Soa6: Valuation of Interest Sensitive Securities - Chapter 3

[ACCtuary]

On page 46, the drift is defined in terms of a function theta. What the $#@! is theta. Where is it defined? We see the notation theta used for two different things:

<ul>
decay of time value of option (Panjer 6.7)
number of shares owned in a dynamic hedging strategy
[/list]
However I suspect this theta is different.

[Exam Slave]

Later on the same page, theta is discussed. Page 48 is the "below" referred to.

It gets ugly, and I'll bet that JAM has a better method for attempting to rmember this stuff.
I'll go look for a reasonable past exam question, although it might be from exam 230.

[Songbird]

Don't confuse them. This theta is a function specific to this situation, unrelated to other "theta"s you have read elsewhere. They could've used f or g or h, but probably followed the notation in the original papers. Basically, the drift rate function here is composed of 2 parts: 1) a function called theta 2) a "pull" on the interest rate so that it does not get unrealistically high

[AOnly]

I just noticed that it says politely after theta is mentioned in the model that it is an "unknown function". How cheerful :wink:

And no. JAM does not shed much light on this.

[Songbird]

I think the "unknown" here tells you that it is a function to be "calibrated" using market information. After calibration, it will become "known" and you can use the model to value other securities. Models are usually like that: you have a few things you need to calibrate first before you can use it. If you are clear which are those things, it'll be easier to follow.

[Dr T Non-Fan]

This unknown-ness causes trouble making an exam question that can be answered in 10 minutes.

[ACCtuary]

I think songbird is very sophisticated based on this and other posts. Big mathematical background, obviously experienced at this sort of thing.

Anyway my only point is to pay a compliment. I guess I'm not antieverything


<font size=-1>[ This Message was edited by: AntiEverything on 2002-02-13 15:42 ]</font>

[smarty pants]

Theta is not to be defined, but to be adored.

[Songbird]

Hey thanks AntiEverything! :smile:
Good luck in your exams.
smarty pants: I think you're referring to "the tao" and not "the-ta". :razz:

[ACCtuary]

Here's a thought - this model and the one in chapter 34 of Fabozzi are both models of the term structure of interest rates. Are you allowed to create just any model and then use it for valuation?

Obviously there are equilibrium (no arbitrage) conditions, and requirement that you match the existing market securities prices.

I just don't see this model Vasicek as being very testable - I guess you could do a drift term, or determine which type of branch you have.

What advantage does Vasicek give you in modelling the term structure that the model in Fabozzi does not?

That's what's on my mind. It is not my intention to have others "do the work for me" on this one.

[ACCtuary]

It is taking me a long time to get through chapter 3,5, and 8. Maybe I should skim and then come back. Sometimes later chapters give insight.

[Songbird]

Vasicek vs Binomial: one is trinomial while the other is, well, binomial. :razz:
To appreciate their differences, look at the underlying interest rate movement process that they model. Both are discrete approximations of a continuous process. Vasicek is the discrete approximation of a rather general stochastic process with a lot of parameters (flexibility). This point is emphasized in the text when it is shown how this model and the explicit form of the finite difference method for the stochastic PDE are consistent. The binomial tree on the other hand, approximates a simpler process: constant volatility lognormal, with varying mean. (ln r ~ binomial/normal at the limit)
The advantage is also mentioned in the text. A more limited version of the Vasicek model created arbitrage opportunities. This is because the model failed to model reality (of interest rate movements) accurately. Similarly, the binomial model will also have this weakness (constant volatility is not realistic enough).
As to what question can come out in the SOA exams, from my experience:
1) Most questions require a lot of depth of knowledge.
2) Heavy formula question usually only appear in the MC part, while the essay does not call for any monster formula. However, your understanding will be tested. I mean REALLY tested.
IMHO:
For the Vasicek model, I do agree it would be very difficult to set a suitable calculations exam question. Skip it if you don't have the time, but take note that the Vasicek model is very useful and very common in practice, so it's importance should favor it in the exam. Chapters 1,2 and 3 should be read early, while 5 and 8 after all other books are done.

[New at pd]

Another to add to the list:

In the case of the binomial model, the tree is created, and the term structure of interest rates is deduced from the short rates of the tree.

The goal of the Vasicek model is to exactly fit the term structure (calculation of theta in this model depends upon yield curve.)

[Songbird]

New at c6:
I'm afraid I don't agree. For any model, the term structure is deduced from the market, and the model calibrated to the term structure. In exams or exercises you may be given the binomial tree first and asked to "deduce" the term structure, but in practice the term structure is given by the market, not up to you to "deduce".

[New at pd]

Then I guess I don't know what JAM is discussing, on page D-20. Any hints?