Case 013Options and volatility tradingHard

An equally weighted index of ten stocks has implied volatility 18% while each member's implied is 30%. Compute the implied correlation and the sign of P&L for selling index volatility and buying member volatility if realised correlation is 0.25.

The situation

Taruvara Capital looks at a sector index made of ten stocks in equal weights. Three-month options on the index trade at 18% implied volatility, and options on each of the ten members trade at 30%. To keep the arithmetic clean, treat all members as identical.

The desk proposes a {term('dispersion trade', 'Selling volatility on an index while buying volatility on its members, a position that profits when the members move independently and loses when they move together.')}: sell index volatility with a vega of Rs 10 lakh per vol point, and buy member volatility sized so that the member legs break even if members realise their 30% implied. Its forecast is that realised correlation over the three months will be 0.25.

Your task

What correlation is the market implying, what is the sign and rough size of the P&L if realised correlation is 0.25, and what is the risk?

Quick check

Roughly what correlation does an 18% index against 30% members imply?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The index implies a correlation of about 0.29; if members realise 30% and correlation realises 0.25, the index realises about 17.1% and the trade makes money. Short index vol gains about 0.9 vol points, roughly Rs 9 lakh on Rs 10 lakh of vega. The trade is short correlation: in a sell-off where correlation jumps to 0.6, the index realises 24% and the short leg loses about Rs 60 lakh.

Step 1How do you back out a correlation from two volatilities?

Write the index variance in pieces. With equal weights of one tenth, each member contributes its own variance times one hundredth, and each of the 90 pairs contributes the correlation times the product of their volatilities. Index variance is one tenth of the member variance, plus nine tenths of it times the correlation, so the only unknown is the correlation. Think of ten rowers in one boat: each one's wobble mostly cancels, and how far the boat swings depends on how often they lean the same way.

The relationship
σI2=σ2[1n+(1−1n)ρ]  ⇒  324=90+810 ρ  ⇒  ρ≈0.289\sigma_I^2 = \sigma^2\left[\tfrac{1}{n} + \left(1-\tfrac{1}{n}\right)\rho\right] \;\Rightarrow\; 324 = 90 + 810\,\rho \;\Rightarrow\; \rho \approx 0.289
\sigma_Iindex implied volatility, 18%, so variance 324 in percent squared
\sigmamember implied volatility, 30%, so variance 900
nnumber of equally weighted members, 10
\rhoaverage pairwise correlation
What it says in wordsThe index variance is a fixed piece from the members plus a piece proportional to correlation, and solving gives about 0.29.
Index variance: what the members bring, and what correlation addsIf correlation were 090, vol 9.5%Actual index, 18%90234324If correlation were 1900, vol 30%Green: members on their own, 900 / 10. Lime: the correlation part, 810 x correlation.Implied correlation = 234 / 810 = 0.289
Of the index's variance of 324, only 90 comes from the members on their own; the other 234 must come from correlation, and since full correlation would add 810, the implied correlation is 0.289.
Step 2What happens to the trade if correlation realises at 0.25?

Hold the members at their 30% implied, so the long member legs roughly break even, and recompute the index. At a correlation of 0.25 the index realises 17.1%, below the 18% it was sold at, so the short index leg makes about 0.9 vol points, roughly Rs 9 lakh on Rs 10 lakh of vega. The P&L does not depend on which way the market moves. It depends only on whether the members move together more or less than the 0.29 the index price assumed.

The dispersion trade is a bet on realised correlation10%15%20%25%30%implied index vol 18%breakeven ρ 0.289ρ 0.25: 17.1%, gain 9 lakhsell-off, ρ 0.6: 24%, loss 60 lakhseller of index vol gainsseller of index vol loses0.00.20.40.60.81.0Realised average correlation; member volatility held at 30%
With members realising 30%, the index realises 9.5% at zero correlation and 30% at full correlation, crossing its 18% implied at 0.289; at 0.25 the short index leg gains about Rs 9 lakh, and in a sell-off at 0.6 it loses about Rs 60 lakh.
Step 3Who is on the other side, and why does the gap usually exist?

Index options are bought by institutions protecting whole portfolios, which bids up index volatility. Single-stock options are sold by investors writing calls against holdings and by structured products, which pushes member volatility down. Both flows push implied correlation above what usually realises, so selling it earns a premium most of the time, which is the same insurance logic as selling index skew. The premium exists because the trade loses exactly when correlation spikes, in a sell-off, when every stock falls together.

Close with the risks beyond correlation. Members do not all realise 30%: if one stock has a takeover and others are quiet, the member legs carry their own P&L. Vega is not constant as prices move, so the clean variance arithmetic holds best with variance swaps; with options, the trade needs delta-hedging and rebalancing. And size for the tail: the loss in a correlation spike is about 7 times the gain in the base case.

Where candidates lose it

The common loss is dividing the volatilities or the variances, 0.6 or 0.36, and calling it correlation. That ignores the diversified piece the members contribute even at zero correlation, the 90 of the 324.

The second is describing the trade as long volatility because it buys ten options. It is short correlation: its P&L turns on whether stocks move together, and the candidate who cannot name the sell-off as the losing scenario has missed the point of it.

What the interviewer asks next

  • How would you size the member legs so the trade has no net vega?
  • With unequal weights and member vols, how does the implied correlation formula change?
  • Why might a desk prefer variance swaps to options for this trade?
  • Members realise 35% instead of 30%, correlation still 0.25. What happens to the P&L?

Company names and figures are illustrative.