How to Price American-Style Options

The options that we have considered so far are know as American-style options. This simply means that the option contract can be exercised at any time before expiration. European-style options are contracts that can be exercised only at expiration. American-style and European-style are also known as vanilla options. There are many other types of option contracts structured in arbitrarily complex ways. Bermuda-style options, for example, can be exercised only at pre-determined dates, usually monthly.

We have seen that the price of an option can change dramatically as the underlying asset price changes and as the expiration date gets closer. It is therefore very important to understand what determines such changes. The best way to estimate the price of an American-style option is to use an approximation known as the binomial model. I am going to show how it works with an example.

First of all, the time to expiration is broken down into a number of steps. Walking forward from the present time at each step the underlying security price is moved up or down by a given amount. This amount is calculated using the estimated volatility of the stock and all possible price moves are calculated.

Volatility plays the leading role in understanding and pricing options because options are insurance contracts and volatility is the risk one is insuring against. Volatility is essentially a measure of the expected range of changes in a security returns, or, to be more accurate, the deviation of returns from their mean. So, if, for example, the mean return of an asset is 0, a 10% volatility means that returns are likely to fluctuate between a -10% loss and +10% gain, annualized. The key here is the qualification "likely". If returns were perfectly random, that "likely" could be quantified as a probability of 68.3%. What one is insuring against with an option is really the event of returns being outside the volatility of 10%. In this case, the probability of such an event is 1-0.683=31.7%. Under the same assumption of perfect random returns, a volatility of 10% means that the likelihood of returns staying within -20% and 20% annualized increases to 95.4% and therefore the probability of a larger than 20% gain or loss goes down to 4.6%.

So, if at each step the underlying asset price can move either up or down by a certain amount, the number of calculated prices is going to grow geometrically as one steps away from the present and closer to expiration. If, at step 0 we have just one price, the present price, at step 1 we have the current price plus the calculated change and the current price minus the calculated change, at step 2 we have each of step 1 two prices moved up and down, giving four prices, at step 3 we have eight prices, and so on and so forth...

The result looks like a tree with branches that keep bi-furcating at each step in time. That's why this model is called the bi-nomial (tree) pricing model.

At expiration, the last step of the binomial tree, option prices must be equal to their intrinsic value. In order to determine the price of the option at the present time, that is at step 0 of the binomial tree, the option price at each step must be calculated walking back from the last step toward the first step, each time using the known option prices of the previously calculated step.

The main problem with this approach is that volatility, being a measure of risk, is only an estimate. One way to estimate it is by considering the past price history. This is what is called historical volatility. In practice, though, the volatility implied by option prices, that is the risk perceived by the option market, is usually quite different than the historical volatility.

A good place to compare the two volatility measures is IVolatility. This chart shows the historical volatility of GE (blue curve) in the past six months, calculated using 30 trailing days of historical data, together with the implied volatility (yellow curve), that is the volatility that best fits the prices of GE options expiring in one of the following three months (front options). Clearly, during the first half of January, historical volatility was much lower than implied volatility.



The following picture from Hoadley.net shows a 10 step binomial tree calculation estimating the price of GE-BG on January 6, using the appropriate values for dividend yield and risk-free interest rate and using historical volatility as the estimate for volatility:



The resulting option price is $0.73 which is quite lower than $1.05, the closing price of GE-BG on that day.

Of course, if instead of the historical volatility, I use the implied volatility I am going to get the much better estimate of $1.02:



However, that's the result of a circular argument since the implied volatility is precisely that value of volatility that best fits the option prices for the front months on the day that I am considering.

If I increase the number of steps I do not get a significantly different estimate:



Clearly, since volatility is the only calculation parameter that is not known with certainty, a good volatility estimate is paramount when pricing an option contract.

Historical volatility is not the only measure of volatility available. The following chart shows several measures of volatility based on price data for GE in the past six months:




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