When considering the premium or discount of stock returns, several questions arise: (1) What is the best measure of the volatility of stock returns to use when applying the Longstaff model? (2) What actual return volatilities do we observe for NYSE and NASDAQ stocks? (3) Is volatility persistent?

Measuring Volatility

The application of Longstaff’s model requires a measure of stock return volatility. Using historical volatility as an estimate of future volatility is the most common approach. For example, the Financial Accounting Standards Board, in Statement of Financial Accounting Standards No. 123 (revised), Share-Based Payment, requires U.S. companies to consider historical volatility as a key factor in forecasting volatility for the purpose of valuing and expensing stock options. But analysts have several alternatives to using historical volatilities. For instance, if option contracts on the stock are traded, the implied volatility from these options can be used as the volatility estimate in Longstaff’s model. In addition, various volatility models can be used to forecast future volatility (see Figlewski 1997). Nevertheless, the strong persistence in volatility that researchers have found suggests that the historical volatility of a stock’s returns is a reliable forecast of its future volatility.

We used daily returns to calculate historical stock return volatilities.3 This procedure provides a relevant measure of fluctuations in stock prices for several ...

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