Case 048Volatility, options and convertiblesHard

The Sethu 50 index has implied volatility of 15%, while its members' average implied volatility is 25%. What average correlation does that imply, and how would you trade a view that realised correlation will be lower?

The situation

The Sethu 50 is an equal-weighted index of 50 stocks. One-year options on the index trade at an implied volatility of 15%. One-year options on its members trade at an average implied volatility of 25%, and the spread across members is narrow enough to treat them all as 25%.

Your analysis of the last three years says the members have moved together less than the options imply: on your numbers, average correlation is likely to be around 0.25 over the next year. The fund can trade options on the index and on every member.

Your task

What average correlation are the options implying, how would you trade the view that correlation will be lower, how do you size the legs, and what is the risk?

Quick check

Roughly what average correlation do the two volatilities imply?

Worked solution

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

30-second answerThe answer to give first

The options imply an average correlation of about 0.36, or 0.35 once each stock's own variance is removed. To bet that correlation will be lower, sell index volatility and buy volatility on the members, a dispersion trade. If the members realise 25% and correlation 0.25, the index realises only about 12.9%, and the short index leg earns about 2.1 volatility points. The risk is a sell-off, when correlation jumps and the short index leg loses heavily.

Step 1Why is an index less volatile than its members?

A choir sounds steady even when individual singers drift, because their small errors cancel; if they all drift the same way, the whole choir goes off key. An index is volatile only to the extent that its members move together, so index variance is roughly the average correlation times the members' variance. Flip that around and the options market tells you the correlation it expects: (15 / 25) squared = 0.36. In a 50-stock equal-weighted index each stock also adds a little of its own variance, and removing it gives 0.35. That number is the implied correlationThe average correlation between an index’s members that makes the index option price consistent with the members’ option prices..

The relationship

\sigma_I^2 \approx \bar\rho\,\bar\sigma^2 \;\Rightarrow\; \bar\rho \approx \left(\frac{15\%}{25\%}\right)^2 = 0.36 \qquad \bar\rho = \frac{\sigma_I^2 - \sum w_i^2\sigma_i^2}{\bar\sigma^2 - \sum w_i^2 \sigma_i^2} \approx 0.35

sigma_I	index implied volatility, 15%
sigma bar	average member implied volatility, 25%
rho bar	the average pairwise correlation the prices imply
w_i	each member's weight, 1/50

What it says in wordsIndex variance is roughly correlation times member variance, so the ratio of the two squared gives the correlation the market is pricing.

The index's 15% implied volatility against 25% for its members means the market expects them to move together with an average correlation of about 0.36, or 0.35 after removing each stock's own variance.

Step 2How do you trade lower correlation?

Sell volatility on the index and buy volatility on the members, a dispersion tradeSelling index volatility and buying volatility on the index members, which profits when the members move more independently than the index price assumes.. If the members move as much as expected but less together, the members' volatility is paid for fairly while the index turns out calmer than its price, and the short index leg makes money. At a correlation of 0.25 with members at 25%, the index realises about 12.9%, 2.1 points below the 15% you sold. With index vega of Rs 10 lakh per volatility point, that is about Rs 21 lakh.

Size the member leg to remove the bet on volatility in general. If every volatility rises by the same proportion and correlation stays put, the index leg loses on 15 points and the members gain on 25. Buying member vega of 15 / 25 = 0.6 times the index vega, Rs 6 lakh against Rs 10 lakh, makes the book roughly neutral to a uniform rise in volatility, so what remains is mainly a bet on correlation. The table checks it: members up from 25% to 30% with correlation unchanged gives a result of about zero.

With members at 25% volatility, index volatility rises from about 3.5% at zero correlation to 25% at full correlation, so the 15% index price implies 0.35; a realised correlation of 0.25 gives an index volatility of about 12.9%, while a crash with members at 40% and correlation 0.7 gives about 33.6%.

Scenario over the year	Members realise	Correlation	Index realises	Profit, Rs lakh
Your view	25%	0.25	12.9%	+21
All volatility up a fifth	30%	0.35	18.0%	+0
Sell-off	40%	0.70	33.6%	-96

Short Rs 10 lakh of index vega and long Rs 6 lakh of member vega earns about Rs 21 lakh if correlation falls to 0.25, roughly nothing if all volatility rises together, and loses about Rs 96 lakh in a sell-off where correlation jumps to 0.7.

Step 3What is the risk, and why is the trade priced this way?

The risk is a sell-off. When markets fall hard, stocks fall together: correlation jumps, the index's volatility rises towards its members', and the short index leg loses far more than the member leg gains, about Rs 96 lakh in the table. That is why index options usually trade rich to their members: investors pay for index puts as crash insurance, and the dispersion trader is selling them that insurance. The trade earns small amounts in calm years and loses large amounts in crises. Size it for the crisis, keep it short-dated or buy some deep index puts as a hedge, and note the practical costs: fifty option positions on the member side cost more to trade and to rebalance than one on the index.

Where candidates lose it

The common loss is dividing the volatilities, 15 / 25 = 0.6, and calling that the correlation. Correlation lives in variances, so the ratio must be squared; 0.6 overstates it by two thirds.

The second is describing dispersion as a free lunch because implied correlation usually exceeds what is realised. The premium exists because the trade loses heavily in sell-offs; an answer that does not size for that case has not understood what it is selling.

What the interviewer asks next

How would you express the view with variance swaps instead of options?
The members' implied volatilities are very different from one another. How does that change the trade?
Why might implied correlation fall just before a sell-off?
How would you hedge the crash risk and what would it cost?

Company names and figures are illustrative.