Estimating the Stocks Standard Deviation

Estimating the standard deviation of the stock's returns is more difficult and more important than estimating the risk-free rate. The Black-Scholes model takes as its input the current, instantaneous standard deviation of the stock. In other words, the immediate volatility of the stock is the riskiness of the stock that affects the options price. The Black-Scholes model also assumes that the volatility is constant over the life of the option.7 There are two basic ways to estimate the volatility. The first method uses historical data, while the second technique employs fresh data from the options market itself. This second method uses options prices to find the options market's estimate of the stock's standard deviation. An estimate of the stock's standard deviation that is drawn from the options market is called an implied volatility. We consider each method in turn.8

Historical Data. To estimate volatility using historical data, we compute the price relatives, logarithmic price relatives, and the mean and standard deviation of the logarithmic price relatives. Letting PR, indicate the price relative for day t so that PR, = PJP, _,, we give the formulas for the mean and variance of the logarithmic price relatives as follows:

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