Analysis, Geometry, and Modeling in Finance: Advanced Methods in
- Type:
- Other > E-books
- Files:
- 1
- Size:
- 3.06 MB
- Tag(s):
- analysis geometry modeling finance advanced option pricing
- Uploaded:
- Dec 4, 2013
- By:
- mr.finance
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.