Implied against realized: a price against a measurement
Implied volatility is the number that makes the model give the market price. Realized volatility is how much the stock actually moved. An options trader's job is the gap between them.
- Price a straddle with 0.8 × S × σ × √T and read the one-sigma move off it
- Back implied volatility out of a straddle price
- Annualize a daily standard deviation and compare it with implied
- Compute a vega P&L and say which way to trade a gap
The intuition
An at-the-money straddle, a call and a put at the same strike, pays the absolute move at expiry. The expected absolute move of a normal variable is about 0.8 of a standard deviation, so the straddle costs about 0.8 × S × σ × √T. On a $100 stock at 25% implied vol with 91 days to go, the one-sigma move is $12.48 and the straddle costs about $9.99. That price is a forecast of how much the stock will move, in dollars.
Now measure. If the stock's daily returns had a standard deviation of 1.2% over the last three months, its realized volatility was 1.2% × √252 = 19%: six points below the 25% the options charge. Whether that is a trade depends on what comes next, not on what came before: implied is forward-looking and realized is backward-looking, and a quiet three months does not include next week's earnings. Buy the straddle when you expect more movement than the price implies; sell it when you expect less.
Straddle ≈ 0.8 × S × σ × √T; one-sigma move by expiry = S × σ × √T. Implied vol ≈ straddle ÷ (0.8 × S × √T). Realized annual vol = daily standard deviation × √252. Straddle vega per vol point ≈ 0.8 × S × √T × 0.01 per share. Realized above implied: buy vol. Below: sell vol.
Why it works
- The conventions here: an at-the-money straddle on a non-dividend stock at a zero rate, priced with the approximation 0.8 S σ √T; T is calendar days ÷ 365. Realized volatility annualizes daily returns with √252 trading days. One contract is 100 shares.
- Why 0.8. E|Z| = √(2 ÷ π) ≈ 0.8 for a standard normal. The straddle pays |S_T − K|, whose expectation is about 0.8 of the one-sigma move.
- Vol is the unit traders quote in. A $3 option on a $50 stock and a $3 option on a $400 stock are different bets; 25% vol means the same thing on both. Implied vol is the price, converted into a comparable unit.
- Vega is the sensitivity to that price. Differentiate the straddle in σ: 0.8 S √T per unit of vol, so per point multiply by 0.01. It grows with √T: long-dated options are vega trades, short-dated ones are gamma trades.
- Two day counts. Option time uses 365 calendar days because the option lives through weekends. Realized volatility uses 252 because returns are only measured on trading days.
- The gap is a view, not a fact. Options can look cheap because a volatile period is ending, or rich because the market expects an event the past three months did not contain. The trade is a forecast of future realized volatility.
| √T: √(91 ÷ 365) | 0.4993 |
| One-sigma move by expiry: $100 × 25% × 0.4993 | $12.48 |
| Straddle: 0.8 × $12.48 | $9.99 |
| Vega per vol point: 0.8 × $100 × 0.4993 × 0.01 | $0.399 a share |
| Implied vol rises 3 points on 10 contracts: $0.399 × 3 × 1,000 | $1,198 |
| Daily standard deviation 1.2%: realized 1.2% × √252 | 19.0%, 6 points below implied |
At 19% expected realized against 25% implied, the trade is to sell the straddle and delta-hedge. Its main risk is a jump: short gamma loses with the square of the move.
The formulas
The straddle pays the absolute move; about 0.8 of one standard deviation.
The price, converted into vol.
Trading days for returns.
Per share; multiply by the vol change and the shares.
Worked example
Compute the one-sigma move once; the straddle is 0.8 of it. The follow-up explains the 0.8.
See it move
Same stock and options. Change the implied volatility, the days to expiry, how much the stock has actually been moving, and a change in implied vol.
- Raise the implied volatility. The straddle and the one-sigma move rise, and realized minus implied falls.
- Raise the daily standard deviation. Realized volatility and the gap rise; nothing about the options changes.
- Add days to expiry. The straddle, the one-sigma move and the vega per point all rise with the root of time.
- Move implied vol up by more points. The vega P&L rises in proportion.
Run it backwards
The straddle is offered at a known price. What implied volatility is the market charging?
Divide the price by 0.8 × S × √T. That is what 'implied' means: the vol that gives the price back.
The follow-up is why traders quote in vol rather than dollars: it is comparable across strikes, expiries and underlyings.
Traps
Say it in the interview
“The stock has been realizing 19% and the options trade at 25%. What is the trade, and what is the risk?”
Check yourself
4 fresh questions, with new numbers. Answer each one correctly to finish the chapter. Get one wrong and you will see the full working, then you can try it again with new numbers.
Answers within 1% are marked right. Type the number; $, %, x and M are fine. First tries count toward Learned: the topic is Learned once every chapter is done and 75% of first tries were right.
- Straddle ≈ 0.8 S σ √T; it is the expected absolute move, priced.
- Implied is a price; realized is a measurement; the trade is a forecast of the second.
- 365 days for option time, 252 for annualizing returns.
- Vega per point ≈ 0.8 S √T × 0.01; it grows with √T.