Teacher Docent Dario Gasbarra, meeting hours fri 12–14 A407, e-mail: Dario.Gasbarra@rni.helsinki.fi

Description : The Director of Esco has received the option to buy at the end of next year Esco shares at the price of 100 euro/share. If at the end of next year the price of a share is higher than 100 euro, the Director will get the price difference as his reward. If the price of share is below 100 euro, the options of the Director will be worthless. However the company has to count the options given to the Director in the budget of the current year. Since the budget is counted in euro, it is necessary to define the price of an option. In the lectures we tell what is the price.

Prerequisites: There are not formal requirements. The necessary concepts from probability theory, stochastic analysis and functional analysis will be given during the lectures.

Lectures thu 10–12, fri 10–12 B322, Tutorials ma 10–12, C122, given by M.Sc. Mikko Pakkanen . The lectures will take all the 1st and 2nd periods, beginning from thursday 6.9.

Exam : The material will be divided into two parts. The first exam will be between the 1st and 2nd periods on the discrete time theory. The first middle term exam will take place on friday 19.10 at 9-12 am in room C123 , on the discrete time theory.

The final exam is on tuesday 18.12 in room A111 at 12-16.

We have also written some notes about Some basic facts from martingale theory , (last update 17.10.2007).

and on Föllmer pathwise Ito calculus (in finnish), see also Hans Föllmer original article Calcul d'Ito sans probabilites. Séminaire de probabilités de Strasbourg, 15 (1981), p. 143-150 (in french).

Contents: We will follow closely the lectures written by Tommi Sottinen :

Part I: One step
1. Arbitrage: Options; One step pricing model; Expectation and risk-neutral measure; I Fundamental Theorem of Asset pricing, static case
2. Pricing financial derivatives: Buyer and Seller prices; Completeness; II Fundamental Theorem of Asset pricing, static case ; The use of financial derivatives.
   Part II: Discrete time
   
3. Markets and martingales: Market efficiency hypothesis: the equilibrium of market forces produces martingales. Investment strategies and arbitrage.
4. Binomial model: Arbitrage and completeness; European options; American options.
5. Fundamental theorems of mathematical finance: General discrete time model; I Fundamental Theorem: absence of arbitrage; II Fundamental Theorem: completeness
   Part III: Continuous time
   
6. Towards continuous time: Efficient markets in continuous time; From Binomial model to grometric Brownian motion
7. Brownian motion and stochastic integrals: Properties of Brownian motion; Ito integral; Predictable representation and change of measure
8. Black–Scholes-model: Absence of arbitrage and completeness; European options; Sensitivity Parameters
9. Interest rate modelst: short rate models; forward rate models; change of numeraire.

Literature

1. Alvarez, L. ja Koskinen, L. Rahoituksen teoriaa ja sovelluksia aktuaareille. Vakuutusvalvontavirasto, 2007.
2. Björk T. Arbitrage Theory in Continuous Time. Oxford 2004.
3. Elliott R.J., Kopp P.E Mathematics of financial markets. Springer Finance 2005.
4. Föllmer, H. ja Schied, A. Stochastic finance. An introduction in discrete time. de Gruyter Studies in Mathematics, 27. Walter de Gruyter & Co., Berlin, 2002.
5. Lamberton, D. ja Lapeyre, B. Introduction to stochastic calculus applied to finance. Chapman & Hall, London, 1996.
6. Shiryaev, A. Essentials of stochastic finance. Facts, models, theory. Advanced Series on Statistical Science & Applied Probability, 3. World Scientific Publishing Co., Inc., River Edge, NJ, 1999.
7. Shreve, S. Stochastic calculus for finance. I. The binomial asset pricing model. Springer Finance. Springer-Verlag, New York, 2004.
8. Shreve, S. Stochastic calculus for finance. II. Continuous-time models. Springer Finance. Springer-Verlag, New York, 2004.
9. Sottinen, Tommi Rahoitusteoria, luentomoniste.
10. Williams R. Introduction to the Mathematics of Finance. AMS 2006.