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Modelling of Bonds, Term Structure & Interest Rate Derivatives - MAT00009M

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• Department: Mathematics
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2022-23

Related modules

Module will run

Occurrence	Teaching period
A	Spring Term 2022-23

Module aims

The module introduces the probabilistic concepts and techniques necessary for modelling the dynamics of interest rates. The mathematical theory of interest rates is complex because it has to cover random behaviour of several different rates simultaneously, while remaining consistent with no-arbitrage restrictions. Additionally, to be realistic, models need to allow calibration of the parameters to real data. The complexity stems from the fact that in general interest rates depend on running time and maturity time, so are stochastic processes of two time variables, each with a very specific role. Discrete models will be constructed based on tree structures. In full generality the theory of partial stochastic differential equations is needed to investigate continuous-time models.

There is no such thing as the best or universally accepted model of the term structure of interest rates. Hence this module shows a variety of approaches and much time is devoted to the study of their relationships. One crucial issue is concerned with fitting the model to the data, called calibration. Pricing interest rate derivative securities is of great importance since they represent the majority of derivatives traded in financial markets.

Module learning outcomes

At the end of the module you should be able to:

• construct arbitrage-free models of interest rates and the term structure of bond prices in the binary tree model and price interest rate derivatives within such a model;
• be able to price various derivative securities (such as caps, floors, swaps) written on bonds as underlying securities;
• understand the features of various models describing the dynamics of interest rates and be able to see the connections between them both in discrete and continuous time framework;
• price interest rate derivatives within some selected continuous time models, including short-rate and Heath-Jarrow-Morton models;
• demonstrate skills necessary for practical implementation of the techniques, in particular, be able to calibrate selected models.

Module content

1. The concept of the term structure of interest rates. Methods of constructing long horizon term structure (bootstrapping STRIPS) .
2. Tools describing the dynamics of bond prices: yields, forward rates, short (instantaneous) rates. Money market account.
3. Fixed and floating coupon bonds. Interest rate swaps.
4. Risk management in the case of parallel shift in interest rates. Applications of duration and convexity for immunisation of bond portfolios. Problems with non-parallel shifts of term structure and tools necessary in this case.
5. Necessity of developing a theory of random interest rates. Construction of binomial trees for bond prices, yields and forward rates. No arbitrage principle and its consequences concerning admissible models. Risk neutral probabilities and   their dependence on maturity. 
6. Presentation of short term models (Merton, Vasicek, CIR) .
7. Outline of Heath-Jarrow-Morton forward rate model in discrete and continuous time framework.
8. Model calibration and pricing derivatives (calls and puts on bonds, callable bonds, caps, floors, caplets and floorlets).
9. Time allowing: LIBOR model.

Indicative assessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Special assessment rules

None

Indicative reassessment

Task	% of module mark
Closed/in-person Exam (Centrally scheduled)	100

Module feedback

Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.

Indicative reading

D. McInerney and T. Zastawniak, Stochastic Interest Rates, Cambridge University Press, 2015

M. Capinski and T. Zastawniak, Mathematics for Finance, Chapters 10-11, Springer-Verlag, London 2003.

R. Jarrow, Modelling Fixed Income Securities and Interest Rate Options, McGraw-Hill, New York 1996.

T. Bjork, Arbitrage Theory in Continuous Time (Second Edition), Oxford University Press, Oxford 2004.

M. Musiela and M. Rutkowski, Martingale Methods in Financial Modelling (Second Edition), Part II, Springer-Verlag 2007.

D. Brigo and F. Mercurio, Interest Rate Models - Theory and Practice:With Smile, Inflation and Credit (Second Edition), Springer-Verlag 2007.