Jan R. M. Röman

First we will give a short introduction to pricing via arbitrage and the central limit

theorem. Then a number of different binomial models are discussed. Binomial models

are important, both for the understanding and since they are widely used to calculate

the price and Greeks for a number of different derivatives. The reason is that they are

general and can handle all kinds of derivatives, e.g., European, Bermudan and

American options. We also discuss the numerical approach of the binomial models,

finite element methods and Monte-Carlo simulations.

Then, an introduction to probability theory and stochastic integration is given. After

this introduction we are ready to study partial differential equations and the solution

of the Black-Scholes equation. A number of generalizations to Black-Scholes are

given, such as stochastic volatility and time dependent parameters. We also discuss a

number of analytical approximations for American options and why we can't use

Black-Scholes PDE on American options with early exercise.

A short introduction to Poisson processes is also given. Then we study diffusion

processes, Martingale representation and Girsanov theorem. Before we end up with a

general guide to pricing via Black-Scholes we give an introduction to exotic options

wheatear derivatives and volatility models.

This course is followed by Analytical Finance II where we will go into the fix income