Extreme value statistics and recurrence intervals of NYMEX energy futures volatility

Energy markets and the associated energy futures markets play a crucial role in global economies. It is of great theoretical and practical significance to gain a deeper understanding of extreme value statistics of the volatility of energy futures traded on the New York Mercantile Exchange (NYMEX). 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 $P_q\left(\tau \right)\sim e^{{\left(\mathit{a\tau }\right)}^{-\gamma }}$, 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 $\overline{\tau }$. These findings are significant under the Kolmogorov–Smirnov test and the Cramér–von Mises test. We show that the empirical estimations are in nice agreement with the numerical integration results for the occurrence probability W_q(Δt|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 scale as a power law of the preceding interval $\overline{\tau }\left({\tau }_0\right)/\overline{\tau }\sim {\left({\tau }_0/\overline{\tau }\right)}^{\beta }$, 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. Our findings shed new lights on the behavior of large volatilities and have potential implications in risk management of energy futures.     

The Debreu private ownership economy with complementary commodities and prices

Let E_p be a Debreu private ownership economy in which there are some complementary commodities. It means that all commodity bundles are contained in the proper subspace V   of commodity–space [R]l(l∈{1,2,…})${\mathfrak{[R]}}^l\left(l\in \left\{1,2,\dots \right\}\right)$. The production plans maximizing the producers' profits do not have to satisfy the dependency in the qualities of commodities seen in the consumers' plans. It may cause no-existence an equilibrium in economy E_p. The competitive leads, however, to adjustment the production plans to improve the consumers' satisfaction. The procedure of change the production system covering the consumers' requirements is presented. As a result, the model of the private ownership economy with complementary commodities and prices, being the simplification of the initial model, is elaborated. Consequently, the necessary condition for the existence of an equilibrium in the economy E_p is proved.