Volatility: from past prices and implied by a price
1. The idea
Of the formula’s inputs, only the volatility \(\sigma\) cannot be read off a screen. It is the standard deviation of the share’s yearly return with continuous compounding (Level 4), and there are two ways to find it.
From past prices. Take the closing prices of \(n + 1\) trading days in a row, \(S_0, S_1, \dots, S_n\), and each day’s return with continuous compounding, \(u_i = \ln(S_i/S_{i-1})\), where \(\ln\) is the natural logarithm. Their sample standard deviation \(s\) is the volatility over one day. Prices move much more while the exchange is open than while it is closed, so time is counted in trading days: a year has about 252 of them, and a standard deviation grows with the square root of time, so
\[ s = \sqrt{\frac{1}{n-1}\sum_{i=1}^{n}\big(u_i - \bar u\big)^2}, \qquad \sigma = s\sqrt{252}, \]
where \(\bar u\) is the mean of the \(n\) returns. This estimate is the historical volatility.
An example. Closing prices of $30.00, $30.45, $29.85, $30.20 and $30.75 give the returns \(0.014889\), \(-0.019901\), \(0.011657\) and \(0.018048\), with a mean of \(0.0061733\). Their squared differences from the mean add up to \(0.000926912\), so \(s = \sqrt{0.000926912/3} = 0.017578\) and \(\sigma = 0.017578 \times \sqrt{252} = 0.2790\), or 27.90% a year. Four returns give only a rough estimate; more data gives a better one, as long as the share has not changed.
Implied by a price. The formula’s price rises with the volatility, from the lowest price of Level 5 towards the highest. So for a market price between the two, exactly one volatility makes the formula give that price: the implied volatility. No formula gives it; we find it by trial. Try a volatility and compare the formula’s price with the market price: raise the volatility if the price is too low, lower it if the price is too high, and repeat. Traders quote options by their implied volatility, which shows how much the market expects the share to move.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The volatility on charts
First, the first day’s return.
Now move through the days. Each day’s return is the step from one closing price to the next; the volatility measures how far these returns are from their mean.
For an implied volatility, first the formula’s price at a volatility of 30%, for the call of the worked example below.
Now move the volatility. The formula’s price rises with it and crosses the market price at the implied volatility.
4. One worked example, then your turn
In the worked example, we find the volatility that a call’s price implies.
Now a historical volatility with hints, one step at a time.
And one on your own: the volatility that a put’s price implies.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer from past prices and one implied by a price. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: