1. komponenta

Lecture type	Total
Lectures	60

* Load is given in academic hour (1 academic hour = 45 minutes)

COURSE AIMS AND OBJECTIVES: The aim is to explain financial market models in discrete and continuous time, and introduce probabilistic methods for a precise mathematical description and understanding of these models.

COURSE DESCRIPTION AND SYLLABUS:
f. Dynamic discrete models. Model description: financial assets, dynamic portfolios, arbitrage. Absence of arbitrage and martingale measure; fundamental theorem. Derivatives and replicating portfolio. Complete market models. Cox-Ross - Rubinstein's model. Introduction to American Options. The optimal stopping problem and American options. Snell envelope, stopping times and Markov chains. Application to American options in the CRR model.
g. Brownian motion and Ito's calculus. Brownian motion. Martingales with continuous time. It's integral. It's formula. Stochastic differential equations.
h. Black-Scholes model. Model description. Partial differential equations in the Black - Scholes model. Probability change, martingale representation. Pricing and hedging of European and exotic derivatives. Bonds and forwards and futures contracts.

1. Introduction to Stochastic Calculus Applied to Finance, D. Lamberton, B. Lapeyre, Chapmann&Hall, 1996.
2. Martingale Methods in Financial Modelling, M. Musiela, M. Rutkowski, Springer Verlag, 1997.
3. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, S. Shreve, Springer Verlag, 2004.
4. Stochastic Calculus for Finance II: Continuous Time Models, S. Shreve, Springer Verlag, 2004.
5. Financial Calculus, W. A. Baxter, A.Rennie, Cambridge University Press, 1996.
6. Risk - Neutral Valuation: Pricing and Hedging of Financial Derivatives, N. H. Bingham, R. Kiesel, Springer Verlag, 1998.
7. Arbitrage Theory in Continuous Time, T. Bjork, Oxford University Press, 1999.
8. Introduction to the Economics and Mathematics of Financial Markets, J. Cvitanić, F. Zapatero, MIT Press, 2004.
9. Stocahstic Finance: An Introduction in Discrete Time, H. Follmer, A. Schied, W. de Gruyter, 2002.
10. Options, Futures, and Other Derivative Securities, 5 th edition, J. C. Hull, Prentice Hall, 2002.
11. Brownian Motion and Stochastic Calculus, 2 nd edition, I. Karatzas, S. Shreve, Springer Verlag, 1991.

Enrollment :
Passed : Financial markets
Passed : Stochastic processes

Mandatory course - Regular study - Financial and Business Mathematics