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

Abstract

Introduction A Brief Course in Financial Mathematics Derivative products Back to basics Stochastic processes Ito process Market models Pricing and no-arbitrage Feynman-Kac's theorem Change of numeraire 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 Hormander'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 lambda-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 Schrodinger equations Schrodinger Semigroups Estimates and Implied Volatility Wings Introduction Wings asymptotics Local volatility model and Schrodinger equation Gaussian estimates of Schrodinger 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.

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