Options and volatility · O6 · 9 min
Volatility: implied, realised and the risk premium
Turn a volatility figure into a daily move, work out a vega P&L, and say whether options look cheap or dear.
In the news: “The VIX jumped to 30”: what that says about how much shares are expected to move each day.
The essentials
- Realised volatility
- How much a price actually moved: the standard deviation of daily returns, scaled to a year by about 16 (√252 trading days). Typical daily moves of 1.6% make of about 25%.
- Implied volatility and the premium
- The volatility built into option prices. It has tended to sit above the volatility that followed, because buyers pay for crash insurance: that gap, the , is why sellers usually win.
- Rule of 16
- A typical daily move is about yearly volatility ÷ 16: 16% means about 1% a day. A of 20 (the 30-day implied volatility of S&P 500 options) prices daily index moves of about 1.25%.
- Vega
- The change in an option’s value for a 1-point rise in implied volatility: positive for bought options, largest at the money and for long expiries. A book with of −£230 a point makes £920 if implied falls 4 points.
- Index against single-stock volatility
- Shares in an index partly cancel, so index volatility is lower: stocks at 30% with a of 0.25 make an index at about 15%. In a sell-off correlation jumps and index volatility rises towards the stocks’.
Volatility in numbers
Daily move ≈ σ ÷ 16; vega P&L = vega × Δσ
- σ
- volatility a year, in %
- 16
- about √252, the trading days in a year
- Δσ
- change in implied volatility, in points
How it works
1. Measuring how much a price moved
Take a share’s daily moves: +1%, −1%, +2%, −2%. Square them (1, 1, 4, 4), average them (2.5) and take the square root: about 1.6% a day. Scale that to a year by multiplying by √252, about 16, because a year has about 252 trading days: roughly 25%. That yearly figure is .
2. Implied against realised
Realised volatility looks back at what happened. looks forward: it is the volatility built into today’s option prices. Comparing the two is the heart of options trading. If you expect the share to move less than implied volatility says, the options look dear (traders say rich); if more, cheap.
3. The rule of 16
Run the scaling backwards: a typical daily move ≈ yearly volatility ÷ 16. That is the . Volatility of 16% means daily moves of about 1%; 32% means about 2%. About two days in three should move less than that figure.
4. The VIX
The is the implied volatility of S&P 500 options over the next 30 days. So a VIX of 20 prices daily index moves of about 20 ÷ 16 = 1.25%, and a VIX of 30 about 1.9%. When headlines say the VIX jumped, options on US shares have got dearer.
5. Vega
How much an option’s value changes when implied volatility rises one point is its . Bought options have positive vega, sold options negative. Vega is largest at the money and grows with time to expiry, so long-dated options are mostly bets on volatility. P&L from a volatility move ≈ vega × the change in points.
6. Sellers are usually paid
Over long periods, implied volatility has tended to sit above the volatility that followed: buyers pay for protection, and sellers want paying for the risk of a crash. That gap is the . It is why selling options usually makes money, and why it loses heavily when markets crash.
7. An index is calmer than its shares
The shares in an index don’t all move together, so their separate moves partly cancel. For many shares with volatility σ and average correlation ρ (how closely they move together, from 0 to 1), index volatility ≈ σ × √ρ. Shares at 30% with a correlation of 0.25 make an index at about 30% × 0.5 = 15%.
8. Correlation jumps in a sell-off
In a sell-off, shares tend to fall together, so correlation jumps towards 1 and index volatility rises towards the volatility of the single shares. That is one reason index options carry a premium.
Try it
Raise volatility by one point and the option’s value rises by about the Vega figure. Longer expiries have more vega.
P&L per share against the stock priceAt expiryToday, 1 year to expiry
- Premium paid
- 7.97
- time value 7.97
- Breakeven
- 107.97
- Delta
- +0.54
- Gamma
- +0.020
- per 1 point
- Theta
- −0.011
- per day
- Vega
- +0.40
- per vol point
| Stock price | P&L at expiry | P&L today |
|---|---|---|
| 70 | −7.97 | −7.72 |
| 80 | −7.97 | −6.78 |
| 90 | −7.97 | −4.38 |
| 100 | −7.97 | 0.00 |
| 110 | 2.03 | 6.33 |
| 120 | 12.03 | 14.18 |
| 130 | 22.03 | 23.04 |
Worked example
One-month implied volatility on a share at £40 is 30%, and you expect it to realise 22%. You sell 50 at-the-money contracts (100 shares each) with a vega of £0.046 a share, and hedge the delta.
- Daily move priced: 30 ÷ 16 ≈ 1.9%, about £0.75 a day on £40.
- Your forecast: 22 ÷ 16 ≈ 1.4%, about £0.55 a day. The options look dear.
- Your vega: −£0.046 × 50 × 100 = −£230 per point.
- Implied volatility falls 4 points the next day: 4 × £230 = £920 profit, before gamma and theta.
The options price 1.9% a day against your 1.4%; with vega of −£230 a point, a 4-point fall in implied volatility makes £920.
Check yourself
8 quick questions on the explanation above.
Practice
New numbers every set. Misses and guesses go into your review deck.
In the interview
Answer out loud in under a minute, then compare.
What does a VIX of 20 mean in daily moves?
Why is implied volatility usually above realised?
Why is index volatility lower than the average single-stock volatility?
Remember
Daily move ≈ volatility ÷ 16
Implied volatility is the price, realised is what happens, and a hedged option wins or loses by the gap.