1

IDENTIFYING SYSTEMATIC JUMP RISK IN FUTURES

Sijesh C. Aravindhakshana

B. Wade Brorsenb*

a Saudi Aramco, Corporate Planning, Dhahran, Eastern Province, Kingdom of Saudi

Arabia

b Department of Agricultural Economics, Oklahoma State University, Stillwater, OK,

USA

Correspondence to: B. Wade Brorsen Department of Agricultural Economics Oklahoma State University Stillwater, OK 74078-6026 Tel.: (405) 744 6835 Fax: (405) 744 8210 Email: wade.brorsen@okstate.edu JEL Classification: D81, G13, G32, Q13

Running Head: Identifying Systematic Jump Risk

A later version of this paper was published as

Aravindhakshan, S., and B.W. Brorsen. 2014. “Identifying Jumps and Systematic Risks in Commodity Futures Markets.” Review of Futures Markets. 21(3):Article 3.

Helpful comments from Francis Epplin, Tim Krehbiel, and the NCCC-134 audience are gratefully acknowledged. This research topic was suggested by Antonio Câmara. The research was partially supported by funding from the Oklahoma Agricultural Experiment Station and the National Institute of Food and Agriculture Hatch Project OKL02766.

2

Identifying Systematic Jump Risk in Futures

A variety of multivariate jump-diffusion models have been suggested as models of asset prices. This article extends the literature on (joint) mixed jump-diffusion processes in futures markets by using the CRB index futures to represent systematic risk in commodity prices. We derive (joint) mixed bivariate normal distributions and likelihood functions for estimating the parameters of jump-diffusion processes. Likelihood ratio tests are used to select among nested models. The empirical results show the presence of jumps and significant systematic risk in commodity returns. Jumps were frequent with a jump per one to eight days. The approach offers an alternative method of computing value-at-risk measures while allowing for nonnormal distributions.

Keywords: diffusion-jump; futures markets; risk; tail dependence

Previous studies show that futures returns have occasional large movements that result in

asymmetric and leptokurtic distributions (Hudson, Leuthold and Sarassoro 1987; Hall,

Brorsen, and Irwin 1989; Stiegert and Brorsen 1996; Koekebakker and Gudbrand 2004).

Discontinuous jumps in asset prices and time-varying volatility models are the two main

approaches used to model extraordinary discrete price movements (Eraker 2004). We

focus on discontinuous jump processes, which are captured with a jump diffusion (JD)

process. The JD process combines a continuous diffusion process and discrete jumps that

are typically assumed to follow a Poisson process.

In this article, we adapt the consumption-based capital asset pricing model to

estimate a single-index model (see Sharpe 1963; Turvey et al. 1988) that is appropriate to

model the risk of a portfolio of futures positions. We conduct tests to select the most

suitable jump diffusion (JD) process to model commodity futures prices and to estimate

the systematic risk associated with the continuous and discontinuous components. We

consider alternative stochastic processes that are nested in Câmara’s (2009) two counters

of jumps model and estimate the parameters of the distributions. The continuous

movements are modeled using a normal distribution and the discrete movements are

3

modeled using a Poisson jump process. This joint JD process includes correlated and

uncorrelated jumps and models the interaction between futures returns for individual

commodities and commodity market returns as measured by CRB Index futures returns.

In Merton’s (1976) JD process, price changes occur at discrete points in time. The

two weaknesses of the JD process are: (1) the model has only a single counter1 of jumps,

and (2) the jump component represents only non-systematic risk. Empirical studies show

high correlation between individual stock price volatility and market volatility (Jarrow

and Rosenfeld 1984; Jorion 1988). If jumps in an individual asset are correlated with

jumps in the overall market, then contrary to the maintained hypothesis of Merton’s

model, jump risk is systematic and cannot be diversified. Amin and Ng (1993) derived an

option pricing formula to account for stock return volatility that is both systematic and

stochastic. In their model, the number of jumps in the consumption and asset price

process are identical and are allowed to be correlated.

Câmara’s (2009) theoretical “two counters of jumps” model is a joint JD process

of aggregate consumption and stock price. In his model, the jumps are separated into

upside (bubbles) and downside (crashes) jumps with different intensity and distributional

characteristics. Theoretically, Câmara and Heston (2008) generalize Merton (1976) and

Amin and Ng (1993) by using the two counters of jumps model in a univariate model. In

his extended model, Câmara (2009) included correlated jump parameters in stock price

and aggregate consumption. Câmara’s model, however, has a potential estimation

problem as it includes sixteen jump parameters along with other parameters that represent

continuous price movements. As per Kou (2002), having so many parameters in the

1 Mathematically a counter process defines the number of arrivals that have occurred in the interval (0,t).

4

model makes calibration difficult. In addition, if the jump magnitudes are small, the

separation of jumps from continuous co-movements and estimation of parameters

becomes less precise (Todorov and Bollerslev 2010). To circumvent this empirical

problem, the current paper includes four JD processes with fewer parameters and that are

nested in Câmara’s (2009) two counters of jumps model. The criteria2 that are used in the

selection of models are: (1) the model should be able to explain the asymmetric and

leptokurtic nature of returns, (2) the model should have an economic interpretation and

practical implications, and (3) the likelihood function of the distribution should be

mathematically tractable to compute the parameter estimates.

In Merton’s (1976) model, jumps are firm specific and are not correlated with

stocks in general (i.e., with the market). The 1987 stock market crash makes clear that

extreme events can influence all asset prices and market events. Similarly, commodities

and commodity futures prices are systematically related to macroeconomic measures, and

the risk associated with jumps cannot be diversified (Hilliard and Reis 1999). The nature

of the risk associated with jumps is not explicitly stated in Câmara’s (2009) model.

Restricting the two counters of jumps model to having no jumps in consumption and a

single counter of jumps in stock price results in the JD economy of Merton (1976) where

jumps are non-systematic. The JD process of Amin and Ng (1993) with systematic jumps

can be obtained by assuming a single counter of jumps in consumption and stock price. In

the capital asset pricing model (CAPM), only systematic risk is rewarded. The current

paper includes a joint JD process that also represents the extended one factor model of

Todorov and Bollerslev (2010) with returns associated with continuous and discontinuous

2 For detailed description of criteria, see Kou (2002).

5

price moves. The model also allows testing whether the betas associated with these price

moves are the same.

Agricultural commodities have inelastic demand and their supply is highly

dependent upon weather. Supply of agricultural inputs, trade policies, and

macroeconomic factors also influence the price. Some discrete incidence of large price

changes is confined to a single commodity. For example, the news of a single cow testing

positive for Bovine Spongiform Encephalopathy (BSE) resulted in an unprecedented

movement in the futures price of live cattle and was mostly confined to the cattle

industry, not the entire commodity market, and this price move is an example of a non-

systematic jump. Because grains and soybean meal are widely used as livestock feed,

shocks in wheat and soybean markets affect live cattle prices. But these effects may be of

lower magnitude with a gradual increase. However, weather events or incidence of

diseases have little influence on industrial commodities like copper. On the other hand,

influences of macroeconomic variables, like exchange rate fluctuations, affect prices of

all commodities and are systematic even though the effect varies among the commodities.

As Todorov and Bollerslev (2010) argue, categorizing systematic and non-

systematic jumps is important for accurate estimation of derivative prices. Our article

extends Amin and Ng (1993) by adding an uncorrelated jump to individual futures prices.

We extend the futures literature by specifying a joint JD process with a single counter of

jumps in aggregate market prices and two counters of jumps in futures prices. For the

futures JD process, the jumps are defined with separate Poisson processes to represent

systematic and non-systematic jumps. The model includes as a special case the Todorov

and Bollerslev (2010) theoretical framework that extends the generic CAPM model to

6

have two separate betas to represent the systematic risk attributable to continuous and

discontinuous price moves.

JD processes that model systematic risk have several practical applications in

portfolio and credit risk management. As the jumps are correlated with a large number of

assets, the presence of systematic risk reduces the gains from diversification and

substantially increases the probability to lose while holding highly levered positions (Das

and Uppal 2004). According to Duffie and Pan (2001), systematic movements also

influence credit risk, since all credit instruments exhibit correlated default risk. Duffie

and Pan (2001) use a jump-conditional value at risk (VaR) weighted by the probability of

a given number of jumps for the analytical approximation of VaR. Jumps across the

assets are systematic even in the commodity markets, which makes it extremely difficult

for traders to hedge the risk during large correlated price movements. Pan (2002) finds

jump-risk premia in S&P 500 options.

Maximum likelihood is used to estimate the parameters of the diffusion process

using wheat, soybean, live cattle, copper, and the Commodity Research Bureau (CRB)

index of daily futures prices. The Amin and Ng (1993) and Câmara (2009) models are

extensions of the consumption-based representative agent framework. In empirical

analysis, market prices are used instead of consumption growth (Amin and Ng, 2003).

The CRB index of futures prices is used to represent the commodity market. The

empirical results show the presence of downside jumps and systematic jump risk in

commodity futures prices. Amin and Ng’s (1993) model fits the data better than other JD

processes considered here. The differences in beta estimates of continuous and

discontinuous price moves are not statistically significant for agricultural commodities

7

and fail to support Todorov and Bollerslev’s (2010) extended CAPM framework. On the

other hand, the results for copper support Todorov and Bollerslev’s model with separate

betas.

The models developed are potentially useful in calculating value at risk. Currently

firms are left with two primary choices in calculating value at risk. The most common

method is to assume lognormality in spite of considerable evidence against lognormality.

The second alternative is using Gaussian or multivariate-t copulas. Gaussian copulas can

handle non-normal distributions, but they still impose linear correlations. Our approach

would allow higher correlations for the extreme jumps. A firm using this approach would

want to use an index that matches the firm’s positions rather than the CRB Index that is

used here.

THEORETICAL MODEL

In this section JD processes are formally defined. We begin with a univariate JD

with one counter of jumps, expand the model to two counters of jumps, and then define a

bivariate model that can capture correlation in the jumps of the futures prices and the

CRB index. All JD processes included are restricted models of Câmara’s (2009) two

counters of jump process. We now outline the stochastic process of asset price followed

by the JD process with a single counter and then with two counters of jumps. Let the

asset price ) be a stochastic process with no discrete jumps so it can be represented as

1 exp ln2

,

where is the current futures price, ~ 0, is standard Brownian motion, is

the instantaneous expected price change, and is the instantaneous variance. The asset

8

price is assumed to be non-negative and the period return (logarithmic price relative)

ln / is normally distributed as ~ , where 0.5 is the

mean (drift) and is the variance.

Univariate Mixed JD Process with Single Counter of Jumps

The number of extraordinary price changes follows a Poisson counting process

with mean arrival rate (intensity) and jump size , . The magnitude of jumps can

be either positive (upside jumps) or negative (downside jumps). The JD process that

includes both continuous and discontinuous changes in prices is

2 exp ln ∑ , .

In equation (2), the Brownian motion is a continuous process, and the Poisson process is

discontinuous. , , and , are assumed to be independent. The model reduces

to geometric Brownian motion when there are no jumps. As per Jorion (1988), the

probability density function of a mixed univariate normal distribution and a single

counter of jumps is

3 !

1

2exp

2.

Univariate Mixed JD Process with Two Counters of Jumps

Câmara (2009) postulates a bivariate model with two separate Poisson distributed

events to represent upside jumps and downside jumps. A univariate version of this richer,

potentially more realistic and less restricted JD process with two counters of jumps is

9

4 exp ln ∑ , ∑ , ,

where , and , represent the upside jump and downside jumps,

respectively, is the Poisson counter process for upside jumps with intensity , and

is the Poisson counter process of downside jumps with intensity . In the price

(logarithmic price relative) series, both jumps are normally distributed and can have

different distributions each with different mean and variance. These price jumps can be

represented as , ~ , and , ~ , where, and are the

means and and are the variances of upside and downside jumps respectively.

For empirical estimation, the mean of upside jumps is restricted to be positive and

the mean of downside jumps is restricted to be negative. In this model, the jump

magnitudes are correlated if cov , , , 0when , and all other random

variables are independent. The intuition behind this assumption is that the jumps with the

same indexes could be due to the same cause. The mixed bivariate normal pdf of returns

with two counters of jumps is

5 !

∞

!1

2 2min ,

∞

exp2 2min ,

,

. where represents the covariance of upside and downside jumps in returns. The

covariance part in this mixed distribution merits more explanation. The continuous

10

diffusion component (Brownian motion) is assumed independent of all jumps3. The

variance of the jump component of the mixed jump diffusion process is

5

, , ,

, 2 , , , 2min , .

As per the additive rule of covariance cov , cov , cov , and by

induction it can be stated as

cov ∑ , , ∑ , ∑ ∑ cov , , , .

The minimum operator is to ensure equal number of jump magnitudes to

carry out the estimation of covariance4. A JD process with separate upside and downside

jumps is not a new concept in the financial literature. Kou (2002) proposed a double

exponential jump diffusion (DEJD) model. In the DEJD model, the jumps are generated

by a single Poisson process, and the upside and downside jump magnitudes are drawn

from two independent exponential distributions. Later Ramezani and Zeng (2007) posited

a Pareto-Beta jump-diffusion (PBJD) model in which the jumps are generated by two

independent Poisson processes, and the jump magnitudes are drawn from a Pareto

distribution for upside jumps and a Beta distribution for downside jumps. In DEJD and

3 and , and , and , , and , , and , , and

, are independent. In general the continuous components , the discrete components and , and the jump magnitudes are independent. The jumps , and , are allowed to be correlated. These assumptions are mentioned in subsequent models. 4 The sample covariance of two variables X and Y each with sample size n is cov , ∑ ̅ / .

11

PBJD, the distributions are univariate, and the relationship with the market is not defined

as in Câmara’s (2009) model. But, unlike Câmara (2009), they are able to restrict all

positive jumps to be positive and all negative jumps to be negative. Since Câmara’s

model is based on normality, it only restricts the mean of upside jumps to be positive and

the mean of downside jumps to be negative.

Multivariate Mixed JD Process with Systematic Jumps

The JD processes that are described in the above subsections are univariate

processes of asset prices. In these models, the influence of aggregate consumption on

asset price is not defined and these models implicitly assume non-systematic jumps.

Câmara’s (2009) two counters of jumps model as a joint JD process of asset price and

aggregate consumption is

6 exp ln2 , ,

7 exp ln2 , , ,

where is the Poisson counter process attributed to the incidence of upside jumps

with intensity , is the Poisson counter process attributed to the incidence of

downside jumps with intensity parameter , is the current level of consumption,

is the consumption Brownian motion, is the instantaneous expected growth rate of

consumption without any jumps, is the variance of consumption without any jumps,

and , and , are the upside and downside jump magnitudes in aggregate

consumption. In this model, the aggregate consumption Brownian motion and asset price

12

Brownian motion are correlated. The model also allows jump components to be

correlated5.

Restricting Câmara’s (2009) model to a single counter of jumps achieves the JD

economy of Amin and Ng (1993). In their model, the jumps in the asset price are

correlated with the jumps in aggregate consumption. The arrival of information in both

series is defined by a single Poisson counting process with intensity and the

number of jumps in both series is the same. The joint mixed JD process with a single

counter of jumps is

8 exp ln2 ,

9 exp ln2 , .

Aggregate consumption growth is not directly observable and is difficult to

estimate. To bypass this, an index of market prices is used instead of consumption growth

(Amin and Ng 1993). The CRB index of futures prices is used to represent the market.

The joint mixed density function, defined as an extension of the univariate distribution is

(10) , ,

!

1

2 ∗ 1

exp 2 1

(10a)

5 For more details of (auto)correlations between the jumps, see Câmara (2009).

13

2

∗

(10b) ,

where and are the instantaneous mean and variance of market returns without any

jumps, is the mean and is the variance of jumps in the market returns, / 1

is the square of Mahalanobis distance6 of the multivariate distribution, is the correlation

coefficient between the two returns, is the correlation between returns conditional on

zero jumps, and is the correlation between the jumps in both returns.

Multivariate Mixed JD Process with Systematic and Non-systematic Jumps

The generic one factor model that generalizes the popular CAPM model can be

stated as , where i={1,…,N}, is the returns on the ith asset, is the

systematic risk factor, and is the idiosyncratic risk that is uncorrelated with .7

Todorov and Bollerslev (2010) separate the systematic risk factor associated with

continuous and discontinuous price moves. The extended one factor model can

be stated as , i={1,…,N }, where and represent the

systematic risk attributable to continuous and discontinuous price moves, respectively.

The current paper introduces a new diffusion process that separates the systematic risk

6 If has a multivariate normal distribution with covariance matrix and mean vector , the density is given by

| | / ′| | / . The exponent term

′| | is the squared generalized distance from to otherwise known as Mahalanobis distance (Rencher, 2002). 7 For more details see Todorov and Bollerslev (2009).

14

and non-systematic risk associated with jumps in futures prices. In this model, the

uncorrelated jump represents only non-systematic discontinuous price moves whereas in

Todorov and Bollerslev’s (2010) model, represents the idiosyncratic risk of both

continuous and discontinuous price moves. The joint diffusion process that explains

systematic jump risk is

(11) exp ln2 , ,

12 exp ln2 , ,

where , represents the jump in asset price that is correlated with the market jumps

, , and the Poisson counting process that is attributed to the incidence of correlated

jumps is with intensity . The non-systematic jump magnitude in the asset prices

( , is modeled as the second Poisson counting process in the futures prices with

an intensity parameter . The correlated jump and uncorrelated jump are analogous to

and respectively. Hence, in equation (11), systematic and non-systematic asset price

jumps are uncorrelated cov ∑ , , ∑ , 0 . The corresponding joint

mixed bivariate normal distribution is

13 , ,

! !

1

2 12 1

15

(13a)

2

(13b) ,

where and are means, and are the variance of correlated and uncorrelated

jumps. The betas associated with continuous price moves and discontinuous price moves

are ⁄ and ⁄ respectively. The likelihood expressions

for the distribution functions are given in the Appendix.

DATA AND PROCEDURE

Summary statistics are estimated and normality tests are performed for all returns series.

The parameters of four increasingly general JD processes are estimated using numerical

maximization of likelihood functions. Univariate mixed normal distributions are

estimated separately for each returns series with a single counter of jumps and with two

counters of jumps. Likelihood ratio (LR) tests are used to select among nested models.

The parameters of a joint mixed bivariate distribution with correlated single counter of

jumps in both returns, and the model with separate systematic and non-systematic jumps

in commodity futures prices are also estimated. Proc NLMIXED in SAS is used to

estimate the parameters (SAS 2009). To maximize the likelihood function, the infinite

sum has to be truncated (Jorion 1998). In the current paper, the infinite sum is truncated

at ten. An extensive Monte Carlo simulation is performed to verify accuracy and

reliability of statistical programs and procedures. Data are simulated (100,000

16

observations) from the JD models with known parameter values. The maximum

likelihood optimization using the simulated data yielded estimates nearly identical to the

true values.

The Chicago Board of Trade (CBOT) daily settlement futures prices are used for

a period of 18 years (June 1990 to December 2008). The wheat futures prices are for hard

red winter wheat from the Kansas City Board of Trade (2010). The number of futures

maturity months for wheat and copper is five. Live cattle have six maturity months and

soybeans have seven. In calculating the daily returns, differencing is performed before

splicing to avoid creating outliers at the rollover.

A synchronous data set of the Thomson Reuters/Jefferies CRB index is

constructed for the same time period. The CRB index is composed of four groups of

nineteen components that include petroleum products, metals, and agricultural

commodities. In the current study, CRB index futures represent the commodity market

(aggregate consumption) and the logarithmic price relatives as market returns (aggregate

consumption growth). Actively traded futures are selected based on the volume traded

and rollover dates are different for each commodity. The daily closing price of index

futures is used to estimate trading day to trading day changes in the value of the

underlying assets. All return series include 4,513 observations of daily prices to provide

enough degrees of freedom to use tests that are asymptotically valid. The beginning date

of the data series is January 1990, and the ending date is December 2008. To reduce

scaling problems, differences in log daily prices are multiplied by 100 and thus expressed

in percentage terms.

17

EMPIRICAL RESULTS

Table 1 presents summary statistics and tests of normality. The positive excess

kurtosis in all series indicates the presence of fat tails. All series except wheat are

negatively skewed. The magnitudes of skewness and kurtosis are higher for the CRB

index and copper. The normality tests (Kolmogorov-Smirnov D statistic) reject the null

hypotheses of normality at conventional levels of significance. The estimates of second

and third moments and results of normality tests suggest that a JD process may provide a

better fit for the returns than a normal distribution.

The estimated parameters and standard errors of jump diffusion processes with a

single counter of jumps are displayed in table 2. The nested normal distribution with no

jumps is rejected using a likelihood ratio test. In most commodities, the means of returns

and jumps are not significantly different from zero, which is consistent with previous

research that rarely finds a risk premium in futures markets. The magnitudes of volatility

of jumps are higher than those of the diffusion process. Except for wheat, the means of

jumps are negative. The negative jump mean is expected since skewness is negative. In

most cases, however, the mean values of jumps are not statistically significant.

The jump intensity parameters that indicate the presence of large movements

in futures returns are statistically significant. The frequency of jumps vary widely across

the commodities. During the period of analysis, the live cattle market jumps almost every

day 1.13 . Normally a question arises - is it possible that the live cattle market

jumps every day? The daily price movements of live cattle futures are tightly restricted to

be within a range of 3 cents per pound. Table 2 shows that the volatility of jumps (0.596)

in live cattle futures is the lowest among the commodities selected for this study, which is

18

consistent with live cattle having the least kurtosis. From these results, the inference is

that JD models with single counters of jumps are not as good for futures with narrow

price limits. Wheat and soybean contracts have higher price limits (60 cents per bushel

and 70 cents per bushel respectively) and jumps occur in these markets once in six days

and three days respectively. The copper futures and CRB index have no daily price limits,

and the market jumps once in three and eight days respectively.

The parameter estimates of the normal distribution with two counters of jumps are

presented in Table 3. Having four additional parameters associated with the jump

components changes the mean and variance of the diffusion processes. With most

commodities, the variance of the diffusion process decreases with the addition of new

parameters. The estimated variances associated with jumps are higher than those of the

diffusion processes. With copper, live cattle, and the CRB index, upside jumps occur

more often than downside jumps. The constraints that upside jumps have a positive mean

and downside jumps have a negative mean are active in two cases. This result casts some

doubt on Câmara’s assumption that with two counters of jumps one would reflect

negative jumps and one would reflect positive jumps. In comparison with the single

counter of jumps model, very few parameters were statistically significant for wheat and

soybeans. The results of the likelihood ratio test and BIC criterion for selecting the

suitable JD process show that only copper favors the two counters of jumps over the

single jump model.

Table 4 presents the parameter estimates of a bivariate normal distribution

(equations (8), (9), and (10)) with a single counter of jumps in commodity futures and

CRB index futures. All variables except the mean of returns and jumps in some

19

commodities are statistically significant. The magnitudes of parameter estimates are close

to univariate distribution with single counter of jumps. The intensity of simultaneous

jumps is higher in live cattle and soybean futures. Results in Table 4 show that the

estimates of correlation and the beta parameters are statistically significant in all return

series. The magnitudes of correlation are higher for jumps than for the diffusion process.

The results given in Table 4 support the theoretical assumption of higher risk associated

with extreme movements. Except in wheat, the magnitude of beta associated with jumps

is higher than continuous price movements. The difference in beta is statistically

significant in copper futures. In other words, the return series of copper support

Todorov’s model with separate betas for continuous and discontinuous price movements.

Table 5 discusses the estimated bivariate distribution with added non–systematic

jumps in futures returns. With all commodities, the magnitude of mean and variance of

the continuous component changes when an additional jump is added. The jump intensity

coefficients are statistically significant for correlated jumps in all commodities. For

uncorrelated jumps, is significant for copper and live cattle. Uncorrelated jumps are

more frequent than correlated jumps in all commodities. The estimated degree of

correlation between jumps in CRB index and commodities remains higher than with

diffusion movements. For all commodities, the estimates of correlation coefficients are

statistically significant. The correlated jumps are more frequent for agricultural

commodities such as wheat and soybean. Correlated jumps are less frequent in copper

(once in eight days) and live cattle (once in six days). The mean of coefficients for

correlated jumps is negative and shows that simultaneous movements are mostly crashes.

Overall, results show that correlated movements are mostly crashes and occur less

20

frequently than uncorrelated movements. The additional non-systematic jump in the

commodity futures makes no significant change in the magnitudes of beta estimates

except in wheat futures. The results of t-tests indicate that the differences in the estimated

betas are not statistically significant in the agricultural commodities. The difference in

beta is significant for copper and shows that the beta of continuous and jump components

differs. A likelihood ratio test and BIC are employed to select among the nested models.

The LR test rejects the null hypothesis of imposed restrictions 0,

0, and 0 and shows that the bivariate distribution with systematic and non-

systematic jumps fits the data better than the bivariate model with single counters of

jumps in all commodities except soybeans.

CONCLUSIONS

Câmara’s (2009) two counters of jumps model generalizes Merton’s (1976) JD

process that incorporates discontinuous jumps in asset price and the Amin and Ng (1993)

model. The current paper contributes to the existing literature on the distribution of

changes in futures prices by employing a mixed bivariate normal distribution and

estimates the parameters using agricultural, industrial, and CRB index futures. This paper

also extends Câmara’s (2009) model by defining jumps as systematic and non-systematic.

The results show that commodity futures returns are not normally distributed, and

the JD process fits better than the normal distribution. On average, the jumps are crashes

and are systematic. Among the univariate processes, the single counter of jumps model is

suitable for agricultural commodities (wheat, live cattle, and soybeans) and two counters

21

of jumps model is suitable for copper. A similar pattern is exhibited with bivariate JD

processes.

The difference between the betas of jump and continuous components is

significant for copper. Thus, value-at-risk measures for a portfolio that contains copper

and attempt to model the nonnormality with a flexible distribution and a constant

correlation such as with a Gaussian copula will be inaccurate since such measures would

miss the different correlation in copper between the jump and the continuous component.

The correlation of the individual futures contracts with the CRB index is positive and

substantial. Thus, value-at-risk measure that ignored the correlation among commodities

would greatly underestimate the value at risk.

Câmara’s (2009) distinction between negative and positive jumps does not prove

workable. For some commodities, when the mean of jumps is restricted to be positive or

negative, the restriction is active and the estimated mean is zero. Likelihood ratio tests

favor a general model with one jump for the CRB index and the individual commodities

having a jump process that is correlated with the CRB index as well as a second jump

that is unrelated to the CRB jump. The correlations with the jump tend to be higher than

with the diffusion process, which suggests that correlations are higher for extreme events.

22

REFERENCES Amin, K. I., and Ng, V. K., 1993, Option valuation with systematic stochastic volatility.

The Journal of Finance 48, 881-910.

Câmara, A., 2009, Two counters of jumps. Journal of Banking and Finance, 33, 456-463.

Câmara, A., and Heston, S.L., 2008, Closed-form option pricing formulas with extreme events. Journal of Futures Markets, 28, 213-230.

Commodity Research Bureau (CRB), 2010, CRB Data Center. http://www.crbtrader.com/crbindex/crbdata.asp Accessed January 2010.

Das, S. R., and Uppal, R., 2004, Systemic risk and international portfolio choice. The Journal of Finance, 59, 2809-2834.

Duffie, D., and Pan, J., 2001. Analytical value-at-risk with jumps and credit risk. Finance

and Stochastics, 5, 155-180. Eraker, B., 2004, Do stock prices and volatility jump? Reconciling evidence from spot

and option prices. The Journal of Finance, 59, 1367-1403.

Hall, J. A., Brorsen, B.W., and Irwin, S.H., 1989, The distribution of futures prices: A test of the stable Paretian and mixture of normals hypotheses. The Journal of Financial and Quantitative Analysis, 24, 105-116.

Hilliard, J. E., and Reis, J.A., 1999, Jump processes in commodity futures prices and options pricing. American Journal of Agricultural Economics, 81, 273-286.

Hudson, M. A., Leuthold, R.M., and Sarassoro, G.F., 1987, Commodity futures price changes: Recent evidence for wheat, soybeans and live cattle. Journal of Futures Markets, 7, 287-301.

Jarrow, R. A., and Rosenfeld, E.R., 1984, Jump risks and the intertemporal capital asset pricing model. The Journal of Business, 57, 337-351.

Jorion, P., 1988, On jump processes in the foreign exchange and stock markets. Review of Finanial Studies, 1, 427-445.

Kansas City Board of Trade (KCBT), 2010, Daily Wheat Price. Accessed from http://www.kcbt.com/daily_wheat_price.asp. Accessed January 2010.

Koekebakker, S., and Gudbrand, L., 2004, Volatility and price jumps in agricultural futures prices-Evidence from wheat options. American Journal of Agricultural Economics, 86, 1018-1031.

Kou, S. G., 2002, A jump-diffusion model for option pricing. Management Science, 48, 1086-1101.

23

Merton, R. C., 1976, Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics, 3, 125-144.

Pan, J., 2002, The jump-risk premia implicit in options: Evidence from an integrated time-series study. Journal of Financial Economics, 63, 3-50.

Ramezani, C., and Zeng, Y., 2007, Maximum likelihood estimation of the double exponential jump-diffusion process. Annals of Finance, 3, 487-507.

Rencher, A. C., 2002, Methods of Multivariate Analysis. 2nd ed. New Jersey:Wiley-Interscience.

SAS, 2009, SAS System under MICROSOFT Windows. Release 9.1. Cary NC: SAS Institute Inc.

Sharpe, W.F., 1963, A simplified model for portfolio analysis. Management Science, 2, 277-293.

Stiegert, K.W., and Brorsen, B.W., 1996, The distribution of futures prices:diffusion-jump versus generalized beta-2. Applied Economics Letters, 3, 303-305.

Todorov, V., and Bollerslev, T., 2010, Jumps and betas: A new framework for disentangling and estimating systematic risks. Journal of Econometrics, 157, 220-235.

Turvey, C.G., Driver, H.C., and Baker, T.G., 1988, Systematic risk in farm portfolio selection. American Journal of Agricultural Economics, 70, 831-836.

 

APPENDIX

MAXIMUM LIKELIHOOD FUNCTIONS

The appendix describes the maximum likelihood functions used in estimation. To a

considerable extent, the expressions of likelihood functions closely follow Jorion (1988). Five

different sets of density function parameters are estimated. If , , … , are continuous

random variables that represent the normally distributed wheat futures returns ~ , ,

then the logarithm of the likelihood function as a function of parameter vector ,

for a normal distribution can be written as

1 2ln 2

1exp

2.

With a single counter of jumps, the log likelihood function for the mixed jump-diffusion

process (equation 3) with parameter vector , , , , can be written as

2 2ln 2 ln

!1

exp2

.

With two counters of jumps, the logarithm of the likelihood function for the mixed jump-

diffusion process (equation 5) with parameter vector , , , , , , , ,

can be written as

3 2ln 2

ln!

!

1

2 min ,exp

2 2min ,

With a bivariate normal distribution, the logarithm of the likelihood function for the mixed

jump-diffusion process (equation 6) with parameter vector , , , , , , ,

can be written as

4

ln 2

ln!

1

1exp

2 1

2

.

With a bivariate normal distribution and separate systematic and non-systematic jump risk, the

logarithm of likelihood function for the mixed jump-diffusion process (equations (9), (10),

and (11)) with parameter vector , , , , , , , , , , , is

5 ln 2

ln!

!

1

2 1exp

2 1

∗ and

∗.

Table 1. Summary statistics and normality tests for the commodity futures and CRB index returns Parameters Wheat Copper Live cattle Soybeans CRB index Minimum change in daily value -8.397 a -12.067 -6.358 -7.411 -6.406 Maximum change in daily value 8.794 11.742 3.708 6.433 5.429 Mean change in daily value -0.035 0.035 0.002 -0.004 0.000 Variance 2.515 2.703 0.752 1.945 0.556 Skewness 0.072 -0.291 -0.182 -0.249 -0.580 Excess kurtosis 2.092 5.551 1.994 2.684 7.069 Kolmogorov-Smirnov D statistic 0.033

(<0.01)b 0.063

(0.010) 0.042

(0.010) 0.056

(<0.01) 0.062

(<0.01) Note: The beginning date of the data series is January 1990, and the ending date is December 2008. The total number of observations is 4,513. a The returns are expressed in percentages. b P-values are in the parentheses.

Table 2. Parameter estimates of the distribution with single counters of jumps Parameter Symbol Wheat Copper Live Cattle Soybean CRB Index Mean of diffusion -0.060*

(0.027) 0.036

(0.024) 0.029

(0.019) 0.045*

(0.021 0.020*

(0.010) Volatility of diffusion 2 1.733**

(0.103) 1.110**

(0.087) 0.226**

(0.026) 0.863**

(0.049) 0.295**

(0.015) Intensity of jumps 0.169**

(0.048) 0.300**

(0.053) 0.845**

(0.104) 0.297**

(0.041) 0.133**

(0.025) Volatility of jumps 2 5.338**

(1.084) 5.124**

(0.738) 0.596**

(0.054) 3.639**

(0.421) 1.888**

(0.308) Means of jumps 0.151

(0.134) -0.004 (0.086)

-0.031 (0.023)

-0.165* (0.075)

-0.153 (0.073)

Days between jumps 5.893**(1.675)

3.335** (0.596)

1.130**(0.133)

3.362**(0.370)

7.506**(1.406)

Negative log likelihood 9655.5 8246.5 5634.5 7635.0 4681.5 Bayesian information criterion 16956.0 16535.0 11311.0 15312.0 9405.0 Likelihood ratio test of normality 258** 753** 261** 536** 799** a The data are day to day changes in the logarithm of futures prices expressed in percentages. b Standard errors are in the parentheses * significant at 5% level ** significant at 1% level

Table 3. Parameter estimates of the distribution with two counters of jumps Parameter Symbol Wheat Copper Live Cattle Soybean CRB IndexMean of diffusion

-0.054* (0.025)

-0.042 (0.037)

0.025 (0.022)

0.045 (0.031)

0.017* (0.009)

Volatility of diffusion

1.482**(0.210)

0.744**(0.095)

0.197** (0.030)

0.897**(0.060)

0.247** (0.022)

Intensity of upside jumps

0.089 (0.060)

0.721**(0.149)

0.535* (0.245)

0.034 (0.043)

0.280** (0.078)

Intensity of downside jumps

0.422 (0.377)

0.055* (0.025)

0.484* (0.208)

0.212**(0.072)

0.004** (0.002)

Volatility of upside jumps

7.471 (4.126)

1.641**(0.308)

0.557** (0.145)

1.134 (1.347)

0.835** (0.189)

Volatility of downside jumps

1.199 (0.927)

7.142**(5.182)

0.754** (0.212)

3.447**(0.663)

0.620 (0.985)

Mean of upside jumps

0.187 (0.217)

0.192* (0.084)

0.084 (0.137)

2.552 (1.326)

-

Mean of downside jumps

- -1.125* (0.558)

-0.140 (0.148)

-0.633* (0.526)

-3.955* (0.482)

Covariance of jumps

-0.375 (3.924)

5.196 (3.037)

-0.362* (0.200)

-1.615 (2.053)

0.128 (2.076)

Days between upside jumps

11.236 (7.625)

1.386**(0.287)

1.868* (0.856)

29.840 (37.840)

3.568** (0.176)

Days between downside jumps

2.36 (2.114)

18.127* (8.153)

2.067* (0.892)

4.700**(1.595)

231.38** (90.25)

Negative log likelihood value 8456.5 8226.5 5630.0 7633.5 4670.0 Bayesian information criterion 16988.0 16528.0 11337.0 15343 9416.2

a The data are day to day changes in the logarithm of futures prices expressed in percentages. b Standard errors are in the parentheses. * significant at 5% level. ** significant at 1% level.

Table 4. Parameter estimates of bivariate distribution with single counters of jump

Parameter Symbol Wheat Copper Live Cattle Soybean Mean of futures diffusion -0.047

(0.026)0.043

(0.023) 0.031*

(0.016)0.050*

(0.020)Mean CRB index futures diffusion 0.020

(0.010)0.023*

(0.010) 0.032**

(0.012)0.031**

(0.011)Mean of futures jump 0.057

(0.103)-0.031 (0.090)

-0.066(0.037)

-0.155*(0.064)

Mean of CRB index jump -0.087(0.047)

-0.078* (0.040)

-0.075*(0.029)

-0.089**(0.033)

Volatility of commodity futures diffusion 1.700**(0.076)

1.235** (0.073)

0.397**(0.036)

0.874**(0.050)

Volatility of CRB index futures diffusion 0.263**(0.015)

0.260** (0.013)

0.214**(0.017)

0.225**(0.014)

Intensity of jumps 0.216**(0.036)

0.265** (0.039)

0.435**(0.077)

0.347**(0.045)

Volatility of commodity futures jump 4.230**(0.560)

5.273** (0.643)

0.808**(0.089)

3.020**(0.328)

Volatility of CRB index futures jump 1.282**(0.188)

1.010** (0.132)

0.718**(0.107)

0.881**(0.099)

Correlation between futures and CRB diffusions 0.381**(0.019)

0.238** (0.022)

0.143**(0.026)

0.429**(0.020)

Correlation between jumps 0.491**(0.038)

0.476** (0.035)

0.196**(0.037)

0.533**(0.028)

Beta of discontinuous (jump) components 0.892**(0.078)

1.088** (0.088)

0.208**(0.040)

0.987**(0.061)

Beta of continuous components 0.968**(0.051)

0.517** (0.049)

0.194**(0.036)

0.845**(0.045)

Difference in Betas -0.076(0.095)

0.571** (0.112)

0.014(0.065)

0.142(0.089)

Days between jumps (days/jump) 4.630**(0.778)

3.771** (0.556)

2.302**(0.412)

2.881**(0.376)

Negative log likelihood value 14555.5 12605.5 10264.5 11701.5

Bayesian information criterion 29206.0 25446.0 20622.0 23495.0

Table 5. Parameter estimates of bivariate distribution with systematic and non-systematic jumps

Parameter Symbol Wheat Copper

Live Cattle

Soybean

Mean of futures diffusion -0.097a (0.056) b

-0.005(0.037)b

-0.034 (0.019)

-0.035(0.032)

Mean CRB index futures diffusion process

0.017** (0.010)

0.018*(0.009)

0.019* (0.010)

0.027**(0.010)

Intensity of correlated jumps 0.175** (0.032)

0.133**(0.022)

0.160** (0.030)

0.252**(0.323)

Intensity of uncorrelated jumps

0.424

(0.341)0.653**(0.197)

0.788** (0.122)

0.283(0.323)

Mean of correlated futures jump

-0.019* (0.137)

-0.252(0.170)

-0.112 (0.071)

-0.229*(0.102)

Mean of uncorrelated futures jump

0.155 (0.136)

0.114 (0.066)

-0.017 (0.029)

0.067 (0.135)

Mean of CRB index futures jump

-0.100

(0.057)-0.121

(0.070)-0.127* (0.063)

-0.108*(0.044)

Volatility of futures diffusion 1.410** (0.164)

0.809**(0.111)

0.224** (0.027)

0.795**(0.121)

Volatility of CRB index futures diffusion process

0.275** (0.015)

0.299**(0.012)

0.283** (0.0151)

0.249**(0.016)

Volatility of correlated futures jump

4.289** (0.637)

7.196**(1.006)

0.676** (0.129)

3.336**(0.428)

Volatility of uncorrelated futures jump

1.037 (0.571)

1.333**(0.304)

0.528** (0.064)

0.982(0.911)

Volatility of CRB index futures jump

1.527** (0.246)

1.823**(0.270)

1.631** (0.268)

1.146**(0.185)

Correlation between futures and CRB index diffusions

0.431 (0.032)

0.326**(0.0312)

0.196** (0.030)

0.483**(0.042)

Correlation between jumps 0.522 (0.044)

0.566**(0.042)

0.356** (0.069)

0.567**(0.038)

Days between correlated jumps

5.699** (1.049)

7.539**(1.235)

6.239** (1.185)

3.966**(0.748)

Days between uncorrelated jumps

2.358 (1.897)

1.531**(0.464)

1.268** (0.197)

3.535(4.040)

Beta of continuous components

0.973** (0.049)

0.535**(0.042)

0.175** (0.026)

0.967**(0.065)

Beta of discontinuous (jump) components

0.875** (0.079)

1.125**(0.095)

0.229** (0.041)

0.864*(0.042)

Difference in Betas

-0.099 (0.106)

0.589**(0.111)

0.055 (0.056)

0.103(0.089)

Negative log likelihood 12672.5 12583.0 10221.0 11698.0BIC 25463.0 25284.0 20560.0 23514.0

a The data are day to day changes in the logarithm of futures prices expressed in percentages. b Standard errors are in parentheses. * significant at 5% level. ** significant at 1% level.