One factor interest rate models

[Calcium]

Hi there,

I'm hoping someone can definitively tell me which one factor models do and do not satisfy the following desirable properties for the short rate:

1. Can produce a range of yield curves
2. Can be fitted well to historical data
3. Can be calibrated to current bond prices/yield curve

I feel this is not explained well in the notes (if at all, aside from a mention that Hull White can be better calibrated to current observed prices).

Moreover, I've seen conflicting information e.g. for the Hull-White model with respect to its ability to fit well to historical data, the examiner's report for September 2011 question five says 'yes', whereas Mock A marking schedule says 'no'...

Thanks.

 

[captain_calculus]

I agree, it would be a great idea to have a list of advantages and disadvantages for each model in the notes.

In the meantime can anyone help with this?

Thanks

 

[James123]

Hi,

To add to this I am confused with how to work out if a model meets these criteria...

I am looking at April2014 Paper Q5(i)

dr = (mu)rdt + (sigma)dZt

How suitable is this model for the short rate?

The solution states:
1 - Interest rates may be positive
2 - the model is computationally tractable
3- the model wont give a realistic range of yield curves
4- It wont fit historical data well
5 - It cannot be calibrated to current market data
6- It is not very flexible
7- It is arbitrage Free

Can someone explain how you know these about the model??? Am I missing something obvious? As the solution does not even hint at any calculations need to be done for this - ie. you should be able to instinctively tell...

Thanks in advance

 

[Whippet1]

Yes, I don't think these are properties are all obvious.

1. Even if mu is positive and r(t) will tend to drift upwards, a long run of negative values for dZt could still make r(t) go negative.

2. It's tractable because it is a one-factor model, i.e. it has only one Zt term, and so is simpler than models with two or more factors.

3-6. Are also all likely to be true because it's a one-factor model, i.e. a simple and not very realistic model.

7. I'm not sure from looking at the SDE why it should be arbitrage-free?

 

[Jammy]

I have the same doubt. Could someone clear up how we arrive at the given conclusions for properties 1 and 7?

 

[Mark Mitchell]

For 1. the Examiners Report actually says "Interest rates may not be positive". So, Whippet1's explanation is perfectly correct.

For 7. comparing the SDE in (i), with the SDE for the Vasicek model given later in the question, we see that these match up if we set mu in the Vasicek model SDE equal to 0, and then equate "a" in the Vasicek model with minus "mu" in the first model. Then the model asked about in (i) is really just a Vasicek model in disguise. Since we're told in the notes that the Vasicek model is arbitrage-free (though this is not proved for us), then the model in (i) is arbitrage-free too.