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Course, academic year 2023/2024
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Stochastic Models in Finance 1 - NMFP505
Title: Stochastické modely ve financích 1
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: winter
E-Credits: 5
Hours per week, examination: winter s.:2/2, C+Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: English, Czech
Teaching methods: full-time
Teaching methods: full-time
Guarantor: doc. RNDr. Jan Večeř, Ph.D.
Class: M Mgr. FPM
M Mgr. FPM > Povinné
M Mgr. PMSE > Volitelné
Classification: Mathematics > Financial and Insurance Math.
Pre-requisite : {One course of advanced Theory of Probability}
Incompatibility : NMFM505
Interchangeability : NMFM505
Annotation -
Last update: doc. RNDr. Martin Branda, Ph.D. (19.12.2020)
This course covers modern finance theory based on the no-arbitrage principle. In order to prevent an existence of a risk-free profit for any market agent, the prices must be martingales under the probability measures corresponding to the reference asset. As a consequence, the prices of financial contract must satisfy certain partial differential equations in the case of diffusion models. The course illustrates these results for the most common financial contracts in various markets, such as stock, interest rate and exchange rate markets. Examples of real data analysis using Python are given.
Literature -
Last update: doc. RNDr. Martin Branda, Ph.D. (19.12.2020)

Vecer, J.: Stochastic Finance, CRC Press, 2011.

Shreve, S.: Stochastic Calculus for Finance II, Springer 2004.

Teaching methods -
Last update: RNDr. Jitka Zichová, Dr. (13.05.2023)

Lecture + exercises.

Syllabus -
Last update: doc. RNDr. Martin Branda, Ph.D. (02.01.2021)

1. Basic financial contracts (options and futures), assets, price of an asset relative to another reference asset. Portfolio, portfolio value and development of self-financing portfolio.

2. Arbitrage, martingales and martingale measures and the First Fundamental Theorem of Asset pricing. Change of the numeraire.

3. Binomial model of price evolution, valuation and replication of financial contracts in a binomial model.

4. Diffusion models. Stochastic integration. Geometric Brownian motion. Stochastic differential equation.

5. Girsanov's theorem and martingale measures in diffusion models. Completeness of the market, Second Fundamental Theorem of Asset Pricing.

6. Representation of a continuous martingale by a stochastic integral, hedging and replication.

7. Black-Scholes formula. Valuation of options. Feynman-Kac formula, BS equation, replication strategy for simple claims.

8. Applications to real financial data. Automatic processing of financial data, valuation of contracts in real time.

9. Exchange rates, contracts on currencies.

10. Interest rate contracts. LIBOR, forward LIBOR, floorlets, caplets, swaps, swap rate and swaptions.

11. Forward interest rate, Heath-Jarrow-Morton model. Immediate interest rate application (Vašíček, Cox-Ingersoll-Ross).

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