Stochastic integration:

Recapitulation of prerequisites from analysis, measure theory and probability, Brownian motion, continuous time martingales, stochastic integral, Itô formula, stochastic differential equations.

Pricing of financial derivatives:

Black-Merton-Scholes model, derivatives, arbitrage and hedging in general, model completeness, change of measure and Girsanov theorem, parity equations.

Interest rate models:

Bonds and interest, some classical martingale models, pricing of interest rate options.

The lecturer can also include other current topics from recent scientific periodicals in the course.

# Financial mathematics 2

T. Björk: Arbitrage Theory in Continuous Time, 2nd edition, Oxford Univ. Press, Oxford, 2004.

S. E. Shreve: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, New York, 2004.

D. Lamberton, B. Lapeyre: Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall/CRC, Boca Raton, 2000.

J. C. Hull: Options, Futures, and Other Derivative Securities, 6th edition, Pearson/Prentice Hall, Upper Saddle River NJ, 2006.

B. Øksendal: Stochastic Differential Equations: An Introduction with Applications, 6th edition, Springer, Berlin, 2006.

Modern market models are based on stochastic calculus. The course starts with a short introduction of stochastic integration which is needed for understanding the continuous time models in financial mathematics. Stochastic differential equations present on one hand the means for modeling the financial markets, interest rates and portfolios and on the other hand the tool for their efficient study, which leads to optimal stoping problems and to stochastic control theory.

Knowledge and understanding:

Understanding of mathematical models, which are used in mathematical finance, and the means for their treatment.

Application:

The acquired knowledge is both: directly transferable and it also serves as a base for combining mathematical knowledge with economical content.

Reflection:

The area itself, and hence also the course, combines various mathematical disciplines: from linear algebra to partial differential equations.

Transferable skills:

The acquired knowledge is directly applicable in financial institutions, e.g. banks, insurance companies... The content of the course contributes to the sharpening of the ability of mathematical modeling.

Lectures, exercises, one's own seminar assignment

one's own seminar assignment

exam

Grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Realizing irreducible semigroups and real algebras of compact operators. Journal of mathematical analysis and applications, ISSN 0022-247X. [Print ed.], 2008, vol. 348, no. 2, str. 692-707. [COBISS-SI-ID 14899289]

BERNIK, Janez, MASTNAK, Mitja, RADJAVI, Heydar. Positivity and matrix semigroups. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2011, vol. 434, iss. 3, str. 801-812. [COBISS-SI-ID 15745625]

BERNIK, Janez, MARCOUX, Laurent W., RADJAVI, Heydar. Spectral conditions and band reducibility of operators. Journal of the London Mathematical Society, ISSN 0024-6107, 2012, vol. 86, no. 1, str. 214-234. [COBISS-SI-ID 16357721]

Mihael Perman:

- PERMAN, Mihael, WELLNER, Jon A. An excursion approach to maxima of the Brownian bridge. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 2014, vol. 124, iss. 9, str. 3106-3120.
- PERMAN, Mihael. A decomposition for Markov processes at an independent exponential time. Ars mathematica contemporanea, ISSN 1855-3966. [Tiskana izd.], 2017, vol. 12, no. 1, str. 51-65.
- PERMAN, Mihael, ZALOKAR, Ana. Optimal hedging strategies in equity-linked products. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], 2018, vol. 344, str. 601-607.