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# Interest Rate Modelling - MAT00105M

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• Department: Mathematics
• Module co-ordinator: Dr. Zaq Coelho
• Credit value: 20 credits
• Credit level: M
• Academic year of delivery: 2024-25
• See module specification for other years: 2023-24

## Module summary

The module introduces the probabilistic concepts and techniques necessary for modelling the dynamics of interest rates.

• None

## Module will run

Occurrence Teaching period
A Semester 2 2024-25

## Module aims

The module introduces the probabilistic concepts and techniques necessary for modelling the dynamics of interest rates.

## Module learning outcomes

At the end of the module students 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;

• price various derivative securities (such as caps, floors, swaps) written on bonds as underlying securities;

• work with the features of various models describing the dynamics of interest rates and explain 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 term structure of interest rates. Methods of constructing long horizon term structure (bootstrapping STRIPS).

2. 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. The necessity of developing a theory of random interest rates.

5. 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. Short rate models in continuous time: Merton, Vasicek, Cox-Ingersoll-Ross.

7. Heath-Jarrow-Morton forward rate model in discrete and continuous time.

8. Model calibration and pricing derivatives (calls and puts on bonds, callable bonds, caps, floors, caplets and floorlets).

9. LIBOR model of simple forward rates.

## Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Interest Rate Modelling
3 hours 100

None

### Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Interest Rate Modelling
3 hours 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.

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.

The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.