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Register a new userExtreme value statistics and recurrence intervals of NYMEX energy futures volatility
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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 $tau$ between consecutive volatilities exceeding a given threshold $q$. We find that the recurrence intervals are distributed as a stretched exponential $P_q(tau)sim e^{(atau)^{-gamma}}$, where the exponent $gamma$ 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 $bartau$. These findings are significant under the Kolmogorov-Smirnov test and the Cram{e}r-von Mises test. We show that empirical estimations are in nice agreement with the numerical integration results for the occurrence probability $W_q(Delta{t}|t)$ 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 $bartau(tau_0)/bartau sim (tau_0/bartau)^beta$, 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.
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We study the volatility time series of 1137 most traded stocks in the US stock markets for the two-year period 2001-02 and analyze their return intervals $tau$, which are time intervals between volatilities above a given threshold $q$. We explore the probability density function of $tau$, $P_q(tau)$, assuming a stretched exponential function, $P_q(tau) sim e^{-tau^gamma}$. We find that the exponent $gamma$ depends on the threshold in the range between $q=1$ and 6 standard deviations of the volatility. This finding supports the multiscaling nature of the return interval distribution. To better understand the multiscaling origin, we study how $gamma$ depends on four essential factors, capitalization, risk, number of trades and return. We show that $gamma$ depends on the capitalization, risk and return but almost does not depend on the number of trades. This suggests that $gamma$ relates to the portfolio selection but not on the market activity. To further characterize the multiscaling of individual stocks, we fit the moments of $tau$, $mu_m equiv <(tau/<tau>)^m>^{1/m}$, in the range of $10 < <tau> le 100$ by a power-law, $mu_m sim <tau>^delta$. The exponent $delta$ is found also to depend on the capitalization, risk and return but not on the number of trades, and its tendency is opposite to that of $gamma$. Moreover, we show that $delta$ decreases with $gamma$ approximately by a linear relation. The return intervals demonstrate the temporal structure of volatilities and our findings suggest that their multiscaling features may be helpful for portfolio optimization.
We investigate scaling and memory effects in return intervals between price volatilities above a certain threshold $q$ for the Japanese stock market using daily and intraday data sets. We find that the distribution of return intervals can be approximated by a scaling function that depends only on the ratio between the return interval $tau$ and its mean $<tau>$. We also find memory effects such that a large (or small) return interval follows a large (or small) interval by investigating the conditional distribution and mean return interval. The results are similar to previous studies of other markets and indicate that similar statistical features appear in different financial markets. We also compare our results between the period before and after the big crash at the end of 1989. We find that scaling and memory effects of the return intervals show similar features although the statistical properties of the returns are different.
The distribution of the return intervals $tau$ between volatilities above a threshold $q$ for financial records has been approximated by a scaling behavior. To explore how accurate is the scaling and therefore understand the underlined non-linear mechanism, we investigate intraday datasets of 500 stocks which consist of the Standard & Poors 500 index. We show that the cumulative distribution of return intervals has systematic deviations from scaling. We support this finding by studying the m-th moment $mu_m equiv <(tau/<tau>)^m>^{1/m}$, which show a certain trend with the mean interval $<tau>$. We generate surrogate records using the Schreiber method, and find that their cumulative distributions almost collapse to a single curve and moments are almost constant for most range of $<tau>$. Those substantial differences suggest that non-linear correlations in the original volatility sequence account for the deviations from a single scaling law. We also find that the original and surrogate records exhibit slight tendencies for short and long $<tau>$, due to the discreteness and finite size effects of the records respectively. To avoid as possible those effects for testing the multiscaling behavior, we investigate the moments in the range $10<<tau>leq100$, and find the exponent $alpha$ from the power law fitting $mu_msim<tau>^alpha$ has a narrow distribution around $alpha eq0$ which depend on m for the 500 stocks. The distribution of $alpha$ for the surrogate records are very narrow and centered around $alpha=0$. This suggests that the return interval distribution exhibit multiscaling behavior due to the non-linear correlations in the original volatility.
In this chapter, we consider volatility swap, variance swap and VIX future pricing under different stochastic volatility models and jump diffusion models which are commonly used in financial market. We use convexity correction approximation technique and Laplace transform method to evaluate volatility strikes and estimate VIX future prices. In empirical study, we use Markov chain Monte Carlo algorithm for model calibration based on S&P 500 historical data, evaluate the effect of adding jumps into asset price processes on volatility derivatives pricing, and compare the performance of different pricing approaches.
Being able to predict the occurrence of extreme returns is important in financial risk management. Using the distribution of recurrence intervals---the waiting time between consecutive extremes---we show that these extreme returns are predictable on the short term. Examining a range of different types of returns and thresholds we find that recurrence intervals follow a $q$-exponential distribution, which we then use to theoretically derive the hazard probability $W(Delta t |t)$. Maximizing the usefulness of extreme forecasts to define an optimized hazard threshold, we indicates a financial extreme occurring within the next day when the hazard probability is greater than the optimized threshold. Both in-sample tests and out-of-sample predictions indicate that these forecasts are more accurate than a benchmark that ignores the predictive signals. This recurrence interval finding deepens our understanding of reoccurring extreme returns and can be applied to forecast extremes in risk management.
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