Mathematical Finance：
Introduction to continuous time Financial Market models

Dr. Christian-Oliver Ewald
School of Economics and Finance， University of St.Andrews

Abstract
These are my Lecture Notes for a course in Continuous Time Finance which I taught in the Summer term 2003 at the University of Kaiser-slautern. I am aware that the notes are not yet free of error and the manuscrip needs further improvement. I am happy about any comment on the notes.

Contents
1 Stochastic Processes in Continuous Time
1.1 Filtrations and Stochastic Processes
1.2 Special Classes of Stochastic Processes
1.3 Brownian Motion
1.4 Black and Scholes’ Financial Market Model
2 Financial Market Theory
2.1 Financial Markets
2.2 Arbitrage
2.3 Martingale Measures
2.4 Options and Contingent Claims
2.5 Hedging and Completeness
2.6 Pricing of Contingent Claims
2.7 The Black-Scholes Formula
2.8 Why is the Black-Scholes model not good enough ?
3 Stochastic Integration
3.1 Semi-martingales
3.2 The stochastic Integral
3.3 Quadratic Variation of a Semi-martingale
3.4 The Ito Formula
3.5 The Girsanov Theorem
3.6 The Stochastic Integral for predictable Processes
3.7 The Martingale Representation Theorem
4 Explicit Financial Market Models
4.1 The generalized Black Scholes Model
4.2 A simple stochastic Volatility Model
4.3 Stochastic Volatility Model . . . . . .
4.4 The Poisson Market Model . . . . . .
5 Portfolio Optimization
5.1 Introduction
5.2 The Martingale Method
5.3 The stochastic Control Approach