Institute of Applied Mathematics, METU, Ankara, Turkey

Institute of Applied Mathematics
Middle East Technical University
Ankara, Turkey
Telephone: +90 312 210 2987
Fax: +90 312 210 2985
wwwiam@metu.edu.tr

Motivation: Stochastic interest rates are an essential ingredient for the modeling of the financial markets of fixed income and credit markets (of bonds, credits, etc.) and their derivatives (forwards, swaps, credit default swaps etc.). Interest rates of bonds usually contain a “risk-free” component related to the money market and swap market (the term structure) and a further add-on (spread) component related to the default risk. While the former components may be described by multidimensional diffusion process, the latter require the inclusion of jumps, first of all in order to describe the default process (as a pure jump process), and furthermore in order to account for the high volatility in credit default markets (by modeling the default intensity as a jump diffusion process).

The course gives a unified approach to stochastic modeling of both, interest and credit. The content of the course covers in particular following topics: zero rates, zero coupon bonds; term structure; bootstrapping; no arbitrage pricing; relative pricing; change of numeraire; affine term structure; short rate models; diffusion processes; deterministic drift extension; tree-approximation; early exercise pricing; swap numeraire; implied volatility surfaces; LIBOR forward market rate model (LMM); calibration to caps; approximations for the swap rate; instantaneous correlations; calibration of correlations; local and stochastic volatility extensions; credit default process; intensity process; calibration of CDS rates and spreads.

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Aims

A first aim of the course is to obtain insight into stochastic modeling of the common types of interest rates as state variables. We will consider both, short rate models (e.g. V, HW, CIR, CIR++) models) and market forward rate models (e.g. LMM). After getting familiar with the structure of these models, we will consider techniques and approximations for calibrating and solving the SDEs in complete markets. Analytic and numerical methods for the computation of zero bond prices and option prices will be presented. Furthermore pricing and hedging of some of the most common derivatives such as European and Bermudan swaptions will be considered. In order to account for volatility dynamics, we also consider local and stochastic volatility extensions of these models. A second aim is to exhibit analogous intensity based models for interest products with counterparty default risk, such as defaultable bonds, and credit default swaps.

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Learning Outcomes

Upon successful completion of the course, it is expected that the student will be familiar with stochastic short rate models and analogous extended models for default intensity, common versions of the LIBOR market model, calibration of the models, application to pricing of common interest rate and credit derivatives. The student should be aware of the underlying assumptions for pricing and hedging approaches in interest and credit markets.

The inclined research student will be prepared to start mathematical research on the topics mentioned in the course.

During the exercises and in subsequent research projects, the student will learn how to apply the models in practical examples, using the programming standard (Excel/VBA) of the financial industry.

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Suggested Textbooks

Lecture notes will be prepared during the course. Some of the core content of this course is covered in:

D. Brigo and F. Mercurio, Interest Rate Models, Theory and Practice, Springer, 2001

Some of the general mathematical foundations concerning markets, pricing, and hedging are covered in:

M. Musiela, M. Rutkowski, Martingale Methods in Financial Modeling, Springer, 1997
I. Karatzas, S. E. Shreve, Methods of Mathematical Finance, Springer, 1998

Additional recommendation for research papers on the covered topics will be given during the course.

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Outline

Complete markets, no arbitrage pricing, martingale measure (1 weeks)
Bootstrapping of the yield curves (1 week)
Short rate models: Affine term structure, deterministic drift extension, prices of zero bonds and options (2 weeks)
Calibration and tree-approximations of short rate models (2 week)
Pricing of coupon bonds, swaps, caps, and swaptions (1 week)
LIBOR forward market rate model (1 weeks)
Calibration of the LMM (2 weeks)
Local and stochastic volatility extensions (1 weeks)
Modeling of the default intensity process (1 week)
Pricing of defaultable bonds and CDS (1 week)
Calibration of CDS prices (1 week)

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