衍生证券与差分法

Part ⅠPartial Differential Equations in Finance

1 Introduction

1.1 Assets

1.2 Derivative Securities

1.2.1 Forward and Futures Contracts

1.2.2 Options

1.2.3 Interest Rate Derivatives

1.2.4 Factors Affecting Derivative Prices

1.2.5 Functions of Derivative Securities

Problems

2 Basic Options

2.1 Asset Price Model and Ito's Lemma

2.1.1 A Model for Asset Prices

2.1.2 Ito's Lemma

2.1.3 Expectation and Variance of Lognormal Random Variables

2.2 Derivation of the Black-Scholes Equation

2.2.1 Arbitrage Arguments

2.2.2 The Black-Scholes Equation

2.2.3 Final Conditions for the Black-Scholes Equation

2.2.4 Hedging and Greeks

2.3 Two Transformations on the Black-Scholes Equation

2.3.1 Converting the Black-Scholes Equation into a Heat Equation

2.3.2 Transforming the Black-Scholes Equation into and Equation Defined on a Finite Domain

2.4 Solutions of European Options

2.4.1 The Solutions of Parabolic Equations

2.4.2 Solutions of the Black-Scholes Equation

2.4.3 Prices of Forward Contracts and Delivery Prices

2.4.4 Derivation of the Black-Scholes Formulae

2.4.5 Put-Call Parity Relation

2.4.6 An Explanation in Terms of Probability

2.5 American Option Problems as Linear Complementarity

Problems

2.5.1 Constraints on American Options

2.5.2 Formulation of the Linear Complementarity Problem in Plane

2.5.3 Formulation of the Linear Complementarity Problem in Plane

2.5.4 Formulation of the Linear Complementarity Problem on a Fuute Domain

2.5.5 More General Form of the Linear Complementarity

Problems

2.6 American Option Problems as Free-Boundary Problems

2.6.1 Free Boundaries

2.6.2 Free-Boundary Problems

2.6.3 Put-Call Symmetry Relations

2.7 Equations for Some Greeks

2.8 Perpetual Options

2.9 General Equations for Derivatives

2.9.1 Models for Random Variables

2.9.2 Generalization of Ito's Lemma

2.9.3 Derivation of Equations for Financial Derivatives

2.9.4 Three Types of State Variable8

2.9.5 Uniqueness of Solutions

2.10 Jump Conditions

2.10.1 Hyperbolic Equations with a Dirac Delta Function

2.10.2 Jump Conditions for Options with Discrete Dividends and Discrete Sampling

2.11 More Arbitrage Theory

2.11.1 Three Conclusions and Some Portfolios

2.11.2 Bounds of Option Prices

2.11.3 Relations Between Call and Put Prices

Problems

3 Exotic Options

3.1 Introduction

3.2 Barrier Options

3.2.1 Knock-Out and Knock-In Options

3.2.2 Closed-Form Solutions of Some European Barrier Options

3.2.3 Formulation of American Barrier Options

3.2.4 Parisian Options

……

Part Ⅱ Numerical Methods for Derivative Securities

References

Index

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