Efﬁcient Estimation of Volatility using High Frequency Data
Gilles Zumbach1 , Fulvio Corsi2, and Adrian Trapletti3
Olsen & Associates Research Institute for Applied Economics Seefeldstrasse 233, 8008 Z¨ rich, Switzerland. u phone: +41-1/386 48 48 Fax: +41-1/422 22 82
February 21, 2002 Keywords: volatility estimators, high-frequency data, incoherent price formation, daily volatility.Abstract
The limitations of volatilities computed with daily data as well as simple statistical considerations strongly suggest to use intraday data in order to obtain accurate volatility estimates. Under a continuous time arbitrage-free setup, the quadratic variations of the prices would allow us, in principle, to construct an approximately error free estimate of volatility by using data at thehighest frequency available. Yet, empirical data at very short time scales differ in many ways from the arbitrage-free continuous time price processes. For foreign exchange rates, the main difference originates in the incoherent structure of the price formation process. This market micro-structure effect introduces a noisy component in the price process leading to a strong overestimation of volatilitywhen using naive estimators. Therefore, to be able to fully exploit the information contained in high frequency data, this incoherent effect needs to be discounted. In this contribution, we investigate several unbiased estimators that take into account the incoherent noise. One approach is to use a ﬁlter for pre-whitening the prices, and then using volatility estimators based on the ﬁlteredseries. Another solution is to directly deﬁne a volatility estimator using tick-by-tick price differences, and including a correction term for the price formation effect. The properties of these estimators are investigated by Monte Carlo simulations. A number of important real-world effects are included in the simulated processes: realistic volatility and price dynamic, the incoherent effect,seasonalities, and random arrival time of ticks. Moreover, we investigate the robustness of the estimators with respect to data frequency changes and gaps. Finally, we illustrate the behavior of the best estimators on empirical data.
address: Consulting in ﬁnancial engineering, Ch. Charles Baudouin 8, 1228 Saconnex d’Arve, Switzerland e-mail: firstname.lastname@example.org address:University ofSouthern Switzerland Via Bufﬁ, 13 CH-6904 Lugano, Switzerland e-mail: email@example.com address:Wildsbergstrasse 31, CH-8610 Uster, Switzerland e-mail: firstname.lastname@example.org
Volatility enters as an essential ingredient in many ﬁnancial computations, like portfolio optimization, option pricing or risk assessment. Although these computationsdepend critically on the value of the volatility, the estimation of volatility is often done using daily data. Yet, as the volatility measures intuitively “how much the prices jitter”, there is a gain in using data at higher frequencies. The relevant statistical concept corresponding to our intuition is the minimal sufﬁcient statistic: this is the smallest subset of empirical values needed toevaluate a statistical estimate without loosing information. As a simple example, let us consider a random walk with constant drift µ and volatility σ. Given one realization of the random walk, what are the minimal sufﬁcient statistics to estimate µ and σ? For the drift µ, the answer is that only the start and end points of the random walk are needed. This implies that there is no gain in using highfrequency data for estimating the drift. For the volatility σ, the answer is that the absolute value of every increment is needed (there is no information in the sign of the increments regarding the volatility). This means that all the increments help in getting a better estimate for the volatility, and any thinning of the original data implies a loss of information. Hence, to estimate volatility...
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