Chapter 14

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Chapter 14

Postby wlfgngpck » Sat Sep 03, 2011 11:05 pm

I've just finished reading through and re-reading through Chapter 14, and have begun to attempt the exercises, and I seem to be completely lost in this chapter. Since I think interest rate derivatives are probably one of the hardest topics in this book, hopefully my questions will be of some use to everyone here.

For one, the displaced diffusion model seems a bit strange to me, and I think it's because I'm not understanding it (in light of exercise 14.1, which I am unable to solve). As you have written in the book, we have:

df = sigma (f+a)dW, where a is some constant.

Question: Is there really no drift in this model?

I ask because in exercise 14.1, it asks for the drift of f (in the measure associated to that particular ZCB), but if the drift is 0 to start with then it seems like this is a non-starter. I've followed through the method you employ in section 14.3, but if I assume that df = sigma (f+a)dW it seems like there's really just nowhere to go (it implies things are 0 that I know shouldn't be).

(Note: As I'm writing this, I do think back to chapter 6, where we didn't make any assumptions on the drift except that it was constant times S(t) and I guess that could include 0....is that still what's going on here also?)
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Re: Chapter 14

Postby mj » Sun Sep 04, 2011 5:56 am

well if you take the ZCB for the payment time of f as numeraire then the drift is zero, otherwise it is non-zero.

This is the same as for the zero a case.
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Re: Chapter 14

Postby wlfgngpck » Sun Sep 04, 2011 6:20 am

Ok certainly, because in that case we have a martingale. But in your presentation on page 355, there is no mention of the measure or numeraire at all. You simply say:

"In particular, we put df = sigma (f+a)dW, where a is the constant displacement coefficient." The way this is written suggests that this is the process we are assuming as a real-world process, since no numeraire is mentioned. So I'm guessing then that we are assuming in that setting that the numeraire is the ZCB for the payment time of f?

It seems like the point of that exercise is to re-work through the argument in section 14.3, but changing the process by turning df into d(f+a) and turning every f in that process to f+a, and then deriving the drift following the same method. (Is this right?)

But then you bring up precisely my concern in your response. The numeraire you suggested is the ZCB expiring at its reset date....wouldn't that exactly be the ZCB for the payment time of f? And wouldn't this imply the drift is 0? (This certainly can't be the answer, but this is where I'm getting tangled up here).
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Re: Chapter 14

Postby mj » Sun Sep 04, 2011 6:27 am

forget about real-world measures -- in truth, once you have finished reading your first book on derivatives pricing you'll never seem them again...

reset time = beginning of f
payment time = end of f
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Re: Chapter 14

Postby wlfgngpck » Sun Sep 04, 2011 6:36 am

"reset time = beginning of f
payment time = end of f"

And that right there is where I'm screwing things up. This is similar to what you mentioned in Chapter 13, about taking P(T_2) as a numeraire or P(T_1), and that's why in the numeraire with ZCB paying at reset time the drift will be different.

Ok thank you so much!
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