Case 077Volatility tradingWarm up

Devkund Textiles closed at 500, 505, 498, 502, 510, 506, 499, 503, 508, 504 and 507 over eleven days. What is its realised volatility, and how does it compare with 30% implied?

The situation

You sit on the single-stock options desk. Devkund Textiles, a mid-cap, closed at Rs 500, 505, 498, 502, 510, 506, 499, 503, 508, 504 and 507 on eleven consecutive trading days. Its one-month at-the-money options trade at 30% implied volatility.

The desk head hands you the closes and a calculator and asks for the ten-day realised volatility, annualised, and whether the options look cheap or rich on that evidence alone.

Your task

Compute the realised volatility from these closes, annualise it, and say what the gap to 30% implied does and does not tell you.

Quick check

Before computing: the stock moved about 1% a day. Roughly what is that annualised?

Worked solution

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

30-second answerThe answer to give first

Realised volatility is about 18% annualised, well below the 30% implied. The ten daily log returns have a standard deviation of 1.11% a day, which times the square root of 252 is 17.6%. Implied volatility of 30% prices a daily move of 1.89%, nearly twice what the stock delivered. Ten days is a thin sample, but on this evidence the options are rich, and a volatility trader would look for a reason before selling them.

Step 1What exactly is realised volatility?

Think of a commuter whose journey takes 40 minutes on average: the average tells you nothing about how often she is 15 minutes late. Volatility is the spread of outcomes, not the typical one. Realised volatility is the standard deviation of daily returns, scaled to a year by the square root of the number of trading days. The desk uses log returnsThe natural logarithm of today close over yesterday close. For small moves it is almost the same as the percentage change, and it adds cleanly across days. because they add across days, which is what makes the square-root scaling exact rather than approximate.

DayClose, RsLog return
1505+0.995%
2498-1.396%
3502+0.800%
4510+1.581%
5506-0.787%
6499-1.393%
7503+0.798%
8508+0.989%
9504-0.791%
10507+0.593%
Standard deviation1.107%
The ten daily log returns range from minus 1.40% to plus 1.58% with a mean of 0.14%; their sample standard deviation is 1.107% a day.
Step 2How do you get from a daily number to an annual one?

Variance adds. If each day has variance v and the days are independent, 252 days have variance 252v, so the standard deviation grows by the square root of 252, about 15.87. 1.107% a day times 15.87 is 17.6% a year. Run it the other way to read implied volatility: 30% divided by 15.87 is 1.89% a day, the move the option market is pricing. Some desks skip the mean and use the root mean square of returns, which here gives 16.8%; the difference is small, and you should say which convention you used.

Realised: daily returns, their standard deviation, then times the square root of 252-2%-1%+1%+2%0+1.0d1-1.4d2+0.8d3+1.6d4-0.8d5-1.4d6+0.8d7+1.0d8-0.8d9+0.6d101 sd: 1.11% a dayx sqrt(252) = 17.6%30% implied is1.89% a dayred band: the moveimplied vol pricesDaily log returns over ten days, Rs 500 to Rs 507
The ten daily returns have a standard deviation of 1.11% a day, 17.6% annualised, while 30% implied volatility prices a 1.89% daily move shown by the wider red band, so the options are pricing nearly twice the movement the stock produced.
The relationship
σannual=σdaily252=1.107%×15.87≈17.6%\sigma_{\text{annual}} = \sigma_{\text{daily}}\sqrt{252} = 1.107\% \times 15.87 \approx 17.6\%
sigma dailysample standard deviation of the ten log returns, 1.107%
252trading days in a year; the square root is 15.87
What it says in wordsAnnual volatility is daily volatility times the square root of the number of trading days, because variance, not volatility, adds across days.
Step 3What does the gap to implied actually tell a trader?

Implied volatility is a forward-looking price; realised is a backward-looking measurement. The gap, 30% against 18%, is the premium the market is charging for the next month. Implied usually sits above realised, because option sellers are paid for the risk of a jump, so a gap on its own is not a mispricing. The question to ask is why: results due, an index rebalance, a block sale rumour. If nothing is scheduled, selling the one-month straddle and hedging the delta daily earns the gap, and you would size it knowing ten days of history is a thin sample of a stock that can gap 10% on a bad print.

Where candidates lose it

The first loss is annualising with 252 rather than its square root, which turns 1.1% a day into an absurd 279%. Variance scales with time; volatility scales with its square root.

The second is calling the options mispriced because 30% exceeds 17%. Implied almost always exceeds recent realised; the trade is in the reason for the gap, not the gap itself.

What the interviewer asks next

  • The desk uses the last 60 days instead of ten and gets 24%. Which number would you trust and why?
  • Devkund reports results in two weeks. How does that change your reading of the 30% implied?
  • If you sold the one-month straddle at 30% and realised came in at 17%, roughly what fraction of the premium would you keep?

Company names and figures are illustrative.