Forward Volatility and Volatility Surface

Has it worked this time? Yes.

Grant.

 

Re Excel file above. Will need to save to disc before opening.

For some reason the files is named "1" without .xls suffix.

Grant.

 

Grant, in both formulas in cells B3 and B8 you have SQRT and parens a bit messed up. B3 should be:

=SQRT((E2^2*C2-D2^2*B2)/(C2-B2))

(B3 just happens to turn out correct because denominator = 1 in your example)

 

Nonprophet,

Thank you for the correction.

As anticipated, the problem seems due to insufficiently understanding Excels order of precedence. The change in the parentheses in the incorrect example was an attempt at isolating the various elements to force correct calculation. I assumed this was correct as the calcs for both examples are interchangeable, ie copy B8 to B3 and the answer is 7.48%; copy B3 to B8 and the answer is 27.69%.

This was a problem of which I was aware when I first started using Excel over 15 years ago, specifically when trying to verify the figures for Black-Scholes – I would cross-reference various examples from different books – Kolb, Bookstaber, Cox & Rubinstein (I don’t think anyone recognises these names anymore).

Fifteen years and I still haven’t got a grip of it. That’s shaken me. Could have cost a fortune. Sobering thought.

Thanks, once again.

Now, apart from highlighting errors does forward volatility provide additional information over plain vanilla implieds or is it relevant only to exotics?

Grant.

 

Grant, When you write

SQRT(a)/b

then of course b is not included in SQRT.

Your earlier forvol formula also is not correct. It should be:

Forward vol = SQRT((IV2^2*T2-IV1^2*T1)/(T2-T1))

so SQRT(a/b).

It seems you did translate the wrong formula correctly into XL.


In never used forward vol but I guess it is based on a statistical relation between standard deviations of subsets. If so, this formula actually might be incorrect. In words it says:

the variance of the combined set is the weighted mean of the subset variances PLUS the weighted variance of the subset means.

I *think* I miss the 2nd part in your formula. I can look into it if you can tell where you found this formula.

Perhaps a statistician can weigh in?

 

Profitaker,

I've just realised you've also answered my queries on T2W. Well, well.

Here is the link for the formula (bottom of page):

http://www.risklatte.com/exotics/hotNotes/hotNotes017.php

"the variance of the combined..." That's exactly how I saw it (cough).

Grant.

 

In my opinion that formula is incorrect, or incomplete.

The basis of the formula, I assume, is the equation

Vol2^2 = (Vol1^2*T1 + Vol3^2*T3 )/T2 + ((M1–M2)^2*T1 + (M3–M2)^2*T3)/T2

I couldn’t find a quick web reference but it is standard statistics stuff.

Here I write Vol instead of IV (I’d better write StDev). For example, Vol2 is the complete “IV” over 90 days, Vol1 over the first 30 days and Vol3 over the last 60 days, so Vol3 would be the Forward Vol you’re looking for.

M1, M2 & M3 are respective MEANS of the data series (of the log price changes).

The righthand part of the right side of the equation (the complete M-part) is a problem because we don’t know M2 and M3. So why not simply ignore this part? Then solve what’s left over for Vol3 (the Forward vol) and you end up with the risklatte formula.

But based on this the formula is obviously incorrect. However, I checked a few stock price series and the missing part usually turns out very small, so I guess the formula just serves as a practical estimation.

Now you still don’t know its relevance to (vanilla) options but at least you know it’s “wrong”

 

Panzerman,

I wonder if you would clarify a few points, please?

When you refer to “normalise(d)” data, how does one generate relative strikes and iv’s? Could you provide an example?

“Consistency in time” . Assume we are comparing 30, 60 and 90–day options. Would all expirations be adjusted by a ‘factor of duration’ (Taleb, Dynamic Hedging, p151, 1st edn) for 90-days, ie sqrt(90/days to expiration)?

Re volatility surfaces. Please refer to the attached Doc. file (you amy ahve to scroll up the screen).

First column = strikes, 2nd = implied (derived from last trade and contemporaneous underlying), 3rd = change from previous average (I’ve also updated my data to include per strike change from previous).

The Average figure at the foot of the iv column is self-explanatory.

Re the sum of positive (negative) shift and Net. I’m just playing around with figures here. Do these figures reveal anything about the skew? Is there a more appropriate/common measure?

Comments from others, as usual, are also welcome.

Thank you.

Grant