# Numerical methods for financial mathematics

2022/2023
Programme:
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)

Algorithms for option pricing in discrete models. Monte Carlo Methods for European options.
Simulation methods of classical law. Inverse transform method. Computation of expectation.
Variance reduction techniques. Tree methods for European and American options. Conver-
gence orders of binomial methods. Estimating sensitivities. Numerical algorithms for portfolio
insurance. Tree methods and Monte Carlo methods for Exotic options (barrier options, asian options, lookback options, rainbow options).
American Monte Carlo methods.
Finite difference methods for the Black-Scholes partial differential equation.

J. Hull. Options, Futures, and Other Derivatives. Prentice Hall, 2011.
N. H. Bingham, R. Kiesel. Risk-Neutral Valuation: Pricing and Hedging of Financial Derivatives. Springer Finance, 2004.
P. Glasserman. Monte Carlo Methods in Financial Engineering. Springer, 2003.

Objectives and competences

The course covers the chapter of mathematical finance that deal with numerical methods for pricing of derived financial instruments of all kinds.
Since the content is of great practical importance we expect that also specialists from financial practice will present their work experience during the course.
With individual presentations and team work interactions within seminar/project activities students acquire communication and social competences for successful team work and knowledge transfer.

Intended learning outcomes

Knowledge and understanding:
Understanding of mathematical models used in finance. Mathematical tools necessary in modelling.
Application:
The knowledge is directly usable in practice, it is also the source for combing of mathematical theories with finance.
Reflection:
The subject connects many mathematical topics, specially those of numerical methods and probablity theory, with application.
Transferable skills:
The knowledge is directly applicable in everyday practice in financial institutions.

Learning and teaching methods

Lectures, exercises, homeworks, consultations, seminars

Assessment

Type (examination, oral, coursework, project):
seminar work
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Janez Bernik:
BERNIK, Janez, RADJAVI, Heydar. Invariant and almost-invariant subspaces for pairs of idempotents. Integral equations and operator theory, ISSN 0378-620X, 2016, vol. 84, iss. 2, str. 283-288. [COBISS-SI-ID 17449049]
BERNIK, Janez, POPOV, Alexey I. Obstructions for semigroups of partial isometries to be self-adjoint. Mathematical proceedings of the Cambridge Philosophical Society, ISSN 0305-0041, 2016, vol. 161, iss. 1, str. 107-116. [COBISS-SI-ID 17690457]
BERNIK, Janez, MARCOUX, Laurent W., POPOV, Alexey I., RADJAVI, Heydar. On selfadjoint extensions of semigroups of partial isometries. Transactions of the American Mathematical Society, ISSN 0002-9947, 2016, vol. 368, no. 11, str. 7681-7702. [COBISS-SI-ID 17801049]

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]

Marjeta Krajnc:
GROŠELJ, Jan, KRAJNC, Marjetka. C[sup]1 cubic splines on Powell-Sabin triangulations. Applied mathematics and computation, ISSN 0096-3003. [Print ed.], 2016, vol. 272, part 1, str. 114-126. [COBISS-SI-ID 17608537]
KOZAK, Jernej, KRAJNC, Marjetka, VITRIH, Vito. Parametric curves with Pythagorean binormal. Advances in computational mathematics, ISSN 1019-7168, 2015, vol. 41, iss. 4, str. 813-832. [COBISS-SI-ID 1537835204]
KRAJNC, Marjetka, POČKAJ, Karla, VITRIH, Vito. Construction of low degree rational motions. Journal of Computational and Applied Mathematics, ISSN 0377-0427. [Print ed.], 2014, vol. 256, str. 92-103. [COBISS-SI-ID 1024532052]

Mihael Perman:
PERMAN, Mihael. An excursion approach to Ray-Knight theorems for perturbed Brownian motion. Stochastic Processes and their Applications, ISSN 0304-4149. [Print ed.], 1996, let. 63, str. 67-74. [COBISS-SI-ID 7621465]
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]
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. [COBISS-SI-ID 17154393]
PERMAN, Mihael. A decomposition for Markov processes at an independent exponential time. Ars mathematica contemporanea, ISSN 1855-3974. [Spletna izd.], 2017, vol. 12, no. 1, str. 51-65. [COBISS-SI-ID 17677145]

Jaka Smrekar:
FORSTNERIČ, Franc, SMREKAR, Jaka, SUKHOV, Alexandre. On the Hodge conjecture for q-complete manifolds. Geometry & topology, ISSN 1465-3060, 2016, vol. 20, no. 1, str. 353-388. [COBISS-SI-ID 17622361]
SMREKAR, Jaka. CW towers and mapping spaces. Topology and its Applications, ISSN 0166-8641. [Print ed.], 2015, vol. 194, str. 93-117. [COBISS-SI-ID 17413721]
SMREKAR, Jaka. Turning a self-map into a self-fibration. Topology and its Applications, ISSN 0166-8641. Print ed.], 2014, vol. 167, str. 76-79. [COBISS-SI-ID 16943705]