M2MO: Modélisation Aléatoire, Finance et Data Science

Master en statistique, probabilités et finance - Université Paris 7 - Paris Diderot

Courses Group Quantitative Finance Advanced modeling in interest rate

Advanced modeling in interest rate

Lecturer: Z. Grbac
Period: Term 3
Hourly Volume: 3 hours per week


The aim of this course is to provide an overview of the fixed-income markets and the methods used for modeling and pricing of interest rate derivatives. 

The terminology fixed-income market designates a sector of the global financial market on which various interest rate-sensitive instruments, such as bonds, forward rate agreements, swaps, swaptions, caps/floors are traded. Fixed-income instruments represent the largest portion of the global financial market, even larger than equities. Developing realistic and analytically tractable stochastic models for the dynamics of the term structure of interest rates is thus of utmost importance for the financial industry. From the theoretical point of view, interest rate modeling presents a mathematically challenging task, in particular due to the high-dimensionality, possibly even infinite, of the modeling objects. In this sense interest rate models substantially differ from equity price models. Moreover, the credit crisis in 2007?2008 and the Eurozone sovereign debt crisis in 2009? 2012 have heavily impacted the fixed-income market and irreversibly changed the way it functioned in practice, as well as the way in which the interest rate models were developed.


The focus of the course will firstly be on the various classical interest rate models and their properties. For each modeling class, the derivative pricing methods will be studied and calibration issues discussed. Then we will shift the focus to the post-crisis interest rate modeling and show how to extend the classical setup to obtain post-crisis models (known as multiple curve models). In a different vein, models taking into account issues such as realistic modeling of the caplet volatility smile and negativity of interest rates will be presented.  We will conclude with some recent developments related to the LIBOR/Euribor reform and the pricing of derivatives based on the backward-looking rates. 







1. Introduction to fixed-income markets and overview of the most liquidly traded interest rate derivatives. Interest rate curves and their main properties. New challenges in interest rate modeling: interest rate modeling after the 2007-2008 financial crisis (multiple curve interest rate models), negativity of interest rates, LIBOR and Euribor reform. 

2. Short rate models. Calibration to the initial term structure observed in the market: Hull-White Vasicek model, models with deterministic shift (G2++, CIR ++). Option pricing in short rate models.

3. General affine short rate models. Extension to post-crisis short rate models and examples. Calibration. 

4. HJM framework. Examples of volatility structures and special cases. Markov property of the short rate. Swaption pricing using the Jamshidian decomposition. 

5. Multiple curve HJM model 

6. Libor Market Model and its calibration. Black's formula, caplet implied volatility and the problem of volatility smile in the Gaussian interest rate models.  Extensions to stochastic volatility (SABR model)

7. Recent developments in interest rate modeling. Models for negative interest rates. LIBOR replacement and models for backward-looking rates.





1. D. Brigo et F. Mercurio (2006). Interest Rate Models: Theory and Practice (2nd edition), Springer.

2. D. Filipovi? (2009). Term Structure Models, Springer.

3. Z. Grbac et W. J. Runggaldier (2016). Interest Rate Modeling: Post-Crisis Challenges and Approaches. Springer.

4. M. Jeanblanc, M. Yor et M. Chesney (2009). Mathematical Methods for Financial Markets, Springer.

5. M. Musiela et M. Rutkowski (2005). Martingale Methods in Financial Modelling (2nd edition), Springer.


A comprehensive bibliographic list including the relevant research articles will be provided at the first lecture.