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7/29/2019 The Dynamics of Commodity Prices
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The Dynamics of Commodity Prices
Chris Brooks and Marcel Prokopczuk
ICMA Centre University of Reading
May 2011
ICMA Centre Discussion Papers in Finance DP2011-09
Copyright 2011 Brooks and Prokopczuk. All rights reserved.
ICMA CentreUniversity of ReadingWhiteknightsPO Box 242 Reading RG6 6BA UKTel: +44 (0)1183 787402 Fax: +44 (0)1189 314741Web: www.icmacentre.ac.ukDirector: Professor John Board, Chair in FinanceThe ICMA Centre is supported by the International Capital Market Association
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The Dynamics of Commodity Prices
Chris Brooks and Marcel Prokopczuk
May 19, 2011
Abstract
In this paper we study the stochastic behavior of the prices and volatilities
of a sample of six of the most important commodity markets and we compare
these properties to those of the equity market. We observe a substantial degree
of heterogeneity in the behavior of the series. Our findings show that it is
inappropriate to treat different kinds of commodities as a single asset classas is frequently the case in the academic literature and in the industry. We
demonstrate that commodities can be a useful diversifier of equity volatility
as well as equity returns. Risk measurement and options pricing and hedging
applications exemplify the economic impacts of the differences across commodities
and between model specifications.
JEL classification: G10, C32
Keywords: Commodity prices, stochastic volatility, jumps, Markov Chain MonteCarlo
Both authors are members of the ICMA Centre, University of Reading. Contact: MarcelProkopczuk, ICMA Centre, Henley Business School, University of Reading, Whiteknights, Reading,RG6 6BA, United Kingdom. e-mail: [email protected]. Telephone: +44-118-378-4389.
Fax: +44-118-931-4741.
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I. Introduction
Following the seminal work of Samuelson (1965), it is now widely accepted that
commodity prices fluctuate randomly. Understanding the nature of this stochastic
behavior is of crucial importance for decision makers engaging in commodity markets.
Yet traditionally, with the probable exception of gold, commodities have not been in
the focus of investors. However, interest has grown enormously over the last decade
for a variety of reasons. First, following the relatively poor performances of stocks and
Treasuries, investors have sought previously unexplored asset classes as potential new
sources of returns. Second, the low correlation of commodity returns with equities
and their ability to provide a hedge against inflation make them useful additions
to portfolios. The liberalization of numerous markets has also increased corporates
requirements for hedging. This increased investment and hedging interest has led to a
fast growth of the commodity derivatives market.
The aim of this paper is to study the stochastic behavior of commodity prices, both
from an individual perspective but also concerning the cross-market linkages between
commodities, and between commodities and the equity market. As such, this is the
first piece of research to comprehensively apply a range of stochastic volatility models
to commodities from several market segments. A good understanding of commodity
prices behavior and their interdependences as well as their relation to the equity market
is important for investors, producers, consumers, and also policymakers.
We study the following issues for six major commodity markets. First, we investigate
whether volatility does indeed behave stochastically and we estimate several models for
the returns and volatility processes. Second, we examine the volatility of volatility, its
persistence, and whether changes in prices and volatility are correlated. We additionally
allow for jumps in both prices and volatility. We also investigate the linkages between
the commodity markets considered. We determine whether volatilities are correlated
across markets and whether the prices or the volatilities of different commodities jump
at the same time. Moreover, we analyze the same questions for the linkages between
each commodity and the equity market.
Our main findings are as follows. First, we find that, within the stochastic volatility
framework, the models that allow for jumps provide a considerably better fit to the data
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than those which do not, although there is little to choose between the models allowing
for jumps in returns only and those allowing for jumps in both returns and volatility.
Second, we observe alternate signs in the relationships between returns and volatilities
for different commodities negative for crude oil and equities, close to zero for gasoline
and wheat, and positive for gold, silver, and soybeans. We attribute these differences
to variations in the relative balances of speculators and hedgers across the markets.
We also find evidence of considerable differences in both the intensity and frequency
of jumps, although all commodities are found to exhibit more frequent jumps than the
S&P 500. We conclude that commodities have very different stochastic properties, and
therefore that it is suboptimal to consider them as a single, unified asset class. To
analyze the economic implications of the differences across commodities and betweenmodel specifications, as exemplars, we employ four important applications focused on
risk measurement, and options pricing and hedging.
Regarding the stochastic behavior of prices, equity markets, and especially equity
index markets, have received a great deal of attention. The non-normality of equity
returns has been documented extensively. Motivated by the poor performance of
the Black-Scholes-Merton options pricing formula that produced the well known
smile phenomenon, researchers have extended the simple Brownian motion in variousdirections. Merton (1976) is probably the first to suggest adding a discontinuous jump
component to the continuous Brownian price process, and Ball and Torous (1985)
empirically confirm the merits of this approach. Scott (1987) and Heston (1993)
suggest modelling volatility stochastically, and the latter study is able to derive a
semi-closed form solution for European option prices in the environment where volatility
is stochastic. The two ideas are combined by Bates (1996) and Bakshi et al. (1997),
who propose employing stochastic volatility and price jumps to improve the description
of the assets price process with the aim of enhancing the accuracy of options pricing.
Finally, Duffie et al. (2000) develop a general affine framework for asset prices, and
suggest a model with stochastic volatility, price jumps, and, additionally, jumps in
volatility. Andersen et al. (2002), Chernov et al. (2003), Eraker et al. (2003), and
Eraker (2004) study various versions of stochastic volatility models, with the latter
two concluding that jumps in prices and volatility improve the description of S&P 500
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price dynamics as well as the pricing of options written on the index. Asgharian and
Bengtsson (2006) use this class of models to study the cross-market dependencies of
various equity index markets.
Compared to this array of research on equity markets, state-of-the-art empirical
studies on commodity markets are sparse. Of the few examples there are of such
studies, Brennan and Schwartz (1985), Gibson and Schwartz (1990), Schwartz (1997),
and Schwartz and Smith (2000) study the ability of continuous-time Gaussian factor
models with constant volatility to describe the stochastic behavior of the futures
curve, mostly considering the crude oil market. Sorensen (2002) and Manoliu and
Tompaidis (2002) apply the model of Schwartz and Smith (2000) to the agricultural
and natural gas futures markets, respectively. Paschke and Prokopczuk (2009) studythe cross-market dependence structure of prices in the energy segment only. Casassus
and Collin-Dufresne (2005) mainly study the nature of risk premia in four different
commodity markets; they also allow for the possibility of discontinuous price jumps,
which Aiube et al. (2008) conclude are important in the crude oil market. The possibility
of stochastic volatility in commodities is considered by Geman and Nguyen (2005)
for the soybean market, and by Trolle and Schwartz (2009), for crude oil. Finally, a
three-factor incorporating prices, interest rates and the convenience yield is constructedby Liu and Tang (2011) to capture the stochastic relationship between the convenience
yield level and its volatility for industrial commodities.1
Papers considering multiple commodity markets and their dependencies rely mainly
on simple return analyses. For example, Erb and Harvey (2006) and Gorton and
Rouwenhorst (2006) study the benefits of investing in commodity markets; Kat and
Oomen (2007a) and Kat and Oomen (2007b) conduct statistical analyses of commodity
returns. By contrast, we evaluate the cross-linkages between commodities and between
commodities and equities in a more formal, rigorous framework.
The remainder of this paper is structured as follows. Section II describes the models
estimated and the procedures employed to obtain the parameters, while Section III
1There is an enormous number of papers that model time-varying volatilities and correlations withina discrete time GARCH-type framework. In the commodities area, many of these are on the oil marketand focus on the objective of determining effective hedge ratios. A full survey of such work is beyondthe scope of this paper, but relevant studies include Serletis (1994), Ng and Pirrong (1996), Haigh andHolt (2002), Pindyck (2004), Sadorsky (2006), Alizadeh et al. (2008), and Wang et al. (2008).
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presents and discusses the data and the empirical results. Section IV analyzes the
economic implications of employing different models in terms of Value-at-Risk and
options values and hedging errors. Finally, Section V concludes, while further details
of the Markov Chain Monte Carlo procedure are presented in an appendix.
II. Models and Estimation
A. Spot Price Models
The first model specification employed includes stochastic volatility only, and is
therefore denoted SV. The log spot price Yt = log St is assumed to follow the dynamics
dYt = ()dt +
VtdWYt (1)
where WYt is a standard Brownian motion.2 To capture potential seasonal effects in
the price dynamics, we assume a trigonometric function for the drift component in (1)
() = + sin(2( + )) (2)
where [0, 1] denotes the time fraction of the year elapsed. The average drift rateis denoted by . The parameter > 0 controls the amplitude of the function and
therefore captures the strength of the seasonal effect, whereas governs the periodicity
of the process drift capturing the form of the seasonality. For markets that do not
show any seasonal behavior we set = 0, yielding a constant drift. For the volatility
Vt, we consider the square-root process
dVt = ( Vt)dt +
VtdWVt (3)
where WVt is a second Brownian motion with dWYt dW
Vt = dt. The variance process is
mean-reverting towards the long-run mean with speed . The parameter captures
the volatility of volatility (vol-of-vol) and the kurtosis of returns increases for higher
2Alternatively, one could specify the spot price dynamics as mean-reverting process. We have donethis, however, the empirical results were inferior to the non-stationary Brownian motion.
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values of. The correlation between returns and volatility captures the skewness of the
returns distribution. Positive values impliy a skew to the right, and negative values a
skew to the left.
This specification guarantees the positiveness of price and volatility at all times.
Except for the seasonal adjustment, it is identical to the model proposed by Heston
(1993).
Empirical evidence suggests that a pure continuous specification of the spot price
cannot capture all salient features observed in financial data, and in particular the
possibility of rapid price movements, i.e. jumps. To allow for jumps in the price
dynamics, we follow Bates (1996) and add a Poisson process NYt to specification (1),
yielding the SVJ model
dYt = ()dt +
VtdWYt + ZtdN
Yt . (4)
The intensity Y of NYt is assumed to be constant and the jump sizes are assumed to
be generated by a normal distribution, i.e. Zt N(Y, 2Y). Assuming that jumpsoccur infrequently but are relatively large (which is the natural perception of jumps as
opposed to the diffusion components in prices), the jump component will mainly affect
the tails of the return distribution. The specifications for the variance process and the
seasonality adjustment remain unchanged.
Bakshi et al. (1997) and Bates (2000) find that although the inclusion of jumps
in returns helps to describe the behavior of equity prices and the pricing of options,
the model is still severely misspecified. Jumps in returns are transient, however, and
hence a more persistent component is needed. Therefore, Duffie et al. (2000) introduce
a model specification allowing for jumps in both returns and volatility. The return
process remains identical to (4), but the variance process is amended by the jump
process NVt , i.e.
dVt = ( Vt)dt +
VtdWVt + CtdN
Vt . (5)
We assume that prices and volatility jump simultaneously, i.e., NVt = NYt , which is
commonly denoted as a SVCJ model. This assumption is motivated by the idea that
periods of stress when prices jump are often accompanied by high levels of uncertainty,
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resulting in a jump of volatility. However, the jump sizes are not equal. Following
Duffie et al. (2000), we assume the variance jump size to be exponentially distributed,
i.e. Ct exp(V). To allow for dependence between the jump sizes in returns andvolatility, the jumps in returns are conditionally normally distributed with Zt
|Ct
N(Y + Ct,
2Y). Due to the inclusion of the jump component in the variance process,
the long-run mean changes to + V/.
B. Estimation Approach
In this section, we briefly outline the Markov Chain Monte Carlo (MCMC)
estimation approach we use to estimate all models. MCMC belongs to the class of
Bayesian simulation-based estimation techniques. The main advantage of the MCMC
methodology is the fact that it allows us to estimate the unknown model parameters and
the unobservable state variables, i.e. the volatility, the jump times and the jump sizes,
simultaneously in an efficient way. Jacquier et al. (1994) show how MCMC methods
can be used to estimate the parameters and latent volatility process of a stochastic
volatility model. Johannes et al. (1999) extend this approach for jumps in returns and
finally, Eraker et al. (2003) estimate the model including jumps in returns and volatility
via MCMC methods.3
In order to be able to estimate the models, it is necessary to express them in
discretized form. Using a simple Euler discretization, the SVCJ model is given as4
Yt = Ytt + ()t +
VttYt + ZtJt (6)
and
Vt = Vtt + ( Vtt)t + VttVt + CtJt. (7)The innovations Yt and
Vt are normal random variables, i.e.
Yt N(0, t), and
Vt N(0, t) with correlation . The jump times Jt take the value one if a jumpoccurs and zero if not, i.e. Jt Ber(t). The discretized versions of the SVJ and SVmodels are obtained by dropping the respective jump components. In the following, we
3For an excellent introduction to MCMC estimation techniques, see Johannes and Polson (2006).4As we work with daily data, the discretization bias is negligible.
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set t = 1, i.e. one day.
The general idea of the MCMC methodology is to break down the high dimensional
posterior distribution into its low dimensional complete conditionals of parameters
and latent factors which can be efficiently sampled from. The posterior distribution
p(, V , J, Z, C |Y) provides sample information regarding the unknown quantities giventhe observed quantities (prices). By Bayes rule we have
p(, V , J, Z, C |Y) p(Y|V,J,Z,C, )p(V,J,Z,C|)p() (8)
where Y is the vector of observed log prices; V, J, Z, and C contain the time
series of volatility, jump times and jump sizes, respectively; is the vector of
model parameters; p(Y|V,J,Z,C, ) is usually called the likelihood, p(V,J,Z,C|)provides the distribution of the latent state variables, and p() the prior, reflecting the
researchers beliefs regarding the unknown parameters. To keep the influence of the
priors small, we specify extremely uninformative priors.
As p(, V , J, Z, C |Y) is high dimensional, it is not possible to directly sample fromit. Therefore, it is necessary to simplify the problem by breaking down the posterior
distribution into its complete conditional distributions which fully characterize the joint
posterior. Whenever possible, we use conjugate priors which allow us to directly sample
from the conditional. If this is not possible, we rely on a Metropolis algorithm. For
more details on the precise specifications, see the Appendix.
The output of the simulation procedure is a set of G draws
{(g), V(g), J(g), Z(g), C(g)}g=1:G, that forms a Markov Chain and converges top(, V , J, Z, C |Y). Estimates of the parameters, the volatility paths, and the jumpsizes are obtained by simply taking the mean of the posterior distribution. For the
jump times, one additional step is required. As each draw of J is a set of Bernoulli
random variables, i.e. taking on the value one or zero, the mean over all draws will
provide a time series of jump probabilities. To obtain estimates for the jump times,
we follow Johannes et al. (1999) and identify the jump times by choosing a threshold
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probability, i.e. we estimate the jump times J as
Jt =
1 if p(Jt = 1) >
0 if p(Jt = 0) (9)
The threshold is chosen such that the number of jumps identified corresponds to the
estimate of the jump intensity .
III. Data and Empirical Results
A. Data
In this paper, we employ daily spot price data for a variety of commodities traded
in the US. The series are chosen to reflect the relative importance of those markets,
and also to ensure the availability of as long a span of high quality data as possible.
We consider two energy commodities, namely crude oil (CL) and gasoline (HU). From
the metal markets, we include gold (GC) and silver (SI) in the study. Finally, we
consider soybeans (S) and wheat (W) as representatives of the agricultural commodities
market. All commodity data are obtained from the Commodity Research Bureau.5
Additionally, we employ S&P 500 index data (obtained from Bloomberg) to enable us
to put the results into the perspective of the existing literature on equity dynamics and
to investigate the relationship of commodity and equity markets in Subsection D. The
data period covered is more than 25 years, spanning January 1985 to March 2010, and
yielding 6290 observations per commodity and for the S&P 500.
Table 1 provides descriptive summary statistics and Figure 1 shows time series of
the six commodity markets considered. Several points are worth noting. Comparedto the S&P 500, the mean return is smaller for all commodities. The lowest average
return is observed for the wheat market, which is barely positive, and the highest for
the gold market. The standard deviation is higher than the S&P 500 for all but the
gold market. The highest levels of volatility are observed for the energy commodities,
with annualized values of 42.82 % and 44.13 %, which is more than twice the volatility
of 16.00 % and 18.92 % observed in the gold and the equity markets. The kurtosis
5
See www.crbtrader.com.
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is, however, the highest for the S&P 500, while the lowest values are observed in the
agricultural markets. However, these values are still higher than can be explained by a
simple normal distribution. The smallest and largest returns are observed for the energy
markets, both twice as large as for the S&P 500, which is to some extent surprising as
the kurtosis levels are substantially lower. The gold and soybeans markets exhibit the
smallest range between the minimum and maximum values.
TABLE 1 AND FIGURE 1 ABOUT HERE
B. Univariate Analysis
Table 2 reports the estimation results for the SV model, i.e. the stochastic volatility
model without jumps. All but the parameters affecting the mean equation and some
of the correlations are statistically significant. The estimates for the speed of mean
reversion, , are of similar size in all commodity markets and are also comparable
to those of the S&P 500. Only the estimate for the silver market is somewhat higher,
indicating a slightly less persistent volatility process. The long term variance levels, , ofall but the gold market are significantly higher than the corresponding equity estimates.
The S&P 500 annualized long term volatility is 17.58 % (=
252), which is very closeto the unconditional volatility, and also of similar size as in other studies. 6 The highest
levels of annualized long term volatility are observed for the crude oil and gasoline
markets, at 36.00 % and 37.61 % respectively. The difference across the commodity
markets is considerable. The vol-of-vol parameters, , are highly significant, which
confirms the stochastic nature of volatility in the commodity markets considered. Again,with the exception of the gold market, higher values than for the equity market are
observed, implying heavier tails and greater deviance from normality. Moreover, there
are substantial differences across markets for example, the crude oil series exhibits a
vol-of-vol which is twice as big as for soybeans.
Figures 2 plots the fitted volatility process in annualized percentages obtained from
the SV model for the six commodities. It is evident that the crude oil and gasoline
6
For example, Eraker et al. (2003) report 15.10 % for the period 1985 to 1999.
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series are the most variable overall, while gold in particular is the least. The oil-based
products showed two particular spikes in volatility during 1986 and 1990, both of which
can be tied to economic events that occurred at those times. In March 1986, Saudi
Arabia changed its policy and significantly expanded production to increase market
share; other OPEC members followed, leading to price drops in both crude oil and
gasoline. August 1990 marked the start of the first gulf war when Iraq invaded Kuwait.
Furthermore, we can observe the effect of Hurricane Katrina hitting the US gulf coast
in August 2005, leading to a short-term supply bottleneck in gasoline (although it had
little noticeable effect on the price of crude oil). Wheat was the only commodity in
the sample that showed an upward trend in volatility from the late 1990s onwards.
Moreover, it is also of interest to note that the Lehman default of September 2008increased levels of volatility and uncertainty across all markets.
TABLE 2 AND FIGURE 2 ABOUT HERE
The estimates for the correlation of the underlying and the variance process, , are
most interesting, as we can observe different signs for different markets. The crude
oil market shows a negative correlation; the gasoline and wheat markets (almost) zero
correlations; and the gold, silver, and soybean markets significant positive correlations.
For equity markets, is usually found to be negative, which is confirmed by the estimate
of -0.57. The negativity of in equity markets is usually explained by the leverage effect,
as discussed by Black (1976). However, this argument does not apply to commodity
markets. A possible explanation for the different correlations might be the fraction
of hedgers and speculators in the markets. Assuming that hedging activities reduce
volatility, whereas speculation activity increases volatility, this line of argument would
indicate that in the crude oil market, the fraction of hedgers over speculators increases
with rising prices. In the metal and the soybean markets, the opposite would hold true
that is, more hedging takes place for lower prices, while speculation dominates in
times of higher than average prices.
Table 3 provides the parameter estimates for the SVJ model, i.e. the model including
Poisson price jumps. The rates of mean reversion, , substantially decrease in all cases
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and are now lower for every commodity compared to the S&P 500, implying a higher
persistence of the volatility process. The long term volatility levels (of the diffusion
component) decrease, which is a technical effect, as part of the price variation is now
captured by the jump component. Interestingly, however, the change for the S&P 500
is minimal compared to the commodity markets. The vol-of-vol parameters, , also
decrease for the same reason in that part of the excess kurtosis is now captured by the
jump component. Again, the changes in the commodity markets are much bigger than
for the equity case, indicating that the jump component plays an even bigger role in
these markets. The correlation estimates become more pronounced compared to the
SV model, i.e. the negative values decrease, whereas the positive values increase.
TABLE 3 ABOUT HERE
The jump intensity estimates, , lie between 0.022 for wheat and 0.088 for silver,
providing evidence of significant differences between markets. In the oil, gasoline, and
wheat markets, jumps occur about six times per year. In the soybeans and gold market,
we annually observe 12 to 13 jumps on average, whereas the silver price jumps about 22
times per year. These numbers are significantly larger than those observed for equity
indices. In our sample period, we estimate the S&P 500 to jump only 1.8 times per year
(Eraker et al. (2003) estimated 1.5 times per year for their sample period). The mean
jump sizes, J, are negative for all markets, although only significant in some instances.
Taking into account the standard deviation of jump sizes, J, one can observe a huge
variation. A two-sigma interval covers almost as much probability mass on the positive
line as on the negative. The most notable cases are the two energy markets. A plus
two-sigma event in these two markets corresponds to a +14 % and +12 % (daily!) price
jump, respectively. Analogously, a minus two-sigma event corresponds to price drops
of -16 % and -14 %. The results for the other commodity markets are qualitatively
similar, but less extreme. For the equity market, on the other hand, the mean jump
size is significantly negative, and 75 % of the probability mass lies on the negative line.
Lastly, Table 4 reports the estimates for the SVCJ models, i.e. the models including
contemporaneous jumps in prices and volatility. The speed of mean-reversion, , reverts
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back to values comparable to or even higher than in the SV model, indicating a lower
persistence when including jumps. This makes intuitive sense as the inclusion of jumps
induces the need for higher mean-reversion speeds directly after jumps (as only positive
jumps are allowed). The long-term volatility levels further decrease, as part of the
variation is now captured by the volatility jump component. We still observe significant
differences across markets, ranging from 10 % for the gold market to 27.5 % for the
gasoline market. The equity market lies, at 14 %, closer to the lower border of this
interval. The vol-of-vol parameter, , slightly decreases for most markets; the silver
market is a notable exception, where increases from 0.16 to 0.21. The correlation of
the diffusion components mostly decrease in absolute terms; as prices and volatility are
always jumping contemporaneously, part of the correlation is now captured by the jumpcomponents. Interestingly, the jump intensities decrease significantly in some instances,
such as the gold, silver, and soybean markets, whereas it remains almost unchanged in
the crude oil market. Changes in the average price jump and its volatility are mostly
relatively small. This is opposed to the corresponding result for the S&P 500, where the
mean jump size almost doubles. The jump dependence parameter is not estimated
very precisely, a phenomenon which has already been observed by Eraker et al. (2003).
As the values are close to zero, only modest dependence between the jump sizes isobserved. The average volatility jump size lies between 0.51 for the gold, and 4.04 for
the gasoline market.
TABLE 4 ABOUT HERE
Although model choice is not the main focus of this paper, in order to compare the
fit of the three different models to the data, we make use of the Deviance Information
Criterion (DIC) proposed by Spiegelhalter et al. (2002) and in particular applied to
stochastic volatility models by Berg et al. (2004). The DIC can be regarded as a
generalization of the Akaike Information Criterion (AIC), and trades off adequacy and
complexity in a similar fashion.7 As for the AIC, smaller DIC values indicate a better
model fit.
7AIC equals DIC in the special case of flat priors.
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The DIC scores are reported in Table 5. First, we can observe that the fit of the
SV model is by far the worst for every market, providing evidence of the benefits of
including a price jump component. Comparing the DIC scores of the SVJ and the
SVCJ models is less conclusive as most of the scores are close to each other for a
given commodity. However, the SVJ values are always lower for all commodity markets
whereas the SVCJ obtains the lowest score for the S&P 500. As the values for the SVJ
and SVCJ models are always close together and it is therefore difficult to make a final
decision on which model to prefer, we will, in the spirit of robustness, use both in the
subsequent analysis.
TABLE 5 ABOUT HERE
C. Cross-Commodity Market Analysis
The MCMC estimation approach employed not only provides us with parameter
estimates but also with estimates for the latent state variables, i.e. volatility, jump
times, and jump sizes. We can use this information to analyze the cross-marketdependence structure of these state variables.8 In a first step, we calculate the
correlations of the differences in volatility, i.e.
Vt
Vt1, to analyze the extent
to which the volatilities in the various markets move together. Table 6 reports these
correlations, as well as return correlations to enable us to put the volatility results into
perspective.
TABLE 6 ABOUT HERE
As expected, return correlations are highest between related commodities belonging
to the same market segment. However, this correlation is still far from perfect. In
particular, the moderate degree of correlation between crude oil and gasoline of 0.49
8Alternatively, one could try to estimate a multivariate version of the model employed in order toanalyze these dependencies. This approach is, however, computationally infeasible as it would involvea 14x14 covariance matrix when considering all seven markets together.
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is interesting. The strongest correlation is observed for gold and silver with a value of
0.68. Return correlations between commodities of different segments are weak or close
to zero, never exceeding 0.11.
Looking at the volatility correlations, one can identify a similar pattern. There are,
however, some differences. For example, the volatility correlation between soybeans and
wheat is substantially smaller than the corresponding return correlation, indicating that
the prices move more closely together than the volatilities in these markets. Another
interesting point is the correlation of crude oil with the agricultural commodities.
The returns are mildly correlated, indicating some dependence across markets; the
volatilities correlations are, on the other hand, close to zero. Comparing the results
for the SVJ and the SVCJ model, it is interesting to observe that for some instances,the estimated correlation is almost identical (e.g. CL-HU or S-W), whereas in other
instances, the results changes substantially (e.g. GC-SI). This might be a consequence
of the fact that the number of jumps identified in the gold and silver markets changes
substantially between the two model variants (this can be seen from the changes in the
estimated value of ).
Next, we analyze the simultaneous jump probabilities for each pair of commodities.
To do this, we simply count the numbers of simultaneous jumps and divide it by thesample size T, i.e., we calculate the following quantity:
Tt=1 J
it J
jt
T. To put these numbers
into perspective, we also compute the probability of a simultaneous jump assuming
independence which is given by the product of the two jump intensities. Table 7 provides
the results.
TABLE 7 ABOUT HERE
We can observe that the simultaneous jump probabilities of commodities belonging
to the same segment (i.e. the pairs CL-HU, GC-SI, and S-W) are about 4-10 times
higher than one would expect under independence. For all other pairs, the probabilities
are very similar, indicating that jumps of non-related commodities are independent
of each other. Compared to the results for the dependence in volatility, there are
some interesting differences to be observed. The simultaneous jump probability
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between soybeans and wheat is halved from 0.46 % to 0.21 % when including jumps in
volatility. This is in contrast to the volatility correlation, which remains almost equal.
Furthermore, one can observe a substantial decrease of simultaneous jump probabilities
for most cases, whereas the correlations of volatilities remain rather constant, or even
increase, when including volatility jumps.
D. Commodity and Equity Markets
We now investigate the dependencies between the individual commodity markets and
the equity market, i.e. the S&P 500. Table 8 reports the return and volatility
correlations. It is well known that return correlations between commodities and equities
are quite low, which is the reason why commodities are often considered to be a
diversifier for a traditional portfolio of stocks and bonds. From Table 8, it can be
seen that the same holds true for volatility, i.e. equity and commodity volatilities are
almost uncorrelated. Consequently, commodities not only serve as a return diversifier,
but as a volatility diversifier at the same time.
TABLE 8 ABOUT HERE
Table 9 displays the daily simultaneous jump probabilities in equity and commodity
markets as well as the corresponding probabilities assuming independence. For the SVJ
model, the greatest difference is observed for the gold (0.13 % vs 0.04 %) and oil (0.10 %
vs 0.02 %) markets, indicating that jumps in these two commodies are related to jumps
in the equity market. When considering the results for the SVCJ model, one can see
that the joint jump probabilities are all very small. However, compared to the previousresults for the SVJ model, the differences from the case assuming independence become
more pronounced, indicating that the likelihood of a simultaneous jump in volatility
is higher than a simultaneous jump in prices. This is especially true for the soybeans
market.
TABLE 9 ABOUT HERE
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When a simultaneous jump occurs, it might be that the prices in both markets
jump in the same, or alternatively, in opposite directions. Moreover, the jumps might
be more or less pronounced than average jumps. To investigate this issue, we compute
the average jump sizes for each commodity market conditional on the occurrence of a
jump in the equity market and vice versa. Table 10 reports the results. Interestingly, the
average jump size in the equity market is much more negative than the unconditional
average jump size of2.59 (SVJ model). Surprisingly, this effect is strongest for thesoybean and wheat markets, i.e. a jump in the agricultural markets induces an average
jump size which is 2-3 times bigger than normal. A consideration of the average jump
sizes in the commodity markets given that the equity market jumps reveals that the
energy commodities tend to jump upward if the equity market jumps downwards. Thisis different to their unconditional average jump size which is negative. The other
commodity jumps react less to jumps in the equity market; the results for the SVCJ
model are qualitatively similar but with some differences in terms of strength of the
effects.
TABLE 10 ABOUT HERE
IV. Economic Implications
A. Risk Management: Value-at-Risk
To analyze how the differences of the commodity dynamics impact upon economic
quantities of interest, we first consider a simple risk management application and
compute the Value-at-Risk (V aR) for a position in each commodity.9 The V aR
estimates are obtained via Monte Carlo simulation. For each commodity, we simulate
100,000 price (and volatility) paths using the parameter estimates obtained in the
previous section. The time horizon for the analysis is one year; to avoid negative
volatilities due to the discrete implementations of the models, we choose a time step of
1/10th of a day (and rescale the daily parameters accordingly).
9For a recent consideration of the impact of the distribution of commodity returns and model choice
on Value-at-Risk, see Cheng and Hung (2011).
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TABLE 11 ABOUT HERE
Table 11 reports the V aRs for various confidence levels. The V aR is calculated as
the difference of the initial value of the position, which is normalized to 100, and the%-quantile of the profit and loss distribution. First, one can observe that the V aR
increases when jumps are included in the model, which is to be expected since jumps
increase the kurtosis of the return distribution. For all commodities, we can observe
an increase when including contemporaneous jumps in volatility. Interestingly, this is
not the case for the S&P 500. Here, slightly decreased V aR values are obtained. Most
likely, this result is a consequence of the fact that, in this case, the jump intensity in the
SVJ model is almost twice as high as for the SVCJ model. This demonstrates that in
terms of price risk management, the inclusion of jumps in volatility is crucial for most
commodities considered but less important for the S&P 500.
When comparing the V aR values across commodities, one can immediately notice
substantial differences. As expected, gold exhibits by far the lowest values. The energy
commodities exhibit the highest values, a result which is mainly driven by their overall
volatility levels, the higher standard deviations of jumps and the negative correlations
of returns and volatility compared to the other assets. It is interesting to see that the
two agricultural commodities exhibit quite different behavior, however, with silver lying
somewhere in the middle. On one hand, silvers strong positive correlation of returns
and volatility acts as a risk reducer; on the other hand, it exhibits a high jump intensity
and a high volatility of jump sizes which tend to increase the V aR.
B. Options Valuation
One of the advantages of the continuous time models employed in this study is that
semi closed-form options pricing formulas exist. These formulas are based on Fourier
analysis, which was initially proposed by Heston (1993). Duffie et al. (2000) provide
general solutions for all affine models and for the SVCJ model in particular.
In the following, we use the model parameters estimated in the previous section to
analyze the implications for the value of European call options. As we have estimated
all parameters from historical price data under the physical measure, we need to make
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an assumption regarding the market price of volatility and jump risk. Broadie et al.
(2007) review the most recent literature on the sign and size of these risk premia for the
S&P 500, concluding that the evidence is inconclusive; most studies report insignificant
risk premia from a statistical and/or economic point of view. Due to these findings,
and for simplicity, in the following we assume zero volatility and jump risk premia.10
To make the results comparable, we assume that each underlying currently trades
at S = 100. The current variance level is assumed to be equal to its long run average,
and the risk-free rate is assumed to be zero. Using the estimated parameter values, we
calculate the values of at-the-money (ATM; X = S), in-the-money (ITM; X = 0.95S),
and out-of-the-money (OTM; X = 1.05S) call options where X denotes the strike price.
Table 12 reports the results of this computation. The left columns report the value ofthe option, the right columns express the options value relative to the value of the
corresponding ATM option.
TABLE 12 ABOUT HERE
One can observe large differences across assets. Naturally, options written on the
underlyings with the highest volatility levels are most expensive. The value of the
gold and S&P 500 OTM options is already close to zero (although they are only 5%
out-of-the-money), whereas the values for crude oil and gasoline remain at higher levels.
The differences in jump intensities, jump amplitudes and the correlations of returns
with volatilities lead to some interesting differences in the relative values of ITM and
OTM options across commodities.11 For example, comparing the soybeans and wheat
markets, one can observe that the relative value of ITM options is higher for soybeans,
whereas the relative value of OTM options is higher for wheat. Comparing the prices
for the SV and SVJ with the SVCJ model, one can see that prices are lower for the
latter, i.e. neglecting the possibility of volatility jumps leads to increased option prices.
To further compare the consequences of the different behavior of the commodities
10The results would, of course, change if this assumption were not valid, especially if the nature ofthe risk premia differs across commodities. This is a non-trivial question which we leave for futureresearch.
11When considering implied volatilities instead of relative values, these differences would manifest
themselves in the shape of the volatility smile/skew.
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considered for options valuation, we analyze the values of the most popular exotic
contracts, i.e. barrier options. To save space, we focus our attention on at-the-money
down-and-out call and put options and vary the value of the barrier between 90 %
and 100 % of the current spot level. To take the different levels of volatility in the
markets into account, we normalize the value of the barrier option by the value of a
corresponding plain-vanilla option. Consequently, all resulting values must be between
zero and one.
FIGURE 3 ABOUT HERE
Figure 3 displays the results of these computations. One can observe that, especially
for the down-and-out put options, the value is quite distinct across commodities, even
though we have already adjusted for the absolute volatility level by normalizing with the
values of plain-vanilla options. This is due to the fact that the probability of hitting
the barrier is more sensitive to the volatility and jump level than the simple option
value. Inspecting the figures more closely, one can observe differences between model
specifications. For example, the relative value of the down-and-out put option on wheat
for a barrier of 90 % is around 0.4 for the SV and SVJ models but increases to 0.5 in
the SVCJ model, whereas the value of the corresponding silver option remains at 0.6
for all three models. From this result, one can deduce that the inclusion of jumps in
the volatility dynamics is more relevant for wheat options. A similar observation can
be made for soybeans and the energy commodities.
C. Options Hedging
Lastly, we investigate the consequences of the different stochastic behavior of
commodities and different models in terms of the delta hedging of options. Consider,
for example, the influence of the correlation parameter. Depending on the options
position, one is long or short the underlying and long or short volatility. For example,
a short put position translates into a long position in the underlying and a short
position in volatility. Thus, depending on the sign of the correlation, one is either
naturally diversified or not. By implementing a standard delta hedge, one neglects this
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relationship, which will have consequences on the hedging outcome. Similarly, the jump
frequency as well as the mean and standard deviation of the jump sizes will impact upon
the hedging success.
We set up our simulation analysis as follows. We consider a short position in an
ATM put option with a maturity of one month. The hedging horizon is one week, the
current variance level is assumed to be equal to the long run mean, and interest rates
are assumed to be zero. To investigate the potential hedging error, we assume that the
true price dynamics are given by either the SV, the SVJ, or the SVCJ model and we
calculate the corresponding options value.12 We then consider a standard delta hedging
strategy, i.e. we calculate the options delta using Blacks formula to set up the hedging
portfolio. Next, we simulate the price and volatility dynamics until and calculatethe new option value to obtain the payoff of the hedging portfolio. This procedure is
repeated 10,000 times to obtain a distribution of hedging errors.
TABLE 13 ABOUT HERE
Table 13 reports the mean, standard deviation, skewness, and kurtosis of the hedging
errors. The mean hedging error is negative in all cases and larger (in absolute terms)
for the energy commodities, which is as expected due to the higher volatility levels. The
same observation applies to the standard deviation of the errors. When considering the
skewness and kurtosis, one can observe some interesting differences across commodities
and models. Considering the SV model first, we see that all distributions are skewed
to the left, i.e. big negative outcomes are more likely than big positive outcomes.
Interestingly, the skews for the agricultural commodities and the S&P 500 are bigger
than for crude oil. Moreover, looking at the kurtosis of the hedging errors, it becomes
clear that crude oil exhibits the thinnest tails, i.e. extreme negative (and positive)
outcomes are less likely than for the other markets.
The picture completely changes when introducing jumps in the price and variance
processes. Whereas the mean and volatility of the hedging errors remain at similar
levels, or even decrease, the skewness, and, most importantly, the kurtosis, drastically
12As in the previous section, we assume a risk premium of zero.
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increase. This holds true for all markets, but with different rates. Under the SVCJ
model, the kurtosis of crude oil rockets from 2.6 to more than 38, which is now more
than twice the value observed for the S&P 500. By contrast, the rise in kurtosis for the
soybeans case is least dramatic with an increase from 4.4 to 8.6. Overall, one can make
out substantial differences across commodities and models when considering the third
and fourth moments of the hedging errors, which translate into a significant degree of
model risk when delta hedging in these markets.
V. Conclusions
This paper has examined the stochastic behavior of the prices and volatilities of a sample
of six of the most important commodity markets. Using a Bayesian Markov Chain
Monte Carlo estimation approach, three separate stochastic volatility-type models are
estimated and compared for each commodity price series, and for the S&P 500 by way
of comparison. Fairly intuitively, correlations between the returns are high fore pairs
of commodities from the same sub-class but almost zero across sub-classes. The same
pattern holds for the relationships between the simultaneous jump probabilities: within
market segments, jumps often occur in tandem whereas they are essentially independentacross segments.
We are able to demonstrate that not only are return correlations between
commodities and the stock index low, as is well documented, but the correlations
between commodity and stock volatilities are also low. This is an important result
since it shows that commodities may be an even more useful portfolio constituent
than previously thought as they can act as a volatility diversifier as well, which is
important for options portfolios or any portfolio of securities with embedded contingentclaims. The paper examines whether jumps occur simultaneously across commodities
and equities; there is some evidence that jumps in the two asset classes do occur
together, notably for gold and oil, although the results vary somewhat between models.
Finally, we test the economic impact of employing one stochastic volatility model
rather than another by estimating value at risk, by considering the prices of European
calls and barrier options written on each commodity, and by evaluating the effectiveness
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of delta hedging. Again, we find substantial differences both across commodities
and between models, indicating the heterogeneous nature of this asset class and the
importance of judiciously choosing the most appropriate specification for each individual
series.
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Appendix A: MCMC Estimation Details
This Appendix provides more detailed information on the MCMC estimation procedure.
For a general introduction to MCMC techniques see, e.g., Koop (2003). The Gibbs
sampling technique allows one to draw each parameter and state variable of the joint
posterior sequentially. For many parameters, conjugate priors can be used to derive the
conditional posterior distribution, which is a standard distribution and therefore easily
sampled from. Details are given below:
: we use a standard normal distribution as the prior.
and : we use a truncated normal distribution bounded at zero and
hyperparameters of 0 and 1 as priors for each parameter.
and : as the posteriors for and are not known, we follow the
reparametrization suggested by Jacquier et al. (1994) and use the inverse Gamma
distribution with hyperparameters 1 and 200 as priors.
: we use an exponential distribution with a hyperparameter of 0.05 as a prior,
yielding a truncated normal distribution as a posterior.
: the posterior distribution for is non-standard, and we apply an independence
Metropolis algorithm, drawing from the uniform distribution on the unit interval.
: we use a Beta distribution with hyperparameters 2 and 40.
V: we use a Gamma distribution with hyperparameters 1 and 1.
J: we use a Normal distribution with hyperparameters 0 and 100 as priors.
2J
: we use an Inverse Gamma distribution with hyperparameters 5 and 20 as priors.
: we use a standard normal distribution as a prior.
The posteriors of Jt, Zt and Ct are non-standard but provided in the appendix of
Eraker et al. (2003). The conditional posterior distribution of the variance path V is
also non-standard and given by
p(V|Y,J,Z,C, ) Tt=1
p(Vt|Vt1, Vt+1,Y,J,Z,C, ) (10)
where the posterior function for each Vt is given as
p(Vt
|Vt1, Vt+1,Y,J,Z,C, )
1
Vte
1
2(1+2+3) (11)
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with
1 =(Yt+1 () Jt+1Zt+1)2
Vt(12)
2 =
(Vt
(1
)Vt1
(Yt
()
JtZt)
JtCt)
2
2Vt1(1 2) (13)3 =
(Vt+1 (1 )Vt (Yt+1 () Jt+1Zt+1) Jt+1Ct+1)22Vt(1 2) (14)
We use a random walk Metropolis algorithm to sample from. The volatility of the error
term in this procedure is calibrated such that the acceptance probability is within the
range 30-50%. See Koop (2003) on details of calibrating the Random Walk Metropolis
algorithm. Each parameter (and state variable) is sampled 100,000 times (i.e., G =
100, 000) and we discard the first 30,000 burn-in draws as is standard practice with
MCMC estimations.
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1985 1995 2005
0
50
100
150
Crude oil
1985 1995 2005
0
1
2
3
Gasoline
1985 1995 20050
500
1000
1500
Soybeans
1985 1995 20050
200
400
600
800
1000
1200
Wheat
1985 1995 20050
200
400
600
800
1000
1200
Gold
1985 1995 20050
500
1000
1500
2000
Silver
Figure 1: Price Series
This figure shows the historical price series for the six commodities considered. All figures are in
US dollars.
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1985 1995 20050
20
40
60
80
100Crude oil
1985 1995 20050
20
40
60
80
100Gasoline
1985 1995 20050
20
40
60
80
100Soybeans
1985 1995 20050
20
40
60
80
100Wheat
1985 1995 20050
20
40
60
80
100Gold
1985 1995 20050
20
40
60
80
100Silver
Figure 2: Estimated Volatility
This figure shows the estimated volatility processes for the six considered commodities under the
SV model (annualized in percent).
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90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1
SV: DownandoutCall
90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1
SV: DownandoutPut
90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1
SVJ: DownandoutCall
90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1
SVJ: DownandoutPut
90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1SVCJ: DownandoutCall
90 92 94 96 98 1000
0.2
0.4
0.6
0.8
1SVCJ: DownandoutPut
CL
HU
GC
SI
S
W
SP
Figure 3: Down-and-out Options
This figure displays the value of down-and-out barrier options as a fraction of the corresponding
plain vanilla options value. The values on the x-axis correspond to the knock-out barrier. CL
stands for crude oil, HU for gasoline, GC for gold, SI for silver, S for soybeans, W for Wheat, and
SP for the S&P 500.
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Table1:DescriptiveStatistics
Thistablereportsdescriptivestatisticsforthecontinuouspercentagereturnsusedin
thestudy.CLstandsforcrudeoil,HUforgasoline,GCforgold,SIforsilver,
Sfor
soybeans,WforWheat,andSPfortheS&P500.ThesampleperiodisJanuary2,
1985toMarch31,2010.
CL
HU
GC
SI
S
W
SP
Mean
0.0
186
0.0
189
0.0
204
0.0
165
0.0
076
0.0
008
0.0
311
Std.
Dev.
2.6
975
2.7
803
1.0
076
1.7
621
1.5
226
2.1
192
1.1
920
Skewness
-0.6
996
-0.3
748
0.1
052
-1.1
182
-0.6
291
-0.3
507
-1.3
842
Kurtosis
18.0
155
10.0
3311
1.5
751
17.7
078
8.1
172
9.1
778
32.8
066
Min
-40.0
011
-31.4
158-
7.2
327
-23.6
716
-13.1
820
-20.4
226
-22.8
997
Max
21.2
765
23.0
0151
0.2
073
13.4
045
7.8
667
13.1
596
10.9
572
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Table2:ParameterEstimates:SV
Thistablereportsthemeansandstandarddeviations(inparentheses)oftheposterior
distributionsforeachparametero
ftheSVmodel.
CLstandsforcrudeoil,HUfor
gasoline,GCforgold,
SIforsilv
er,
Sforsoybeans,WforWheat,andSPforthe
S&P500.ThesampleperiodisJa
nuary2,1985toMarch31,2010.
C
L
0.0
575(0.0
230)
0.0
204(0.0
033)
5.1
432(0.4
032)
0.3
676(0.0
247)
H
U
0.0
333(0.0
265)
0.0
182(0.0
031)
5.6
137(0.4
452)
0.3
356(0.0
238)
G
C
0.0
161(0.0
082)
0.0
140(0.0
025)
0.9
697(0.1
243)
0.1
358(0.0
085)
SI
0.0
330(0.0
155)
0.0
291(0.0
041)
2.7
776(0.2
155)
0.3
082(0.0
185)
S
0.0
335(0.0
149)
0.0
157(0.0
029)
2.2
171(0.2
275)
0.1
917(0.0
121)
W
0.0
060(0.0
206)
0.0
197(0.0
033)
3.8
276(0.3
040)
0.2
740(0.0
202)
SP
0.0
334(0.0
101)
0.0
177(0.0
025)
1.2
261(0.1
206)
0.1
597(0.0
089)
C
L
-0.1
266(0.0
512)
-
-
H
U
-0.0
039(0.0
533)
0.0
702(0.0
218)
0.0
528(0.0
793)
G
C
0.3
826(0.0
481)
-
-
SI
0.3
138(0.0
459)
-
-
S
0.3
206(0.0
512)
0.0
460(0.0
118)
0.1
363(0.0
692)
W
0.0
223(0.0
542)
0.0
946(0.0
136)
0.4
186(0.0
425)
SP
-0.5
678(0.0
387)
-
-
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Table3:Par
ameterEstimates:SVJ
Thistablereportsthemeansandstandarddeviations(inparentheses)oftheposterior
distributionsforeachparametero
ftheSVJmodel.
CLstandsforc
rudeoil,HUfor
gasoline,GCforgold,
SIforsilv
er,
Sforsoybeans,WforWheat,andSPforthe
S&P500.ThesampleperiodisJa
nuary2,1985toMarch31,2010.
CL
0.06
54(0.0
221)
0.0
096(0.0
019)
3.9
773(0.4
431)
0.2
106(0
.0159)
-0.2
894(0.0
648)
HU
0.04
13(0.0
274)
0.0
100(0.0
027)
4.5
790(0.6
264)
0.2
182(0
.0222)
-0.0
202(0.0
744)
GC
0.01
59(0.0
081)
0.0
096(0.0
019)
0.7
863(0.1
201)
0.1
001(0
.0059)
0.4
660(0.0
591)
SI
0.05
12(0.0
163)
0.0
125(0.0
025)
2.0
745(0.2
420)
0.1
649(0
.0175)
0.5
507(0.0
653)
S
0.05
55(0.0
160)
0.0
105(0.0
021)
1.9
251(0.2
390)
0.1
439(0
.0105)
0.3
477(0.0
620)
W
0.00
91(0.0
204)
0.0
092(0.0
021)
3.2
832(0.3
668)
0.1
682(0
.0165)
-0.0
443(0.0
727)
SP
0.03
80(0.0
100)
0.0
132(0.0
024)
1.2
217(0.1
401)
0.1
381(0
.0094)
-0.6
417(0.0
369)
J
J
CL
0.02
49(0.0
047)
-1.4
218(0.7
941)
7.7
162(0.7
108)
-
-
HU
0.02
41(0.0
071)
-1.1
414(0.7
866)
6.5
266(0.8
605)
0.0
909(0
.0197)
0.0
369(0.0
568)
GC
0.05
31(0.0
117)
-0.0
492(0.1
471)
1.8
276(0.1
794)
-
-
SI
0.08
79(0.0
163)
-0.4
627(0.1
901)
2.7
129(0.2
476)
-
-
S
0.04
71(0.0
107)
-0.8
150(0.2
658)
2.2
938(0.2
473)
0.0
360(0
.0177)
0.1
464(0.1
169)
W
0.02
25(0.0
064)
-0.7
974(0.6
487)
4.7
285(0.5
420)
0.0
944(0
.0116)
0.4
215(0.0
363)
SP
0.00
71(0.0
023)
-2.5
866(0.8
356)
2.8
435(0.5
423)
-
-
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Table4:ParameterEstimates:SVCJ
Thistablereportsthemeansandstandarddeviations(inparentheses)oftheposterior
distributionsforeachparameteroftheSVCJmodel.
CLstandsforcrudeoil,HUfor
gasoline,GCforgold,
SIforsilv
er,
Sforsoybeans,WforWheat,andSPforthe
S&P500.ThesampleperiodisJa
nuary2,1985toMarch31,2010.
CL
0.0
622(0.02
27)
0.0
173(0.0
041)
2.4
719(0.4
271)
0.2
085(0.0
219)
-0.1
623(0.0
623)
0.0
221(0.0043)
HU
0.0
441(0.02
75)
0.0
200(0.0
041)
3.0
628(0.6
079)
0.2
151(0.0
343)
-0.0
348(0.0
721)
0.0
182(0.0063)
GC
0.0
124(0.00
82)
0.0
185(0.0
032)
0.4
098(0.0
700)
0.0
961(0.0
080)
0.3
391(0.0
553)
0.0
172(0.0047)
SI
0.0
283(0.01
59)
0.0
374(0.0
076)
1.2
995(0.2
321)
0.2
110(0.0
213)
0.2
583(0.0
594)
0.0
305(0.0098)
S
0.0
338(0.01
57)
0.0
262(0.0
046)
1.0
821(0.1
656)
0.1
447(0.0
158)
0.2
960(0.0
657)
0.0
174(0.0056)
W
0.0
164(0.02
07)
0.0
240(0.0
045)
1.7
065(0.2
534)
0.1
373(0.0
249)
-0.0
146(0.0
791)
0.0
177(0.0038)
SP
0.0
421(0.01
00)
0.0
225(0.0
041)
0.7
874(0.0
787)
0.1
264(0.0
098)
-0.5
831(0.0
395)
0.0
041(0.0012)
J
J
V
CL
-1.5
291(1.19
32)
7.9
167(0.7
397)
-0.0
170(0.4
167)
2.3
176(0.9
789)
-
-
HU
-2.0
728(1.70
65)
6.7
416(0.9
186)
0.1
425(0.2
745)
4.0
436(1.3
707)
0.0
851(0.0
204)
0.0
277(0.0684)
GC
-0.3
294(0.51
97)
2.7
819(0.3
301)
0.5
546(0.6
923)
0.5
108(0.1
470)
-
-
SI
-0.6
968(0.63
75)
3.9
967(0.4
951)
-0.0
099(0.3
746)
1.5
656(0.6
734)
-
-
S
-0.7
981(0.84
80)
2.8
019(0.3
667)
-0.0
612(0.2
793)
1.7
129(0.6
099)
0.0
456(0.0
124)
0.1
455(0.0715)
W
-1.6
252(0.97
08)
4.9
274(0.5
244)
0.0
478(0.2
229)
3.0
470(0.7
534)
0.0
902(0.0
109)
0.4
144(0.0393)
SP
-4.4
478(0.94
11)
2.2
486(0.5
251)
0.0
519(0.2
012)
2.7
594(0.7
914)
-
-
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Table 5: Model Comparison
This table reports the DIC scores for the model specifications examined: SV, SVJ,
and SVCJ. CL stands for crude oil, HU for gasoline, GC for gold, SI for silver, S for
soybeans, W for Wheat, and SP for the S&P 500. The sample period is January 2,
1985 to March 31, 2010.
SV SVJ SVCJ
CL 27,934 26,756 26,779
HU 29,169 28,444 28,534
GC 16,096 14,634 14,871
SI 23,683 21,458 21,776
S 21,569 20,657 20,992
W 25,395 24,609 24,725
SP 17,437 16,969 16,819
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Table 6: Correlations of Returns and VolatilitiesThis table reports return and volatility correlations. Panel A reports the correlations
of returns. Panel B displays the correlations of changes in volatility calculated from
the estimated volatility process of the SVJ model. Panel C displays the correlations
of changes in volatility calculated from the estimated volatility process of the SVCJ
model. CL stands for crude oil, HU for gasoline, GC for gold, SI for silver, S for
soybeans, W for Wheat, and SP for the S&P 500. The sample period is January 2,
1985 to March 31, 2010.
Panel A: Returns
CL HU GC SI S W
CL 1.0000HU 0.4894 1.0000
GC 0.0539 0.0104 1.0000
SI 0.0404 0.0060 0.6849 1.0000
S 0.1094 0.1098 0.0006 -0.0106 1.0000
W 0.0969 0.0755 0.0030 -0.0157 0.3941 1.0000
Panel B: SVJ Volatilities
CL HU GC SI S W
CL 1.0000HU 0.3084 1.0000
GC -0.0084 0.0410 1.0000
SI -0.0257 0.0090 0.6375 1.0000
S -0.0574 -0.0157 -0.0028 0.0124 1.0000
W 0.0368 0.1163 0.0340 0.0440 0.1146 1.00001
Panel C: SVCJ Volatilities
CL HU GC SI S W
CL 1.0000
HU 0.3051 1.0000
GC 0.0392 0.0244 1.0000
SI 0.0118 0.0274 0.4175 1.0000
S -0.0058 0.0041 0.0067 0.0282 1.0000
W -0.0003 0.0265 0.0081 0.0314 0.1045 1.0000
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Table 7: Simultaneous Jump Probabilities
This table reports the simultaneous jump probabilities of returns in the lower
triangular. The upper triangular reports probabilities assuming independence.
CL stands for crude oil, HU for gasoline, GC for gold, SI for silver, S for
soybeans, W for Wheat, and SP for the S&P 500. The sample period is January
2, 1985 to March 31, 2010.
Panel A: SVJ
CL HU GC SI S W
CL - 0.06% 0.12% 0.21% 0.12% 0.06%
HU 0.40% - 0.12% 0.21% 0.11% 0.05%
GC 0.17% 0.14% - 0.45% 0.24% 0.11%
SI 0.17% 0.25% 2.23% - 0.43% 0.20%
S 0.08% 0.13% 0.32% 0.62% - 0.11%
W 0.08% 0.05% 0.22% 0.29% 0.46% -
Panel B: SVCJ
CL HU GC SI S W
CL - 0.04% 0.04% 0.08% 0.03% 0.04%
HU 0.35% - 0.03% 0.06% 0.03% 0.03%
GC 0.08% 0.05% - 0.06% 0.03% 0.03%
SI 0.10% 0.06% 0.65% - 0.06% 0.06%
S 0.03% 0.06% 0.06% 0.08% - 0.03%
W 0.03% 0.05% 0.02% 0.06% 0.21% -
38
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Table 8: Correlation of Equity and CommoditiesThis table reports the correlation of equity (S&P 500) and commodity market
returns and the correlations of volatility. The first row presents the correlations
between the returns of the S&P and the commodity; the second and third rows
present the correlations of the first diffferences in volatility between the S&P
and the commodity for the SVJ and SVCJ models respectively. CL stands for
crude oil, HU for gasoline, GC for gold, SI for silver, S for soybeans, W for
Wheat, and SP for the S&P 500. The sample period is January 2, 1985 to
March 31, 2010.
CL HU GC SI S WReturns 0.0298 0.0459 -0.0418 -0.0630 0.0792 0.0700
Volatility: SVJ 0.0398 0.0433 0.0470 0.0429 -0.0151 0.0418
Volatility: SVCJ 0.0289 0.0592 0.0403 0.0369 0.0299 0.0358
39
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Table 9: Simultaneous Jump Probabilities for Equity and CommoditiesIn parentheses we report the simultaneous jump probabilities assuming indepen-
dence. Upper part: SVJ, lower part SVCJ. CL stands for crude oil, HU for
gasoline, GC for gold, SI for silver, S for soybeans, W for Wheat, and SP for
the S&P 500. The sample period is January 2, 1985 to March 31, 2010.
SVJ CL HU GC SI S W
SP 0.10% 0.02% 0.13% 0.10% 0.05% 0.03%
(0.02%) (0.02%) (0.04%) (0.07%) (0.04%) (0.02%)
SVCJ CL HU GC SI S W
SP 0.03% 0.03% 0.05% 0.06% 0.06% 0.03%
(0.01%) (0.01%) (0.01%) (0.02%) (0.01%) (0.01%)
40
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Table 10: Conditional Jump Sizes in Equity and Commodity Markets
xx stands for the respective commodity. CL stands for crude oil, HU for
gasoline, GC for gold, SI for silver, S for soybeans, W for Wheat, and SP for
the S&P 500. The sample period is January 2, 1985 to March 31, 2010.
SVJ CL HU GC SI S W
SP/xx -3.60 -4.43 -4.95 -5.70 -6.51 -8.32
xx/SP 3.87 14.84 0.34 -1.11 -2.03 -2.82
SVCJ CL HU GC SI S W
SP/xx -4.39 -5.41 -5.45 -4.87 -5.05 -5.06
xx/SP 9.11 7.01 0.02 -0.54 -2.47 -5.07
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Table 11: Value-at-Risk
This table reports the Value-at-Risk (V aR) estimates for a horizon of one year
and a confidence level of based on a Monte Carlo simulation with 100,000replications. CL stands for crude oil, HU for gasoline, GC for gold, SI for
silver, S for soybeans, W for Wheat, and SP for the S&P 500.
Panel A: SV
V aR10% V aR5% V aR1% V aR0.1%
CL 26.78 36.96 53.53 69.31
HU 32.03 41.27 56.60 70.09
GC 13.53 17.66 25.28 33.51
SI 21.45 27.89 39.02 49.91
S 18.44 24.41 35.03 46.37
W 30.99 38.72 51.61 64.33
SP 13.60 20.71 34.19 48.80
Panel B: SVJ
V aR10% V aR5% V aR1% V aR0.1%
CL 32.79 42.77 58.42 73.08
HU 35.23 43.75 58.26 71.01
GC 13.98 18.00 25.12 32.75SI 25.50 31.32 41.16 50.64
S 21.68 27.45 37.41 47.59
W 33.01 40.27 53.11 64.59
SP 17.03 24.27 37.83 51.66
Panel C: SVCJ
V aR10% V aR5% V aR1% V aR0.1%
CL 35.03 44.50 60.22 73.94
HU 39.52 48.84 63.71 76.18GC 15.44 19.98 28.54 37.98
SI 28.12 34.97 46.26 57.14
S 23.42 30.07 42.02 54.34
W 35.62 43.37 56.73 69.09
SP 15.28 22.43 36.10 50.85
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Table 12: Option Prices
This table reports the option prices for a horizon of one month. The price of
the underlying is normalized to 100. The strike price is set to 95 (ITM), 100
(ATM), and 105 (OTM). The left part (Absolute) reports the options price,
the right part (Relative) reports the options price relative to the ATM option
price. CL stands for crude oil, HU for gasoline, GC for gold, SI for silver, S
for soybeans, W for Wheat, and SP for the S&P 500.
Panel A: SV
Absolute Relative
ITM ATM OTM ITM ATM OTM
CL 7.01 4.07 2.11 172% 100% 52%
HU 7.14 4.26 2.32 168% 100% 55%
GC 5.23 1.77 0.38 295% 100% 22%SI 6.01 3.00 1.31 200% 100% 44%
S 5.80 2.69 1.03 215% 100% 38%
W 6.51 3.53 1.67 185% 100% 47%
SP 5.49 1.99 0.35 276% 100% 17%
Panel B: SVJ
Absolute Relative
ITM ATM OTM ITM ATM OTM
CL 7.15 4.20 2.21 170% 100% 53%
HU 7.17 4.28 2.33 167% 100% 54%
GC 5.24 1.77 0.37 295% 100% 21%
SI 6.02 2.98 1.25 202% 100% 42%
S 5.81 2.68 0.99 217% 100% 37%
W 6.51 3.51 1.64 185% 100% 47%
SP 5.54 2.06 0.38 269% 100% 18%
Panel C: SVCJ
Absolute Relative
ITM ATM OTM ITM ATM OTMCL 6.65 3.56 1.66 187% 100% 47%
HU 6.79 3.80 1.89 179% 100% 50%
GC 5.12 1.38 0.20 372% 100% 14%
SI 5.82 2.60 0.94 223% 100% 36%
S 5.53 2.19 0.63 253% 100% 29%
W 6.07 2.87 1.09 211% 100% 38%
SP 5.38 1.71 0.19 314% 100% 11%
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Table
13:HedgingError
Thistablereportsthemean,standarddeviation(std),skewness(skew),and
kurtosis(kurt)ofthehedgingerrorwhendelta-hedginganATM
optionusing
Blacksformula.Theoptionhasamaturityofonemonth,an
dthehedging
horizonisoneweek.CLstandsforcrudeoil,HUforgasoline,GCforgold,
SI
forsilver,
Sforsoybeans,Wfo
rWheat,andSPfortheS&P500
.
PanelA:SV
CL
HU
GC
SI
S
W
SP
M
ean
-0.4
441
-0.4
864
-0
.1964
-0.3
395
-0.3
018
-0
.3955
-0.2
319
Std
0.8
305
0.8
618
0.3
510
0.6
464
0.5
272
0.7
072
0.3
781
S
kew
-1.0
725
-1.3
783
-1
.4260
-1.3
480
-1.4
964
-1
.4902
-1.4
691
Kurt
2.5
727
4.1
403
4.2
224
4.1
102
4.4
456
4.6
943
4.1
992
PanelB:SVJ
CL
HU
GC
SI
S
W
SP
M
ean
-0.4
896
-0.4
745
-0
.1967
-0.3
341
-0.3
028
-0
.3990
-0.2
309
Std
0.8
772
0.7
944
0.3
237
0.5
446
0.4
823
0.6
497
0.3
659
S
kew
-3.4
585
-2.5
196
-2
.1526
-2.4
347
-1.8
658
-2
.5869
-2.0
724
Kurt
22.9
281
12.2
694
8.5
793
10.5
993
5.7
959
13
.1202
8.3
335
PanelC:SVCJ
CL
HU
GC
SI
S
W
SP
M
ean
-0.4
037
-0.4
062
-0
.1444
-0.2
634
-0.2
192
-0
.2997
-0.1
873
Std
0.8
425
0.7
275
0.2
879
0.4
967
0.3
785
0.5
319
0.3
034
S
kew
-4.7
612
-3.4
210
-3
.5175
-2.9
353
-2.1
156
-3
.7515
-2.4
437
Kurt
38.8
046
24.8
022
23
.9426
16.6
159
8.5
904
25
.8874
15.2
599