NYMEX能源期货波动率的极值统计与重现区间
Extreme value statistics and recurrence intervals of NYMEX energy futures volatility
摘要：
能源市场和相关的能源期货市场在全球经济中发挥着至关重要的作用。本文研究了四种纽约商品交易所（NYMEX）能源期货日波动率时间序列的回归区间的统计特性，回归区间定义为连续波动率超过给定阈值$q$的等待时间$τ$。我们发现，复发间隔分布作为一个拉伸指数$P_q(τ)\sim e^{(aτ)^{-γ}}$，其中的指数$γ$随着$q$的增加而减少，并有没有标度行为的分布不同阈值$q$后，复发间隔与平均复发间隔$\barτ$标度。这些发现在Kolmogorov-Smirnov检验和Cram{é}r-von Mises检验下是显著的。我们表明，经验估计是在很好的协议与数值积分结果的发生概率$W_q(Δ{t}|t)$的下一个事件高于阈值$q$在一个（短）的时间间隔后，经过一段时间$t$从最后一个事件以上$q$。我们还调查了记忆效果的复发间隔。结果表明，大、小复发间隔的条件分布不同，复发间隔的条件均值与前一个间隔$\barτ(τ_0)/\barτ\sim (τ_0/\barτ)^β$成幂律关系，表明复发间隔具有短期相关性。去趋势波动分析和去趋势移动平均分析进一步揭示了复发间隔具有长期相关性。我们确认，“集群”的波动率复发间隔是由长期的相关性众所周知的是存在于波动率。
英文摘要：
Energy markets and the associated energy futures markets play a crucial role in global economies. We investigate the statistical properties of the recurrence intervals of daily volatility time series of four NYMEX energy futures, which are defined as the waiting times $τ$ between consecutive volatilities exceeding a given threshold $q$. We find that the recurrence intervals are distributed as a stretched exponential $P_q(τ)\sim e^{(aτ)^{-γ}}$, where the exponent $γ$ decreases with increasing $q$, and there is no scaling behavior in the distributions for different thresholds $q$ after the recurrence intervals are scaled with the mean recurrence interval $\barτ$. These findings are significant under the Kolmogorov-Smirnov test and the Cram{é}r-von Mises test. We show that empirical estimations are in nice agreement with the numerical integration results for the occurrence probability $W_q(Δ{t}|t)$ of a next event above the threshold $q$ within a (short) time interval after an elapsed time $t$ from the last event above $q$. We also investigate the memory effects of the recurrence intervals. It is found that the conditional distributions of large and small recurrence intervals differ from each other and the conditional mean of the recurrence intervals scales as a power law of the preceding interval $\barτ(τ_0)/\barτ\sim (τ_0/\barτ)^β$, indicating that the recurrence intervals have short-term correlations. Detrended fluctuation analysis and detrending moving average analysis further uncover that the recurrence intervals possess long-term correlations. We confirm that the "clustering" of the volatility recurrence intervals is caused by the long-term correlations well known to be present in the volatility.
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