## yearfractions in equations and a question about swaps

This forum is to discuss the book "the concepts and practice of mathematical finance" by Mark Joshi.

### yearfractions in equations and a question about swaps

Hi

I have again two questions which boggles my mind.

First question is simple, but I cannot find an answer for this from anywhere in the text. This is because in book one always makes the simplifying assumption that dates T are assumed to be nice whole numbers, even though they are dates in practise.

So take an example. Assume vol(t) means a forward rate volatility, and the forward rate has been quoted in actual/360 basis (for example).

If there is an integral which needs to be computed

int_{T_0}^{T_p} vol(t)^2 dt = int_{0}^{T_p-T_0} vol(t-T_0)^2 dt

vol here is yearly volatility, but what exactly does T_p-T_0 mean?
I have two alternatives:
It is the euclidean distance between the two dates T_0 and T_p in YEARS.
If this is the case, what does a year mean? Computer stores dates in serial numbers, where usually the difference between two dates is a number which represents the days between the dates. What is the method to convert days to years? Divide by 365?

It is the daycount adjusted dayfraction between dates T_0 and T_p quoted in act/360 basis.

I have a feeling that the second one is true, but I want to make sure.

More generally in equations where there is the difference between two dates, which is the right way to think what the actual number is? Is it always the euclidean distance (in years) or the daycount adjusted alternative, and the right adjustment needs to be figured out from the occasion?

Second question in swaps:
The market convention in EUR market is to assume the floating leg in swaps to be 6M interest quoted in actual/360 basis. The fixed leg pays annually, and the interest is being made in the bond 30/360 basis.

Does this means that only the fixed leg pays interest with 30/360 convention and floating leg pays interest with actual/360 basis? Or is it that also the floating leg pays interest in 30/360 basis even though the interest rates are based on actual/360 basis?

Thanks in advance!
Lauri

Posts: 11
Joined: Fri Jun 13, 2008 12:37 pm

day count fractions are only relevant when computing pay-offs, the volatility is the vol of a rate how that rate is defined doesn't affect it.

I would think that
the fixed leg pays interest with 30/360 convention and floating leg pays interest with actual/360 basis. This isn't really my expertise. I think James and Webber covers this stuff.
mj
Site Admin

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

Hello Mark

Thanks for the second question answer, Ill check the reference you gave.

For the first question: In practise the forward rate expiry maturity pairs are not equally spaced because of holidays and other kind of calendar stuff that practioners take into account :)

Of course it affects the integral values if you take day count fractions into account.
But from what you said I understood that the right way is to think that the indices T_i are pure time points in time frame (in years), and one takes this moment as the origin. Other time points are measured with respect to this day in years and when transforming the days into years one uses the normal euclidean distance and each year has 365 days.
Is this right?
Lauri

Posts: 11
Joined: Fri Jun 13, 2008 12:37 pm

yes

ultimately you best benchmark is whether the final model reproduces caplet prices correctly -- if you get it wrong, it won't...
mj
Site Admin

Posts: 1380
Joined: Fri Jul 27, 2007 7:21 am

Return to The concepts and practice of mathematical finance

### Who is online

Users browsing this forum: No registered users and 0 guests