Lesson 6 of 20 · 7 min
Input four: volatility
Volatility is how far the underlying is expected to wander, and it is the input the price is most sensitive to.
Volatility is the annualised standard deviation of the underlying's returns. In plainer terms: a single number summarising how far the price is expected to move around, in either direction, over a year.
A 25 percent volatility means the model expects the price to end the year within 25 percent of where it started, up or down, about two thirds of the time. Convert it to daily terms by dividing by the square root of the number of trading days: 25 percent divided by the square root of 252 is about 1.6 percent a day. That conversion is worth remembering, because it turns an abstract annual figure into something you can compare with what you observe.
Volatility's effect on price is direct, large, and identical in sign for calls and puts. In the base case, the at-the-money call is worth $1.88 at 15 percent volatility, $3.02 at 25 percent, and $4.73 at 40 percent. The put moves the same way: $1.55, $2.69, $4.40. Higher volatility raises both, because both have truncated payoffs and both benefit from a wider distribution.
That last point catches people. Volatility is not directional. It does not know or care whether the wandering is upward or downward, and raising it makes every contract on the chain more expensive at once. A holder can be entirely correct about direction and still lose money because this input fell, which is the mechanism behind volatility crush in lesson 16.
Volatility is also the only one of the five inputs that cannot be looked up. The other four are observable facts; this one is an estimate, which is why lesson 9 exists.
Key terms
- Volatility (σ)
- Annualised standard deviation of returns. A summary of how far the underlying is expected to wander, without regard to direction.
- Realised volatility
- What the underlying actually did, measured from historical prices. Backward-looking.
- Implied volatility
- The volatility figure that makes the model's price equal the market price. Forward-looking and inferred, not measured.
Worked example
One input, doubled premiums
Base case: underlying 100, strike 100, 30 days, rate 4%. Only volatility changes.
- 10% volatility
- Call $1.31, put $0.99
- 15% volatility
- Call $1.88, put $1.55
- 25% volatility
- Call $3.02, put $2.69
- 40% volatility
- Call $4.73, put $4.40
- 50% volatility
- Call $5.87, put $5.54
- Daily move implied by 25%
- about 1.6%
The base case.
25% ÷ √252.
From 10% to 50% volatility the call price rises 4.5 times, with the underlying, the strike and the calendar all unchanged.
Check yourself
Implied volatility on a chain rises from 25% to 40% while the underlying does not move. What happens to the calls and to the puts?
Saved in this browser only. No account, nothing sent anywhere.
Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.