Topics in financial mathematics 2

2022/2023
Programme:
Financial Mathematics, Second cycle
Year:
1 ali 2 year
Semester:
first or second
Kind:
optional
Group:
M5
ECTS:
6
Language:
slovenian, english
Hours per week – 1. or 2. semester:
Lectures
2
Seminar
1
Tutorial
2
Lab
0
Content (Syllabus outline)

Lecturer can choose amon the following and some other current topics:
Portfolio management: mean-variance model.
Markowitz theory. Volatility of returns and its measurement. Arbitrage pricing. CAPM model.
One and multifactor models. Bayesian models.
Black-Litterman algorithm. One period and multiperiod models. Pricing in continuous time.
Mathematical models for high frequency trading.
Consumption and investment: definitions, optimization problems, general equilibrium, side conditions, incomplete markets.
Stochastic optimization: stochastic control theory, Malliavin calculus. Viscosity solutions.

Readings

I. Aldridge: High frequency trading: A practical guide to algorithmic strategies and trading systems. Wiley, 2013.
M. Capinski, T. Zastawniak, Mathematics for Finance, An Introduction to Financial Engineering, London, Springer, 2. izdaja, 2011.
D. G. Luenberger, Investment science, New York, Oxford University Press, 2. izdaja, 2013.
E. J. Elton, M. J. Gruber, S. J. Brown, W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, New York, Wiley, 8. izdaja, 2009.
G. Da Prato, Introduction to stochastic analysis and Malliavin calculus, Pisa : Edizioni della Normale, 2. izdaja, 2008.
D. Nualart, The Malliavin calculus and related topics, Berlin, Heidelberg, New York: Springer, 2006.

Objectives and competences

The course covers topics im mathematical finance in which economic reasoning is combined with advanced mathematical tools.
Some of them are based on previous courses and give additional interpretation, some contribute to understanding of the risks.
Since the content is of great practical importance we expect that also specialists from financial practice will present their work experience during the course.

Intended learning outcomes

Knowledge and understanding:
Understanding of mathematical models used in mathematical finance and the mathematical tools used in solutions.
Application:
The knowledge and skills acquired are directly transferable and can also serve for combining mathematical reasoning with economic topics.
Reflection:
The subject of the course, hence the course itself, combines numerous mathematical skills starting from linear algebra to partial differential equations.
Transferable skills:
The knowledge and skills acquired are immediately applicable in financial institutions such as banks and insurance companies. The content alsoserves to deepen the ability to use mathematical models.

Learning and teaching methods

Lectures, exercises, consultations, seminars

Assessment

Individual seminar
Exam on the course content
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Tomaž Košir:
BERNIK, Janez, DRNOVŠEK, Roman, KOKOL-BUKOVŠEK, Damjana, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. On semitransitive jordan algebras of matrices. Journal of algebra and its applications, ISSN 0219-4988, 2011, vol. 10, iss. 2, str. 319-333. [COBISS-SI-ID 15908697]
KOŠIR, Tomaž, OBLAK, Polona. On pairs of commuting nilpotent matrices. Transformation groups, ISSN 1083-4362, 2009, vol. 14, no. 1, str. 175-182. [COBISS-SI-ID 15077977]
BERNIK, Janez, DRNOVŠEK, Roman, KOŠIR, Tomaž, LIVSHITS, Leo, MASTNAK, Mitja, OMLADIČ, Matjaž, RADJAVI, Heydar. Approximate permutability of traces on semigroups of matrices. Operators and matrices, ISSN 1846-3886, 2007, vol. 1, no. 4, str. 455-467. [COBISS-SI-ID 14492761]
Mihael Perman:
PERMAN, Mihael, WELLNER, Jon A. On the distribution of Brownian areas. Annals of applied probability, ISSN 1050-5164, 1996, let. 6, št. 4, str. 1091-1111. [COBISS-SI-ID 7101017]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities and decompositions for general perturbed risk processes. Annals of applied probability, ISSN 1050-5164, 2004, vol. 14, no. 3, str. 1378-1397. [COBISS-SI-ID 13168985]
HUZAK, Miljenko, PERMAN, Mihael, ŠIKIĆ, Hrvoje, VONDRAČEK, Zoran. Ruin probabilities for competing claim processes. Journal of Applied Probability, ISSN 0021-9002, 2004, vol. 41, no. 3, str. 679-690. [COBISS-SI-ID 13207641]