## The Annals of Applied Probability

### Estimating Variance From High, Low and Closing Prices

#### Abstract

The log of the price of a share is commonly modelled as a Brownian motion with drift, $\sigma B_t + ct$, where the constants $c$ and $\sigma$ are unknown. In order to use the Black-Scholes option pricing formula, one needs an estimate of $\sigma$, though not of $c$. In this paper, we propose a new estimator of $\sigma$ 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 $\sigma$, 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.

#### Article information

Source
Ann. Appl. Probab., Volume 1, Number 4 (1991), 504-512.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aoap/1177005835

Digital Object Identifier
doi:10.1214/aoap/1177005835

Mathematical Reviews number (MathSciNet)
MR1129771

Zentralblatt MATH identifier
0739.62084

JSTOR