1st Edition

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

By Pierre Henry-Labordère Copyright 2009
    402 Pages 30 B/W Illustrations
    by Chapman & Hall

    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.


    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.


    Pierre Henry-Labordere

    … 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