The Option trading model

When the CBOE did start listing options, it was a slow start. Originally, only twelve of the biggest companies had options. There were only Calls ( Puts weren’t listed until 1977). Option chains were calculated overnight by mainframe computers and printed out. Option traders walked around the trading floor with printouts of hours-old data and had to do arithmetic to price options as the stock prices moved.

Today, thousands of stocks and ETFs have listed options. Some of the information that is instantly available seems quite fantastic. For example, for any option strategy, the probability of profit, as well as the probability of making the maximum profit, can instantly be calculated. Just how do they know that?

The fact is that all of those values are estimated, using an option pricing formula, or model; and that those estimates are based on many layers of assumptions. These assumptions include the following:

Markets do not allow the existence of any arbitrage opportunity (i.e., there is no way to make a riskless profit).

It is possible for the trader to borrow and lend cash at a known, constant risk-free interest rate.

It is possible for the trader to buy, sell or sell short any number of shares of stock, including fractional shares.

Stock and option transactions do not incur any fees or costs (i.e., frictionless market).

The stock price follows a geometric Brownian motion with constant drift and volatility.

This means that on average every stock price should oscillate around a line that has an upward slope equal to the risk-free interest rate. The size of the oscillation is measured as the stock’s volatility. The future price of every stock should fall within a normal distribution (bell curve) that is centered on that slightly upward-sloping line.

Well now. That’s quite a list of assumptions. None of them can really stand up to close inspection. And yet a whole industry is built around the idea that the calculations based on all those faulty assumptions are accurate, or at least useful.

And strangely enough, they are. It has been empirically shown in several studies that observed option prices are “fairly close” to the model’s calculations. Close enough, in fact, that the original Black-Scholes model from 1973, with some extensions (to account for dividends and a few other things), is still in use.

One thing that accounts for the model’s robustness is the wonderful idea of “implied volatility.” This idea is embedded in the model, and provides an escape hatch that gives it an out when option prices actually trade at prices that are different from what it predicts.

The model calculates a theoretical price for every option. This calculation takes as inputs only the current stock price, option strike price, time to expiration, the risk-free interest rate, dividend amount, if any, and the stock’s historical volatility. Underpinning the idea that a calculated price based on these inputs is valid, are all the assumptions listed above.

When an option sells for a price that is different from the theoretical price, then something must be different. It can’t be the current stock price, option strike price, time to expiration, risk-free interest rate, or dividend amount. All those things are fixed. All that remains is volatility. If people are paying more for an option than its theoretical value, they must believe that volatility going forward will be different from historical volatility. If they’re paying more than the theoretical price, they must expect bigger price moves; if less, smaller ones. They “imply” what they believe future volatility will be, by the amount they are willing to pay (and sellers are willing to accept) for the options. We just solve the formula backward, to determine what level of volatility would have to be fed into it, for it to calculate a theoretical price that is equal to the observed price.

turns out that this over- or under-pricing compared to historical volatility has patterns of its own, that can persist for periods of time from days to years. Actual volatility of a stock’s price (which becomes historical volatility when we look back at it) grows and shrinks according to the market’s mood. Implied volatility grows and shrinks too, loosely related to actual volatility but constantly rising above and falling below it. Yet miraculously, one thing that remains true is that implied volatility has mean-reverting tendencies. When it gets too large, it usually drops back toward its average. When it gets too small, it eventually climbs back up. This happens reliably enough that we can base our selection of option strategies on it.