Case 056Options and volatility tradingCore

A stock's one-month at-the-money implied volatility is 32% and its three-month is 24%, with results due in two weeks. Back out the move the options imply for the results day, and decide whether the event is priced rich.

The situation

Charvika Derivatives makes markets in options on a mid-cap stock that reports quarterly results in two weeks. The one-month at-the-money option, 21 trading days to expiry, trades at 32% implied volatility. The three-month option, 63 trading days, trades at 24%. Both expiries include the results day; the next results after that fall beyond three months.

The stock's last eight results-day moves were +5.2%, -3.8%, +6.9%, -4.1%, +2.7%, -7.5%, +3.3% and -4.4%. The head trader asks what move the options are pricing for the results day and whether that is too much.

Your task

Split each option's total variance into normal days and the results day, back out the implied results-day move, compare it with history, and say whether the event is rich.

Quick check

Roughly what standard deviation of move do the options imply for the results day alone?

Worked solution

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

30-second answerThe answer to give first

The options imply a results-day move of about 7.6% standard deviation, against a historical 5.0%, so the event looks rich. The three-month option implies normal days at about 18.8% volatility. Twenty of those leave 57.4 of the one-month option's 85.3 per cent squared for the results day. With the historical move instead, one-month volatility would be about 25%, not 32%.

Step 1Why can you not read the event straight off the 32%?

Think of a month's rainfall. If a city usually gets a little rain each day but one known day brings a cyclone, the monthly total is mostly that one day, and dividing the total by thirty hides it. Implied volatility is an annualised average over every trading day to expiry, so a single big day inflates the short option far more than the long one. That is why the one-month option trades at 32% while the three-month trades at 24%: both contain the same results day, but the three-month spreads it over three times as many days. The kink in the term structureImplied volatility plotted against time to expiry. A short expiry well above a longer one usually signals a known event inside the short window. is the event showing through.

Step 2How do you split the variance between normal days and the results day?

Work in total variance, the implied volatility squared times the fraction of a year to expiry, because variances of separate days add and volatilities do not. The one-month option carries 0.32 squared x 21/252 = 85.3 per cent squared; the three-month carries 0.24 squared x 63/252 = 144.0. Assume every ordinary day has the same volatility and the results day replaces one of them. The difference between the two options is 42 ordinary days, so their volatility is about 18.8% a year, and twenty such days account for 27.9 of the one-month total. The remaining 57.4 belongs to the results day: a standard deviation of 7.6%, or an expected absolute move of about 6.0%.

The relationship

\sigma_E^2 = \sigma_{1m}^2 \tfrac{21}{252} - \sigma_n^2 \tfrac{20}{252} = 85.3 - 27.9 = 57.4\ (\%^2), \quad \sigma_E = 7.6\%

\sigma_E	standard deviation of the stock's move on the results day
\sigma_{1m}	one-month implied volatility, 32%
\sigma_n	volatility of an ordinary day, 18.8%, backed out using the three-month option

What it says in wordsWhatever variance the one-month option holds beyond twenty ordinary days is the market's price for the results day, here a 7.6% move.

Of the one-month option's 85.3 per cent squared of variance, about 67% comes from the single results day, which is why the short expiry trades at 32% while the three-month, carrying the same event over far more days, trades at 24%.

Step 3Is a 7.6% move rich against what the stock has done?

The last eight results-day moves have a root mean square of 5.0% and an average absolute size of 4.7%. The options are pricing a results day about half as large again as the stock's own history, and in variance, the quantity an option seller is paid for, more than twice as large: 57.4 against 25.0. Put the historical move back in and the one-month option would be worth about 25.2% volatility. On those numbers the event is rich, and the trade that isolates it sells the one-month and buys the three-month, a calendar spread that is short the event and roughly flat the ordinary days.

None of Charvika's last eight results-day moves exceeded 7.5%, and their root mean square of 5.0% sits well inside the 7.6% standard deviation the options now imply for the coming results day.

Say the limitations, because the head trader will. Eight observations are a thin history; one 15% move would change the picture. The market may know something this quarter that the history does not, such as a pending regulatory decision or a guidance change. And the split assumes ordinary days in the next month are as volatile as those over the next three, which a quiet three-month option can understate. The conclusion is a lean toward selling the event, sized for the chance that this quarter is the exception.

Where candidates lose it

The common answer compares 32% with 24% and calls the short option expensive without asking why. The gap is not mispricing by itself; it is the results day, and the real question is whether the move hidden in it is too large.

The second miss is subtracting volatilities instead of variances. Volatilities of separate days do not add; their squares do, which is why every step here runs in per cent squared and only the final answer goes back to a square root.

What the interviewer asks next

How would you build a position that is short the results day but close to flat on ordinary days?
If the stock gaps 10% on results, roughly what does a short one-month straddle lose?
After results, where would you expect the one-month implied volatility to trade?

Company names and figures are illustrative.