PhD Course in Continuous-Time Finance
The Interest Rate Bit (April 7, 8 and 22 with RP)


ex-ante

Where: CBS (as usual). More specifically
Monday April 7: 10.30-17.30 at "Dalgas Have" in room DSØ-074
Tuesday April 8: 9-16 at "Julius Thomsens Plads" in room Js06 (morning) and Js13 (afternoon).
Tuesday April 22: 10.30-[some time in the afternoon] at "Solbjerg Plads" in room SPs13
We'll look at how to price options on coupon bonds. This turns out to be more involved than options on ZCBs. See the old exam-question and its answer below.
I will talk about LIBOR market models (articles below, but we can do it more compactly), discuss pricing of caps (Bjørk Ch. 19) and swaptions (see last year's PhD-course exam), and present an application to the evaluation of capped floating rate loans.
Here is a more elaborate plan


ex-post or "what was covered"

Monday April & Tuesday April 8:

I covered Chapters 15 to 19 in Bjørk without major ommisions.
The first approximately 8 pages of Chapter 19 is described differently in my "Numeraire"-note below, but once we get to Theorem 19.8 we're "in 'sync". Decide for yourself which presentation you like best. Mine has more "consider the following odd construction"-arguments, but has the advantage of being able to draw on knowledge of measure changes and Girsanov's theorem. I handed out copies of Bjørk's Swedish lecture notes about this. (So of course Bjørk knows these results; his presentation is a peadagogical choice.)

A note with some more details of the proof of (the important & useful) Proposition 15.5.

Tuesday April 22:

I talked about options on coupon bearing bonds (see the old exam-question + the solution below). The argument works only in the 1-D case. I mentioned a nice article by Claus Munk about what to do in a multi-dimensional case. (But I remembered incorrectly; the article does in fact NOT contain numerical examples for the 1-D case.) The idea goes back to Jamshidian altough my presentation is slightly different and more in line with Geman, El Karoui & Rochet "Changes of Numeraire, Changes of Probability Measure and Option Pricing", J. of Applied Probability 1995 (no electronic version).
Simple forward rates (called LIBORs; see also Bjørk 15.2) floating rate bonds & swaps (Bjørk 15.3). Then I looked at option contracts relates to LIBOR, mainly caplets (and caps) and very breifly about swaptions. In the lognormal LIBOR market model we obtain simple formula for caplet-prices. This is covered in the last "other course" exam question below. The last part of this exercise looks at cute (if I may say so myself) construction of capped floation rate loans.
My slides.



Working on your own

Looking at old exam exercises is the best preparation for you-know-what (but not when).
Some good test exercises from Bjørk to try yor hand at:
3.3. Lemma 3.15 is a very important result. By considering real and imaginary parts seperately you should be able to prove (3.44). To put the final nail in the coffin you need to know that a) the characteristic function for a random variable X is the (complex) function t->E(exp(i*X*t)) b) the characteristic function uniquely determines the distribution c) if X~N(0,sigma^2) the E(exp(i*X*t)) = exp(-0.5*t^2*sigma^2)
4.1-4 +17.1: Vasicek calculations.
4.12: Feynman/Kac relevant for stochatic interest rates.
Work through Section 4.2 on Geometric Brownian Motion. How does Prop. 4.12 change of we're standing at some time point 0 < s < t? What about the multidimensional case?
17.5. After (or before) that read Section 17.4.4 about fitting the Vasicek or Hull/White model.
17.7
17.8
18.1
18.2
19.1-2
19.3



Literature

Change of numeraire

A little note I've made myself (in Danish).
A recent Journal of Derivatives article by Benninga, Bjørk & Wiener about change of numeraire.

Libor Market Models

The models appeared virtually simultaneuously in articles by
Miltersen, Sandmann and Sondermann in Journal of Finance
Brace, Gatarek and Musiela in Mathematical Finance
Jamshidian in Finance and Stochatics

A very useful extension by Andersen and Andreasen appeared in Aplied Mathematical Finance: Here's a working paper version.

Expectation Hypotheses

Will not be covered.

Mortgage Backed Securities

We will not go into much deatil. But here is a classic article by Richard Stanton

Exercises and some Answers & Notes

PhD-course exam 2000
PhD-course exam 2001 (My part only.) My part was about pricing options on coupon-bearing bonds. I've made a little note with a possible answer.
PhD-course exam 2002 (My part only.)
Another set of take-home-exam-questions I've made for a Bjørk based course. And a solution. It is the last exercise that is particularly relevant here. Here is little note I recently made with a toy-model that that exploits the construction of the capped floating rate loan.