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**Subject 10. The critical role of volatility**

We are interested in the

*future*(not past or current) volatility over the life of the option. Volatility is an extremely important variable to price options. In fact, with the possible exception of the cash flows on the underlying, volatility is the only variable that cannot be directly observed and easily obtained in the pricing of options.There are two basic ways to estimate the volatility.

**Historical Data**We compute the price relatives, logarithmic price relatives, and the mean and standard deviation of the logarithmic price relatives. We then use the standard deviation as our volatility measure.

Note that as three inputs (the interest rate, the time to maturity, and the standard deviation) to the Black-Scholes-Merton model depend on the unit of time, we should express all three variables in the same time units. Generally, one year is the most convenient common unit of time.

The historical estimate of the volatility is based on what happened in the past. To get the best estimate, we must use a lot of prices, but that means going back farther in time. The farther back we go, the less current the data become, and the less reliable our estimate of the volatility.

**Implied Volatility**There are five inputs to the Black-Scholes-Merton model, which the model relates to a sixth variable, the option price. With a total of six variables, any five imply a unique value for the sixth. The technique of implied volatility uses known values of five variables to estimate the standard deviation. The estimated standard deviation is an implied volatility because it is the value implied by the other five variables in the model.

Consider the following example of an implied standard deviation based on a call option. We assume that X = $100, S

_{0}= $100, r(c) = 0.1, price of the call option = $5, and the option has 90 days remaining until expiration. To find the implied standard deviation, we need to find the standard deviation that is consistent with these other values. To do this, we can compute the Black-Scholes-Merton model price for alternative standard deviations. We adjust the standard deviation to make the option price converge to its actual price of $5.

In our example we first try σ = 0.1, which gives a call price of $3.41. This price is too low. Next, σ = 0.2 results in a call price of $5.24, which is too high. We keep doing this until we try σ = 0.187, which results in a call price of $5.00.

Therefore, if the option is selling for about $5, we say that the market is pricing it at a volatility of 0.187. This number represents the market's best estimate of the true volatility. Unfortunately, a circularity exists in the argument. If one uses the Black-Scholes-Merton model to determine if an option is over- or underpriced, the above procedure assumes that the market correctly prices the option. Nonetheless, the implied volatility is a source of valuable information on the uncertainty in the underlying, and option traders use it routinely.

### Study notes from a previous year's CFA exam:

h. determine the historical and implied volatilities of an underlying asset;