Analysis, Geometry, and Modeling in Finance: Advanced Methods in torrent

Features
-Presents original ideas never before published in a financial mathematics book
-Demonstrates how differential geometry, spectral decomposition, and supersymmetry -can be used as new tools in finance
Covers practical issues from the industry, such as the calibration of stochastic -Libor market models
-Contains recent results on stochastic volatility models
-Uses Mathematica® and C++ for numerical implementations
-Provides end-of-chapter problems, including some based on recently published research papers

Summary
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
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.