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

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

Lectures mo 10–12, tu 12–14 C124, Tutorials we 12–14, B321. during weeks 36-42 and 44-50, beginning from thursday 1.9.

Exam : The material will be divided into two parts. A middle term exam was held on tu 21.10.08 on the discrete time theory.

Here are the problems ( suomeksi ) and solutions (suomeksi) of the final exam held on tu 16.12.08.

There will be another final exam on monday 22.12 at 10.00-14.00 in classroom D123.

You can check your exam points and final grade from this list .

Course description:

Part I: Discrete time.

Introduction: the B-S (Bond and Stock) market and the problem of Option pricing.

Essential concepts from probability: random variables, expectation, L^p spaces, change of measure, conditional expectation, Bayes formula.

Arbitrage pricing theory for one-period market: the 1st fundamental theorem: separating hyperplane theorems in finite and infinite dimension, characterizations of arbitrage, risk-neutral measure, Shiryaev's constructive argument, change of numeraire.

Hedging and pricing of contingent claims, market completeness and 2nd fundamental theorem. Incomplete markets. Upper and lower prices.

Multiperiod models : Fundamentals of stochastic process. Integration by parts formula.

Basics of martingale theory in discrete time. Doob decomposition. Martingale transforms. Change of measure and likelihood processes, martingale representation property.

1st fundamental theorem: martingale characterization of arbitrage. Martingale representation property and completeness of financial markets: 2nd fundamental theorem.

Stopping times, Optimal Stopping problems and american options.