Mathematical models of financial derivatives markets II

elective courses

Prerequisites: Introduction to stochastic analysis.

Financial Engineering and Mathematical models of financial derivatives markets are desirable, though not a prerequisite.

The course will desribe: Interest rate securities. Models of short rate. HJM model. Interest rate derivatives (FRA, caps, floors, swaptions etc.). Market models. Callibration of models to market data.

1. Basic definitions of interest-rate derivatives. Martingale pricing. A change of numeraire toolkit.

2. Short rate models: Vasicek model, Hull-White model, CIR model. Affine models. Pricing derivatives in short-rate models. Calibration to market data.

3. Forward-rate models. Model HJM and its properties. Market model and the derivation of the Black pricing formula.

D. Brigo, F. Mercurio – Interest Rate Models – Theory and Practice, Springer, 2006.

J. Jakubowski, A. Palczewski, M. Rutkowski, Ł. Stettner – Matematyka finansowa, instrumenty pochodne. WNT, Warszawa 2006.

M. Baxter – General interest-rate models and the universality of HJM, w Mathematics of Derivative Securities, M. Dempster, S. Pliska Eds., Cambridge University Press 1997, str. 315--335.

D. Filipovic Term-Structure Models. A Graduate Course, Springer, 2009.

Student:

1. knows basic interest rate derivatives, understands the principle of martingale valuation of derivatives, knows the method of changing the numeraire as a derivative valuation technique;

2. knows basic stochastic models of short-term interest rate: Vasicek, Hulla-White, CIR and affine models; knows the basic properties of these models;

3. knows the methodology for the valuation of derivatives in short-term rate models;

4. knows how to calibrate short-term rate models to market data;

5. knows the basic stochastic model of the forward rate - the HJM model and its properties and limitations;

6. knows what the market model of the forward rate is; he knows the proof of Black's formula for caps.

Social competence:

1. understands the problem of stochastic interest rate modeling and the associated modeling of difficulties;

The result of the exam consists of the results from class (for solving homeworks, active participation) - 1/3 and the results of the written exam

consisting of problems and theoretical questions (2/3). Opportunity to improve the grade of the exam during the oral exam.