High-Frequency Financial Econometrics
نویسندگان
چکیده
This book is fundamentally about the estimation of risk. At an intuitive level, risk is easy to understand: given an asset with a current price of say $100, what is the likelihood that at some future time—the risk horizon—its price will be less than $90? Or more than $120? And how does the probability of observing a price below $90 change as a function of the risk horizon? Such heuristic notions can be formalized in several different ways. Suppose that the (log) price of the asset follows Xt “ X0 ` σWt where Wt is a standard Brownian motion and σ ą 0 is a parameter quantifying the risk of X, which needs to be estimated. Suppose also that the price is observed at a sequence of fixed points in time, say at the end of each trading day. Then a simple measure of risk is the squared change in the observed price from one day to the next, possibly averaged over several adjacent intervals to obtain a smoothed estimate—this is, of course, the (daily) variance. As the number of intervals increases, and assuming the statistical properties of the underlying price process remain unchanged, this estimator eventually yields the true value of σ (or in fact its square). But asymptotic results can be considered in another sense: suppose that the total length of the observation interval is fixed (say one day) but the distance between observations within the interval becomes increasingly small—that is, one is in the realm of high-frequency data. Then another way to estimate risk on the interval is to sum the squared incremental price changes. This estimator is called the realized volatility, and it is this estimator, along with its variants and applications, that form the main focus of the book. Formally, if the observation interval has length T and the distance between observations is ∆ then the realized volatility is:
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