The Annals of Applied Probability

Estimating Variance From High, Low and Closing Prices

L. C. G. Rogers and S. E. Satchell

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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

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

First available in Project Euclid: 19 April 2007

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Primary: 62M05: Markov processes: estimation
Secondary: 62P20: Applications to economics [See also 91Bxx] 90A12 60J60: Diffusion processes [See also 58J65] 60J65: Brownian motion [See also 58J65]

Black-Scholes formula option pricing Brownian motion Wiener-Hopf factorisation


Rogers, L. C. G.; Satchell, S. E. Estimating Variance From High, Low and Closing Prices. Ann. Appl. Probab. 1 (1991), no. 4, 504--512. doi:10.1214/aoap/1177005835.

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