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2307
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Crude oil, as a key global commodity, has experienced
considerable price level variation in the boom preceding the global financial
crisis in 2008 and the ensuing Great Recession. A major oil price shock in
2008 was caused by constraints on the production of crude oil paired with
low elasticity of demand (for details, see Hamilton (2009) and
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Kilian (2009)).
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This shock, while being caused by fundamentals, was clearly exacerbated by
financial speculation and ‘financialization’ of commodities. Variation in oil
price levels has been accompanied by wide variations in the volatility of returns.
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2817
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In the futures markets, returns exhibit heavy tails, autocorrelation,
and volatility clustering, leading to significant challenges in modeling their
first and second moments.
Both the International Monetary Fund (IMF) and the Federal Reserve
Board (see
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Alquist et al. (2011) and
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IMF 2005 p. 67; 2007, p. 42) use futures
prices as the best available proxy for the market expectations of the spot
crude oil price.
Like many financial series, commodity futures prices are likely to exhibit
randomwalk behavior.
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4551
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The high frequency data allows us to test various models for oil futures
returns using a straightforward Generalized Method of Moments (GMM)
estimator that matches sample moments of the realized volatility to the corresponding population moments of the integrated volatility in the spirit of
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Bollerslev and Zhou (2002).
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These models are then compared, in terms of
overall fit of the data and forecast accuracy statistics, over the full sample.
The model with stochastic volatility and jumps is also tested over a subsample (January 2006–December 2012) to address structural stability (as in
1
Andersen, Benzoni and Lund (2002)).
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5238
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Key findings include the importance
of both jumps and stochastic volatility in oil futures returns and the apparent
unimportance of leverage as a modeled component.
The wider applicability of this method of estimation to other markets is
outside the scope of this paper, but an interesting topic for future research.
2. Review of the literature
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Schwartz (1997), Schwartz and Smith (2000), Casassus and CollinDufresne (2005)
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propose multifactor models for energy prices where returns are only
affected by Gaussian shocks only, but constrain volatility to be constant.
Pindyck (2004) examines the volatility of energy spot and futures prices, estimating the standard deviation of their first differences.
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5469
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Review of the literature
Schwartz (1997), Schwartz and Smith (2000), Casassus and CollinDufresne (2005) propose multifactor models for energy prices where returns are only
affected by Gaussian shocks only, but constrain volatility to be constant.
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Pindyck (2004)
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examines the volatility of energy spot and futures prices, estimating the standard deviation of their first differences. Askari and Khrichene (2008) fit jumpdiffusion models to futures on Brent crude oil.
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5693
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Pindyck (2004) examines the volatility of energy spot and futures prices, estimating the standard deviation of their first differences. Askari and Khrichene (2008) fit jumpdiffusion models to futures on Brent crude oil.
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Schwartz and Trolle (2009)
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propose a multifactor stochastic volatility model for pricing
futures and options on light sweet crude oil trading on the NYMEX. Using
daily data, they present evidence that taking account of stochastic volatility
improves pricing, but they consider the inclusion of jumps to be less important.
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6014
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Schwartz and Trolle (2009) propose a multifactor stochastic volatility model for pricing
futures and options on light sweet crude oil trading on the NYMEX. Using
daily data, they present evidence that taking account of stochastic volatility
improves pricing, but they consider the inclusion of jumps to be less important.
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Vo (2009)
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estimates a multivariate stochastic volatility model using
daily data on the West Texas Intermediate (WTI) crude oil futures contracts
traded on the NYMEX and finds that stochastic volatility plays an important
role.
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6246
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Vo (2009) estimates a multivariate stochastic volatility model using
daily data on the West Texas Intermediate (WTI) crude oil futures contracts
traded on the NYMEX and finds that stochastic volatility plays an important
role.
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Larsson and Nossman (2011)
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find evidence for stochastic volatility and
jumps in both returns and volatility daily spot prices of WTI crude oil from
1989 to 2009.
The role of volatility as a measure of uncertainty of oil price futures is
2
stressed by Bernanke (1983), Pindyck (1991) and Kellogg (2010) who show
that this measure of uncertainty is extremely relevant for firms’ investment
decisions.
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6491
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Larsson and Nossman (2011) find evidence for stochastic volatility and
jumps in both returns and volatility daily spot prices of WTI crude oil from
1989 to 2009.
The role of volatility as a measure of uncertainty of oil price futures is
2
stressed by
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Bernanke (1983), Pindyck (1991) and
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Kellogg (2010) who show
that this measure of uncertainty is extremely relevant for firms’ investment
decisions.
Our contribution lies in the use of the information on volatilty of oil
futures returns provided by high frequency, intraday data while focusing on
the role of volatility as measure of variability and uncertainty of oil price
forecasts.
3.
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9480
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The Box–Pierce portmanteau (orQ) test for white noise
rejects its null for both series. The daily returns series exhibits significant
ARCH effects at 1, 5, 10 and 22 lags, while no evidence of ARCH effects is
found in the realized volatility series.
4. Estimation method
Following
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Bollerslev and Zhou (2002),
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who use continuously observed
futures prices on oil, we build a conditional moment estimator for stochastic
volatility jumpdiffusion models based on matching the sample moments of
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realized volatility with population moments of integrated volatility.
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12466
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As we discuss below, there is no
empirical support for leverage, in that the parameter expressing the effect of
leverage is never significantly different from zero. Thus, we present here our
findings from the SV and SVJ models.
5.1. Stochastic Volatility model (SV)
We model the returns on futures on crude oil using the
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Heston (1993)
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model. For simplicity, we set the drift of the log price equal to zero.2This
choice is consistent with Alquist et al. (2011) who find that a reasonable
and parsimonious forecasting model for spot oil prices is the random walk
without drift.
2As Bollerslev et al. (2002) suggest, a drift could be easily introduced in the futures
returns equation.
7
dpt=dln(Ft)
=
√
VtdW1t
dVt=κ(θ−Vt)dt
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12587
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Stochastic Volatility model (SV)
We model the returns on futures on crude oil using the Heston (1993) model. For simplicity, we set the drift of the log price equal to zero.2This
choice is consistent with
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Alquist et al. (2011)
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who find that a reasonable
and parsimonious forecasting model for spot oil prices is the random walk
without drift.
2As Bollerslev et al. (2002) suggest, a drift could be easily introduced in the futures
returns equation.
7
dpt=dln(Ft)
=
√
VtdW1t
dVt=κ(θ−Vt)dt+σ
√
VtdW2t
E(dW1tdW2t) = 0
In this model, there are two orthogonal Wiener processes,dW1tanddW2t,
driving the evolution of returns and
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15544
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All five estimated parameters
9
of the model are very precisely estimated and take on sensible values from
an analytical perspective.
In order to better motivate the concept of jumps in the futures returns
process, we employ nonparametric methods to identify those periods when
“extreme events” may have occurred. Following
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Tukey (1977),
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we consider
extreme events to be those periods when the onetradingday change in futures returns lay outside the bounds of the “adjacent values” of a conventional
box plot. The adjacent values are defined using 1.5 times the interquartile
range (IQR), or difference between the empirical 75th and 25th percentiles
(p75, p25) of the series.
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16976
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This experiment allows us
to assess the structural stability of the model as within the shorter sample
λ, the frequency of extreme events, is significantly different from zero.
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5.3. Does leverage matter?
As suggested by
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Alquist et al. (2011),
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there is no reason why oil producers
should be concerned about the volatility of the price of oil. The data seem to
suggest that there is no connection between the shocks affecting futures prices
and the shocks affecting the corresponding volatility.
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