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A high frequency ANALYSIS of oil futures returns
Benot Svi, Universit de la Mditerrane Aix-Marseille II and London Business School,benoit.sevi@gmail.com
Overview
The question of which model should be used when modeling the price of crude oil futures is examined in this paper with a particular emphasis on empirical results from very high-frequency data. We consider both the mixture-of-normalshypothesis and the presence of jumps to provide insights on how to model the stochastic behavior of prices to fit the empirical facts.
Andersen et al. (2001a and b, 2003) have shown how normality could be recovered through an adequate standardization of open-to-close returns with a precise measure of volatility. The authors show theoretically that realized volatility built using high-frequency (say 5 minutes) provide an excellent measure of the true (latent) volatility. When realized volatility is then used to standardize returns, standardized returns are shown to be almost normally distributed despite the formal hypothesis that the distribution of returns is normal is rejected with standard tests. We show that this result is also valid for crude oil futures returns. To reach normality, we then follow Andersen et al. (2010) and remove jumps from daily returns in a sequential way. When jumps are removed, standardized returns are closer to normality and the null hypothesis of normality is not rejected at standard statistical thresholds (see also Fleming and Paye, 2011). This supports the hypothesis of a price process subordinated to a volatility process in the Clark (1973) which would be related to the flow of information.
We then use the methodology recently developed in At-Sahalia and Jacod (2010) and Todorov and Tauchen (2010) to investigate the need to consider jumps in the stochastic modeling of crude oil futures prices. These authors develop statistical tools to measure the presence of jumps and estimate their activity. Using these instruments we are able to answer the following questions: should jumps be part of the stochastic modeling of the oil futures prices? Are jumps rather small with infinite activity or large with small activity? Is a Brownian motion more adapted to the stochastic modeling of oil futures prices or is a pure-jump model a good model? Our results indicate that large jumps should be part of the stochastic modeling of oil futures prices thus confirming results in recent papers where authors use a parametric approach (Lee et al., 2010). We also show that a Brownian motion is part of the model and is more adequate than a pure-jump process.
Methods
To reach normality for the standardized returns, we need to consider jump-adjusted returns as in Andersen et al. (2010). We follow these authors and their procedure of sequential jump identification to possibly identify several significant jumps in the same day and then consider returns adjusted for these identified jumps. The initial methodology to detect jumps is in Barndorff-Nielsen and Shephard (2004, 2006) but we also use the recent procedure (median realized volatility) in Andersen et al. (2008) which do not suffer from the upward bias in finite sample and is more robust to the presence of zero-return. Finally, the time clock has to be changed as in Andersen et al. (2010) and much earlier An and Geman (2000) as trading activity is time-varying and standardization should take into account differences in the flow of information as it has an impact on volatility (see also Oomen (2006) for a discussion of business time vs. calendar time when consider high-frequency returns).
High-frequency returns are then used with the methodology in At-Sahalia and Jacod (2010) and Todorov and Tauchen (2010) who develop formal tests and graphical analysis based on the concept of power variations. The intuition is that different powers of high-frequency returns permit to emphasis the continuous or the jump component in prices.
Note that realized volatility is also computed for robustness checks as in Zhang et al. (2005) to deal with the so-called microstructure noise present in the financial data (Hansen and Lunde, 2006). Our results are robust when this methodology is used in place of the standard realized volatility computation.
Results
Normality is recovered once jumps are considered and time is changed.
Jump diffusion is a good proxy for oil futures returns while a pure-jump process is less realistic in this framework. The theoretical model developed in Hilliard and Reis (1998) does provide pricing elements which are relevant in light of our empirical results as they consider the issue of jumps in commodity prices when pricing derivatives.
Conclusions
Using high-frequency returns to compute daily realized variance allows reaching normality after jumps are removed and time is changed. Beyond normality, it is shown that jumps are part of a satisfying stochastic model for oil futures. This is the first analysis, to our best knowledge, considering the stochastic modeling of oil futures (the most traded commodity contract in the world) using methodologies relying on high-frequency data in a complete nonparametric framework.
References
Ait-Sahalia, Y., Jacod, J., 2009b. Estimating the degree of activity of jumps in high frequency financial data. Annals of Statistics 37, 2202-2244.
ANDERSEN, T.G., BOLLERSLEV, T., DIEBOLD, F.X., EBENS, H., 2001. The distribution of stock return volatility. Journal of Financial Economics 61, 43-76.
ANDERSEN, T.G., BOLLERSLEV, T., DIEBOLD, F.X., LABYS, P., 2001. The distribution of exchange rate volatility. Journal of the American Statistical Association 96, 42-55.
ANDERSEN, T.G., BOLLERSLEV, T., DIEBOLD, F.X., LABYS, P., 2003. Modeling and forecasting realized volatility. Econometrica 71, 579-625.
ANDERSEN, T.G., BOLLERSLEV, T., FREDERIKSEN, P., NIELSEN, M., 2010. Continuous-time models, realized volatilities, and testable distributional implications for daily stock returns. Journal of Applied Econometrics
ANDERSEN, T.G., DOBREV, D., SCHAUMBURG, E.,, 2008. Jump-robust volatility estimation using nearest neighbor truncation. Unpublished manuscript.
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2259-2284.
BARNDORFF-NIELSEN, O., SHEPHARD, N., 2004. Power and bipower variation with stochastic volatility and
jumps. Journal of Financial Econometrics 2, 1-37.
BARNDORFF-NIELSEN, O., SHEPHARD, N., 2006. Econometrics of testing for jumps in financial economics using bipower variation. Journal of Financial Econometrics 4, 1-30.
CLARK, P.K., 1973. A subordinated stochastic process model with finite variance for speculative prices. Econometrica 41, 135-155.
FLEMING, J., PAYE, B.S., 2011. High-frequency returns, jumps and the mixture of normals hypothesis. Journal
of Econometrics 160, 119-128.
HANSEN, P.R., LUNDE, A., 2006. Realized variance and market microstructure noise. Journal of Business and
Economic Statistics 24, 127-218.
Hilliard, J. E., Reis, J., 1998. Valuation of Commodity Futures and Options under Stochastic Convenience Yields, Interest Rates, and Jump Diffusions in the Spot. Journal of Financial and Quantitative Analysis 33, 61-86.
LEE, Y.-H., HU, H.-H., CHIOU, J.-S., 2010. Jump dynamics with structural breaks for crude oil prices. Energy Economics 32, 343-350.
OOMEN, R.C.A., 2006. Properties of realized variance under alternative sampling schemes. Journal of Business
and Economic Statistics 24, 219-237.
Todorov, V., Tauchen, G., 2010. Activity Signature Functions for High-Frequency Data Analysis. Journal of Econometrics 154.
ZHANG, L., MYKLAND, P.A., AI T-SAHALIA, Y., 2005. A tale of two time scales: determining integrated volatility with noisy high frequency data. Journal of the American Statistical Association 100, 1394-1411.
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