Prissättning av derivattillgångar

**Valid from:** Autumn 2020**Decided by:** Professor Thomas Johansson**Date of establishment:** 2020-05-19

**Division:** Mathematical Statistics**Course type:** Course given jointly for second and third cycle**The course is also given at second-cycle level with course code:** FMSN25**Teaching language:** English

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 doctoral 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.

*Competences and Skills*

For a passing grade the doctoral 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 doctoral 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.

The course consists of two 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 models formulated in continuous time. The models we focus on are formulated as stochastic differential equations (SDE:s). The theories behind Brownian motion, stochastic integrals, Ito-'s formula, measures changes and numeraires are presented and applied to option theory both for the stock and the interest rate markets. We derive e.g. the Black-Scholes formula and how to create a replicating portfolio for a derivative contract.

- T., B.: Arbitrage Theory in Continuous Time, 3rd ed.. Oxford University Press, 2009.
- S., R.: Derivative Pricing. Avd. Matematisk Statistik, 2010.

**Types of instruction:** Lectures, laboratory exercises, exercises

**Examination formats:** Written exam, written assignments**Grading scale:** Failed, pass**Examiner:**

**Admission requirements:** FMSF10 Stationary Stochastic Processes or FMSF15 Markov Processes. Knowledge corresponding to FMSF05 Probability Theory helps.

**Course coordinator:** Magnus Wiktorsson `<magnus.wiktorsson@matstat.lu.se>`**Web page:** www.maths.lth.se/matstat/kurser/fmsn25/