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# Index constituent weightings and implied volatilities

Discussion in 'Options' started by Grant, Aug 26, 2006.

1. ### Grant

Assume an index is composed of two stocks, A & B.

Prices, weightings are:
A at 100, 75%
B at 50, 25%

Therefore, the current value of the index = 100 x .75 + 50 x .25 = 87.5.

Now assume the implied volatility of A is 25%, B at 15%.

Is the implied volatility of the index 22.5%:

A at .25 implied x .75 weighting (= .1875)
+
B at .15 implied x .25 weighting (= .375)
=22.5%?

Thank you for any assistance.

Grant.

2. ### Profitaker

Depends how stocks A & B were correlated.

If the correlation was +1 then yes, the index would have a volatility of 22.5%.

3. ### Grant

Profitaker,

And a lack of correlation would be a potentail arbitrage?

Grant,

4. ### Profitaker

It depends what the index options were trading at.

If there was no (zero) correlation the index volatility would be 19.12%. If the index options were trading for more than that you'd have an arb by selling the index Vol and buying the component vol, known as a dispersion trade / volatility arb.

If the index options were trading for less than 19.12% you could do the opposite - long index Vol and short component vol (a reverse dispersion trade) though it carries risk and so isn't an arb.

You should determine the implied index correlation (IIC), value between 0 - 1, and then take a view on the likely correlation.

2 weeks ago I ran the numbers for the FTSE100 constituents (using Sept06 options and a nominal IV of 20% where no options on the component stocks exist) and the IIC was 0.44.

5. ### rosy2

you can get a better grasp of the weightings by obtaining the eigenvalues. and to determine if there is a profitable arbitrage you would need to know if the vol is mean reverting

6. ### ElectricSavant

Profitaker,

If this is too personal would you name the minimums...

Michael B.

7. ### Profitaker

Rosy

Mean reversion of vol is irrelevant in this case. All what matters is the difference between the index IV and the components IV. Ditto for eigenvalues (whatever that is).

ES

I trade the UK FTSE100 options and (so far) only from the reverse dispersion side. A 100% replication isn't possible or practical, so it becomes slightly subjective when choosing sector and individual stock exposure. I usually have other options running concurrently in my portfolio so I can't really answer your Q. Not being evasive - it just depends.

incorrect . Using the same logic , if all 500 SPX stocks have an IV of 30 ( and equal weighting) , would the Index's IV be 30 too ?
Never gonna happen

9. ### ElectricSavant

ok thanks Profit...good trading to you...and I sure do appreciate your posts..

Michael B.

10. ### Profitaker

If all stocks were correlated +1 then yes it would.

#10     Aug 26, 2006
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