Financial mathematics 1

Mathematics, First Cycle
3 year
Lecturer (contact person):
Hours per week – 2. semester:
Content (Syllabus outline)

Interest rates, time value of money, term structure.
Bonds, financial derivatives.
Market model: finite sets of assets, discrete time, The Fundamental Asset Pricing Theorems.
Option pricing: definitions, European options, American options, exotic options.
Pricing of European options: Binomial model, Black-Scholes Formula.
Optimal investement: strategies, static model, dynamic model.
American options: American contingent claims, stopping times, Snell enveloppe, buyer's price, seller's price.
Stochastic models of interest rates: discrete models, term rate options.


P. Koch Medina, S. Merino. Mathematical finance and probability: a discrete introduction. Birkhäuser, 2003.
J. Hull. Options, futures and other derivatives. Prentice Hall. 8. izdaja, 2011.
S. E. Shreve. Stochastic calculus for finance 1: The binomial asset pricing model. Springer, 2005.
S. M. Ross, An elementary introduction to mathematical finance : options and other topics. 2. izdaja, Cambridge University Press, 2003.
D.G. Luenberger. Investment science. Oxford University Press, 2. izdaja, 2013.
Z. Bodie, A. Kane, A. Marcus. Investments. 9. izdaja, McGraw-Hill Irwin, Boston, ZDA, 2011.
B. Steiner. Mastering financial calculations: A step-by-step guide to the mathematics of financial market instruments. 2. izdaja, Financial Times Prentice Hall, 2007.
M. Capiński, T. Zastawniak: Mathematics for Finance : An Introduction to Financial Engineering, Springer, London, 2005.
J. Y. Campbell, L. M. Viceira: Strategic Asset Allocation : Portfolio Choice for Long-Term Investors, Oxford Univ. Press, Oxford, 2002.

Objectives and competences

There are some fundamental principles underlying the modern financial methematics. The aim of the course is to present these principles (the law of one price, the no arbitrage condition) in the simplest discrete models. Optimal investment theory leads to market models, the fundamental asset pricing theorem and to option pricing theory. The main topics include binomial model and the Black-Scholes Formula. Stopping times are introduce and pricing of aAmerican claim is presented. Important element of the theory are also stochastic models for interets rates.

Intended learning outcomes

Knowledge and understanding: Understanding of mathematical models that are used in the pricing and hedging on the financial markets. Understanding the relation of model selection and its consequences.
Application: All the methods are directly applicable in the financial markets. They also give a base to study more advanced models.
Reflection: Understanding theoretical concepts in practice.
Transferable skills: The knowledge is directly transferable to the practice in financial institutions, such as banks and insurance companies. Beside the practical aspects also skills of financial modelling are advanced through the course.

Learning and teaching methods

Lectures, exercises, consultations


2 midterm exams instead of written exam, written exam
Oral exam
grading: 5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Mihael Perman:
KOMELJ, Janez, PERMAN, Mihael. Joint characteristic functions construction via copulas. Insurance. Mathematics & economics, ISSN 0167-6687, 2010, vol. 47, iss. 2, str. 137-143. [COBISS-SI-ID 16242777]
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]
Tomaž Košir:
GRUNENFELDER, Luzius, KOŠIR, Tomaž, OMLADIČ, Matjaž, RADJAVI, Heydar. Finite groups with submultiplicative spectra. Journal of Pure and Applied Algebra, ISSN 0022-4049. [Print ed.], 2012, vol. 216, iss. 5, str. 1196-1206. [COBISS-SI-ID 16183385]
BUCKLEY, Anita, KOŠIR, Tomaž. Plane curves as Pfaffians. Annali della Scuola normale superiore di Pisa, Classe di scienze, ISSN 0391-173X, 2011, vol. 10, iss. 2, str. 363-388. [COBISS-SI-ID 15928409]
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]
Janez Bernik:
BERNIK, Janez, MASTNAK, Mitja. Lie algebras acting semitransitively. Linear Algebra and its Applications, ISSN 0024-3795. [Print ed.], 2013, vol. 438, iss. 6, str. 2777-2792. [COBISS-SI-ID 16553561]
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]
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]