Interest rate models 2012-2013
(5374INRM6Y)

### Contents

Changing interest rates constitute one of the major risk sources for banks, insurance companies, and other financial institutions. Modeling the term-structure movements of interest rates is a challenging task. This course gives an introduction to the mathematics of term-structure models in continuous time. It includes practical aspects for fixed-income markets such as day-count conventions, duration of coupon-paying bonds and yield curve construction; arbitrage theory; short-rate models; the Heath-Jarrow-Morton methodology; consistent term-structure parametrizations; affine diffusion processes and option pricing with Fourier transform; LIBOR market models; and credit risk. The focus is on a mathematically straightforward but rigorous development of the theory.

### Prerequisites

Measure theory, stochastic processes at the level of the course Measure Theoretic Probability, knowledge of stochastic integrals (key words: continuous time martingales, progressive processes, Girsanov transformation, stochastic differential equations) at the level of Stochastic Integration, knowledge of principles of financial mathematics, for instance at the level of Stochastic Processes for Finance.

### Literature

Damir Filipovic, Term-Structure Models, Springer, ISBN 978-3-540-09726-6. Buy it!. The "Contents" of this course are taken from the back cover of the book. See also a list of errata, also available here.

### Related course

Levy Processes and Stochastic Volatility

Peter Spreij

### Homework

Strict deadlines: the lecture after you have been given the assignment, although serious excuses will always be accepted.

### Schedule

Fall semester: Thursdays, 11:00-13:00, Room D1.162 (Science Park); see the map of Science Park and the travel directions. First lecture on 6 September 2012.

### Examination

The final grade is a combination of the results of the take home assignments and the oral exam. To take the oral exam, you make an appointment for a date that suits your own agenda. If it happens that you'd like to postpone the appointment, just inform us that you want so. This is never a problem! The only important matter is that you take the exam, when you feel ready for it. What do you have to know? The theory, i.e. all important definitions and results (lemma's, theorems, etc.). Optional: you may prepare three theorems together with their proofs. You select your favorite ones! Criteria to consider: they should be interesting, non-trivial and explainable in a reasonably short time span. You will be asked to present one or two of them. I am unavailable in the periods December 7-19, December 21, December 26 - January 8, January 21-23.

### Programme

 1 Lecture: Chapter 2 (highlights) Homework: Make exercises 2.1, 2.3(a), 2.5, 2.7; read also the parts of Chapter 2 that have been skipped during class. 2 Lecture: Parts of Chapter 3, survey of some practical methods, Theorem 3.1 in more detail Homework: Make exercises 3.1, 3.2, 3.5 (but don't spend an unreasonable amount of time on them; you are allowed to work in pairs), and look at some examples (get the gist of them!). 3 Lecture: Chapter 5 (highlights), Chapter 6, Lemma 6.1 very briefly; Open question: how to use the last lines of the proof of Proposition 5.2? Homework: Make exercises 5.1, 5.2, 5.4, and if you really want: 10.12 (optional) 4 Lecture: Sections 6.2 -6.4, section 6.5 briefly mentioned, Section 7.1 up to Proposition 7.1 Homework: Make exercises 6.5, 6.6, 7.1(a) (prove first for continuous semimartingales the product rule $\mathcal{E}(X)\mathcal{E}(Y)=\mathcal{E}(X+Y+\langle X,Y \rangle)$ and derive from this a quotient rule) 5 Lecture: (plan) remainder of Chapter 7, Chapter 8 in 5 minutes. Homework: Make exercises 7.2 or 7.3 (you choose) and 7.9(a). Read Sections 8.1 and 8.2 (the non-mathematical part). 6 Lecture: Sections 9.1-9.3 (highlights), conclusions of Section 9.5 mentioned; Theorem 10.1 briefly with a sketch of half of the proof. Homework: Make exercises 9.1, and 9.6(b) or 9.5 (you may use Proposition 9.4.1, but it is perhaps not necessary and you can depart immediately from (9.4)); read the second half of the proof of Theorem 10.1. 7 Lecture: Section 10.2 except Theorem 10.3 and Corollary 10.1 (perhaps next time), section 10.7.1 Homework: Exercise 10.1 (I think you can do without Lemma 10.12; you can always verify that the given function is indeed a solution; moreover, there should be (1-2*u_1*(T - t)) rather than (1-2*u_2*(T - t)), 10.2 8 Lecture: Section 10.3 Homework: Make exercises 10.4, and 10.5 or 10.6; take notice of the contents of Theorem 10.3 and Corollary 10.1 9 Lecture: Sections 10.4 and 10.5 Homework: Read Corollary 10.4 and make exercise 10.6; choose an additional exercise from 10.7, 10.15 and 10.16 10 Lecture: Sections 11.2-11.4 Homework: Make exercise 11.3 or exercise 11.4 (probably you can use here some ideas from the proof of Lemma 11.2), and make exercise 11.5 or give a proof of Corollary 11.1 11 Lecture: Remainder of Chapter 11, Homework: Make exercise 11.7 or 11.8 and (optional!) 11.6. 12 Lecture: Sections 12.2 and 12.3 Homework: Read section 12.1 and make exercises 12.2 and 12.3 13 Lecture: Sections 12.3.1 - 12.3.4 Homework: Make exercises 12.4 (it is sufficient to write down the Riccati equations and compare your work to the expressions on page 87; don't do lengthy computations!), 12.5 (optional!) and 12.6 (also prove the identity in the proof)