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# forward vol

Discussion in 'Options' started by erasmus, Jan 27, 2010.

1. ### erasmus

You can calculate the forward volatility between tenor 1, T1, and tenor 2, T2, using the formula below.

Consider: T2> T1

Vol T2-T1= (((VolT2^2*)*T2-(VolT1^2*)*T1)/(T2-T1))^(1/2)

What happens when Vol T1 > Vol T2 ? You can get situations where you have a negative in the sqrt function?

Apparently then by going short the Vol T1 (option) and long the vol T2 (option) you can arbitrage? Can someone plz clarify in more detail?

Thanks

2. ### Martinghoul

Look on Wilmott, there are answers aplenty there...

3. ### MasterAtWork

Hi,

you have a lower boundary for vol2

vol2=vol1*SQRT(T1/T2)

There is no way Vol2 be lower that level. If it were, that would tells you that volatility between T1 and T2 gets negative. How would it be ?

Masteratwork.

4. ### Martinghoul

In the absence of vol of vol, it implies negative fwd variance, which is arbitrageable. In the presence of vol of vol, the equation doesn't have to hold. So be careful when trying to apply this logic in the presence of skew to non-ATM options.

5. ### MasterAtWork

HI Martinghoul

What do you mean with "In the presence of vol of vol, the equation doesn't have to hold" ?
The equation is based on the definition of the variance. I May miss something but I don't see where vol of vol is involved ?

6. ### Martinghoul

Lemme find you a paper written by someone better than me. If I can't, I'll figure out how to say it meself.

7. ### MasterAtWork

'would be nice

8. ### erasmus

I found this one

Although i am still not clear on the arbitrage arguement (from a practical perspective).

On the side something unrelated:

If you know 1week atm vol traded at 10%, where should 6 day theoretically trade? If you consider 365 * daily variance = 10% then you know the daily variance. Then plug that into:

7 * daily variance = 2 * (weekend variance) + 5 * (business day variance)

Assuming weekend variance = 0, Then you know the business day variance. Then for 6 day vol consider that as 2 weekend days and 4 business days so:

6 * daily variance = 4 * (business day variance)

Then using the business day variance previously found, solve for the daily variance in this last equation and then annualize it. But there is something wrong with this method as it doesnt give the same answer as simply doing

SQRT(4/5) * 10%

And im pretty sure the last short cut is correct

9. ### erasmus

That should be 365 * daily variance = (10%)^2

10. ### Martinghoul

MAW, I have been trying to find some specific stuff on the matter. I have found a few links, but they don't address the question specifically (there's a few papers that talk about it on JStor, but I can only see abstracts).

My understanding of this is that calculation of standard deviation as a square root of variance has all sorts of issues, especially in the presence of autocorrelation.

#10     Feb 2, 2010
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