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

15


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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).

16


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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

17


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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.

18


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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

19


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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.

20


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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

21


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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.

22


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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)

23


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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.

24


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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).

30


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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)

-

-

33


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Table3:Par

ameterEstimates:SVJ


distributionsforeachparametero

ftheSVJmodel.

CLstandsforc

rudeoil,HUfor

gasoline,GCforgold,

SIforsilv

er,




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)

-

-

34


37/46

Table4:ParameterEstimates:SVCJ


distributionsforeachparameteroftheSVCJmodel.

CLstandsforcrudeoil,HUfor

gasoline,GCforgold,

SIforsilv

er,




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)

-

-

35


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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

36


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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

37


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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


42/46

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


43/46

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

41


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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


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


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

42


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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%

43


46/46

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

the dynamics of commodity prices

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