Analysis, Geometry, and Modeling in Finance: Advanced Methods in Options 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.