Fractals in trade duration: capturing long-range dependence and heavy tailedness in modeling trade duration

Several studies that have investigated a few stocks have found that the spacing between consecutive financial transactions (referred to as trade duration) tend to exhibit long-range dependence, heavy tailedness, and clustering. In this study, we empirically investigate whether a larger sample of stocks exhibit those characteristics. We do so by comparing goodness of fit in modeling trade duration data for stable distribution and fractional stable noise based on a procedure applying bootstrap methods developed by the authors with several alternative distributional assumptions in modeling trade duration data. The empirical results suggest that the autoregressive conditional duration model with stable distribution fits better than other combinations, while fractional stable noise itself fits better for the time series of trade duration. Our result is consistent with the general findings in the literature that trade duration is informative and that short trade durations move prices more than long trade duration. In addition, our result confirms the advantage of fractal models in the study of roughness in trade duration and provides some evidence for duration dependence. S. Rachev’s research was supported by grants from the Division of Mathematical, Life and Physical Science, College of Letters and Science, University of California, Santa Barbara, and the Deutschen Forschungsgemeinschaft. W. Sun’s research was supported by grants from the Deutschen Forschungsgemeinschaft. P.S. Kalev’s research was supported with a NCG grant from the Faculty of Business and Economics, Monash University. Data are supplied by Securities Industry Research Center of Asia-Pacific (SIRCA) on behalf of Reuters. The first draft of this paper was presented at the International Conference on High Frequency Finance 2006; the authors would like to thank the conference participants for their valuable comments.     

MULTIVARIATE STABLE FUTURES PRICES

This paper introduces new techniques for modeling financial data under the assumption that the data belong to the domain of attraction of a multivariate stable Pareto law. We provide tail estimators for the index of stability parameter a and the corresponding spectral measure. These estimators are then applied to test the associtation of the individual components and to compute estimates of portfolio risk and the covariation of commodities. A practical example is given using DM-dollar and JY-dollar exchange rates data.     

Limiting distribution of the collision resolution interval

We use the theory of probability metrics to study the asymptotic normality of the collison resolution intervals in the CTM multi-access protocol under general conditions on the number of retransmitted messages and of new arrivals during the collision slots. Our main result establishes stability of the central limit theorem for the CTM algorithm. We provide extensive simulation results investigating the extent to which the mean of the collision resolution interval eventually becomes unstable for increasing values of n , the number of users who initially collide. The normal fit is numerically investigated and is shown to be quite satisfactory and stable with respect to moderate perturbations and n ≥50.     

Financial market models with Lévy processes and time-varying volatility

Asset management and pricing models require the proper modeling of the return distribution of financial assets. While the return distribution used in the traditional theories of asset pricing and portfolio selection is the normal distribution, numerous studies that have investigated the empirical behavior of asset returns in financial markets throughout the world reject the hypothesis that asset return distributions are normally distribution. Alternative models for describing return distributions have been proposed since the 1960s, with the strongest empirical and theoretical support being provided for the family of stable distributions (with the normal distribution being a special case of this distribution). Since the turn of the century, specific forms of the stable distribution have been proposed and tested that better fit the observed behavior of historical return distributions. More specifically, subclasses of the tempered stable distribution have been proposed. In this paper, we propose one such subclass of the tempered stable distribution which we refer to as the “KR distribution”. We empirically test this distribution as well as two other recently proposed subclasses of the tempered stable distribution: the Carr–Geman–Madan–Yor (CGMY) distribution and the modified tempered stable (MTS) distribution. The advantage of the KR distribution over the other two distributions is that it has more flexible tail parameters. For these three subclasses of the tempered stable distribution, which are infinitely divisible and have exponential moments for some neighborhood of zero, we generate the exponential Lévy market models induced from them. We then construct a new GARCH model with the infinitely divisible distributed innovation and three subclasses of that GARCH model that incorporates three observed properties of asset returns: volatility clustering, fat tails, and skewness. We formulate the algorithm to find the risk-neutral return processes for those GARCH models using the “change of measure” for the tempered stable distributions. To compare the performance of those exponential Lévy models and the GARCH models, we report the results of the parameters estimated for the S&P 500 index and investigate the out-of-sample forecasting performance for those GARCH models for the S&P 500 option prices.     

Spot and derivative pricing in the EEX power market

Using spot and futures price data from the German EEX Power market, we test the adequacy of various one-factor and two-factor models for electricity spot prices. The models are compared along two different dimensions: (1) We assess their ability to explain the major data characteristics and (2) the forecasting accuracy for expected future spot prices is analyzed. We find that the regime-switching models clearly outperform its competitors in almost all respects. The best results are obtained using a two-regime model with a Gaussian distribution in the spike regime. Furthermore, for short and medium-term periods our results underpin the frequently stated hypothesis that electricity futures quotes are consistently greater than the expected future spot, a situation which is denoted as contango.     

Approximation of aggregate and extremal losses within the very heavy tails framework

The loss distribution approach is one of the three advanced measurement approaches to the Pillar I modeling proposed by Basel II in 2001. In this paper, one possible approximation of the aggregate and maximum loss distribution in the extremely low frequency/high severity case is given, i.e. the case of infinite mean of the loss sizes and loss inter-arrival times. In this study, independent but not identically distributed losses are considered. The minimum loss amount is considered increasing over time. A Monte Carlo simulation algorithm is presented and several quantiles are estimated. The same approximation is used for modeling the maximum and aggregate worldwide economy losses caused by very rare and very extreme events such as 9/11, the Russian rouble crisis, and the U.S. subprime mortgage crisis. The model parameters are fit on a data sample of operational losses. The respective aggregate and extremal loss quantiles are calculated.     

Smooth monotone covariance for elliptical distributions and applications in finance

Sample covariance is known to be a poor estimate when the data are scarce compared with the dimension. To reduce the estimation error, various structures are usually imposed on the covariance such as low-rank plus diagonal (factor models), banded models and sparse inverse covariances. We investigate a different non-parametric regularization method which assumes that the covariance is monotone and smooth. We study the smooth monotone covariance by analysing its performance in reducing various statistical distances and improving optimal portfolio selection. We also extend its use in non-Gaussian cases by incorporating various robust covariance estimates for elliptical distributions. Finally, we provide two empirical examples using Eurodollar futures and corporate bonds where the smooth monotone covariance improves the out-of-sample covariance prediction and portfolio optimization.     

Option pricing and hedging under a stochastic volatility Lévy process model

In this paper, we discuss a stochastic volatility model with a Lévy driving process and then apply the model to option pricing and hedging. The stochastic volatility in our model is defined by the continuous Markov chain. The risk-neutral measure is obtained by applying the Esscher transform. The option price using this model is computed by the Fourier transform method. We obtain the closed-form solution for the hedge ratio by applying locally risk-minimizing hedging.     

Tempered stable and tempered infinitely divisible GARCH models

In this paper, we introduce a new GARCH model with an infinitely divisible distributed innovation. This model, which we refer to as the rapidly decreasing tempered stable (RDTS) GARCH model, takes into account empirical facts that have been observed for stock and index returns, such as volatility clustering, non-zero skewness, and excess kurtosis for the residual distribution. We review the classical tempered stable (CTS) GARCH model, which has similar statistical properties. By considering a proper density transformation between infinitely divisible random variables, we can find the risk-neutral price process, thereby allowing application to option-pricing. We propose algorithms to generate scenarios based on GARCH models with CTS and RDTS innovations. To investigate the performance of these GARCH models, we report parameter estimates for the Dow Jones Industrial Average index and stocks included in this index. To demonstrate the advantages of the proposed model, we calculate option prices based on the index.     

Momentum strategies based on reward–risk stock selection criteria

In this paper, we analyze momentum strategies that are based on reward–risk stock selection criteria in contrast to ordinary momentum strategies based on a cumulative return criterion. Reward–risk stock selection criteria include the standard Sharpe ratio with variance as a risk measure, and alternative reward–risk ratios with the expected shortfall as a risk measure. We investigate momentum strategies using 517 stocks in the S&P 500 universe in the period 1996–2003. Although the cumulative return criterion provides the highest average monthly momentum profits of 1.3% compared to the monthly profit of 0.86% for the best alternative criterion, the alternative ratios provide better risk-adjusted returns measured on an independent risk-adjusted performance measure. We also provide evidence on unique distributional properties of extreme momentum portfolios analyzed within the framework of general non-normal stable Paretian distributions. Specifically, for every stock selection criterion, loser portfolios have the lowest tail index and tail index of winner portfolios is lower than that of middle deciles. The lower tail index is associated with a lower mean strategy. The lowest tail index is obtained for the cumulative return strategy. Given our data-set, these findings indicate that the cumulative return strategy obtains higher profits with the acceptance of higher tail risk, while strategies based on reward–risk criteria obtain better risk-adjusted performance with the acceptance of the lower tail risk.