1. Historical volatility and implied volatility: they are not the same thing

Derivatives 3: Volatility

The previous levels have treated the option as a contract: the right to buy or sell, the premium, the part that is intrinsic value and the part that is time value. It is time to open the box that time value comes out of. Inside there is a single variable that decides almost everything, and it is called volatility. The problem is that the word is used for two things that do not even belong to the same category.

Volatility is amplitude, not direction

The volatility of an asset is the dispersion of its returns: the standard deviation of those returns, expressed as a percentage a year.

The first thing to take on board is that it has no sign. An asset that rises 30% over the year by advancing two tenths a day has extremely low volatility. Another that ends where it started, after twelve months of 4% daily jumps, has an enormous one. It does not know whether you have won or lost: it measures how much the ground moves, not where it is taking you.

The historical one is counted, not opined on

Historical volatility, also called realized volatility, is calculated by looking backwards: you take a period's closing prices, turn them into daily returns, work out their standard deviation and annualize it by multiplying by the square root of the number of sessions in a year, roughly 252.

It is an arithmetical fact: two people with the same data and the same window get the same number. The only negotiable thing is the window, and that is no small matter: the ten-session historical and the one-year historical of the same asset can differ by a factor of three, and both are equally correct.

What 20% a year really means

This is the arithmetic worth carrying in your head, because it turns an abstract number into something recognizable on a screen. Since the square root of 252 is almost exactly 16, to go from annual volatility to a typical daily move you just divide by 16. With an asset at 20% a year:

• One day. 20 divided by 16 gives 1.25%. That is the typical daily move, not the maximum: two sessions out of three will fall inside that band, one in three will step outside it and one in twenty will exceed double, 2.5%.
• One week. Five sessions. The conversion scales with the square root of time, so it comes to around 2.8%.
• One month. About 21 sessions. That works out at roughly 5.8%.
• One quarter. About 63 sessions, a quarter of the year. It comes out at half the annual move, 10%.

Look carefully at that last line, because it is the trap most people fall into: quadrupling the horizon does not quadruple the expected move, it only doubles it. A month does not move twelve times less than a year; it moves about three and a half times less. Time enters volatility through the square root, not in direct proportion.

The implied one is backed out of the price

The classic valuation model, the one Black, Scholes and Merton published in the early seventies, is a machine that takes six inputs and returns a theoretical price: price of the underlying, strike price, time to expiry, interest rate, dividends and volatility. Five of those six can be looked up. The sixth, the volatility there will be between today and expiry, does not exist yet.

But there is something that can be observed: the price the option is trading at right now. So the machine is turned around. Instead of feeding in a volatility to get a price, you feed in the market price and ask what volatility would have to go in for the formula to return exactly that price. The number that comes out of that inversion is the implied volatility.

It is worth being precise, because almost the whole level depends on it. Implied volatility is not a measurement of the world: it is a translation of the price. Saying that March options have an implied of 24 is saying they are expensive to that extent, only in units that let you compare different assets, strikes and expiries. It is a price dressed up as a percentage.

Two numbers that do not share a nature

Summed up in four differences that should never be mixed up:

• Direction in time. The historical looks backwards. The implied looks forwards.
• Nature. The historical is a fact. The implied is a collective opinion that somebody is paying real money for.
• Origin. The historical comes out of the underlying's price series. The implied, out of the options market's order book.
• Stability. With the window fixed, yesterday's historical will never change. The implied changes every second.

From which something uncomfortable follows: comparing today's implied with the historical of the last thirty days is useful as context, but it is not comparing like with like. The clean comparison would be against the volatility realized over the next thirty days, a number that will not be known until a month from now. That gap is the heart of the next module.

A real case: 2017 against March 2020

2017 was one of the calmest years on record in the American stock market. The VIX, which is the thirty-day implied on the S&P 500, averaged around 11 and posted closes below 10 in November, all-time lows. The volatility realized by the index was just as low. Divided by 16: people were paying for a typical daily move of seven tenths of a point.

On March 16, 2020 that same index closed at 82.69, its high, and the S&P 500 fell close to 12% in the session. A VIX of 82 corresponds to a typical daily move above 5%: seven times that of 2017.

These are not two different markets. It is the same index, with the same companies inside it, thirty months apart. When somebody tells you volatility is at 30, ask which of the two, and over what window.