Aim
The student should get a thorough understanding and insight in the economical and mathematical considerations which underlie the valuation of derivatives on financial markets. The student should get knowledge about and ability to handle the models and mathematical tools that are used in financial mathematics. The student should also get a thorough overview concerning the most important types of financial contracts used on the stock- and the interest rate markets and moreover get a solid base for understanding contracts that have not been explicitely treated in the course.

Knowledge and understanding
For a passing grade the student must

• understand the fundamental economical concepts : Financial contract/Contingent claim, Self financing portfolio, Arbitrage, Replicating portfolio/Hedge and Complete market,

• understand the tools and concepts from stochastic calculus: martingales, Itô's formula, Itô isometry, Feynman-Kac representation, change of measure (Girsanov transformation) and change of numeraire,

• understand how the basic financial contracts work and how they relate to each other, e.g., European and Asian options, Forward contracts, zero coupon bonds, coupon bond, LIBOR and interest rate swap.

Skills and abilities
For a passing grade the student must

• use the fundamental financial concepts to express relations between various financial contracts,

• use the tools and concepts from stochastic calculus to price financial contracts assuming specific models for the underlying assets. This especially includes the ability to use, derive and understand the Black-Scholes formula as well as the ability of extending it to similar contracts,

• use Monte Carlo methods to price financial derivatives. Here the student should be able to use various variance reduction techniques such as antithetic variables, control variates and importance sampling. This part of the course is assessed in the home assignments and compulsory computer exercises.

Judgement and approach
For a passing grade the student must

• apply a mathematical point of view on financial contracts,

• from a financial and a mathematical perspective, judge what a reasonable valuation of a financial contract should fulfil.

Contents
The course consists of three related parts. In the first part we will look at option theory in discrete time. The purpose is to quickly introduce fundamental concepts of financial markets such as free of arbitrage and completeness as well as martingales and martingale measures. We will use tree structures to model time dynamics of stock prices and information flows.

In the second part we will study alternative models formulated in continuous time. The models we focus on are formulated as stochastic differential equations (SDE:s). Most of the second part is devoted to the probability theory required to understand the SDE models. This includes, e.g., Brownian motion, stochastic integrals and Itô's formula.

Finally, in the third part we study various applications of the theory from part two. Here we come back to option theory and derive, e.g., the Black-Scholes formula. After that we will study the bond market and interest rate derivatives.

Literature
Björk, T.: Arbitrage Theory in Continuous Time, 2nd Ed., 2004.
Rasmus, S.: Derivative Pricing, Avd. Matematisk Statistik, 2006.