1st Edition
Analysis, Geometry, and Modeling in Finance Advanced Methods in Option Pricing
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.
Introduction
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
Introduction
Libor market models
Markovian realization and Frobenius theorem
A generic SABR-LMM model
Asymptotic swaption smile
Extensions
Solvable Local and Stochastic Volatility Models
Introduction
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
Introduction
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
Introduction
Functional integration
Functional-Malliavin derivative
Skorohod integral and Wick product
Fock space and Wiener chaos expansion
Applications
Portfolio Optimization and Bellman–Hamilton–Jacobi Equation
Introduction
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
References
Index
Problems appear at the end of each chapter.
Biography
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 2011aThe 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 2010This 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 2009The book by Pierre Henry-Labordère is a quite a tour de force—Advanced 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 FinanceWhen 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