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November, 1991 Estimating Variance From High, Low and Closing Prices
L. C. G. Rogers, S. E. Satchell
Ann. Appl. Probab. 1(4): 504-512 (November, 1991). DOI: 10.1214/aoap/1177005835


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.


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L. C. G. Rogers. S. E. Satchell. "Estimating Variance From High, Low and Closing Prices." Ann. Appl. Probab. 1 (4) 504 - 512, November, 1991.


Published: November, 1991
First available in Project Euclid: 19 April 2007

zbMATH: 0739.62084
MathSciNet: MR1129771
Digital Object Identifier: 10.1214/aoap/1177005835

Primary: 62M05
Secondary: 60J60 , 60J65 , 62P20 , 90A12

Keywords: Black-Scholes formula , Brownian motion , option pricing , Wiener-Hopf factorisation

Rights: Copyright © 1991 Institute of Mathematical Statistics


Vol.1 • No. 4 • November, 1991
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