Introduction
A Brief Course in Financial Mathematics
Derivative products
Back to basics
Stochastic processes
It process
Market models
Pricing and no-arbitrage
FeynmanKac’s theorem
Change of numraire
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 Hrmander’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 Schrdinger equations
Schrdinger Semigroups Estimates and Implied Volatility Wings
Introduction
Wings asymptotics
Local volatility model and Schrdinger equation
Gaussian estimates of Schrdinger 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 BellmanHamiltonJacobi Equation
Introduction
Hedging in an incomplete market
The feedback effect of hedging on price
Nonlinear BlackScholes 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.
