue 2011a

The author presents in his book powerful tools and methods, such as differential geometry, spectral decomposition, super symmetry, and others that can be also applied to practical problems in mathematical finance.
—Adriana Hornikova, Technometrics, August 2010

This is an extraordinary monograph, one of the few not to be missed by anybody deeply interested in stochastic financial modeling. It demonstrates in a rather striking manner how concepts and techniques of modern theoretical physics … may be applied to mathematical finance and option pricing theory. … it also presents original ideas never before published by researchers in finance. The monograph builds an original bridge to connect analysis, geometry, and probability together with stochastic finance, a bridge supported by both very advanced mathematics and imagination. Mathematica and C++ are used for numerical implementation and many end-of-chapter problems lead the reader to recently published papers.
EMS Newsletter, September 2009

The book by Pierre Henry-Labordère is a quite a tour de forceAdvanced Methods in Option Pricing might appear to some as an understatement. One finds in this opus many gems from theoretical physics (non-Euclidean geometry, super-symmetric quantum mechanics, path integrals, and functional derivatives) applied to financial time series modeling and option pricing theory. Some of them are in fact known in the financial literature under different names; one of the most useful aspects of this book is a precise dictionary that should allow different communities to interact more easily. The advanced methods proposed by Pierre Henry-Labordère are beautiful and fascinating and will probably help to attract still a larger number of brilliant minds to financial mathematics, both in academic circles and in trading rooms.
—Jean-Philippe Bouchaud, Chairman, CFM Professor, École Polytechnique, and Editor-in-Chief, Quantitative Finance

When facing complex problems that arise in the real world, one should always remember that real answers to real questions may require imagination. This book is the manifest prototype of this timeless principle.
—Peter Carr, Head of Quantitative Financial Research, Bloomberg LP, and Director of the Masters Program in Mathematical Finance, Courant Institute, New York University, USA