Estimating Variance From High, Low and Closing Prices

The log of the price of a share is commonly modelled as a Brownian motion with drift, .Bt+ct, where the constants c and . are unknown. In order to use the Black-Scholes option pricing formula, one needs an estimate of ., though not of c. In this paper, we propose a new estimator of . based on the high, low, and closing prices in a day's trading. This estimator has the merit of being unbiased whatever the drift c. In common with other estimators of ., the approximation of the true high and low values of the drifting Brownian motion by the high and low values of a random walk introduces error, often quite a serious error. We shall show how a simple correction can overcome this error almost completely.