Elementary Stochastic Calculus, with Finance in View

The following sections are included:

• Preface

• Contents

This book grew out of lecture notes for a course on stochastic calculus for economics students. When I prepared the first lectures I realized that there was no adequate textbook treatment for non-mathematicians. On the other hand, there was and indeed is an increasing demand to learn about stochastic calculus, in particular in economics, insurance, finance, econometrics. The main reason for this interest originates from the fact that this mathematical theory is the basis for pricing financial derivatives such as options and futures. The fundamental idea of Black, Scholes and Merton from 1973 to use Itô stochastic calculus for pricing and hedging of derivative instruments has conquered the real world of finance; the Black–Scholes formula has been known to many people in mathematics and economics long before Merton and Scholes were awarded the Nobel prize for economics in 1997–

The following sections are included:

• Basic Concepts from Probability Theory
  • Random Variables
  • Random Vectors
  • Independence and Dependence

• Stochastic Processes

• Brownian Motion
  • Defining Properties
  • Processes Derived from Brownian Motion
  • Simulation of Brownian Sample Paths
  • Notes and Comments

• Conditional Expectation
  • Conditional Expectation under Discrete Condition
  • About σ-Fields
  • The General Conditional Expectation
  • Rules for the Calculation of Conditional Expectations
  • The Projection Property of Conditional Expectations
  • Notes and Comments

• Martingales
  • Defining Properties
  • Examples
  • The Interpretation of a Martingale as a Fair Game
  • Notes and Comments

The following sections are included:

• The Riemann and Riemann–Stieltjes Integrals
  • The Ordinary Riemann Integral
  • The Riemann–Stieltjes Integral

• The Itô Integral
  • A Motivating Example
  • The Itô Stochastic Integral for Simple Processes
  • The General Itô Stochastic Integral

• The Itô Lemma
  • The Classical Chain Rule of Differentiation
  • A Simple Version of the Itô Lemma
  • Extended Versions of the ltô Lemma
  • Notes and Comments

• The Stratonovich and Other Integrals

The following sections are included:

• Deterministic Differential Equations

• Itô Stochastic Differential Equations
  • What is a Stochastic Differential Equation?
  • Solving Itô Stochastic Differential Equations by the Itô Lemma
  • Solving Itô Differential Equations via Stratonovich Calculus

• The General Linear Differential Equation
  • Linear Equations with Additive Noise
  • Homogeneous Equations with Multiplicative Noise
  • The General Case
  • The Expectation and Variance Functions of the Solution

• Numerical Solution
  • The Euler Approximation
  • The Milstein Approximation
  • Notes and Comments

The following sections are included:

• The Black–Scholes Option Pricing Formula
  • A Short Excursion into Finance
  • What is an Option?
  • A Mathematical Formulation of the Option Pricing Problem
  • The Black and Scholes Formula
  • Notes and Comments

• A Useful Technique: Change of Measure
  • What is a Change of the Underlying Measure?
  • An Interpretation of the Black-Scholes Formula by Change of Measure
  • Notes and Comments

The following sections are included:

• Appendix
  • Modes of Convergence
  • Inequalities
  • Non-Differentiability and Unbounded Variation of Brownian Sample Paths
  • Proof of the Existence of the General Itô Stochastic Integral
  • The Radon–Nikodym Theorem
  • Proof of the Existence and Uniqueness of the Conditional Expectation

• Bibliography

• Index

• List of Abbreviations and Symbols