Jiang L. Mathematical Modeling and Methods of Option Pricing

World Scientific, 2005. – 344 p. – ISBN: 9812563695, 9789812563699From the unique perspective of partial differential equations (PDE), this self-contained book presents a systematic, advanced introduction to the Black–Scholes–Merton’s option pricing theory. A unified approach is used to model various types of option pricing as PDE problems, to derive pricing formulas as their solutions, and to design efficient algorithms from the numerical calculation of PDEs. In particular, the qualitative and quantitative analysis of American option pricing is treated based on free boundary problems, and the implied volatility as an inverse problem is solved in the optimal control framework of parabolic equations.Risk Management and Financial Derivatives
Risk and Risk Management
Forward Contracts and Futures
Options
Option Pricing
Types of Traders
Arbitrage-Free Principle
Financial Market and Arbitrage-Free Principle
European Option Pricing and Call-Put Parity
American Option Pricing and Early Exercise
Dependence of Option Pricing on the Strike Price
Binomial Tree Methods Discrete Models of Option Pricing
An Example
One-Period and Two-State Model
Binomial Tree Method of European Option Pricing (I)
Non-Dividend-Paying
Binomial Tree Method of European Options (II)
Dividend-Paying
Binomial Tree Method of American Option Pricing
Call-Put Symmetry
Brownian Motion and I to Formula
Random Walk and Brownian Motion
Continuous Models of Asset Price Movement
Quadratic Variation Theorem
Ito Integral
Ito Formula
European Option Pricing Black-Scholes Formula
History
Black-Scholes Equation
Black-Scholes Formula
Generalized Black-Scholes Model (I) Dividend-Paying
Options
Generalized Black-Scholes Model (II) Binary Options and Compound Options
Numerical Methods (I) Finite Difference Method.
Numerical Methods (II) Binomial Tree Method and Finite
Difference Method
Properties of European Option Price
Risk Management
American Option Pricing and Optimal Exercise Strategy
Perpetual American Option
Models of American Options
Decomposition of American Options
Properties of American Option Price
Optimal Exercise Boundary
Numerical Method (I) Finite Difference Method
Numerical Methods(II) Line Method
Other Types of American Options
Multi-Asset Option Pricing
Stochastic Models of Multi-Assets Pricing
Black-Scholes Equation
Black-Scholes Formula
Rainbow Options
Basket Options
Quanto Options
American Multi-Asset Options
Path-Dependent Options (I) – Weakly Path-Dependent Options
Barrier Options
Time-Dependent Barrier Options
Reset Options
Modified Barrier Options
Path-Dependent Options (II) – Strongly Path-Dependent Options
Asian Options
Model and Simplification
Valuation Formula for European-Style Geometric Average Asian Option
Call-Put Parities for Asian Options
Lookback Option
Numerical Methods
Implied Volatility
Preliminaries
Dupire Method
Optimal Control Method
Numerical Method