Term structure model

I'm struggling with some of the parts that are left as an exercise to the reader in Chapter 15 of CT8.

In particular, p10: "We can use an argument similar to the original derivation of the Black-Scholes model..."

I assume this means the 5-step method not the one in Ch12, as it involves a risk-neutral measure.

So comparing to the 5-step method:
- we have assets B(t,T) which we model with a diffusion process as on p12
- we have the risk-free force of interest r(t) which we now also model as a diffusion process p10
Substituting exp(-integral(r(t)dt)) for exp(-rt) at appropriate points in the previous 5-step model, everything goes along fine.

But then I get to step 5, where I find I'm trying to replicate zero coupon bonds using zero-coupon bonds and cash. Surely that can be done by just holding one unit of B(t,T) at all times t ?!

Also, I'm not sure what the "market price of risk" arguments are for, except that they show that the risk-neutral measure is independent of the maturity time T of the bond we're looking at. Why is it a problem that r(t) is not a tradeable asset? r was not a tradeable asset before, it was constant.

Am I on the wrong track completely here? I'd be grateful for any hints!

 

Hi Louisa

Chapter 15!!! Wow!!!! Did you know that the exams aren't until April?
I'm afraid I haven't quite got that far in the notes yet. Well done!

 

Well not really :-/ I haven't done the exercises yet, just skimming ahead to the interesting bits. It takes me lots of time to assimilate things - and to find people to answer awkward questions!
Are you studying it? How's it going?

 

Yup - my first go at this & CT6. So far CT8 feels more attainable than CT6, but I fear I may be being lulled into a false sense of security.

Got through the first part alive and just getting into the more abstract stuff. My first run through the notes is usually in-depth 'cos otherwise I get to the later bits and don't have the pre-knowledge from the first chapters. The downside of this method is that by March I've not read the first part for 3 months and have completely forgotten it all!

 

Louisa, after you've done some question practice you'll have a much clearer understanding of Chapter 15. Look at past exam questions on this topic, it's quite repetitive and I think you may be going in too deeply on the detail. With this subject I specifically felt that a broad understanding of the course notes and lots of past exam questions is the key to passing. Don't get cought up in the mathematical detail (I am thinking especially chapter 6-8 and 15)

Good luck.

 

Hi Erik, and thanks for the tips.

Is it 'mathematical detail'? Not convinced; the maths is clear: calculate E_Q(whateveritwas), my question is about *why*. If I don't understand why, I won't know what to do in a different-but-similar situation.

Perhaps will become clear on a second pass; or one of my more thorough colleagues can explain in due course (hi OneLastTheorem!)

Or anyone know if any of the recommended books describe this model?

 

Louisa, I presume you are talking about valuing a bond option? If so, the correct matching portfolio is:

Phi units of a cash bond B_t ( = exp(int_0^t{r_s ds}) )

Psi units of a discount bond beginning at time t that matures at time T (denote its value at time t to be P(t,T) )

The maths gets pretty nasty compared to Black-Scholes for shares. Refer to Baxter & Rennie for some details.

Also if you are _really_ interested, look up option pricing under the Affine Term Structure model, which generalises the nasty maths for pricing bond options. This is way beyond CT8 though, so I think you are best not to worry about it...

 

It's simpler than that - they're just valuing the zero-coupon bond itself B(t,T).
I was probably making it more complicated than it should be - all you need to show is that the discounted bond price is martingale? So I guess it's not as bad as shares and options?
L

 

This question is doing my head in too - especially after I spent ages making sure I understood the 5-Step/Martingale Method for Share Derivatives.

Using the formula for bond-prices on page 10 of Chapter 15, I can show that the discounted Bond Price is a martingale.

But wouldn't it be a martingale under the real-world probability measure too?

And what are the other steps in showing that this formula is the price of the bond?

I'm assuming it's quite a bit harder than the method for option-prices - otherwise why not go through it in the notes after all the details of the proof in chapter 14?

It could be that I'm missing something simple, or perhaps the answer is way beyond CT8. If it's the latter, does anyone know if this is covered in any later subject?

Cheers,



Sam

 

This quote from http://www.quantnotes.com/fundamentals/bonds/interestrate2.htm
seems to help explain what's going on.
"A bond is treated as a derivative of interest rate; much like an option is a derivative of the underlying stock. The difference however, is that the stock can be traded openly on the market. Interest rate is not an asset, it is more like a market parameter, and can not be traded. Hence, the standard Black-Scholes argument (i.e. to hedge the derivative with the underlying), breaks down. The only instrument that can be used to hedge a bond is another interest rate derivative; another bond of perhaps longer maturity than the bond that we are trying to price. But this seems counter-intuitive. How can you price an interest rate instrument (e.g. bond or bond option) using another instrument (bond) which you do not know how to price in the first place? This is kind of like a 'chicken and egg' situation, and is why you need a market price of risk. The market price of risk introduces a basis; a start point. The argument is as follows. Forget about the 'chicken and egg' situation, and assume that the current market of bonds is perfect. Therefore, assume that the prices of bonds (driven by demand and supply) are arbitrage free, so that the instrument that you are hedging with is well defined and not an unknown. The model and PDE has to incorporate this 'basis', so we use the market price of risk to force the model to accept it. In short, use market price of risk as a calibration parameter to fit the model to the market (which we assume is perfect and arbitrage free), and to ensure that the values of instruments output by the model/PDE closely resemble the actual market prices of those same instruments. "

So it sounds like for bonds, the market price of risk is key; and the no-arbitrage assumption is used to show it's constant. Then "risk" is used as the underlying tradeable thing to apply the Black Scholes hedging argument to.

Don't have the Acted notes to hand, but that sounds consistent with what I remember from them?

I wouldn't expect the bond price to be martingale under the real world measure. What makes you think it is Sam (if you're still around)?

 

Hi Louisa,

Thanks for the link. Did you sit CT8 this time? If so, how did it go? It went alright for me, but not brilliant - could probably go either way.

I think I get what your argument means.

I wonder if you can think of it like this: we can create a replicating protfolio for a Bond1 of term T1 (which takes the place of the derivative in the original Black-Schole argument) using the risk-free asset and a Bond2 of term T2 (which takes the place of the underlying share in the original B-S argument).

Then we can work out the price of Bond1 in terms of Bond2. If we know the price of Bond2 from Market data then (if there is no arbitrage) we can get a theoretical price for Bond1?

But will this only work if we have a one-factor model for the short interest rate?

Once again I may be way off.

I'm thinking of getting a financial economics text book because I really want to understand the subject (I know it probably seems like I need to get out more when I say that, but it's good to study something interesting for a change). Can anyone recomend a good text book?



Sam

ps. I was talking about the discounted price being a martingale. Here's what I meant (using share-prices because this is probably wrong for bonds after all).

For example the expected discounted future share price discounted at the risk-neutral rate of return is the current price if we use the risk-neutral measure to work out the expected value. But I think we could use exctly the same argument using the actual drift of the share price and the real-world probability measure.

I was asking why we couldn't use this in our arguments on how to price derivatives. Even though I could follow the maths for the Black-Scholes PDE and for the 5-step method, I couldn't think of a good explanation in words for things like this. I think I understand it a bit better now though.

 

That seems right that it would only work for a one-factor model, the one factor giving the risk which has a market price. Otherwise we'd have more mu's and sigma's hanging around and it'd get messier?

No, I got it last time - but still interested as I'm planning to attempt ST6 next time. I've just got the Baxter&Rennie book examstudent mentions - looks really helpful as well as concise. Hull on the other hand is 750 large pages of dense print long, and looks scary.

I don't see how that would work. We have a cash bond that returns at the risk-free rate of return r to build portfolios with, but nothing that grows at the drift of the real-world measure.

L

 

No, not quite. You need to use state price deflators to do this, i.e.

E_Q(exp(-rt) S_t) = S_0

and

(xi_0)^-1 E_P(xi_t exp(-rt) S_t) = S_0

where xi_t is the Radon-Nikodym derivative.

As an exercise, try calculating xi_t for the Black-Scholes model of equity prices and see if you can prove the above are the same.

 

Thanks for your replies examstudent, Louisa and Gareth.

I'll have a go at that state-price deflator approach in continuous time. I didn't have time to look at that in any detail before the exam.

I hope I don't sound stupid and/or over-ambitious when I try to talk about things like this. I did realise that I couldn't use exactly the same approach under the real-world measure. I just thought that you might be able to do something similar by defining martingales under the real-world measure - and I was trying to work out what would be different to make it work.

Thanks for your help anyway - and sorry these posts are getting a long way off the "term structure of interest rates" topic.

I'd certainly consider doing ST6. I'd have to be quite careful before starting it because I know it's quite hard. Not sure what my current employers would think to it though since I work in Life Insurance and Pensions.



Sam