Source: Ann. Appl. Probab. Volume 18, Number 2
(2008), 813-823.
In earlier studies, the estimation of the volatility of a stock using information on the daily opening, closing, high and low prices has been developed; the additional information in the high and low prices can be incorporated to produce unbiased (or near-unbiased) estimators with substantially lower variance than the simple open–close estimator. This paper tackles the more difficult task of estimating the correlation of two stocks based on the daily opening, closing, high and low prices of each. If we had access to the high and low values of some linear combination of the two log prices, then we could use the univariate results via polarization, but this is not data that is available. The actual problem is more challenging; we present an unbiased estimator which halves the variance.
References
Alizadeh, S., Brandt, M. and Diebold, F. (2002). Range-based estimation of stochastic volatility models. J. Finance 57 1047–1091.
Ball, C. A. and Torous, W. N. (1984). The maximum likelihood estimation of security price volatility: Theory, evidence, and an application to option pricing. J. Business 57 97–112.
Barndorff-Nielsen, O. E., Hansen, P. R., Lunde, A. and Shephard, N. (2007). Designing realised kernels to measure the ex-post variation of equity prices in the presence of noise. Technical report, Univ. Oxford.
Barndorff-Nielsen, O. E. and Shephard, N. (2004). Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica 72 885–925.
Beckers, S. (1983). Variance of security price returns based on high, low and closing prices. J. Business 56 97–112.
Brandt, M. W. and Diebold, F. X. (2006). A no-arbitrage approach to range-based estimation of return covariances and correlations. J. Business 79 61–74.
Broadie, M., Glasserman, P. and Kou, S. G. (1997). A continuity correction for discrete barrier options. Math. Finance 7 325–349.
Broto, C. and Ruiz, E. (2004). Estimation methods for stochastic volatility models. J. Economic Surveys 18 613–649.
Brunetti, C. and Lildholdt, P. M. (2002). Return-based and range-based (co)variance estimation, with an application to foreign exchange markets. Technical Report 127, Center for Analytical Finance, Univ. Aarhus.
Garman, M. and Klass, M. J. (1980). On the estimation of security price volatilities from historical data. J. Business 53 67–78.
Ghysels, E., Harvey, A. C. and Renault, E. (1996). Stochastic volatility. In Statistical Methods in Finance (G. S. Maddala and C. R. Rao, eds.). Handbook of Statist. 14 119–191. North-Holland, Amsterdam.
Kunitomo, N. (1992). Improving the Parkinson method of estimating security price volatilities. J. Business 65 295–302.
Parkinson, M. (1980). The extreme value method for estimating the variance of the rate of return. J. Business 53 61–65.
Rogers, L. C. G. and Satchell, S. E. (1991). Estimating variance from high, low and closing prices. Ann. Appl. Probab. 1 504–512.
Rogers, L. C. G. and Shepp, L. (2006). The correlation of the maxima of correlated Brownian motions. J. Appl. Probab. 43 999–999.
Shephard, N., ed. (2005). Stochastic Volatility: Selected Readings. Oxford Univ. Press.
Yang, D. and Zhang, Q. (2000). Drift-independent volatility estimation based on high, low, open and close prices. J. Business 73 477–491.
Zhang, L., Mykland, P. A. and Aït-Sahalia, Y. (2005). A tale of two time scales: Determining integrated volatility with noisy high-frequency data. J. Amer. Statist. Assoc. 100 1394–1411.
