Tools › Investing › Options › Volatility Cone and Smile › Example
A 20-day realised volatility by hand, the volatility cone of 500 prices, and the smile numbers of an option chain
| Tool | Volatility Cone and Smile |
| Source | Burghardt, G., and Lane, M. (1990). "How to Tell if Options Are Cheap." Journal of Portfolio Management, Vol. 16, No. 2, pp. 72–78 (the cone, as a method; no number of the paper is reproduced). The case is the tool's illustrative default: made-up prices and option quotes, not market data. |
| Level | Undergraduate |
Problem
A price history of 500 daily closes ends 101.59, 100.00. Find the realised volatility of the latest 20 days (daily log returns, sample standard deviation, annualised with 252 trading days), and where it stands among all 20-day windows of the history. Then, for options with 60 days to expiry on the stock at 100 (interest rate 4 %, dividend yield 1 %, continuously compounded): a 100 put quoted at 3.32, calls at 105 and 110 quoted at 1.68 and 0.59, puts at 90 and 95 quoted at 0.71 and 1.60, and a 70 call quoted at 29.50. Find the at-the-money implied volatility, the 25-delta risk reversal and the butterfly.
Inputs to Enter
- Closing prices: the 500 illustrative prices of the default (random-number generator, seed 357); implied volatility 22 %
- Advanced mode: the default option chain (70 C 29.50; puts 80 to 100; calls 100 to 120), stock price 100, 60 days to expiry, interest rate 4 %, dividend yield 1 %
Answers
- Latest log return ln(100.00 / 101.59) = −0.015775; the 20 latest returns sum to 0.031904 (mean 0.001595) and their squared deviations to 0.00314260
- Daily standard deviation √(0.00314260 / 19) = 0.012861; realised volatility 0.012861 · √252 = 20.42 % a year
- The 480 windows of 20 days: lowest 6.1 %, quartiles 9.8 / 14.5 / 18.2 %, highest 30.9 %; the entered 22 % is above 90 % of them
- Cone, lowest to highest: 10 days 3.9 to 38.2 %, 60 days 8.4 to 26.4 %, 120 days 9.6 to 22.5 %, 250 days 11.9 to 18.8 %
- Forward price 100 · e^(0.03 · 60/365) = 100.49; the 70 call's lower bound 99.84 − 69.54 = 30.29 exceeds its quote 29.50, so the line is left out
- Implied volatilities: 100 put 22.11 %, 105 call 21.02 %, 110 call 20.28 %, 90 put 25.44 %, 95 put 23.62 %; at the money 22.11 + (100.49 − 100) / 5 · (21.02 − 22.11) = 22.00 %
- 25-delta call (deltas 0.318 and 0.145) 20.73 %; 25-delta put (deltas −0.131 and −0.262) 23.79 %; risk reversal −3.06 points; butterfly (20.73 + 23.79) / 2 − 22.00 = 0.26 points
Notes
The tool's default inputs are this case; press Submit (Advanced mode for the option chain). The rolling statistics are checked against pandas on 20 seeded series and the implied volatilities against a scipy root-finder on 20 seeded chains (test_volatility_cone_smile.php). The cone describes one price history and the smile restates quoted prices through one model; neither says what an option is worth. A calculator for education and analysis, not investment advice.