High-Frequency Financial Econometrics
High-Frequency Financial Econometrics
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DOI：
10.3390/risks4010005
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发表时间：
2016-02
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影响因子：
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通讯作者：
Harley Thompson
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文献类型：
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作者：
Harley Thompson
这本书基本上是关于风险的估计。在直观的层面上，风险很容易理解：假设一项资产的当前价格为100美元，那么在未来某个时候--即风险水平--其价格低于90美元的可能性有多大？或者超过120美元？而观察到价格低于90美元的概率是如何作为风险期限的函数变化的？这样的启发式概念可以用几种不同的方式来形式化。假设资产的(对数)价格遵循Xt“X0‘σWt，其中Wt是标准布朗运动，σą0是量化X的风险的参数，需要估计。还假设价格是在一系列固定的时间点上观察到的，比如在每个交易日结束时。然后，一个简单的风险度量是观察到的价格从一天到第二天的平方变化，可能是几个相邻区间的平均值，以获得平滑的估计--当然，这就是(每日)方差。随着区间数的增加，假设基础价格过程的统计特性保持不变，这个估计器最终得到σ的真值(或者实际上是它的平方)。但渐近结果可以从另一种意义上考虑：假设观测间隔的总长度是固定的(比方说一天)，但是间隔内的观测之间的距离变得越来越小--也就是说，一个人处于高频数据的领域。然后，估计区间风险的另一种方法是对平方增量价格变化求和。这种估计器被称为已实现波动率，正是这种估计器及其变种和应用构成了本书的主要焦点。形式上，如果观察间隔的长度为T，并且观察之间的距离为∆，则已实现的波动率为：
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: