The Mathematics of Derivatives: Tools for Designing Numerical Algorithms

Contents

Preface

Acknowledgments

PART I   The Models

CHAPTER 1

Introduction to the Techniques of Derivative Modeling

1.1 Introduction

1.2 Models

1.2.1 What Is a Derivative?

1.2.2 What Is a Model?

1.2.3 Two Initial Methods for Modeling Derivatives

1.2.4 Price Processes

1.2.5 The Archetypal Security Process: Normal Returns

1.2.6 Book Outline

CHAPTER 2

Preliminary Mathematical Tools

2.1 Probability Distributions

2.2 n-Dimensional Jacobians and n-Form Algebra

2.3 Functional Analysis and Fourier Transforms

2.4 Normal (Central) Limit Theorem

2.5 Random Walks

2.6 Correlation

2.7 Functions of Two/More Variables: Path Integrals

2.8 Differential Forms

CHAPTER 3

Stochastic Calculus

3.1 Wiener Process

3.2 Ito’s Lemma

3.3 Variable Changes to Get the Martingale

3.4 Other Processes: Multivariable Correlations

CHAPTER 4

Applications of Stochastic Calculus to Finance

4.1 Risk Premium Derivation

4.2 Analytic Formula for the Expected Payoff of a European Option

CHAPTER 5

From Stochastic Processes Formalism to Differential Equation Formalism

5.1 Backward and Forward Kolmogorov Equations

5.2 Derivation of Black-Scholes Equation, Risk-Neutral Pricing

5.3 Risks and Trading Strategies

CHAPTER 6

Understanding the Black-Scholes Equation

6.1 Black-Scholes Equation: A Type of Backward Kolmogorov Equation

6.1.1 Forward Price

6.2 Black-Scholes Equation: Risk-Neutral Pricing

6.3 Black-Scholes Equation: Relation to Risk Premium Definition

6.4 Black-Scholes Equation Applies to Currency Options: Hidden Symmetry ...