Module also offered within study programmes:
General information:
Name:
DISCRETE MODELS OF FINANCIAL MARKETS
Course of study:
2019/2020
Code:
AMAT-2-009-MU-s
Faculty of:
Applied Mathematics
Study level:
Second-cycle studies
Specialty:
Insurance Mathematics
Field of study:
Mathematics
Semester:
0
Profile of education:
Academic (A)
Lecture language:
English
Form and type of study:
Full-time studies
Course homepage:
 
Responsible teacher:
dr hab. Capiński Maciej (mcapinsk@agh.edu.pl)
Module summary

Discrete models of financial markets.

Description of learning outcomes for module
MLO code Student after module completion has the knowledge/ knows how to/is able to Connections with FLO Method of learning outcomes verification (form of completion)
Social competence: is able to
M_K001 The student will understand the limitations of mathematical models and the difficulties in making financial mathematics models relevant to real markets. MAT2A_U08, MAT2A_U15 Activity during classes,
Examination
Skills: he can
M_U001 The student will be able to use the CRR model to price basic derivatives (European/American calls and puts). MAT2A_U10, MAT2A_U16 Activity during classes,
Examination
Knowledge: he knows and understands
M_W001 The student will know how the basic properties of option prices follow from the "no arbitrage formula" MAT2A_U03, MAT2A_U14, MAT2A_W07, MAT2A_W01, MAT2A_W04 Activity during classes,
Examination
M_W002 The student will know how the "no arbitrage" priciple can be used to derive pricing formulae in discrete markets for basic derivatives. MAT2A_U03, MAT2A_U14, MAT2A_U01 Activity during classes,
Examination
Number of hours for each form of classes:
Sum (hours)
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
30 30 0 0 0 0 0 0 0 0 0 0
FLO matrix in relation to forms of classes
MLO code Student after module completion has the knowledge/ knows how to/is able to Form of classes
Lecture
Audit. classes
Lab. classes
Project classes
Conv. seminar
Seminar classes
Pract. classes
Zaj. terenowe
Zaj. warsztatowe
Prace kontr. przejść.
Lektorat
Social competence
M_K001 The student will understand the limitations of mathematical models and the difficulties in making financial mathematics models relevant to real markets. + - - - - - - - - - -
Skills
M_U001 The student will be able to use the CRR model to price basic derivatives (European/American calls and puts). + - - - - - - - - - -
Knowledge
M_W001 The student will know how the basic properties of option prices follow from the "no arbitrage formula" + - - - - - - - - - -
M_W002 The student will know how the "no arbitrage" priciple can be used to derive pricing formulae in discrete markets for basic derivatives. + - - - - - - - - - -
Student workload (ECTS credits balance)
Student activity form Student workload
Summary student workload 125 h
Module ECTS credits 5 ECTS
Udział w zajęciach dydaktycznych/praktyka 30 h
Realization of independently performed tasks 88 h
Examination or Final test 2 h
Contact hours 5 h
Module content
Lectures (30h):

1. Binomial model in one step, the concept of the portfolio, the absence of arbitrage, the valuation by replication, uniqueness of martingale measure and its application to the valuation of derivative securities.

2. Trinomial model as the simplest example of incomplete market, the range of prices determined by the family of martingale measures, sub and super-replicating strategies

3. Supplementing the model by adding assets. Condition for completeness in the language of the matrix of prices. Range of prices of derivatives linked to the supplemented market.

4. Many steps. The concept of strategy as a predictable process, the value strategy. Self-financing strategies, necessary and sufficient condition. Discounted prices and strategies. Admissible strategies, the principle of no arbitrage.

5. Binomial model, the valuation of European options. Application of the concept martingale in binomial model with a detailed description of filtration. Option price as an example of a martingale.

6. Markov property. Hedging against risk after issuing the option. Limit passage in the formula for the option price.

7. The first fundamental theorem in one step. Separation Lemma

8. Martingale transform. Representation theorem in binomial model.

9. Version of the theorem for multiple steps. The second fundamental theorem. Many assets, characterization of completeness by adjusting the number of degrees of freedom to the number of assets.

10. American option as Snell envelope. Stopping times, optimality.

11. Stopped processes, properties. Martingale properties of Snell envelope.

12. Examples of optimal stopping times, theorems on maximal and minimal times.

13. Doob decomposition and application to stopping times. American option pricin and hedging.

14. Futures in binomial trees, exotic options.

Additional information
Teaching methods and techniques:
  • Lectures: Treści prezentowane na wykładzie są przekazywane w formie prezentacji multimedialnej w połączeniu z klasycznym wykładem tablicowym wzbogaconymi o pokazy odnoszące się do prezentowanych zagadnień.
Warunki i sposób zaliczenia poszczególnych form zajęć, w tym zasady zaliczeń poprawkowych, a także warunki dopuszczenia do egzaminu:

-

Participation rules in classes:
  • Lectures:
    – Attendance is mandatory: Yes
    – Participation rules in classes: Studenci uczestniczą w zajęciach poznając kolejne treści nauczania zgodnie z syllabusem przedmiotu. Studenci winni na bieżąco zadawać pytania i wyjaśniać wątpliwości. Rejestracja audiowizualna wykładu wymaga zgody prowadzącego.
Method of calculating the final grade:

written and oral exam

Sposób i tryb wyrównywania zaległości powstałych wskutek nieobecności studenta na zajęciach:

Student powinien zgłosić się do prowadzącego w celu ustalenia indywidualnego sposobu nadrobienia zaległości.

Prerequisites and additional requirements:

Prerequisites and additional requirements not specified

Recommended literature and teaching resources:

1) M.Capinski, T.Zastawniak, Discrete Models of Financial Markets, Cambridge University Press (2011)

2) M.Capinski, T.Zastawniak, Mathematics for Finance, Springer, London 2010.

3) S.Pliska, Introduction to Mathematical Finance. Discrete time models, Blackwell, Oxford 1997

4) R.Elliott, P.E.Kopp, Mathematics of Financial Markets, Springer 2006

5) S.Shreve, Stochastic Calculus for Finance I, The Binomial Asset Pricing Model, Springer 2004.2.

Scientific publications of module course instructors related to the topic of the module:

1. M.Capinski, T.Zastawniak, Mathematics for Finance, Springer, London 2010.

2. M.Capinski, T.Zastawniak, Discrete Models of Financial Markets, Cambridge University Press (2011)

3. M.J. Capiński, Hedging conditional value at risk with options : short communication , European Journal of Operational Research (2015) vol. 242 iss. 2, s. 688–691.

4. M.J. Capiński, P.E. Kopp, Portfolio Theory and Risk Management, Cambridge University Press (2014)

5. Capiński, Maciej; Zastawniak, Tomasz;
Numerical methods in finance with C++;
Mastering Mathematical Finance. Cambridge: Cambridge University Press (2012).

Additional information:

None