THREE ESSAYS ON AGRICULTURAL PRICE VOLATILITY AND THE LINKAGES
BETWEEN AGRICULTURAL AND ENERGY MARKETS
By
Feng Wu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Agricultural, Food and Resource Economics
2012

ABSTRACT
THREE ESSAYS ON AGRICULTURAL PRICE VOLATILITY AND THE LINKAGES
BETWEEN AGRICULTURAL AND ENERGY MARKETS
By
Feng Wu
This dissertation contains three essays. In the first essay I use a volatility spillover model
to find evidence of significant spillovers from crude oil prices to corn cash and futures prices,
and that these spillover effects are time-varying. Results reveal that corn markets have become
much more connected to crude oil markets after the introduction of the Energy Policy Act of
2005. Furthermore, crude oil prices transmit positive volatility spillovers into corn prices and
movements in corn prices become more energy-driven as the ethanol gasoline consumption ratio
increases. Based on this strong volatility link between crude oil and corn prices, a new cross
hedging strategy for managing corn price risk using oil futures is examined and its performance
studied. Results show that this cross hedging strategy provides only slightly better hedging
performance compared to traditional hedging in corn futures markets alone. The implication is
that hedging corn price risk in corn futures markets alone can still provide relatively satisfactory
performance in the biofuel era.
The second essay studies the spillover effect of biofuel policy on participation in the
Conservation Reserve Program. Landowners’ participation decisions are modeled using a real
options framework. A novel aspect of the model is that it captures the structural change in
agriculture caused by rising biofuel production. The resulting model is used to simulate the
spillover effect under various conditions. In particular, I simulate how increased growth in
agricultural returns, persistence of the biofuel production boom, and the volatility surrounding

agricultural returns, affect conservation program participation decisions. Policy implications of
these results are also discussed.
The third essay proposes a methodology to construct a risk-adjusted implied volatility
measure that removes the forecasting bias of the model-free implied volatility measure. The risk
adjustment is based on a closed-form relationship between the expectation of future volatility
and the model-free implied volatility assuming a jump-diffusion model. I use a GMM estimation
framework to identify the key model parameters needed to apply the model. An empirical
application to corn futures implied volatility is used to illustrate the methodology and
demonstrate differences between my approach and the model-free implied volatility using
observed corn option prices. I compare the risk-adjusted forecast with the unadjusted forecast as
well as other alternatives; and results suggest that the risk-adjusted volatility is unbiased,
informationally more efficient, and has superior predictive power over the alternatives
considered.

ACKNOWLEDGEMENTS

First and foremost I would like to gratefully and sincerely thank my dissertation
supervisor, Dr. Zhengfei Guan, for his guidance, understanding, patience, and most importantly,
his friendship during my graduate studies. He has supported me throughout my dissertation with
his patience and knowledge whilst giving me the freedom to work in my own way. I appreciate
all his contributions of time, ideas, and funding to make my Ph.D. experience productive and
stimulating. I am also deeply indebted to my major professor Dr. Robert Myers. His insights and
incisive analytical skills sharpened my analysis. I have benefitted tremendously from his
insightful comments and constructive suggestions at different stages of my research. His
encouragement of thinking like an applied economist will never be forgotten. I would also like to
express my admiration and appreciation to my other committee members, Richard Baillie and
Jinhua Zhao, for their helpful feedback on my research. Lastly, I would like to express my
deepest appreciation towards my parents and family who have always been there for me no
matter where I am, for unconditional support and patience.

iv

TABLE OF CONTENTS
LIST OF TABLES ........................................................................................................................ vii
LIST OF FIGURES ..................................................................................................................... viii
KEY TO ABBREVIATIONS ........................................................................................................ ix
CHAPTER 1: VOLATILITY SPILLOVER EFFECTS AND CROSS HEDGING IN CORN
AND CRUDE OIL FUTURES ....................................................................................................... 1
1.1 Introduction .......................................................................................................................... 1
1.2 Trivariate Volatility Spillover Model .................................................................................. 3
1.2.1 Model Specifications .................................................................................................... 4
1.2.2 Parameterizations for Volatility Spillovers .................................................................. 7
1.2.3 Volatility Spillover Ratio ............................................................................................. 9
1.3 Data and Estimation ........................................................................................................... 10
1.4 Spillover Model Results ..................................................................................................... 16
1.4.1 The Constant Spillover Model ................................................................................... 17
1.4.2 The Event Spillover Model ........................................................................................ 21
1.4.3 The Substitution Spillover Model .............................................................................. 24
1.5 Implications for Hedging Strategies................................................................................... 27
1.5.1 Optimal Hedging ........................................................................................................ 27
1.5.2 Comparisons of Hedging Performance ...................................................................... 28
1.6 Conclusions ........................................................................................................................ 31
APPENDIX A Derivation of Equation 1.21 ............................................................................ 33
REFERENCES......................................................................................................................... 35
CHAPTER 2: SPILLOVER EFFECTS OF BIOFUEL POLICY ON THE CONSERVATION
PROGRAM ................................................................................................................................... 38
2.1 Introduction ........................................................................................................................ 38
2.2 Landowner’s Decision under Uncertainty ......................................................................... 42
2.3 Structural Change and Parameter Uncertainty ................................................................... 43
2.4 Linear Complementarity Problem ...................................................................................... 46
2.5 Data and Parameter Estimation .......................................................................................... 50
2.5.1 Data on Farming Returns ........................................................................................... 50
2.5.2 Parameter Estimation ................................................................................................. 51
2.6 Results ................................................................................................................................ 53
2.7 Sensitivity Analysis ............................................................................................................ 59
2.7.1 Length of the High Growth State, 1  ........................................................................ 59
2.7.2 The Effect of Volatility,  ......................................................................................... 61
2.8 Conclusions ........................................................................................................................ 63
APPENDIX B Derivation of Equation 2.12 ............................................................................ 65
APPENDIX C Derivation of Equation 2.14 ............................................................................ 67
APPENDIX D Discretization of the LCP ................................................................................ 70
APPENDIX E Derivation of Equation 2.20 ............................................................................. 75
REFERENCES......................................................................................................................... 78
v

CHAPTER 3: THE RISK-ADJUSTED IMPLIED VOLATILITY AND ITS FORECAST
PERFORMANCE IN CORN FUTURES ..................................................................................... 82
3.1 Introduction ........................................................................................................................ 82
3.2 Model Specifications and Volatility Forecast .................................................................... 86
3.2.1 Price Dynamics .......................................................................................................... 86
3.2.2 The Model .................................................................................................................. 88
3.3 Estimation .......................................................................................................................... 90
3.3.1 Moment Conditions .................................................................................................... 91
3.3.2 Volatility Measures .................................................................................................... 92
3.3.3 The GMM Framework ............................................................................................... 94
3.3.4 Simulation Analysis ................................................................................................... 96
3.4 Empirical Application ........................................................................................................ 97
3.4.1 Data ............................................................................................................................ 98
3.4.2 Estimation Results .................................................................................................... 101
3.4.3 Robustness Analysis ................................................................................................. 103
3.5 Forecast Evaluation .......................................................................................................... 105
3.5.1 Testing the Unbiasedness Hypothesis ...................................................................... 106
3.5.2 Testing Informational Efficiency ............................................................................. 108
3.5.3 Testing Predictive Power ......................................................................................... 109
3.6 Conclusion ....................................................................................................................... 110
APPENDIX F Derivation of Equation 3.10 ........................................................................... 113
APPENDIX G Monte Carlo Experimental Design ................................................................ 116
REFERENCES....................................................................................................................... 119

vi

LIST OF TABLES
Table 1.1. Descriptive Statistics and Correlation Analysis........................................................... 12
Table 1.2. Johansen Cointegration Test ........................................................................................ 13
Table 1.3. Vector Error Correction Model for Oil Futures, Corn Cash, and Futures Prices ........ 15
Table 1.4. Estimation of the Constant Volatility Spillover Model ............................................... 18
Table 1.5. Univariate TGARCH (1, 1) Models for Crude Oil Futures Price ................................ 19
Table 1.6. Summary Statistics of Spillover Ratio for Three Models ............................................ 19
Table 1.7. Estimation of the Event Spillover Effect Model .......................................................... 22
Table 1.8. Estimation of the Substitution Spillover Effect Model ................................................ 25
Table 1.9. In- and Out-of-sample Hedging Performance ............................................................. 30
Table 2.1. Option Value of Participation in the CRP for the Baseline ......................................... 54
Table 3.1. Monte Carlo Simulation Results .................................................................................. 97
Table 3.2. Summary Statistics for Volatilities, 1987-2010 ......................................................... 100
Table 3.3. Parameter Estimates using the RV, BV and MF ....................................................... 102
Table 3.4. Parameter Estimates using the RV_10, BV_10 and MF ........................................... 104
Table 3.5. Parameter Estimates using the RV, BV_R and MF ................................................... 105
Table 3.6. Testing Unbiasedness and Informational Efficiency ................................................. 107
Table 3.7. Testing Predictive Power ........................................................................................... 110

vii

LIST OF FIGURES
Figure 1.1. Movement of the Event Dummy and the Ethanol-Gasoline Consumption Ratio
Variables ....................................................................................................................................... 11
Figure 1.2. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Constant
Spillover Model ............................................................................................................................ 20
Figure 1.3. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Event
Spillover Model ............................................................................................................................ 23
Figure 1.4. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Substitution
Spillover Model ............................................................................................................................ 26
Figure 2.1. Option Value of Land Conversion Opportunity under Two Different Expected
Growth Rates (the Baseline Case) ................................................................................................ 54
Figure 2.2. Farming Threshold Returns ( R * ) versus Growth Rate Expectations ( mt ) under
Different Durations of the High Growth State .............................................................................. 60
Figure 2.3. Farming Threshold Returns ( R * ) versus Growth Rate Expectations ( mt ) under
Different Volatility Parameters ..................................................................................................... 61
Figure 3.1. Realized Volatilities (RVs), Bipower Variations (BVs), and MF Implied Volatilities
(MF) in Annualized Standard Deviation..................................................................................... 101

viii

KEY TO ABBREVIATIONS
BEKK

Baba-Engle-Kraft-Kroner

BS

Black-Scholes

CBOT

Chicago Board of Trade

CRP

Conservation Reserve Program

DM

Diebold-Mariano

EIA

Energy Information Administration

EISA

Energy Independence and Security Act

ERS

Economic Research Service

GBM

geometric Brownian motion

GMM

Generalized Method of Moments

HV

historical volatility

IV

integrated volatility

LCP

linear complementarity problem

MAE

mean absolute errors

MAPE

mean absolute percentage errors

MDM

modified Diebold-Mariano

MF

model-free

MTBE

Methyl Tertiary Butyl Ether

NASS

National Agricultural Statistics Service

NYMEX

New York Mercantile Exchange

QV

quadratic variation

RMSE

root-mean-square-errors
ix

RV

realized volatility

SBC

Schwartz Bayesian Criterion

SD

standard deviations

USDA

U.S. Department of Agriculture

VEC

vector error correction

x

CHAPTER 1: VOLATILITY SPILLOVER EFFECTS AND CROSS HEDGING IN CORN
AND CRUDE OIL FUTURES

1.1

Introduction
U.S. biofuel production has gone through a rapid expansion in response to both higher

energy prices making biofuels more cost effective and government policies designed to reduce
U.S. dependence on imported crude oil for energy needs (Tyner, 2008). Replacing fossil fuels
with biofuels is also regarded by some as having the potential to at least partially address climate
change issues as well as energy security. The expansion in biofuel production is expected to
persist according to recent studies on the economics of biofuels (Liska et al., 2009; Feng, Rubin,
and Babcock, 2010).
Rising biofuel production, particularly the production of corn-based ethanol, is likely to
have made corn and crude oil markets more connected, possibly causing volatility spillovers
from crude oil markets to corn markets. Zulauf and Roberts (2008) measured corn historical and
expected volatilities from 1989 to 2007 and found corn price volatility has increased
substantially over this period. Increased corn price volatility presumably results in greater costs
for managing risks, such as more costly crop insurance premiums, higher option premiums, and
higher hedging costs. Therefore, it is important to understand these new volatility relationships in
order to inform and develop appropriate risk management strategies for corn market participants
in an era of high biofuel production.
Volatility spillovers have been studied extensively in the finance literature (Bekaert and
Harvey, 1997; Ng, 2000; Bekaert, Harvey, and Lumsdaine., 2002; Baele, 2005; and Christiansen,
2007). Many applications separate the shock to an individual country stock or bond return into
three components: local, regional, and global. Ng (2000), for example, analyzes the sources of
1

return volatility of stock markets in the Pacific-Basin and finds transmission from the world and
regional markets. However, volatility spillover effects have rarely been studied among
commodities and may provide interesting insights into the nature of commodity price volatility
and the connections between different markets. In particular, possible volatility spillovers from
crude oil markets into corn markets have become an increasingly important issue given the
growth in biofuel production.
In this study I examine volatility spillover effects from crude oil prices to corn prices,
paying particular attention to the extent to which volatility in corn prices is impacted by external
shocks from crude oil markets. More specifically, I construct a volatility spillover model and
investigate the effect of shocks from the crude oil futures price on corn cash and futures prices. I
compare three model specifications with different assumptions on the spillover effects: constant
spillover parameters in the constant spillover model; differing spillover parameters before and
after the introduction of the Energy Policy Act of 2005 in the event spillover model; and timevarying spillover parameters that are allowed to vary with the ratio of fuel ethanol consumption
to gasoline consumption in the substitution spillover model. The constant spillover model reveals
statistically significant volatility spillovers from crude oil prices into corn prices, with similar
effects on corn cash and futures prices. The event spillover model shows that spillover intensities
have increased significantly since the Energy Policy Act of 2005. In the substitution spillover
model, I further examine the impact of fuel substitution on corn price volatility and show that the
sign and size of a spillover depend on the magnitude of the ethanol-gasoline consumption ratio.
Given evidence of strong linkages between crude oil and corn markets, this study further
examines a new cross hedging strategy to determine whether this would provide improved
hedging performance for corn market participants. This cross hedging strategy allows portfolios

2

of corn cash, corn futures, and crude oil futures. A considerable amount of previous research has
focused on optimal hedging strategies using corn futures (e.g., Baillie and Myers, 1991; Myers,
1991; and Moschini and Myers, 2002). However, previous research has not considered the
possibility of hedging corn cash positions using both corn and crude oil futures simultaneously in
a broader portfolio that allows for volatility spillovers. In this essay I compare the performance
of the cross hedging strategy to that of conventional hedging strategies using corn futures only.
The comparison sheds light on whether corn futures alone could continue to provide effective
risk management for corn market participants in an era of increased biofuel production.

1.2

Trivariate Volatility Spillover Model
The volatility spillover studies of Bekaert and Harvey (1997), Ng (2000), Bekaert,

Harvey and Ng (2005), Baele (2005), and Christiansen (2007) all consider volatility spillover
effects on international stock and bond markets. For example, Ng (2000) develops a two-factor
spillover model in which unexpected shocks to stock returns on any Pacific-Basin market are
influenced by news (shocks) originating from two foreign markets. This literature has focused on
spillovers in the same type of market (i.e. stock or bond market) at different levels (e.g., local,
regional, or global). In this study, however, I investigate spillovers across two types of market,
focusing on the impact of external shocks from the crude oil futures market on corn cash and
1

futures prices. The trivariate volatility spillover model has the general form:

1

Crude oil spot prices could also be included explicitly in my analysis, but my ultimate interest
lies in cross hedging a cash corn position in corn and crude oil futures markets. Therefore I
restrict the analysis to the three key variables required for this purpose—corn spot prices, corn
futures prices, and crude oil futures prices.
3

(1.1)

ot   E (ot I t 1 )   o, t 
 p    E ( p I )   
t t 1   p, t  ,
 t 
 f t   E ( f I )   
  
t t 1   f , t 



(1.2)

 o, t   1

 
 p, t    t

 
 f , t  t



(1.3)

 eo, t I t 1    2


  o, t 0 0 
,
Cov e p, t I t 1    0

 
Ht 

 e f , t I t 1   0


 

0 0   eo , t 


1 0  e p , t  ,

0 1  e f , t 





where ot , pt , and f t denote crude oil futures price, corn cash price, and corn futures price,
respectively; I t 1 is information available at time t  1 ;  o, t ,  p, t , and  f , t are respective
prediction errors in the crude oil futures market, the corn cash market, and the corn futures
market; eo, t is an idiosyncratic structural shock to oil futures; [e p, t , e f , t ]'

is a vector of

idiosyncratic structural shocks to corn cash and futures prices that may be mutually correlated
2
but are both uncorrelated with eo, t ;  o,t is a time varying conditional covariance for eo, t ; and

H t is a time-varying conditional covariance-variance matrix for [e p, t , e f , t ]' . This model clearly
allows oil futures price shocks to spill over into corn markets.

1.2.1 Model Specifications
In equation (1.1), oil futures, corn cash, and corn futures prices are composed of a
conditional expectation that needs to be modeled. Here I specify a vector error correction (VEC)
model for the conditional means:

4

(1.4)

 o, t 
ot    o  n
ot  i  2


p        p     ECT   
i  p, t  ,
 t   p   i  t i   i


f t     i 1 f t  i  i 1

  f 


 f , t 



where  is the first-difference operator;  i ( i  1, , n ) are three-by-three matrices of
coefficients; i are three-by-one vectors of coefficients; and ECTi denotes error correction
terms to capture up to two cointegration relationships. The specification allows for past oil
returns in the mean equations of corn markets and vice versa. In other words, past information
about oil or corn prices can influence current price expectations for any of the prices. I also allow
for possible seasonality in the conditional means, but empirical results show no significant
seasonality so that seasonal variables are not included in the model specification. The specific
form for the error correction model depends on the data and will be discussed in the data analysis
section below.
I assume the crude oil shocks ( eo, t ) follow a conditional normal distribution and allow its
2
conditional variance (  o,t ) to vary over time in response to changing market conditions. Since it

is typically found that negative shocks lead to higher subsequent volatility than positive shocks
of an equal magnitude, I follow Glosten, Jagannathan, and Runkle (GJR) (1993) and allow a
GARCH process that accounts for an asymmetric volatility effect. The full GJR- GARCH(m ,l)
2
for  o,t takes the form:

(1.5)

2
eo, t I t 1 ~ N (0,  o, t ) ,

(1.6)

2
2
2
2
 o, t   0    i eo, t  i   i d t  i eo, t  i   i o, t  i   0 zt    i Qit ,

l

l

m

k

i 1

i 1

i 1

i 1

5

where d t  i ( i  1, , l ) are dummy variables that take a value of one if eo, t i  0 and zero if
eo, t i  0 ; z t is the number of days to contract maturity to allow for time-to-maturity effects;

and Qit ( i  1, , k ) are seasonal dummies..
In equation (1.2), shocks to the corn cash price and corn futures price are allowed to be
driven by an external shock from the crude oil market in addition to a purely idiosyncratic
component. Volatility spillover effects from the crude oil market to corn cash and futures
markets are introduced via the idiosyncratic crude oil shock eo, t while et  [e p, t , e f , t ]' is
assumed to be uncorrelated with eo, t and follow a conditional normal distribution with mean
zero and time-varying covariance matrix H t . There are several possible parameterizations of the
resulting bivariate GARCH process for [e p, t , e f , t ]' including the VECH model of Bollerslev,
Engle, and Wooldridge (1988), the constant correlation model of Bollerslev (1990), the factor
ARCH model of Engle, Ng, and Rothschild (1990), and the Baba-Engle-Kraft-Kroner (BEKK)
model defined in Engle and Kroner (1995). The asymmetric version of the BEKK model (Kroner
and Ng, 1998) is used here to account for a possibly asymmetric volatility effect, due to the fact
that negative shocks might have a bigger impact on future volatility than do positive shocks of
the same size. The generalized BEKK(m,l) model is then specified as:

e f , t ]' I t 1 ~ N (0, H t ) ,

(1.7)

[e p, t

(1.8)

H t  C ' C   Ai' et  i et'  i Ai   Di' t  i t'  i Di   Bi' H t  i Bi  G  zt ,

l

l

m

i 1

i 1

i 1

where C, A, B, and D are lower triangular matrices; G is a symmetric matrix;  is a Kronecker
product; and z t is a strictly exogenous variable, representing the number of days to contract
maturity. The vector t  [ p, t  f , t ]' represents the vector of negative shocks with
6

t  min{0, et } . I also allow for possible seasonality in the covariance structure using dummy
variables. However, later results indicate that the coefficients on seasonal dummies are small and
not statistically significant. Therefore, seasonal variables are excluded from (1.8).

1.2.2 Parameterizations for Volatility Spillovers
Three different parameterizations for the spillover parameters,  t and  t , are introduced.
The first specification assumes that the spillover parameters are constant throughout the entire
sample period and investigates whether there are significant volatility spillovers from crude oil
prices:
(1.9)

t   ,

(1.10) t   .
However, some studies have found that spillovers vary in intensity in response to various
influences, especially to important legislative or policy events. In the present context, the Energy
Policy Act of 2005 is particularly important. This Act established the Renewable Fuel standard
requiring that transportation fuels sold in the US contain a minimum amount of renewable fuel,
2

and also mandates increasing domestic use of renewable fuels to 7.5 billion gallons by 2012.

Subsequent tax incentives, federal and state mandates, and the progressive elimination of MTBE
(Methyl Tertiary Butyl Ether) as an additive in many states, have quickly increased the demand
for biofuels, particularly corn-based ethanol. The linkage between crude oil prices and corn
2

Under the act, for the calendar years 2006 through 2012 the applicable volumes of biofuel that
must be mixed with gasoline sold in the United States increase from 4, 4.7, 5.4, 6.1, 6.8, 7.1, 7.4,
and 7.5 billion gallons, respectively. However, I would argue that these future mandates became
known when the act was passed in 2005, and that rational agents would immediately build these
expectations into current pricing and price expectations. Therefore, I view the date when the act
was passed as the key determinant of the main change in the oil-corn price relationships.
7

prices is thus likely to have strengthened as a result of the Act. Therefore, I allow the Act to have
an impact on spillover effects in the event spillover specification:
(1.11) t   0  1Dt ,
(1.12) t  0  1Dt ,
where Dt is dummy variable which equals 1 for the period after the Act is passed (Jul 29, 2005)
and 0 otherwise.
Third, I further relax the assumption of constant spillovers. Other volatility spillover
studies have found evidence of time-varying correlations across national stock and bond markets
(Ng, 2000; Christiansen, 2007). To explore the possibility of time-varying volatility spillovers
further I allow spillover parameters to vary with certain underlying factors. In view of the
substitution effect between different types of energy, I might expect corn prices and crude oil
prices to become more connected as the ratio of fuel ethanol consumption to gasoline
consumption rises. Therefore I use this ratio as an indicator of the size of spillovers between the
markets. It is likely that rising ethanol consumption, and thus a high consumption substitution
ratio in response to high crude oil prices, will also push up the demand for corn, the major
feedstock of ethanol, thereby increasing the integration of crude oil and corn markets. Thus a
more appropriate way of investigating the behavior of spillover effects may be to allow the
parameters to vary with the ethanol to gasoline consumption ratio. To avoid potential
endogeneity problems, I use lagged consumption ratios to capture the time-varying spillovers.
The specification of the substitution spillover model is therefore:
(1.13) t   0  1Rt 1 ,
(1.14) t  0  1Rt 1 ,

8

where Rt 1 is the consumption ratio of the previous period.

3

1.2.3 Volatility Spillover Ratio
Under the assumption of no correlation between eo, t and [e p, t , e f , t ]' , the conditional
variances of corn cash and futures prices are given by:
11
2
(1.15) E ( 2 , t I t 1 )  H t  t2 o, t ,
p
2
(1.16) E ( 2 , t I t 1 )  H t22  t2 o, t ,
f

where H tij is the element in the i-th row and the j-th column of H t . The signs and significance
of  t and  t determine whether volatility spillover effects from crude oil markets are present in
corn markets. To measure the proportion of the forecast error variance of corn markets that is
accounted for by crude oil volatility spillover effects, I define spillover ratios for corn cash and
futures prices as:
(1.17) SR p, t 

(1.18) SR f , t 

2
 t2 o, t
11
2
H t   t2 o, t

2
t2 o, t
2
H t22  t2 o, t

 [0,1] ,

 [0,1] .

3

Using lagged consumption ratios in the substitution spillover model is mainly intended to avoid
potential endogeneity and thus simplify joint estimation of eqs. (1.2), (1.7), (1.8), (1.13), and
(1.14). However, this treatment may raise issues regarding the specification of dynamics. I
replaced lagged consumption ratios with contemporaneous ratios and re-estimated eqs. (1.2),
(1.13), and (1.14) using an IV regression (the instrument is lagged consumption ratios) with
robust standard errors. Coefficient estimates for consumption ratios are similar to those in the
substitution spillover model.
9

These ratios summarize the relative importance of shocks in crude oil markets on volatility in
corn markets at different points in time.

1.3

Data and Estimation
Weekly data are used in the study, covering the period from January 2, 1992 to June 30,

2009. All price data are the mid-week closing price (Thursday) and include 883 observations.
Corn cash price p t is the average of the low and high bids for #2 yellow corn from mid-states
Terminals, Toledo, Ohio. The prices were obtained from quotes in the Livestock and Grain
4

Market News Portal, USDA. The corn futures price f t is for the nearest expiration contract on
5

the CBOT. The crude oil futures price ot is for the nearest expiration contract on light, sweet
crude oil traded on NYMEX. Fuel ethanol consumption data (in thousands of gallons) are
obtained from the energy review of the Energy Information Administration (EIA). Gasoline
consumption refers to U.S. total gasoline sales/deliveries of prime suppliers and these data are
from the Petroleum Marketing Monthly of EIA. Figure 1.1 plots the movement of the event
dummy and the ethanol-gasoline consumption ratio variables. The ratio exhibits a strong upward
trend with an average increase of 16.84%. Since ratios were higher during the latter half of the
sample, its correlation coefficient with the event dummy is fairly high (0.87). The presence of
such co-movement should not be surprising since the Act also drives the energy substitution ratio.

4

The last trade date for corn futures contracts is the business day prior to the 15th calendar day
of the contract month.
5
For crude oil, each contract expires on the third business day prior to the 25th calendar day of
the month preceding the delivery month. If the 25th calendar day of the month is a non-business
day, trading ceases on the third business day prior to the business day preceding the 25th
calendar day. After a contract expires, the nearest expiration contract for the remainder of that
calendar month is the second following month.
10

Figure 1.1. Movement of the Event Dummy and the Ethanol-Gasoline Consumption Ratio
Variables (―For interpretation of the references to color in this and all other figures, the reader is
referred to the electronic version of this dissertation.‖)
9
8
7
6
5
4
3
2
1
0
92

94

96

98

00

Dummy

02

04

06

08

Ratio(percent)

Table 1.1 presents summary statistics for weekly price series over the sample period and
for three sub-periods, separated by two important dates, the first being July 29, 2005 when the
Energy Policy Act of 2005 was passed, and the second being July 3, 2008 when the crude oil
price climbed to its historical record high. The correlation matrices show that the connection
between corn prices and crude oil prices changes from a weak negative into a strong positive
relationship. The skewness and excess kurtosis measures show that the price change distributions
are asymmetric and fat-tailed. Formal testing rejects the null hypothesis of unconditional
normality at the 5% level of significance.

6

6

The Shapiro–Wilk test is used to test the null hypotheses that each sample series of price
changes is normally distributed. The test statistics for the change of corn cash, corn futures, and
crude oil futures prices are 0.87, 0.84 and 0.86, respectively. The p-values show all the null
hypotheses are rejected at the 5% significant level.
11

Table 1.1. Descriptive Statistics and Correlation Analysis
Descriptive Statistics
Correlation
Jan2, 1992- Jul 28, 2005 (689 obs.)
pt
pt
ft
ot
ft
ot
Mean
1.0000
pt
2.3980
2.4695
25.0246
Std. dev.
0.9809
1.0000
ft
0.5950
0.5369
10.0282
Min
-0.1985
-0.2099
1.0000
ot
1.5800
1.7625
10.7200
Max
5.4300
5.4000
67.4900
Aug 4, 2005-Jul 3, 2008(150 obs.)
pt
pt
ft
ot
ft
ot
Mean
pt
3.4136
3.5877
77.6336
1.0000
Std. dev.
ft
1.3864
1.3927
22.0097
0.9965
1.0000
Min
ot
1.5250
1.8825
50.4800
0.8183
0.8271
1.0000
Max
7.1950
7.5375 145.2900
Jul 10, 2008-Jun 30, 2009(44 obs.)
pt
pt
ft
ot
ft
ot
Mean
pt
4.0865
4.1960
65.1375
1.0000
Std. dev.
ft
0.6022
0.6898
25.4375
0.9903
1.0000
Min
ot
3.1800
3.1825
33.9800
0.8968
0.9263
1.0000
Max
5.7050
5.9775 121.1800
Full Sample (883 obs.)
pt
pt
ft
ot
ft
ot
Mean
1.0000
pt
2.6546
2.7455
35.9604
Std. dev.
ft
0.9329
0.9278
24.9612
0.9904
1.0000
Min
1.0000
ot
1.5250
1.7625
10.7200
0.5849
0.6318
Max
7.1950
7.5375 145.2900
Skewness
-1.0072
-0.9383
-0.9976
Kurtosis
12.9611
13.0195
12.9390
Notes: skewness and kurtosis are presented for the weekly change in the price series.

12

As with most weekly asset price data, Augmented Dickey-Fuller tests and Phillips-Perron
tests could not reject the unit root hypothesis in any of the three price series. Johansen tests with
and without trend specifications show that there is no evidence of cointegration between crude
oil futures and either corn cash or futures prices, which suggests no long-run equilibrium
relationship between crude oil prices and corn prices. Johansen tests also suggest a cointegrating
relationship of order one between corn cash and futures prices, as confirmed in other research
(Beck, 1994). The exact form of the Johansen cointegration test depends on the specification of
deterministic components of the system. Here I include a linear trend to account for carrying
charges and changes in convenience yield over time to maturity. Table 1.2 presents these
Johansen cointegration test results. Neither the trace statistics nor the maximum eigenvalue
7

statistics in the first difference reject one cointegrating relationship at the 5% level. An error
correction model is therefore chosen for the mean equations for corn cash and futures prices.
Table 1.2. Johansen Cointegration Test
Null ( H 0 )

Alternative ( H1 )

max

trace

Corn cash and futures
Rank = 0
Rank ≥ 1
27.08**
33.32**
Rank ≤ 1
Rank ≥ 2
6.24
6.24
Corn cash and crude oil futures
Rank = 0
Rank ≥ 1
17.82
11.16
Rank ≤ 1
Rank ≥ 2
6.65
6.65
Corn futures and crude oil futures
Rank = 0
Rank ≥ 1
18.20
11.46
Rank ≤ 1
Rank ≥ 2
6.74
6.74
Note: ** denotes rejection of the null hypothesis at the 5% significance levels. max critical
values at the 5% significant level under the two null hypotheses are 18.96 and 12.52, while
trace critical values at the same level are 25.32 and 12.25, respectively.

7

Under other trend specifications, the two statistics in the first difference do not reject one
cointegrating rank at the 5% level, either.
13

Given the large number of parameters that would have to be estimated in the spillover
model, I follow a two-step procedure similar to that implemented by Bekaert and Harvey (1997),
8

Ng (2000), and Baele (2005). In the first step the vector error correction model (equation (1.4))
is estimated to obtain estimates of the forecast errors ( o, t ,  p, t ,  f , t ) for oil futures, corn cash,
and futures prices. In the second step, the first stage residuals are used as data in joint maximum
likelihood estimation of the trivariate spillover model (Equations (1.2), (1.7) and (1.8)),
assuming the purely idiosyncratic shock vector has a bivariate conditional normal distribution
with mean zero and time-varying variance matrix. I report the Bollerslev-Wooldridge robust
quasi-maximum likelihood covariance matrix, which is robust to misspecification of the
distribution of the error term. For the event or substitution spillover model, equations (1.11)(1.12) or equations (1.13)-(1.14) are substituted into equation (1.2) and the same maximum
likelihood estimation procedure is implemented.
Based on results from the unit root and cointegration analysis, equation (1.4) is estimated
by restricting one cointegration relationship between corn cash and corn futures prices and zero
11
adjustment parameter for the oil price ( 1  0 ). The estimation is implemented by Engel-

Granger two-step procedure to VEC(10) which renders all residuals white noise and results are
given in Table 1.3. The Q-statistics at lags from 1 to 53 confirm that there is no significant
autocorrelation at the 1% level in the residuals. The results show that the crude oil returns series
has statistically significant autocorrelations at lags 1, 4, and 8. The Wald statistic for the joint
significance of all lagged oil price changes indicates that they have significant explanatory power
for future oil price returns. These results suggest that the oil futures price does not follow a
8

It may be more desirable to estimate the system jointly, but this is often not feasible due to the
large number of parameters to be estimated in spillover models. The two-step method yields
consistent but not necessarily efficient estimates (Ng, 2000).
14

martingale. Sadorsky (2002) suggests this is due to futures markets not being unbiased due to the
existence of a risk premium. Also, it is found that the corn futures price has a similar
characteristic. In the last row of the table, I also give results for estimation of the cointegration
relationship between corn cash and futures prices.
Table 1.3. Vector Error Correction Model for Oil Futures, Corn Cash, and Futures Prices
 o, t 
ot    o  10
ot  i   0 




p        p     ECT   
 p, t 
 t   p   i  t  i   12 


 
i 1
f t    
f t  i   13 

  f 


 f , t 



ot

Coe.

t statistic

pt

Coe.

t statistic

f t

Coe.

t statistic

o

0.0339

0.4716  p

0.0013

0.2966  f

0.0010

0.2337

1,1
1

-0.0794

2,1
-2.1819 1

-0.0065

3,1
-3.0305 1

-0.0073

-3.3807

1,1
2

0.0698

-0.0042

-1.9086

1,1
3

-0.0344

0.0021

0.9531

1,1
4

0.1026

0.0015

0.7016

0.0010

0.4729

0.0000

-0.0133

0.0044

2.0530

-0.0009

-0.3986

-0.0004

-0.2024

0.0001

0.0251

0.3230

3.9047

-0.3748

-4.4816

0.0536

0.6447

0.2500

2.9996

-0.0708

-0.8429

-0.0257

-0.3059

1,1
5
1,1
6
1,1
7
1,1
8
1,1
9
1,1
10
1,2
1
1,2
2
1, 2
3
1,2
4
1, 2
5
1,2
6

0.0107
-0.0204
0.0556
0.1041
0.0121

2,1
1.9076  2
2,1
-0.9416  3
2,1
2.8190  4
2,1
0.2922  5
2,1
-0.5615  6
2,1
1.5345  7
2,1
2.8514  8
2,1
0.3351  9

-0.0101

-0.2802

-1.9699

-1.4182

-0.8845

-0.6300

-1.9306

-1.3842

0.5977

0.4270

0.8436

0.5986

-3.3482

-2.3739

2,
101
2
1 , 2
 2,2
2
2
 3,2
 2,2
4
2
 5,2
2
 6, 2

-0.0047
0.0022
0.0018
-0.0007
0.0013
0.0029
-0.0013
-0.0030

3,1
-2.1765  2
3,1
1.0219  3
3,1
0.8549  4
3,1
-0.3343  5
3,1
0.6085  6
3,1
1.3760  7
3,1
-0.5870  8
3,1
-1.4304  9

-0.0004

-0.1847

0.3988

4.8895

-0.3411

-4.1373

-0.0470

-0.5743

0.2922

3.5555

0.0006

0.0067

0.0616

0.7438

15

3,1
10
3
1 , 2
 3,2
2
3, 2
3
 3,2
4
3, 2
5
 3,2
6

Table 1.3. (Cont’d)
Coe.
ot
1,2
7

0.7887

1,2
8
1, 2
9
1,2
10
1,3
1
1,3
2
1,3
3
1,3
4
1,3
5
1,3
6
1,3
7
1,3
8
1,3
9
1,3
10

pt

Coe.

2,2
0.5557  7

0.0026

t statistic

-0.9950
2.4941

-1.4420

4.3851

3.1326

2.4240

1.7130

2.8348

2.0042

-1.1809

-0.8359

0.9783

0.6899

3.9776

2.8076

0.4389

0.3068

2.2964

1.6324

-3.8311

-2.7160

3.9367

ft
Trend
Cons.

1.7713

-2.0340

pt

-0.7084

2.7638

Coe.
1.0202
-0.0002
-0.0811

2
 8 ,2
2
 9,2
2,
102
2
1 ,3
 2,3
2
2
 3,3
 2,3
4
2
 5,3
2
 6,3
2
 7,3
2
 8 ,3
2
 9,3
2,
103
12
1

f t

Coe.

t statistic

3, 2
0.0311  7

-0.0546

-0.6463

-0.0675

-0.8072

0.1837

2.1900

-0.0968

-1.1524

-0.2125

-2.5490

0.4179

4.9581

-0.0288

-0.3414

-0.2290

-2.7215

0.0903

1.0686

-0.0252

-0.2989

0.0371

0.4353

0.1703

2.0325

-0.1985

-2.3630

0.2600

3.0651

-0.0015

-0.3532

t statistic

-0.1911

-2.3174

0.1386

1.6767

-0.1369

-1.6529

-0.3186

-3.8766

0.3999

4.8137

0.0514

0.6187

-0.2653

-3.1986

0.0444

0.5336

-0.1222

-1.4689

0.0110

0.1307

0.2482

3.0050

-0.1201

-1.4497

0.3006

3.5945

-0.0102

-2.4647

3
 8,2
 3,2
9
3,
102
3
1 ,3
 3,3
2
3,3
3
 3,3
4
3,3
5
 3,3
6
 3,3
7
3,3
8
 3,3
9
3,
103
13
1

t statistic
158.7372
-8.4505
-4.5569

Note:  ij is the element in the i-th row and the j-th column of  l ( l  1,,10 )
l

1.4

Spillover Model Results
The spillover models are estimated under different spillover parameterizations. This

section presents and discusses the estimation results with a BEKK (1, 1) process for H t .

16

1.4.1 The Constant Spillover Model
I first constrain all spillover parameters to be constant over time. The estimation results
9

are presented in Table 1.4. First, D(1,1) is negative and statistically significant at the 1% level,
which supports the existence of asymmetric volatility in the corn cash price. Second, all elements
in matrix G are insignificant at the 5% level, which suggests no time-to-maturity effects in corn
price volatilities. Third, estimated coefficients in the ARCH and GARCH matrices (A and B) are
also significant and have large magnitudes, which suggests volatility persistence.
In the spillover functions I find significant volatility spillover effects from crude oil
prices to corn prices (  and  are positive and statistically significant at the 1% level). The
positive sign shows evidence of increased volatility of corn prices due to spillovers from crude
oil prices. Robust Wald tests reject the joint hypothesis of no spillover effects, H 0 :     0 , at
the 1% level. Also, results do not reject the null hypothesis, H 0 :    , at the 1% level,
suggesting that volatilities for corn cash and futures prices are influenced by crude oil prices to
the same extent.

9

Test results show that the estimated BEKK model captures all dynamics in the means and
variances of corn prices. The estimated residuals are serially uncorrelated and have no GARCH
effects. The null hypothesis that the Ljung-Box Q-statistics of the standardized residuals and the
squared standardized residuals are equal to zero is not rejected.
17

Table 1.4. Estimation of the Constant Volatility Spillover Model
 o, t   1 0 0 eo, t 

 


 p, t    t 1 0 e p, t  ,

  0 1 

 e f , t 




 f , t   t

[e p, t , e f , t ]' I t 1 ~ N (0, H t ) ,
H t  C ' C  A' et 1et' 1 A  B' H t 1B  D't 1t' 1D  G  zt .



C(1,1)
C(1,2)
C(2,2)
A(1,1)
A(1,2)
A(2,2)
D(1,1)
D(1,2)
D(2,2)
B(1,1)
B(1,2)
B(2,2)
G(1,1)
G(1,2)
G(2,2)
Log-likelihood
SBC

Coefficient
0.0088
0.0099
0.0003
0.0003
0.0003
0.0881
0.0746
0.0900
-0.0475
-0.0253
-0.0088
0.9084
0.9040
0.8769
0.0000
0.0000
0.0000

Std. Error
0.0014
0.0014
0.0001
0.0001
0.0001
0.0120
0.0100
0.0132
0.0133
0.0133
0.0175
0.0116
0.0116
0.0148
0.0000
0.0000
0.0000
2342.4140
-5.2405

z-Statistic
6.3661
6.8722
2.5410
2.5391
1.9556
7.3203
7.4230
6.8326
-3.5655
-1.9053
-0.5053
78.2287
77.6592
59.1153
-1.0406
-0.7638
0.2262

p-value
0.0000
0.0000
0.0111
0.0111
0.0505
0.0000
0.0000
0.0000
0.0004
0.0567
0.6133
0.0000
0.0000
0.0000
0.2981
0.4450
0.8210

Before estimating the time series of the spillover ratios from equations (1.17)-(1.18), I
first estimate the conditional variance of the crude oil shock from equations (1.5) and (1.6). The
results of the TGARCH(1,1) model are presented in Table 1.5. As expected, I find time-tomaturity effects and seasonality. But there are no significant asymmetric effects in the volatility
of crude oil futures prices. On average over the sample period, the crude oil volatility spillover
effect makes up from 0 to 23.572% of the conditional variance of the corn cash market shocks,
while it makes up a bigger proportion (between 0 and 30.52%) of the corn futures prices (Table
18

1.6). The mean energy spillover ratios are 2.89% and 3.49% for corn cash and futures prices,
respectively.
Table 1.5. Univariate TGARCH (1, 1) Models for Crude Oil Futures Price
51

2
2
2
2
 o, t   0  1eo, t 1  1d t 1eo, t 1   1 o, t 1   0 zt    i Qit
i 1

Coefficient

0
1
1
1
0

Std. Error

z-Statistic

3.8134
0.2161
0.1172
0.5793

2.3882
0.0470
0.0668
0.0566

1.5968
4.5965
1.7548
10.2351

p-value
0.1103
0.0000
0.0793
0.0000

-0.0422
0.0113
-3.7397
0.0002
Note: seasonal variables are weekly dummies with the total number of 51 and their coefficient
estimates are not reported in the table to save space.

Table 1.6. Summary Statistics of Spillover Ratio for Three Models
Constant Spillover
Event Spillover
Sub. Spillover
Before the Act
After the Act
Cash Futures
Cash Futures
Cash Futures
Cash Futures
Mean
0.0289
0.0349 0.0108 0.0170 0.0953 0.0940 0.0310 0.0344
Median
0.0211
0.0267 0.0078 0.0127 0.0812 0.0780 0.0040 0.0096
Maximum 0.2357
0.3052 0.1068 0.1821 0.2780 0.2884 0.5656 0.4834
Minimum
0.0001
0.0001 0.0000 0.0000 0.0124 0.0100 0.0000 0.0000
Std. Dev.
0.0242
0.0290 0.0100 0.0158 0.0563 0.0550 0.0609 0.0554
In sum, the volatility spillover effect from crude oil prices is statistically significant but
accounts for a relatively small proportion of total volatility in corn prices. Figure 1.2 shows the
time series evolution of spillover ratios for the corn market.

10

There are some relatively high

spillovers after 2005. However, the overall spillover effect throughout the sample period is
relatively small in many periods.

10

Spillover ratios spike in early 2003. This abrupt shock captures the impact of the Iraq War.
19

Figure 1.2. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Constant
Spillover Model
.24

.20

SRP

.16

.12

.08

.04

.00
92

94

96

98

00

02

04

06

08

92

94

96

98

00

02

04

06

08

.25

.20

SRF

.15

.10

.05

.00

20

1.4.2 The Event Spillover Model
The event spillover model accounts for changes brought about by the introduction of the
Energy Policy Act of 2005, allowing spillover parameters to take on different values before and
after the event. The estimation results are presented in Table 1.7. Results show that for the
sample period before the Act was introduced energy volatility spillovers (  0 and  0 ) are
significant at conventional significance levels. However, results also point to a stronger positive
spillover effect after the introduction of the Act. Both 1 and 1 are statistically significant at
the 5% level. The Wald test of the hypothesis H 0 : 1  1 cannot be rejected, suggesting the
crude oil market has similar spillover effects on corn cash and futures markets. It is interesting to
see that the introduction of the Act dramatically increased the connection between the crude oil
market and the corn market. The crude oil price now plays a more important role in corn price
movements. As a result, corn producers are now being exposed to an additional source of
uncertainty from energy spillovers. A closer examination shows that the average spillover ratio
in the cash corn market before the introduction of the Act was 1.08% whereas it became 9.53%
thereafter, a jump of more than 8 times the original level (Table 1.6). The spillover ratio for corn
futures was also 1.70% before the introduction but increased to 9.40% thereafter, a jump of more
than 5 times. Figure 1.3 clearly shows the dramatic changes. The spillover ratio gets close to 28%
at times after the introduction of the Act.

21

Table 1.7. Estimation of the Event Spillover Effect Model
 o, t   1 0 0 eo, t 

 


 p, t    t 1 0 e p, t  ,

  0 1 

 e f , t 



 f , t   t

t   0  1Dt ,
t  0  1Dt ,

[e p, t , e f , t ]' I t 1 ~ N (0, H t ) ,
H t  C ' C  A' et 1et' 1 A  B' H t 1B  D't 1t' 1D  G  zt .

0
1
0
1
C(1,1)
C(1,2)
C(2,2)
A(1,1)
A(1,2)
A(2,2)
D(1,1)
D(1,2)
D(2,2)
B(1,1)
B(1,2)
B(2,2)
G(1,1)
G(1,2)
G(2,2)
Log-likelihood
SBC

Coefficient
0.0054
0.0098
0.0071
0.0081
0.0003
0.0003
0.0002
0.0851
0.0717
0.0865
-0.0443
-0.0243
-0.0063
0.9087
0.9058
0.8784
0.0000
0.0000
0.0000

Std. Error
0.0022
0.0045
0.0022
0.0042
0.0001
0.0003
0.0004
0.2021
0.2624
0.3264
0.4350
0.5588
0.7081
0.0396
0.0250
0.0164
0.0000
0.0000
0.0000
2345.953.464
-5.2331

22

z-Statistic
2.4790
2.1699
3.1814
1.9417
2.1107
1.0262
0.6040
0.4211
0.2733
0.2651
-0.1018
-0.0434
-0.0089
22.9462
36.2360
53.6951
-0.4593
-0.1615
0.2354

p-value
0.0132
0.0300
0.0015
0.0522
0.0348
0.3048
0.5459
0.6737
0.7846
0.7910
0.9189
0.9654
0.9929
0.0000
0.0000
0.0000
0.6460
0.8717
0.8139

Figure 1.3. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Event
Spillover Model
.30

.25

SRP

.20

.15

.10

.05

.00
92

94

96

98

00

02

04

06

08

92

94

96

98

00

02

04

06

08

.30

.25

SRF

.20

.15

.10

.05

.00

23

1.4.3 The Substitution Spillover Model
The substitution spillover model investigates the time-varying impact of the fuel
consumption ratio on volatility spillovers. Results in Table 1.8 provide insights into how the
consumption ratio has impacted volatility spillovers into the corn market. The results show that
the crude oil price does not always have significant spillover effects on corn prices. The sign and
size of the spillover effect depends on the magnitude of the consumption substitution ratio.
Parameter estimates for  0 and  0 are not significantly different from zero, while 1 and 1
are positive and significant at the 1% level. A higher consumption ratio implies that ethanol has
become a more important substitute for gasoline, and thus the connection between oil and corn
markets increases. Looking at the most recent portion of the sample (Figure 1.4), the spillover
remained in the high range in 2008 and spiked at more than 48% when the consumption ratio
rose to close to 8%. It dipped to an average of 15% when the consumption ratio decreased after
2009 during the deep economic recession. At low consumption ratios, the connection between
crude oil and corn markets is essentially cut off as biofuel-based demand for corn is sluggish, but
when the consumption ratio reached its maximum value, crude oil prices accounted for more
than 40% of total corn price volatility (Figure 1.4), which suggests that crude oil markets have
become a much more important factor in explaining corn price movements during these periods.
It is clear that the relative influence of the crude oil market varies with consumption ratios and
changes over time.
I also go one step further to estimate a more general spillover model that nests the event
spillover and consumption substitution specifications. That is, the spillover parameters are
allowed to vary with both the event dummy and the consumption substitution ratio variables. The
results (they are not reported to save space) show that the event dummy coefficient estimates are

24

insignificant while the consumption ratio coefficient estimates are positive and significant at the
1% level. This is not surprising given that the high correlation between two variables. The
consumption substitution ratio captures the effect of the Energy Policy Act of 2005 so that
adding the event dummy variable to the substitution model does not increase explanatory power.
Furthermore, the fit of the model does not improve, as shown by an increased Schwartz Bayesian
Criterion (SBC).
Table 1.8. Estimation of the Substitution Spillover Effect Model

 o, t   1 0 0 eo, t 

 


 p, t    t 1 0 e p, t  ,

 


 f , t  t 0 1 e f , t 




t   0  1Rt 1 , t  0  1Rt 1 ,
[e p, t , e f , t ]' I t 1 ~ N (0, H t ) ,
H t  C ' C  A' et 1et' 1 A  B' H t 1B  D't 1t' 1D  G  zt .

Coefficient

0
1
0
1
C(1,1)
C(1,2)
C(2,2)
A(1,1)
A(1,2)
A(2,2)
D(1,1)
D(1,2)
D(2,2)
B(1,1)
B(1,2)
B(2,2)
G(1,1)
G(1,2)
G(2,2)
Log-likelihood
SBC

Std. Error

z-Statistic

p-value

-0.0006
0.0032
0.0021
0.0027
0.0002
0.0001
0.0001
0.0793
0.0666
0.0803
-0.0562
-0.0423
-0.0334
0.9242
0.9258
0.9035
0.0000
0.0000
0.0000

0.0028
0.0006
0.0028
0.0005
0.0001
0.0001
0.0001
0.0113
0.0085
0.0098
0.0129
0.0122
0.0150
0.0101
0.0082
0.0095
0.0000
0.0000
0.0000
2349.7980
-5.2419

-0.2058
5.1188
0.7604
5.1931
1.4461
1.3917
1.1897
7.0242
7.8549
8.2059
-4.3486
-3.4604
-2.2303
91.8567
112.4387
94.9563
0.0191
0.3959
0.9785

0.8370
0.0000
0.4470
0.0000
0.1481
0.1640
0.2341
0.0000
0.0000
0.0000
0.0000
0.0005
0.0257
0.0000
0.0000
0.0000
0.9847
0.6922
0.3278

25

Figure 1.4. Spillover Ratios for Corn Cash (SRP) and Futures (SRF) Markets – the Substitution
Spillover Model
.6

.5

SRP

.4

.3

.2

.1

.0
92

94

96

98

00

02

04

06

08

92

94

96

98

00

02

04

06

08

.5

.4

SRF

.3

.2

.1

.0

26

1.5

Implications for Hedging Strategies

1.5.1 Optimal Hedging
The additional source of uncertainty resulting from the strong connection between crude
oil and corn prices during some periods may present a new challenge to corn market participants.
Producers and traders may need to re-evaluate their risk management strategy to cope with this
additional source of risk. Naturally, hedgers may be tempted to exploit the information linkage
between crude oil and corn prices by examining a cross hedge strategy of simultaneously using
corn futures and crude oil futures to hedge cash corn price risk. This section examines whether
such a strategy would significantly improve hedging performance.
Consider an investor with a fixed corn cash position who wishes to hedge this cash
position in both corn and crude oil futures markets. I use a one-period portfolio selection
framework. Assume the investor aims to maximize expected utility from the end-of-period
portfolio profit:
(1.19) max bt 1 , at 1 E[U ( t ) I t 1 ]
subject to
(1.20)  t  ( pt  pt 1 )qt 1  ( f t  f t 1 )bt 1  (ot  ot 1 )at 1
where  t is end-of-period profit; initial and end-of-period prices (corn cash, corn and oil futures)
are subscripted by t-1 and t, respectively; qt 1 is a fixed quantity of cash corn purchased; and

bt 1 and at 1 are the quantities of corn and crude oil futures purchased (sold if negative).
Assuming the joint distribution of ( t , pt , f t , ot ) conditional on I t 1 is multivariate normal, the
optimal hedge ratio of the cross hedge can be derived as (see appendix A for details):
q 
 b
1
(1.21)  t 1 t 1  
 at 1 qt 1  mt22 mt33  mt23 mt32

27

m12 mt33  m13 mt32 
t
 t
,
m13 mt22  m12 mt23 
t
 t


where mtij is the element in the i-th row and the j-th column of M t , the conditional covariance
matrix of ( pt , f t , ot )' .
Equations (1.16) and (1.6) specify models for mt22 and mt33 . From the spillover effect
model, I can also infer other elements of the covariance matrix:
2
(1.22) m13  E ( p, t  o, t I t 1 )  t  o, t ,
t
2
(1.23) mt23  E ( f , t  o, t I t 1 )  t  o, t ,
12
2
(1.24) m12  E ( p, t  f , t I t 1 )  H t  t t  o, t .
t

The resulting covariance matrix M t is used to compute the optimal hedge ratio vector for cross
hedging.

1.5.2 Comparisons of Hedging Performance
In order to compare the performance of different hedging strategies, I first break the full
sample into two parts for purposes of in-sample and out-of-sample evaluation. In-sample
evaluation covers the period from January 2, 1992 to March 22, 2007. Out-of-sample data are
from March 29, 2007 to June 30, 2009, and includes 100 observations. Assume an investor holds
one bushel of cash corn (i.e., qt 1 = 1) continuously over the sample period, which is hedged by
selling nearest expiration futures contracts of corn and crude oil. The futures position is adjusted
on a weekly basis conditional on all past information. As the futures contract matures, futures
positions are rolled over into the next expiration contract.

11

11

Performance is evaluated in terms of

Rolling cost is not considered in the performance evaluation because all hedging strategies
involve contract rolling.
28

the effects on the variance of the investor's portfolio return. The performance comparisons are
conducted under three different hedging rules: no hedge, traditional hedge with only corn
futures,

12

and the cross hedge using corn and crude oil. For out-of-sample performance

evaluation, I implement dynamic forecasts. That is, the model is re-estimated each time when a
new data point is observed and included in the estimation, and the optimal hedge ratio is then recalculated. This process continues until 100 forecasts have been generated.
Ex-ante evaluations of hedge performance are used to construct the ex-ante conditional
standard deviation of portfolio returns for each date based on the spillover model.

13

I compute

the percentage reduction in the conditional standard deviation under two hedging rules, the
traditional hedge and the cross hedge, all compared to a no-hedge outcome. The average
percentage reduction is computed over the sample period.

14

In the in-sample case, the cross

hedge reduces the conditional standard deviation by 53.10% compared to no hedging while the
traditional hedge provides similar hedging performance, reducing the conditional standard
deviation by 51.87%. Since traders are more concerned with the future performance than past,
out-of-sample forecasts are a more useful way of analyzing hedging performance and are of
more interest (Haigh and Holt, 2002). In the out-of-sample case, the cross hedge reduces the
conditional standard deviation by 48.10%, slightly better than the traditional hedge performance
reduction of 47.95% (see Table 1.9). Therefore I conclude that the cross hedge strategy performs
marginally better compared to the traditional hedge with only corn futures.
12

I use the spillover empirical model to estimate the components of the traditional hedge
formula. The formula can be found in Myers (1991).
13
The computations are based on the constant spillover model. Similar results can be obtained
from the event and substitution spillover models.
14
I did not compare the ex-ante conditional means of return because they are the same, which is
implied by the assumption that the expected return from holding futures is zero (Myers, 1991).
29

Ex-post evaluations of hedge performance are undertaken by constructing the actual expost portfolio returns using observed prices and comparing their standard deviations over the
relevant period. In the in-sample case, the cross hedge performs slightly better than any other
strategy in terms of risk reduction. The standard deviation falls 42.22% under a cross hedge,
while the traditional hedge provides a 42.21% reduction. In the out-of-sample case, cross
hedging offers a little more risk reduction (72.15%) than the traditional hedge (71.93%).
Similarly, ex-post performance of cross hedge is also slightly improved.
Table 1.9. In- and Out-of-sample Hedging Performance
In-sample
Out-of-sample
s. d.
s. d.
Ex-ante
No hedge
8.5539
14.3200
Biv. hedge 4.1171(-51.87%)
7.4530(-47.95%)
Cross hedge 4.0121(-53.10%)
7.4321(-48.10%)
Ex-post
No hedge
9.7713
28.1070
Biv. hedge 5.6460(-42.21%)
7.8886(-71.93%)
Cross hedge 5.6457(-42.22%)
7.8293(-72.15%)
Note: all standard deviations are scaled by 0.01. Numbers in parentheses are percentage
deviations from the no hedge strategy.

Our results show that a cross hedging strategy using oil and corn futures contributes only
a small improvement over hedging based solely on corn futures. However, this is exactly what I
would expect if the corn futures market is efficient and incorporates all relevant information,
including information contained in oil prices. The fact that crude oil and corn prices are related,
and there is volatility spillover between the markets, does not necessarily distort the relationship
between corn cash and futures prices which may (and should, if the market is operating
efficiently) continue to be strongly correlated. Cross hedging would only be expected to result in
improved hedging performance if corn futures price movements deviate significantly from corn
cash price movements, so that corn futures alone become an imperfect hedging vehicle for corn
30

cash price risk. So I view the fact that cross hedging does not improve hedging performance,
despite the connection and volatility spillover between oil and corn prices, as evidence that the
corn futures market continues to operate efficiently and effectively. In short, the futures market is
not detached from the cash market in a time of increased volatility from the biofuel boom.
Despite the fact that cross hedging does not improve hedging performance our results do
show that with the increasing use and production of biofuel, a stronger connection between crude
oil and corn markets is being established, particularly in times of high oil prices and relatively
high ethanol use. The Energy Independence and Security Act (EISA) of 2007 has mandated
increased biofuel production (36 billion gallons by 2022) so the connection between crude oil
and corn markets will likely continue to grow in the future, the implications of which deserve to
be examined further. In addition, understanding the nature of the spillover and its pattern is
critical for forecasting price movements, for production and investment planning, and for risk
management in a broader context beyond hedging portfolio management.

1.6

Conclusions
This paper investigates the magnitude and changing nature of volatility spillover effects

from crude oil markets to corn markets. I find that volatility spillovers from crude oil prices are
significant and have similar impacts on corn cash and futures prices. The relative importance of
spillovers jumps significantly after the introduction of the Energy Policy Act of 2005. Also, more
substantial volatility spillovers occur in periods of high ethanol-gasoline consumption ratios. The
emerging connection between energy and corn prices is exposing agriculture to additional risk,
which implies agricultural risk management may be more costly and potentially calls for new
strategies for risk management. Given the statistically significant spillovers from the crude oil

31

market, corn market participants perhaps should consider pursuing a cross hedge with crude oil.
This study shows that such a cross hedge has slightly better performance than traditional hedging
strategies, but the improvement is not large, which indicates that the corn futures market
continues to provide a reliable, efficient way to manage the risk of cash corn price movements.
The findings in this study suggest that corn market participants can still rely on corn futures
markets to hedge risk and achieve a relatively satisfactory performance, even when there are
significant spillovers from the energy market.

32

APPENDIX A

33

APPENDIX A: DERIVATION OF EQUATION 1.21
The derivation is based on the relation Cov[ g ( x), y]  E[ g ' ( x)]Cov( x, y) , if the random
variables x and y are jointly normally distributed, and g is a differentiable function. Assuming the
joint distribution of ( t , pt , f t , ot ) conditional on I t 1 is multivariate normal, the first order
conditions are

(A.1)

 p 
qt 1 
 t 
U '  fc 
M t bt 1   

,


U ''  t 
at 1 
 f o 


 t 


p

where M t is the covariance matrix of ( pt , f t , ot )' conditional on the information set; and  t ,

 tf and  to are the conditional expected returns from holding cash and futures positions,
respectively. It is often assumed that the expected return to trading futures is approximately zero
f

(  t =  to =0). Then

(A.2)

( M t1 ) 21 
bt 1 
1

 qt 1 ,

a 
 t 1  ( M t1 )11 ( M t1 ) 31 



where ( M t1 ) ij is the element in the i-th row and the j-th column of the inverse covariance
matrix, M t1 .
The optimal hedge ratio is the proportion of the long cash position which should be
covered by a short position in corn and crude oil futures. So the optimal hedge ratio can be
written as,

(A.3)

 bt 1 qt 1 
1
 a
  22 33
 t 1 qt 1  mt mt  mt23 mt32

m12 mt33  m13 mt32 
t
 t

13 22
mt mt  m12 mt23 
t


34

REFERENCES

35

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Bekaert, G., Harvey, C. R., Lumsdaine, R. L. 2002. Dating the integration of world equity
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37

CHAPTER 2: SPILLOVER EFFECTS OF BIOFUEL POLICY ON THE CONSERVATION
PROGRAM

2.1

Introduction
Over the past several decades, public awareness and concerns of environmental issues,

including environmental degradation and global warming, have prompted the U.S. government
to make the environment a priority policy area and promote policies that address public interests
in these areas, particularly in environmental conservation and greenhouse gas emissions. The
Conservation Reserve Program (CRP), enacted in 1985, is by far the most important U.S.
conservation program in terms of scale and budget. The CRP aims to provide environmental
benefits (e.g., to reduce erosion, improve water quality, establish wildlife habitat, and sequester
carbon) through retiring environmentally sensitive cropland from production, and in exchange,
give participants a dependable source of income in the form of CRP land rental payments. The
program’s economic and environmental benefits have been well documented (see, e.g., Young
and Osborn, 1990; Wu, 2000; Wu, Zilberman, and Babcock, 2001; Wu and Lin, 2010). In its first
twenty years of implementation, the program has prevented an estimated 450 million tons of soil
from erosion and sequestered 50 million tons of carbon dioxide per year (USDA-FSA, 2007).
Another policy initiative that has profound environmental and economic impacts is the
2005 and 2007 Energy Acts (2005 Energy Policy Act and 2007 Energy Independence and
Security Act (EISA)), which set a roadmap for bioenergy production and biofuel mandates in the
United States. This policy was designed to mitigate greenhouse gas emissions and to reduce U.S.
dependency on energy import. The policy has created tremendous demand and supply of biofuels
in the country and caused structural changes in agricultural commodity markets and farming
returns (USDA-ERS, 2007; Collins, 2008; Lipsky, 2008; Frank and Garcia, 2010). Supported by
38

ethanol tax credits and mandates on biofuel consumption under the 2005 and 2007 Acts, annual
ethanol production and, consequently, the acreage of corn, the major feedstock of ethanol in the
U.S. have increased rapidly. While the desire to reduce greenhouse gas emission was an
important driver of the production and consumption of biofuels, the ethanol-driven structural
changes in U.S. agriculture have been pressuring lands out of conservation. As of 2009, CRP
enrollment stood at 31.2 million acres, approximately 5 million acres lower than in 2007
(USDA-FSA, 2010). Exiting conservation programs results in increasing greenhouse gas
emissions from land use changes, thus compromising the initial goal of the conservation program
as well as that of the biofuel policy. This research studies the spillover effect of biofuel policy on
conservation participation and the implications of potential policy options, illustrating the nature,
mechanism, and extent of competition and spillover between government policies.
To analyze the effect of biofuel policy, an understanding of landowners’ conservation
participation decision process is important. When making participation decision, landowners
trade off the cost and benefit of participation; the factors, such as annual program land rental
payments, revenue and cost of crop production, all affect landowners’ participation decision (e.g.,
Parks and Kramer, 1995; Lubowski, Plantinga, and Stavins, 2008; Suter, Poe, and Bills, 2008;
Change and Boisvert, 2009). Biofuel production has triggered a sharp increase in the opportunity
cost of giving up agricultural production. Since the Energy Policy Act was passed in 2005,
farming returns have experienced a sharp increase; and agriculture has entered a high growth
stage, departing from the historical equilibrium growth rate. The prospect and persistence of the
biofuel driven agricultural boom have direct impact on landowners’ decisions. Another important
factor that impact participation decision is risk and uncertainty (Capozza and Li, 1994; Dixit and
Pindyck, 1994; Schatzki, 2003; Isik and Yang, 2004); this is particularly important in the current

39

context of ethanol-driven structural changes. Faced with risk and uncertainty, landowners may
choose to wait for more information about future returns and thus delay participation. Though
participation in the CRP offers landowners a stable stream of cash flows from government
payments over the contracted years, once enrolled in the CRP, the land will be locked up for 10
or 15 years, so the participation decision is essentially irreversible during the fixed contract
period. In particular, landowners who participate in the CRP may be losing out economically due
to higher forgone farming returns in the future, especially in an environment of increasing
ethanol production.
In this study I propose a two-stage Gordon growth process to model the structural change
in farming returns and investigate the impact of biofuel production on conservation participation
in the real options decision framework. Traditionally, the uncertain farming return is assumed to
follow a geometric Brownian motion (GBM), in which the location (mean) of the distribution is
assumed to be fixed (see, e.g., Dixit and Pindyck, 1994). Under the biofuel-driven structural
changes, however, the assumption of constant growth of farming returns would be inappropriate.
It is more realistic to assume that the growth rate of agricultural returns in the biofuel era may be
different than in the past. In this study I assume a two-stage process, in which the current high
growth will eventually settle in to a long-term equilibrium growth as the ethanol industry
matures or as a result of policy changes, for instance, repealing of the ethanol tax credit in the
future.
I further assume that the duration of high growth is random due to exogenous policy
uncertainty, for example, repealing ethanol tax credit or removing ethanol import tariff. Unlike
the traditional GBM model’s assumption of a known growth rate which implies that decision
makers have full information over the entire stochastic process, under the regime of two-stage

40

growth and random duration of high growth, I assume decision makers observe only historical
information and make decisions basing on their expectations, which is more realistic and
consistent with the real world decision process. This incomplete information and expectation
formation are particularly relevant at a time of structural change where past observations only
contain partial information about the future growth rate. This parameter uncertainty causes an
extra layer of uncertainty compared to the GBM assumption and has important implications for
decision making. To our knowledge, this issue has not been addressed in modeling land use
decisions. The model I propose in this study assumes a geometric Brownian motion with a
stochastic growth rate (Gennotte, 1986; Brennan, 1998; Xia, 2001; Abasov, 2005), in which
landowners continuously update their expectations, conditional on new information arriving at
the time of the decision. The expectation formation with information updating is modeled using a
non-linear Kalman filtering approach (Kalman, 1960).
Based on the proposed model, I derive thresholds of farming returns (profit per acre)
under different growth expectations that would make land conversion attractive. The model is
then used to investigate the impacts of the structural change and biofuel production on program
participation using Monte Carlo simulations. I further analyze the sensitivity of the land-use
conversion thresholds to changes in the policy and economic environment and examine the
landowner’s response to changes in market conditions, including the expected duration of the
biofuel boom and the uncertainty surrounding future agricultural returns. The comparative static
analysis of conversion thresholds and the Monte Carlo simulations of program participation rates
provide a number of important new insights with significant policy implications.

41

2.2

Landowner’s Decision under Uncertainty
The CRP provides an annual per-acre rental payment to landowners to take highly

erodible or environmentally sensitive cropland out of production. The payment is fixed over the
contracted period once the land is enrolled. The program also provides cost-share assistance to
participants who establish approved resource-conserving vegetative covers on eligible cropland.
The cost-share assistance can be an amount no more than 50 percent of the participants’ costs in
establishing covers. In some cases, other financial incentive payments (e.g. one-time signup
payments) are also offered. Landowners who decide to enroll in the program must enter into a T
year contract (10 or 15 years).
Consider a landowner facing a decision to convert a unit of land from crop production to
conservation.

15

I assume that the participation decision is made in a continuous-time framework.

Defining the farming return as Rt , the expected discounted farming return forgone over the T
year horizon is:
t T

(2.1)

V ( Rt )  Et

 Rt e

 t

dt ,

t

where  is the continuous discount rate.
After participation, the annual rental payment Q is fixed. The expected land rental
payment received over the T years of the CRP contract is:

15

To accommodate the likelihood of increased biofuel production, I limit my discussion to a
single switching decision from cropland to the CRP. After the contract expires, the landowner is
faced with an opposite land conversion decision – stay in the CRP or conversion back to
production. My model could also be extended to address this problem, but such an extension
would be outside the scope of the current study.
42

t T

(2.2)

V (Q) 

 t
 Qe dt  (1  k )C   

Q(1  e   T )

t



 (1  k )C   ,

where C is the total land cover establishment cost when participating in the CRP, k is the
portion paid by the government as an incentive for participation, and  is the additional financial
incentive, e.g., a one-time sign-up incentive payment. This payment is made as soon as the
contract has been signed and approved.
Denote M ( Rt , Q)  V (Q)  V ( Rt ) as the benefit from land conversion at time t. The
minimum requirement for conversion is that M ( Rt , Q) be positive. The conversion decision can
be characterized as an optimal timing problem given uncertain market conditions, in which
landowners choose when to stop waiting and participate in the CRP. The optimal stopping
problem can be represented as:
(2.3)

J ( Rt )  max T Et [e   (T  t ) M ( RT , Q)] ,

which reflects the landowner’s choice of the optimal conversion time, T  t , to maximize the
discounted expected conversion benefit at t.

2.3

Structural Change and Parameter Uncertainty
The participation problem is similar to an American put option: landowners have an

option to put the land into conservation, with discounted total rental payments over the contract
period as the exercise price. Valuation of American put options usually assumes the underlying
asset return follows a geometric Brownian motion (GBM) (Carey and Zilberman, 2002; Isik and
Yang, 2004):
(2.4)

dRt  Rt dt  Rt dWt ,

43

where  is a drift term,  is a constant volatility parameter, and Wt is a standard Brownian
motion, defined on a complete probability space.
This model incorporates the strong assumption that the parameters of the process are
known. That is, all possible sources of uncertainty that affect the farming return are summarized
in the form of a log-normal distribution with a known, constant growth rate  and a volatility
parameter , which, in practice, must be estimated from historical farming returns data. These
assumptions are often plausible. Given the current structural changes in farming returns, however,
these assumptions would be too restrictive and likely do not hold; and assuming a constant,
known farming return growth rate may result in substantial error (Gennotte, 1986).
It is more realistic to assume that the growth rate of agricultural returns in the biofuel era
may be different than in the past, and that landowners’ expectations about growth vary at
different points in time. To capture the impact of structural change on expectations about future
agricultural returns, I assume the growth rate (  ) is stochastic and unobservable. I use a twostage stochastic Gordon growth process to model  t , assuming an initial high growth (  ) stage,
followed by an average growth (  ) stage as the ethanol industry matures and reaches a long-run
equilibrium. The two-stage assumption is reasonable given the biofuel boom, and the anticipated
technological breakthroughs in more cost-effective production of cellulosic ethanol, the second
generation biofuel.
In the generalized farming return model,  t is specified as:
(2.5)

 t   ,  ,    ,

where  and  are the growth rates of the two states. The duration of state  , denoted T ,
follows an exponential distribution with parameter  :

44

(2.6)

prob(T  t )  exp( t ) .

To estimate  t , I use the Kalman filter information updating algorithm, which is a
recursive algorithm that continuously updates model estimates using new information. The
Kalman filter minimizes the estimated error variance and generates more efficient parameter
estimates compared to other estimation procedures (Kalman, 1960). I allow landowners to update
their information and thus learn about the true parameter distribution with each new farming
return realization. At time t , there is an initial prior about the growth rate. As new farming
returns are observed over time, this prior distribution is updated. Liptser and Shiryayev (1977)
derive the basic equation of optimal nonlinear filtering in a partially observable random process.
Specifically, the conditional probability of the growth rate reverting to the long-term
average  is defined as:
(2.7)

pt  prob( t    tR ) .

Based on the information set  tR available to landowners at time t ,

16

landowners form

expectations about the value of  t ,
(2.8)

mt  E ( t tR )  (1  pt )  pt  .

Given parameter uncertainty, the Brownian motion Wt is also unobserved. I define a
process based on the observed farming return Rt and its expected growth rate mt as:
(2.9)

dW t 

1 1
( dRt  mt dt ) .
 Rt

16

In the landowner’s information set, the agricultural return Rt is observable, while its growth
rate  t is not.
45

Liptser and Shiryayev (1977) show that W t is a standard Brownian motion with respect to the
information set  tR . Rearranging (2.9), I obtain:
(2.10) dRt  Rt mt dt  Rt dW t .
Substituting (2.4) into (2.9) and rearranging yields
(2.11) dW t  dWt 

1



( t  mt )dt .

Landowners seek to extract information on the expected growth rate from observed past
returns, and keep updating the expectation when new return realizations come along. The
conditional mean mt evolves according to (see Appendix B for the derivation):
(2.12) dmt   (   mt )dt 

1



(  mt )(mt   )dW t .

Combining (2.3), (2.10), and (2.12), I arrive at the following decision model:


  (T  t )
M ( RT , Q)]
 J ( Rt , mt )  max T Et [e

(2.13) dRt  Rt mt dt  Rt dW t
.

1
dmt   (   mt )dt  (  mt )(mt   )dW t


In this framework, landowners maximize option value J by choosing the optimal time T
to enter into the CRP contract. Landowners continuously update their estimate of the growth rate
of farming returns based on new return realizations. With the uncertain growth rate, there is no
analytical solution to the decision problem. The problem is therefore formulated as a linear
complementarity problem (LCP), and solved numerically using an implicit finite difference
approach (Wilmott, Howison, and Dewynne, 1993).

2.4

Linear Complementarity Problem
46

Under known regularity conditions, there will be a critical return R * for a given initial
expected growth rate, such that participating in the CRP is optimal if Rt  R * , while continuing
in farming is optimal if Rt  R * . The solution to the participation problem involves finding this
threshold R * . Using Ito’s lemma, I derive a partial differential equation in a continuous region
of the farming return (see Appendix C for the derivation):

1
2

J  Rt mt J R   (   mt ) J m  Rt2 2 J RR
(2.14)



1
2 2

,

(  mt ) (mt   ) J mm  Rt (  mt )(mt   ) J Rm  0
2

2

where J denotes the put option value, J R , J RR denotes, respectively, the first and second
derivatives of J with respect to Rt , and J Rm denotes the cross derivative of J with respect to

Rt and mt .
I solve the optimal stopping problem (2.13) numerically using an LCP algorithm. This
technique is often used to solve American options due to the existence and uniqueness of
solutions (Wilmott, Howison, and Dewynne, 1993), but to our knowledge this technique has not
been used previously to solve an optimal land conversion problem with a structural change in the
return process. Define the left-hand side of (2.14) as HJ :
1 2 2


 Rt mt J R   (   mt ) J m  2 Rt  J RR 

,
(2.15) HJ  J  
 1 (  m ) 2 (m   ) 2 J

t
t
mm  Rt (  mt )(mt   ) J Rm 
 2 2



where J is the opportunity cost of holding the option (i.e., the required return for holding it).
The expression within the square parentheses represents the expected return from waiting due to
the change in the option value. Then, the LCP can be specified as:

47

 HJ  0

(2.16)  J  M ( Rt , Q)
.
 HJ  J  M ( R , Q)  0
t

This LCP describes the strategy with regard to holding versus exercising the option. The
inequality HJ  0 states that the required return for holding the option must be at least as great
as the expected return of holding it. If HJ >0, the required return for holding the option exceeds
the expected return, and it is optimal to exercise the option immediately. If HJ =0, the required
return equals the expected return, and the landowner can either hold the option or exercise it,
depending on the second inequality in (2.16). The inequality J  M ( Rt , Q) indicates that the
option value should be at least as large as the intrinsic value of participating immediately in the
CRP. When J  M ( Rt , Q) , holding the option is optimal. If HJ =0 and J  M ( Rt , Q) , then
landowners are indifferent between participation and non-participation. The equality of

HJ  J  M ( Rt , Q)  0 excludes the possibility of both HJ >0 and J  M ( Rt , Q) .
Because of the uncertainty regarding the growth rate of the farming return in our model,
the optimal participation problem does not have a closed-form solution. However, it is possible
to solve the problem numerically. The numerical algorithm for determining the value of the
option involves the discretization of the LCP in (2.16) using an implicit finite difference method
(see Appendix D for the derivation). For a numerical solution of the LCP, I must specify the
boundary conditions.

17

17

In my model, the LCP is time-independent, or of the elliptic type. However, in practice, it
appears that there is no satisfactory method dealing with elliptic LCPs. Following Abasov (2005),
I introduce an artificial time variable into the partial differential inequalities of the elliptic LCP
and solve the resulting parabolic LCP. Therefore, a terminal condition (with respect to the time
variable) is required to implement this method.
48

Boundary condition 1. It can be seen from (2.10) that once Rt reaches zero, it will stay
there because of dRt  0 . Therefore, it is optimal to exercise the option immediately for
landowners, in which case:
(2.17) J (0, mt , t )  M (0, Q) .
Boundary condition 2. At a very high farming return Rmax , the put option is deeply out
of the money and I can simply set the option value equal to zero:

18

(2.18) J ( Rmax, mt , t )  0 .
Boundary condition 3. Since I are searching for a solution on a rectangular domain in the
space of ( Rt , mt ), I apply two other boundary conditions, mmin   and mmax   . In
19

particular, I use boundary condition BC2 on mmin and mmax (Tavella and Randall, 2000).

Terminal condition. The terminal condition follows from the observation that at t  
there is no uncertainty about the true value of the growth rate. As a result, mt becomes a
constant. Hence, J ( Rt , m, ) is the solution of the one-dimensional problem:

 J ( R , m, )  max E [e   (T  t ) M ( R , Q)]

t
T t
T
(2.19) 
,
dRt  Rt mdt  Rt dW t

which can be solved in closed-form. In this case, the put option value will be (see Appendix E
for the derivation):

18

In the numerical implementation, Rmax is set to be three times the average farming return.

19

The boundary conditions proposed by Tavella and Randall (2002) are to apply the pricing
equation itself as a boundary condition, rather than to appeal to other financial arguments. BC1
postulates a linear dependence of the option value on the underlying variable, while BC2
discretizes the drift and volatility with second order one-sided differential operators. BC2 yields
smaller errors when I compute the option value using a finite difference method.
49



 Rt 
*

M ( R , Q) * 


(2.20) J ( Rt , m, )  
R 

M ( Rt , Q)


where Rt* 

2.5





(   m)V (Q)

  1 (1  e  (   m)T )

and  

Rt  R *
Rt  R *

1
1
 (m   2 )  (m   2 ) 2  2  2
2
2

2

.

Data and Parameter Estimation
To analyze the landowners’ decision, I need to first estimate parameters in the proposed

land-use decision model, which include the growth rates and volatility of farming returns,
namely,  ,  , and  . In addition, I also need to derive the distribution parameter  associated
with the high grown duration.

2.5.1 Data on Farming Returns
Assume that a representative landowner in the U.S. is making a decision to convert
croplands to conservation.

20

Annual per-acre cropland returns are calculated as the weighted

average of the net returns of three major crops — corn, soybean, and wheat.

21

The weight is the

percentage of planted acreage for each crop relative to their total planted acreage. Data on
marketing-year-average prices and national level yields used to calculate the value of production
are from the National Agricultural Statistics Services (NASS), USDA. Per-acre crop production
costs are from the Economic Research Service (ERS), USDA, which include operating costs and

20

USDA defines CRP eligible land as ―cropland (including field margins) … physically and
legally capable of being planted in a normal manner to an agricultural commodity‖.
21
In 1975-2009, combined corn, soybean, and wheat acreage on average accounts for 62% of
the U.S. total cropland.
50

allocated overhead, excluding the opportunity cost of land.

22

The net cropland return is the gross

value of production less crop production costs. Government payments are also important to
landowners when making land-use decisions. Therefore, per-acre government payments are also
estimated and included in cropland returns. National direct government payments by program are
available from the ERS, USDA (payments for conservation programs are excluded in the
calculation).

23

Per-acre government payments received for crop production are calculated by

dividing direct government payments by total cropland acreage reported in the Major land Uses
series of the ERS, USDA. The calculated annual cropland returns from 1975 to 2009 form a
sample of 35 observations.

2.5.2 Parameter Estimation
In July 2005, the 2005 Energy Policy Act was passed.

24

As a result, biofuel production

increases considerably from 2006, which led to an increased demand for and higher prices of
agricultural commodities. Recent studies also recognize biofuel production as the major driver of
a structural change of commodity prices and farming returns. Frank and Garcia (2010) identified
22

The cost and return accounts were categorized into cash and economic costs for all
commodities from 1975 to 1994. Beginning 1995, the accounts were revised to conform to
methods recommended by the American Agricultural Economics Association Task Force on
Commodity Costs and Returns, which recommended that the cost and return accounts be divided
into operating costs and allocated overhead costs. To be consistent throughout, I adjusted the
account item ―taxes and insurance‖ from 1975 to 1994 according to the new method. I took the
difference at the transition (between 1994 and 1995); the ratio of the difference to the balance is
assumed to apply to all previous years and was used in the adjustment. The comparison of
USDA’s former and new cost of production estimation methodology can be found at:
http://www.ers.usda.gov/data/costsandreturns/compare.htm.
23
Payments include receipts from deficiency payments, marketing loan gains, indemnity
programs, disaster payments, and production flexibility contract payments, etc.
24
This Act established the Renewable Fuel standard requiring that transportation fuels sold in
the United States contain a minimum amount of renewable fuel.
51

a structural change in 2006 for agricultural commodity prices using data from 1998 to 2009 and
found that the agricultural markets, especially the corn market, have been more energy-driven
since 2006. A report from USDA evaluated the effects of biofuel production on farm income and
predicted that biofuel production would substantially improve net farm income during 20072016 (USDA-ERS, 2007). Given this evidence I assume that a shift of farming returns to high
growth occurred in 2006 and use annual returns before 2006 to estimate the low growth rate  .
When the farming return follows a GBM, the logarithm of the return can be described as:
(2.21) d ln( Rt )  (   0.5 2 )dt  dWt .
To verify whether ln( Rt ) follows the process described above, (2.21) must be
approximated in discrete time. The discretized version of (2.21) can be written as:
(2.22) ln( Rt )  ln( Rt 1 )  (  0.5 2 )t   t t ,
where  t is a normally distributed random variable with mean 0 and variance 1.
I performed an augmented Dickey–Fuller test with a trend on the logarithm of the return
series to investigate whether the data-generating process is nonstationary. The null hypothesis of
unit root cannot be rejected at standard significant levels. Therefore, the analysis will be
undertaken assuming that the farming return follows a GBM. The maximum-likelihood estimates
of the drift  and the variance  2 for farming return Rt will be     0.5 2 and    , where

 is the mean and  is the standard deviation of the series of ln( Rt )  ln( Rt 1 ) . I obtain  =4%
and   0.26 from this estimation.
Based on (2.6), the expected time of staying in the high growth state is 1 /  , after which
the ethanol industry attains equilibrium and agricultural returns revert to their long-term growth
rate. Tokgoz et al. (2008) predict that the long-run equilibrium of the ethanol industry may be
52

achieved in 2016-17. Therefore, I assume that   1 / 7 , which is equivalent to assuming that the
high growth () stage will continue for an average of 7 more years (2010-2016). The high
growth rate  is computed based on annual farming returns from 2006 to 2009 and the report of
USDA Agricultural Projections to 2016 conducted in 2010. This report shows that although
increases in corn-based ethanol production in the United States are projected to slow, the demand
for ethanol remains high. This affects the production, use, and pricing of farm commodities
throughout the sector. Consequently, although net farm income initially declines from the highs
of 2007 and 2008, it remains historically strong and will rebound according to the projections.
With the projected price and yield data for corn, soybean, and wheat, the production costs, and
25

the proportion of planted areas of these crops, I estimate the higher growth rate  to be 8.5%.
For other parameters, a discount rate  of 6% is used,

26

and the government’s cost

share proportion ( k ) for establishing land cover is 50%. The cost C depends on the land cover.
In general, the cost is $60/acre/year. In the ―continuous contract‖ category (e.g., with wetland
and buffer), participants also receive a sign-up incentive payment (  ) equal to $10/acre/year
upon enrollment in the program. The annual rental payment is set at $65 according to recent CRP
reports and statistics.

2.6

Results
The option value, J ( Rt , mt ) , is determined using the numerical method described in

Appendix D. Figure 2.1 plots the value of the option to participate in the CRP under any two
25

Since the number of observations are small, I use the simple average of annual growth rates to
proxy for  .
26
The discount rate is the same as or close to the rate used in the literature (Ince and Moiseyev,
2002; Insley, 2002; De La Torre Ugarte et al., 2003; Insley and Rollins, 2005).
53

different expected growth rates ( mt  0.049 or mt  0.085 ) for the baseline case where

  8.5% ,   4% ,  =1/7, k =0.5, C =60,   0.26 ,   0.06 , T  15 , Q  65 , and   0 .
This figure shows that the option value increases (decreases) as the farming return or its expected
growth rate decreases (increases).
Figure 2.1. Option Value of Land Conversion Opportunity under Two Different Expected
Growth Rates (the Baseline Case)

Table 2.1 shows the option value on the grid. The highlighted cells represent
combinations of the farming return and its expected growth rate, for which the landowner is
indifferent between waiting and participating immediately in the CRP. The numbers below these
highlighted cells give the value of continuing to hold the option to convert the land, while the
numbers in the upper part of the table give the value of participating immediately in the program.
Together, these combinations specify the critical farming return as a function of the expected

54

growth rate. Farming returns lower than the threshold will make the CRP attractive enough for
landowners to stop waiting and participate in the CRP. On the other hand, farming returns higher
than the threshold make the CRP less attractive, and landowners will choose to wait.
Table 2.1. Option Value of Participation in the CRP for the Baseline
0.04 0.0445 0.049 0.0535 0.058 0.0625 0.067 0.0715 0.076 0.0805 0.085
R/m
0.00 612.9

612.9 612.9

612.9 612.9

612.9 612.9

612.9 612.9

612.9 612.9

0.25 609.7

609.6 609.5

609.5 609.4

609.3 609.3

609.2 609.1

609.1 609.0

0.50 606.4

606.3 606.2

606.1 605.9

605.8 605.7

605.5 605.4

605.2 605.1

0.75 603.2

603.0 602.8

602.6 602.4

602.2 602.0

601.8 601.6

601.4 601.2

























30.25 222.0

214.5 207.0

199.3 191.4

183.4 175.3

167.0 158.5

149.9 141.2

30.50 218.7

211.2 203.6

195.8 187.9

179.9 171.7

163.3 154.8

146.1 137.3

30.75 215.5

207.9 200.3

192.4 184.4

176.3 168.0

159.6 151.0

142.3 134.7

31.00 212.3

204.6 196.9

189.0 181.0

172.8 164.4

155.9 147.3

139.4 132.2

31.25 209.0

201.4 193.5

185.6 177.5

169.2 160.8

152.2 144.2

137.0 129.8

31.50 205.8

198.1 190.2

182.2 174.0

165.7 157.2

149.0 141.8

134.7 127.5

31.75 202.6

194.8 186.8

178.7 170.5

162.1 153.9

146.7 139.5

132.4 125.2

32.00 199.3

191.5 183.5

175.3 167.0

158.8 151.6

144.4 137.3

130.1 123.0

32.25 196.1

188.2 180.1

171.9 163.8

156.6 149.4

142.2 135.0

127.9 120.9

32.50 192.9

184.9 176.8

168.8 161.6

154.4 147.2

140.0 132.9

125.8 118.8

32.75 189.6

181.6 173.9

166.7 159.5

152.3 145.1

137.9 130.8

123.7 116.7

33.00 186.4

179.0 171.8

164.6 157.4

150.2 143.0

135.8 128.7

121.7 114.7

33.25 184.1

177.0 169.8

162.6 155.3

148.1 141.0

133.8 126.7

119.7 112.8

























Note: threshold returns under different growth rate expectations can be easily detected from
highlighted cells.

Our main findings can be summarized as follows. 1) The option value decreases when
farming returns Rt and the expected growth rate mt increase, as illustrated in Figure 2.1; the

55

threshold return Rt* declines when the expected growth rate increases, making conversion more
difficult. 2) The option value increases with  (recall that 1 /  represents the average time of
staying in the high growth state), which means that the longer the high growth state is expected
to persist, the lower is the value of the option. This suggests that the threshold at which the
landowner is likely to enroll in the CRP is lower in a time of extended high growth, and hence
landowners are less likely to participate. These results are intuitive: higher farming returns,
higher expected growth rates, and a longer time of remaining in the high growth state all reduce
the incentive for conservation. This is consistent with the observation of reduced participation in
the last few years when farming returns grew rapidly.
The land-use decisions model developed in this study and the decision thresholds
calculated can be used to simulate CRP participation probability. I conducted Monte Carlo
simulations to investigate the enrollment probability for a piece of land with the baseline
characteristics as discussed above. I use the Euler method to simulate the dynamics of farming
return (eq. 2.10) and growth rate (eq. 2.12). The farming return and growth rate dynamics are
simulated over a 5-year horizon from 2010. The initial farming return in 2009 is assumed to be
$104 per acre;

27

and the initial growth rate expectation is set at 7.30%, which is the mean growth

rate for 2009 according to the dynamics of eq. 2.12, implying that the landowner believes that
the growth rate has a 27% chance of reverting to the long-term average according to eq. 2.8. The
conversion thresholds (i.e., the highlighted boundaries in Table 2.1) are used as the decision
criteria: at a certain growth rate expectation, if the simulated farming return is lower than the

27

Since the land eligible for CRP participation generally has lower productivity, the initial
farming return for CRP land is the general cropland return multiplied by a coefficient of 0.68,
which is the average ratio of rental payments of CRP land to the cash rent of the general cropland,
assuming proportional downward adjustment.
56

threshold anytime in the 5-year horizon, the land would be enrolled into the conservation
program but otherwise would stay in farming. The results of 20,000 Monte Carlo replications
show that in 5 years the land with the baseline characteristics discussed would have a 3.19%
probability of being enrolled in the CRP. To get a feel for the sensitivity of participation to initial
farming returns, I conducted an additional simulation setting the initial farming return at a low
value of $90 per acre, which resulted in a participation rate of 4.96%. A rough estimate of the
elasticity of the participation probability with respect to farming returns is -4.1, which indicates
that participation is very sensitive to changes in initial farming returns.
To evaluate the impact of structural change, I perform another simulation assuming

    4.0% , which is equivalent to assuming the traditional geometric Brownian motion
without parameter uncertainty. Accordingly, the initial farming return in 2009 is set at $66,
which is calculated using the realized farming return in 2005 and an annual growth of 4%,
adjusted for the lower productivity of CRP lands. New conversion thresholds are derived and
used in the Monte Carlo simulation. The simulation result shows that the participation rate rises
to 15.92%, which is five times the baseline case of 3.19%. That is, structural change has reduced
participation probability by 80 percent. This shows that incorrectly ignoring uncertainty in
growth rates surrounding future agricultural returns can have a huge impact on estimated
participation.
Though biofuel production is a major source of structural change, other factors may also
be important. Based on the evidence from the literature, I assume that biofuel production
accounted for 60 percent of the increase in corn prices, 40 percent of the increase in soybean
prices, and 26 percent of the increase in wheat prices (Collins, 2008; Lipsky, 2008). Assuming
the same effect on net returns for each crop and using historical average planted ratios of each

57

crop as weights, biofuel production contributes roughly 40% of the jump in growth from 4% to
8.5%. To separate the impact of biofuel production on CRP participation, I re-simulate the
participation probability excluding biofuel production’s effect on the growth rate. For this
purpose, I reset   6.7% (keeping   4% ) and re-calculate the land conversion threshold
returns under different growth expectations. With an adjusted initial growth rate of 6% (similarly,
assuming the landowner holds the same 27% belief on the probability of the growth rate
reverting to the long-term average) and an adjusted initial farming return of $74 for 2009, the
simulation results show that the participation rate would be 9.92% in the next 5 years, which is
28

more than three times higher than the baseline case of 3.19%.

That is, of the entire structural

change impact (an 80 percent drop from 15.92% to 3.19%), biofuel production accounts for
approximately half of the reduction. Further simulation shows that, to offset the participation loss
due to the biofuel impact, the government would need to raise the per-acre rental payment by
$25, which means an increase of nearly 40%.
The negative effect of the biofuel boom on participation is intuitive. Policies designed to
support biofuel production would create incentives to move resources into biofuels. For example,
the 2005 Energy Policy Act mandates 7.5 billion gallons of renewable fuel to be added to
gasoline by 2012. Subsequent tax incentives and subsidies (e.g. federal tax credits of 51 cents a
gallon for ethanol blenders) have paved the way for the increase in biofuel production. The
growing demand for biofuels has decreased participation in the CRP, as shown by our
simulations. This unintended consequence on conservation programs will counteract the
emission reduction benefits of biofuel and thus need to be taken into consideration in the
evaluation of biofuel programs and policy designs. Also, it raises the question of whether corn
28

The initial farming return of $74 is calculated using the realized farming return in 2005 and an
annual growth of 6%, adjusted for the lower productivity of CRP lands.
58

ethanol production could lead to effective net greenhouse gas emission reductions compared to
the reductions achieved by the CRP itself (Searchinger et al., 2008; Piñ
eiro et al., 2009).

2.7

Sensitivity Analysis
A sensitivity analysis is undertaken to determine landowners’ response to changes in

various parameters. In particular, I analyze landowners’ response to market conditions.

2.7.1 Length of the High Growth State, 1 
In this section I investigate the impact of high growth duration on conservation
participation. Figure 2.2 shows the return boundaries under different expectations of the high
growth duration, namely, 4, 7, 12 and 20 years. For example, if the landowner expects the
growth rate to be at 7.15% and persist for 12 years (corresponding to the 2022 target set in the
2007 energy act), he would be willing to participate in the CRP at a lower, $30.25/acre farming
return. The Monte Carlo simulation result shows that the participation probability in a 5-year
horizon reduces from 3.19% to 2.53% when the high growth duration extends from 7 years to 12
years. Persistence of the biofuel boom has a clear impact on the length of the high growth state
and therefore would lead to decreased conservation participation. Figure 2.2 also shows that the
lower the initial growth rate expectation, the less sensitive is the boundary to the duration of the
high growth state.

59

Figure 2.2. Farming Threshold Returns ( R * ) versus Growth Rate Expectations ( mt ) under
Different Durations of the High Growth State

The durations of biofuel boom and in turn the high growth in farming returns are to a
large extent dependent on a long list of incentives that the State and Federal governments have
instated to support ethanol production since the 1970s and protections from the U.S. trade
barriers. The most prominent among them are the Federal ethanol tax credit and ethanol import
tariff. While the tax credit goes to ethanol refiners, farmers have benefited because with
sufficiently high competition among blenders, the corn price increases by almost the full amount
of the credit (Elobeid et al., 2007). Also, imposing an import tariff raises domestic ethanol and
corn prices. However, to date rising food prices, the desperation to reduce federal budget deficits,
and criticisms to corn-based ethanol from environmental organizations have led to the debate of
eliminating ethanol tax credits and import tariffs. Elobeid and Tokgoz (2008) simulated the

60

effect of removing the tax credit and trade distortions and found that corn demand and prices
would decline substantially. Importantly, the ethanol industry may end its boom sooner and
farming returns may return to its long-run equilibrium in a shorter duration. In that case, I have a
larger  value and would expect an increase in program participation as simulated in Figure 2.2.

2.7.2 The Effect of Volatility, 
I further study the effect of uncertainty on the participation threshold. I keep other
parameters in the baseline unchanged while varying the standard deviation of the logarithm of
farming returns. The impact of changes in volatility is illustrated in Figure 2.3.
Figure 2.3. Farming Threshold Returns ( R * ) versus Growth Rate Expectations ( mt ) under
Different Volatility Parameters

61

Figure 2.3 shows that volatility in the farming return contributes to a decrease in the
participation threshold. As expected, when the volatility of the farming return increases, the
option value increases, and the threshold decreases, indicating the landowner would wait for a
lower farming return realization before committing to land conversion. For example, when
volatility changes from 0.2 to 0.26, the participation threshold decreases from $32.75/acre to
$31.25/acre at the 7.15% growth rate expectation. The decrease in the threshold value may have
a negative effect on participation. However, besides lowering the critical trigger value, higher
volatility may also make the farming return more likely to reach the critical value, which would
have a positive effect on participation. To determine the overall effect of volatility, I examine the
probability of participation with Monte Carlo simulation. The simulation results show that when
volatility increases from 0.20 to 0.26, the probability of participation actually moves up, from
0.63% to 3.19%, which suggests that risk in agriculture would induce conservation. The
implication is that reducing risk and uncertainty in agricultural returns, for example, with
programs on farm income stabilization, may have a negative impact on CRP participation and
environmental conservation.
One could also consider the effect of uncertainty on the reverse land use decision to
convert from CRP land back to crop production. Higher uncertainty in crop returns should
increase the value of the option to convert from CRP back to crop production which should
contribute to an increase in the threshold ( R * ) above which farming returns must go before the
landowner will convert back to farming. As a result, when all other conditions are kept
unchanged, landowners would need a higher realized return from farming in order to convert,
and therefore would be more likely to delay the decision to return to crop production. However,
besides increasing the critical trigger value, higher volatility may also make the farming return

62

more likely to reach the critical value, which would have a positive effect on the overall
probability of converting back to farming. Determining the net effect of increased uncertainty in
farming returns on the net probability of converting from CRP back to crop farming would
require a complete numerical analysis of the reverse conversion decision, which was not
undertaken here.

2.8

Conclusions
Biofuel policies have spurred biofuel production and have been pressuring land out of

conservation programs, offsetting the emission reduction goal of the biofuel policy and
increasing the potential for environmental degradation. This essay proposes a general land-use
decision model to analyze the spillover effect of biofuels policy on conservation program
participation. The proposed model captures the biofuel-driven structural changes in the
agricultural sector and is used to simulate landowners’ conservation participation decision under
different layers of uncertainty and investigates factors that affect this decision. I derived the
threshold conditions that would trigger participation in conservation. The decision model
provides a useful tool for quantifying the spillover effect, determining optimal land use, and for
policy makers to develop informed and more cost-effective environmental policies.
Methodologically, I derived numerical solutions to the land-use decision problem under
parameter uncertainty using a technique that can handle a fairly general class of specifications
for the source of uncertainty. More specifically, I use a nonlinear Kalman filter approach to
addressing parameter uncertainty, which continuously updates information observed by
landowners and estimates the random growth rate of farming returns with minimum error. The

63

real options problem is formulated as a linear complementarity problem and solved with a fully
implicit finite difference approach.
I find a strong negative effect of biofuel production on conservation enrollment. Overall,
the structural changes have reduced the chance of CRP participation by 80 percent, and one half
of this reduction appears to stem from biofuel production. Our simulation results further suggest
that, to offset the participation loss caused by biofuel production, the government would have to
increase the per-acre rental payment by 40%. Our results show that higher farming returns,
higher expected growth rates, and a longer time of remaining in the high growth state all reduce
the incentive for conservation due to the decreasing value of the conservation option for
landowners. I also found that, though higher volatility of agricultural return reduces the decision
threshold of farming returns, its overall effect on participation is positive, which implies that
government programs aiming to reduce farm income risk would have a negative effect on
conservation participation.

64

APPENDIX B

65

APPENDIX B: DERIVATION OF EQUATION 2.12
Let d t  At ( t ,  )dt  Bt ( )dWt , where the unobservable component  t is described
by a continuous-time Markov chain with transition matrix  (t ) and state space E with states

 and  . Let   (t )  prob( t    ) . According to Lipster and Shiryayev (1977) Theorem
t
9.1,   (t ) satisfies the following equation:
(B.1)

A (  ,  )  At ( )
d  (t )   E  (t )  (t )dt    (t ) t
dW t ,
Bt ( )

where At ( )   E At ( ,  )  (t ) and dW t 

A ( )
1
d t  t
dt .
Bt ( )
Bt ( )

In our setup, dRt   t Rt dt  Rt dWt , and pt  prob( t    tR ) , which means that

At ( t ,  )   t Rt , Bt ( )  Rt , and At ( )  pt Rt  (1  pt )Rt . From our Markov system, I


can derive that    and   0 under state space E   ,  . An application of Theorem
9.1 yields:
(B.2)

p (1  pt )(   )
dpt   (1  pt )dt  t
dW t ,



where dW t  dWt 
Plugging pt 

(B.3)

1



( t  mt )dt .

mt  
into the above equation yields:
 

dmt   (   mt )dt 

1



(  mt )(mt   )dW t .

66

APPENDIX C

67

APPENDIX C: DERIVATION OF EQUATION 2.14
The derivation is through Ito’s Lemma. I know that dRt  mt Rt dt  Rt dW t and
dmt   (   mt )dt 

(C.1)

1



(  mt )(mt   )dW t .Taking the square of the first equation obtains:

(dRt ) 2  ( Rt mt dt ) 2  2Rt2 mt dtd W t  Rt2 2 (dW t ) 2 .

I already know that (dt ) 2 and cross product dtd W t are equal to zero in the mean square sense,
and (dW t ) 2  dt , therefore:
(C.2)

E (dRt ) 2  Rt2 2 dt .

Similarly, I obtain:
1

(  mt ) 2 (mt   ) 2 dt ,

(C.3)

E (dmt ) 2 

(C.4)

E (dRt dmt )  Rt (  mt )(mt  )dt .



2

Applying Ito’s Lemma to J ( Rt , mt ) , I can derive a partial differential equation:
(C.5)

dJ ( Rt , mt )  J R dRt  J m dmt 





1
J RR (dRt ) 2  J mm (dmt ) 2  2 J Rm dRt dmt ) .
2

Substituting C.2, C.3, and C.4 into C.5, I obtain:
1 2 2


 Rt mt J R   (   mt ) J m  2 Rt  J RR

 dt .
(C.6) E[dJ ( Rt , mt )]  
 1 (  m ) 2 (m   ) 2 J

t
t
mm  Rt (  mt )(mt   ) J Rm 
 2 2



The Bellman equation for eq. 2.13 is expressed as:
(C.7)

J ( Rt , mt )dt  EdJ ( Rt , mt ) .

Therefore,

68

(C.8)

1 2 2


 Rt mt J R   (   mt ) J m  2 Rt  J RR

  0.
J  
 1 (  m ) 2 (m   ) 2 J

t
t
mm  Rt (  mt )(mt   ) J Rm 
 2 2



69

APPENDIX D

70

APPENDIX D: DISCRETIZATION OF THE LCP
This Appendix describes the numerical approach used to solve the LCP for valuing the
option to participate in the CRP. For a general discussion of numerical methods of option
valuation, refer to Wilmott, Howison, and Dewynne (1993). A finite difference scheme is used in
this paper, which involves reducing a continuous partial differential equation to a discrete set of
difference equations. Two basic finite difference methods are the implicit method and the
explicit method. When applied to the partial differential equation, backward and forward
difference approximations for the derivative of option value with respect to time lead to implicit
and explicit finite-difference schemes, respectively (Wilmott, Howison, and Dewynne, 1993).
The implicit finite-difference method is robust because it can overcome the stability and
convergence problems that plague the explicit finite-difference method. In this paper, I use an
implicit difference scheme—the Crank-Nicolson scheme.
To explain the three-dimensional ( R, m, t ) discretization for eq. 2.15, consider a threedimensional space formed by the farming return R , the growth rate m , and time t . I divide the
R -axis into equally-spaced nodes a distance R apart, the m -axis into equally-spaced nodes a

distance m apart, and the t -axis into equally spaced nodes a distance t apart. This divides the
( R, m, t ) space into a grid, where grid points have the form (sR, nm, it ) . I then compute the

option value J at these points:
(D.1)

R  R0 , R1 , R2 ,, RS , m  m0 , m1 , m2 ,, mN  , t  t 0 , t1 , t 2 ,, t I .
The finite difference method involves replacing partial derivatives by approximations

based on Taylor expansions near the point of interest. At any point on the grid

( R, m, t )  ( Rs , mn , ti ) , the value of the option is denoted as J s,n,i . A formula for approximating

71

the partial derivatives using the implicit difference method can be found in Wilmott, Howison,
and Dewynne (1993). Replacing these derivatives in eq. 2.15 yields a series of difference
equations:
i

a s 1, n 1 J s 1, n 1  a s 1, n J s 1, n  a s 1, n 1 J s 1, n 1  a s, n 1 J s, n 1  a s, n J s, n 
(D.2) HJ i  
 ,
 a s, n 1 J s, n 1  a s 1, n 1 J s 1, n 1  a s 1, n J s 1, n  a s 1, n 1 J s 1, n 1



2
Rs 1 (  mn 1 )(mn 1   )
Rs 1mn Rs 1 2

Where a s 1, n 1  
, a s 1, n 
,
4Rm
2R
2R 2

R (  mn 1 )(mn 1   )
a s 1, n 1  s 1
,
4Rm
a s, n 1  

a s, n 

(  mn 1 ) 2 (mn 1   ) 2
2 2 m 2

(  mn ) 2 (mn   ) 2

a s, n 1  

2 2 m 2

2 m

2

 (mn 1   )
2m

,

R 2 2
,
 s
2R 2

(  mn 1 ) 2 (mn 1   ) 2
2





 (mn 1   )
2m

,

R (  mn 1 )(mn 1   )
a s 1, n 1  s 1
,
4Rm
R (  mn 1 )(mn 1   )
R m
R2  2
a s 1, n   s 1 n  s 1
, a s 1, n 1   s 1
.
4Rm
2R
2R 2
The superscript i on the right-hand side means that all variables within the square parentheses are
evaluated at t i .
The difference equation must also be specified when the growth rate is at its maximum
and minimum values. I use BC2 boundary condition on mmin   and mmax   (Tavella and
Randall, 2000). The one-sided difference for J mm (the second derivative of J with respect to
72

m ) and the forward-backward difference for J Rm (the cross derivative of J with respect to R
and m) are used at mmax   :
i

(D.3)

i

 J s, n  2 J s, n 1  J s, n  2 
 J s, n  J s 1, n  J s, n 1  J s 1, n 1 
J mm  
 , J Rm  
 .
mR
m 2





Then, in eq. 2.15,
i

(D.4)

a s 1, n 1 J s 1, n 1  a s 1, n J s 1, n  a s, n  2 J s, n  2 
HJ i  
 ,
 a s, n 1 J s, n 1  a s, n J s, n  a s 1, n J s 1, n




R (  mn 1 )(mn 1   )
where a s 1, n 1   s 1
,
Rm
2
Rs 1 (  mn )(mn   ) Rs 1mn Rs 1 2
a s 1, n 


,
Rm
2R
2R 2

a s, n  2  

a s, n 1 

a s, n  

(  mn  2 ) 2 (mn  2   ) 2
2 2 m 2

(  mn 1 ) 2 (mn 1   ) 2

 m
2

2

(  mn ) 2 (mn   ) 2
2 2 m 2



 ( mn  2   )
2m

,

R (  mn 1 )(mn 1   )
 s
,
Rm

R 2 2  (mn   ) Rs (  mn )(mn   )
 s


,
2m
Rm
R 2

R m
R2  2
a s 1, n   s 1 n  s 1
.
2R
2R 2
Similarly, the one-sided difference for J mm and the forward-backward difference for J Rm are
used at mmin   :
i

(D.5)

i

 J s, n  2  2 J s, n 1  J s, n 
 J s, n 1  J s, n  J s 1, n 1  J s 1, n 
J mm  
 , J Rm  
 .
mR
m 2





Then, in eq. 2.15,
73

i

(D.6)

a s 1, n J s 1, n  a s 1, n 1 J s 1, n 1  a s, n J s, n 
HJ i  
 ,
 a s, n 1 J s, n 1  a s, n  2 J s, n  2  a s 1, n J s 1, n 



2
R (  mn )(mn   ) Rs 1mn Rs 1 2


where a s 1, n   s 1
,
Rm
2R
2R 2

R (  mn 1 )(mn 1   )
a s 1, n 1  s 1
,
Rm
a s, n  

a s, n 1 

(  mn ) 2 (mn   ) 2
2 m
2

2



(  mn 1 ) 2 (mn 1   ) 2

a s, n  2  

 m
2

2

2
Rs  2



 ( mn   )
2m

R (  mn )(mn   )
,
 s
Rm

R (  mn 1 )(mn 1   )
,
 s
Rm

(  mn  2 ) 2 (mn  2   ) 2
2 2 m 2

R

2

R m
R2  2
, a s 1, n   s 1 n  s 1
.
2R
2R 2

Equations D.2, D.4 and D.6 form a system of equations, which can be written in matrix
form and solved by iteration (Insley, 2002).

74

APPENDIX E

75

APPENDIX E: DERIVATION OF EQUATION 2.20
The simple optimal stopping problem without parameter uncertainty is:

(E.1)

 J ( R , m, )  max E [e   (T  t ) M ( R , Q)]

t
T t
T
.

dRt  Rt mdt  Rt dW t


Using Ito’s lemma, the fundamental differential equation of this optimal stopping problem is an
ordinary differential equation:
(E.2)

1 2 2
Rt  J RR  Rt mJ R  J  0 ,
2

where J R and J RR are the first and second derivatives of J with respect to Rt .
Let R * represent the threshold value, which triggers participation in the CRP. This
ordinary differential equation is solved subject to the following boundary conditions. First, the
continuity condition is:
(E.3)

J ( R * )  M ( R * , Q)  V (Q) 

R * (1  e  (   m)T )
.
 m

Second, the smooth pasting condition is:
(E.4)

J R ( R * )  M R ( R * , Q)  

(1  e  (   m)T ) .
 m

In addition, I have:
(E.5)

J (, m, )  0 ,

which says that when the farming return approaches infinity, the land conversion option is
worthless. The general form of the solution to equations is

(E.6)



R 
M ( R * , Q) t 

 *
J ( Rt , m, )  
R 

M ( Rt , Q)


Rt  R *
Rt  R *

76

,

1
1
 (m   2 )  (m   2 ) 2  2  2
2
2
where  
, which is the negative root of the fundamental
2


quadratic equation


(   m)  V (Q)
1 2

.
  (  1)  m    0 , and the threshold Rt* 
  1 (1  e  (   m)T )
2

77

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78

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81

CHAPTER 3: THE RISK-ADJUSTED IMPLIED VOLATILITY AND ITS FORECAST
PERFORMANCE IN CORN FUTURES

3.1

Introduction
The question of whether implied volatility provides unbiased and informationally

efficient forecasts of future realized volatility has been studied extensively in the empirical
finance and time series econometrics literature. The typical test employed in the literature takes
the regression form:
(3.1)
where

,
is realized volatility over the period to

same period, and

,

is implied volatility over the

is an alternative predictor typically generated from historical

information. The objective of the test is to establish whether implied volatility is unbiased (
and
(

=0

=1) and subsumes all information contained in historical volatility (hereafter HV)
=0). While there is some inconsistency in previous results on informational efficiency,

29

the general result is that the Black-Scholes (hereafter BS) implied volatility, a frequently used
measure in the literature, is a biased forecast of future volatility in the sense that estimated
different from zero and estimated

is

is significantly less than unity. More recent studies

corrects various data and methodological problems in earlier research (see, e.g., Canina and
Figlewski, 1993; Christensen and Prabhala, 1998; Ederington and Guan, 2002; Jiang and Tian,
2005). These studies collectively found that the BS implied volatility is informationally efficient
29

Using the Black-Scholes volatility implied in at-the-money options, the test results in the
literature are mixed. For example, Canian and Figlewski (1993) found the BS implied volatility
does not incorporate information contained in HV while Jorion (1995) and Szakmary et al. (2003)
reported evidence that the implied volatility is a more efficient forecast than HV.
82

but biased. The bias was explained by arguing that tests based on the BS implied volatility are
joint tests of market efficiency and the BS option pricing model, and so necessarily suffer from
bias due to model misspecification. In view of limitations of the BS option pricing model, a
model-free (hereafter MF) implied volatility measure that does not require any particular optionpricing model has recently been proposed in the literature (Britten-Jones and Neuberger, 2000;
Jiang and Tian, 2005; Carr and Wu, 2009). The MF implied volatility is computed from a set of
options with strikes instead of only at-the-money options, and shown by Britten-Jones and
Neuberger (2000) to equal the risk-neutral expectation of future volatility under diffusion
assumptions. Since the MF implied volatility does not incur any model misspecification error, it
would seem to be more likely to be an unbiased estimate of realized volatility than the BS
implied volatility. But Jiang and Tian (2005) found that the MF implied volatility is also biased.
So the existence of bias continues to be a puzzle.
Lamoureux and Lastrapes (1993) were the first to suggest that a risk premium could be
responsible for the bias in implied volatility forecasts. The risk premium explanation has also
been investigated in more recent studies (Chernov, 2007; Doran and Ronn, 2008). For example,
Chernov (2007) argued that implied volatility is derived in a risk-neutral environment while
realized volatility is based on observed market outcomes. Since a risk premium can cause a
disparity between observed and risk-neutral probability measures, it will inevitably result in a
disparity between realized and implied volatilities.
One way to resolve the bias puzzle is to incorporate the possibility of risk premiums into
the implied volatility forecast. Chernov (2007) found that the disparity between realized
volatility and at-the-money implied volatility approximately equals a function of the latent spot
volatility, and he suggested using spot volatility as an additional predictive variable in the test

83

regression. Although at-the-money options are generally more actively traded than other options,
and are certainly a good measure to start with, the test may be still subject to model
misspecification errors because his approximation formula is only valid in a fairly restricted
model for price dynamics.

30

In this research I focus on the MF implied volatility but propose a risk-adjusted modelfree implied volatility that may help correct the bias in traditional implied volatility forecasts. To
make the risk adjustment I convert the risk-neutral model-free implied volatility into a forecast
under an observed probability measure which incorporates a risk premium. This method was first
proposed by Becker, Clements, and Coleman-Fenn (2009). However, their risk adjustment model
was derived only for a general diffusion processes. Here I go further and generate the riskadjusted volatility forecast in the framework of a jump-diffusion model of asset returns. The
jump-diffusion model is more general and capable of better capturing empirically relevant
features of observed asset return dynamics. I derive the risk adjustment model with an exact,
closed-form expression between the expectation of future volatility under the observed
probability measure and the MF implied volatility. The jump-diffusion risk-adjusted model
immediately explains the typical finding of a downward bias for the MF implied volatility.
Similar to prior studies on implied volatility forecasts, the new model indicates that the volatility
risk premium contributes to the forecast bias. But more importantly, jump risk premiums are also
shown to play a role in the forecast bias.
I estimate the model parameters of interest and use them to convert the risk-neutral
measure into its observational equivalent. I rely on standard Generalized Method of Moments
30

Chernov (2007) showed that extending the approximation formula to more full models, for
example, the stochastic volatility with jumps in return and volatility models, hinges on an
additional assumption that the risk-neutral expectation of jump in returns is equal to zero. But
such an assumption contradicts empirical results.
84

(GMM) estimation of the cross conditional moments between risk-neutral and observed
expectations of volatility. The method is easier to implement than other alternatives which rely
on computationally difficult joint estimation of both underlying asset returns and option prices
(see, e.g., Jones, 2003; Aï
t-Sahalia and Kimmel, 2007). Bollerslev, Gibson, and Zhou (2010)
used two basic moment conditions in a stochastic volatility model to extract the volatility risk
premium. I extend the Bollerslev, Gibson, and Zhou framework to a jump-diffusion model to
identify the parameters needed. The extended framework not only generalizes their original
moment conditions but also augments moment conditions that involve the jump component of
the process. Through a simulation experiment, the framework is shown to perform well with a
small sample size of volatility in non-overlapping two-month intervals.
The empirical application addresses corn futures. However, unlike stock index markets,
commodity markets have no available MF implied volatilities, which poses a significant
challenge to estimation.

31

In this study I rely on Jiang and Tian (2005) method to construct the

MF implied volatility using observed corn options. Meanwhile, I extend the estimation model to
highlight the key seasonality feature in agricultural commodities. Using corn futures and options
data from 1986 to 2010, I find that statistically significant risk premiums exist in the corn market,
which supports our risk premium-based explanation of the bias. Furthermore, these estimates are
robust with respect to microstructure market noise that may invalidate our underlying semimartingale assumption at high sampling frequencies.
Finally, I investigate the forecasting ability of corn risk-adjusted (hereafter AD) MF
implied volatility using three criteria: unbiasedness, informational efficiency relative to
alternative forecasts, and superiority in predictive power. Evaluations are conducted against three
31

For example, the VIX Index, an implied volatility measure based on S&P 500 stock
index option prices, is available from January 1990.
85

alternative predictors of volatility: a) HV, b) the BS implied volatility, and c) the risk-neutral MF
implied volatility. Our results support that the AD implied volatility is unbiased while the
unadjusted MF implied volatility is biased, consistent with our theoretical prediction. The results
also provide evidence for informational efficiency of the corn option market because both the
MF implied volatilities subsume information contained in the BS implied volatility and HV.
More importantly, I find that the AD implied volatility provides a more precise forecast
compared to alternative forecasts.
The rest of this paper is organized as follows. In section 3.2, I propose a stochasticvolatility jump-diffusion model and derive the explicit expression between the expectation of
future volatility and the MF implied volatility. In section 3.3, I outline basic moment conditions,
calculate volatility measures, construct the GMM framework to estimate the parameters of
interest, and provide finite sample simulation evidence on the performance of the estimator.
Section 3.4 presents the corn dataset used for this study, reports the empirical estimates from
applying the estimation procedure to corn data set spanning twenty-three years, and evaluates the
robustness of these estimates. In section 3.5, forecast performance of the new implied volatility
is examined. Section 3.6 provides concluding comments.

3.2

Model Specifications and Volatility Forecast

3.2.1 Price Dynamics
Asset prices are assumed to follow a jump-diffusion model and the asset volatility is
allowed to be stochastic. Specifically, the dynamics of logarithmic asset prices
observed probability measure P are:
(3.2)

,

86

under the

(3.3)
where

,

denotes the drift term;

volatility mean;

is speed of volatility mean reversion;

is the volatility of volatility;

and

processes with the correlation coefficient equal to
and distributed independently of

and

is a normally distributed random

and variance

percentage jump size is

The term

are two correlated Wiener

is a Poisson process with intensity

; and

variable with mean

is the long-term

. Consequently, the expected

is the compensation for the instantaneous

change as a result of a jump so that

has its mean equal to 0. The

general specification in equations (3.2) and (3.3) nests many well-known models. The Hull and
White (1987) or Heston (1993) version of the model follows from setting no jump and
or

. The Bates (2000) version is also a special case of our model if a square-root

volatility process is specified.
Assuming no arbitrage, the corresponding dynamics under a risk-neutral probability
measure Q are:

32

(3.4)

,

(3.5)
where

,
with

being a volatility risk premium;

;

and

are two

new correlated Wiener processes under measure Q with the same correlation coefficient .

32

The detailed derivation can be found in Appendix A of Pan (2002).
87

is

a new Poisson process with intensity

; and

has a new mean

but its variance remains unchanged at

. Note that the differences in

jump parameters are related to jump risk premiums (including jump size and intensity premiums),
but I do not explicitly present the premiums in the model since the adjustment can be done
without completely specifying each premium.

33

Under arbitrage-free assumptions, the expected

return from holding futures contracts should be zero under the Q measure, that is,
. This can be easily verified by Ito’s lemma applied to equation (3.4).

3.2.2 The Model
Under the specifications of eq. (3.2), the variance over the trading period

is

measured by the quadratic variation (hereafter QV):
(3.6)

,
34

where the integrated volatility

(hereafter IV),

, denotes the contribution

from the continuous price path, while the second term accounts for the variance contributed from
the jumps. The expected QV under the P measure is:
(3.7)

,

33

The explicit relationship between model parameters under the observed and risk-neutral
probability measures can be found in Doran and Ronn (2008).
34
Following the recent literature I will interchangeably use variance and volatility here and
throughout the paper unless specifically noted.
88

where

, is the expected squared jump value. The QV can be estimated by a

nonparametric estimator, namely, the realized volatility (hereafter RV), which will be discussed
in the section on volatility measures.
Next I introduce the MF implied volatility. Suppose call options with a continuum of
strike prices for a given maturity are traded on an underlying asset. Britten-Jones and Neuberger
(2000) defined the MF implied volatility from the time to

as the integral of call option

prices over an infinite range of strike prices
(3.8)
where

,
denotes the price of a call option maturing at time

with an strike price

Obviously, unlike the conventional concept of implied volatility, the MF implied volatility
does not involve any specific underlying option pricing model. Britten-Jones and Neuberger
(2000) showed that the MF implied volatility is the risk-neutral expected IV under diffusion
assumptions. Jiang and Tian (2005) demonstrated that the relationship is still approximately
valid even if the underlying asset price process accommodates jumps. In the following, I will
derive an exact expression between the expected QV and the MF implied volatility in our jumpdiffusion models. The expression facilitates transforming the risk-neutral MF implied volatility
into its equivalent in the observed measure. First I derive an explicit theoretical relationship
between the MF implied volatility and the risk-neutral expected IV. As shown in Appendix F, the
exact equation is
(3.9)
where

,
is a composite parameter associated with jumps. This equation

generalizes the derivation by Britten-Jones and Neuberger (2000), because the MF implied
89

volatility would be equal to the risk-neutral expected IV in the absence of jumps (

). When

there are jumps, the MF implied volatility becomes a jump-adjusted risk-neutral expected IV.
Utilizing the results in Bollerslev and Zhou (2006), I derive the critical expression
between the expected QV and the MF implied volatility (see Appendix F),
(3.10)

,

where
,

and
.
The bias puzzle can be immediately explained by eq. 3.10. Two potential sources of bias can be
identified when using the MF implied volatility to forecast future volatility. First, the negative
volatility risk premium

reduces the degree of mean reversion in the risk-neutral volatility

process relative to that of the actual volatility process (

, which in turn causes the ratio

to become less than unity; Second, the jump risk premiums have effects on the intercept via
and

. In a simulation study Doran and Ronn (2008) demonstrated that the volatility risk

premium can explain the difference between the BS implied volatility and RV at the at-the-money
point, while the jump risk premiums are responsible for the bias in out-of-the-money options.
Their conclusion echoes our findings because the MF implied volatility aggregates information
across options.

3.3

Estimation

90

Eq. 3.10 shows that the risk-neutral MF implied volatility can be linked in a closed-form
expression to the expected QV under the observed measure. The expression thus provides a
fundamental tool to transform the MF implied volatility into its risk-adjusted counterpart. Instead
of identifying each separate model parameter needed for the adjustment, it is more convenient to
directly estimate the key parameters (
composite parameters for jumps (

in the stochastic volatility model, along with the
, because this process avoids the joint estimation of both

the underlying asset return and specific option pricing models. In particular, let
denote the parameter vector of interest. I estimate
on futures and options. Since the affine structure of

using the joint time-series data
allows us to generate a

rich set of moment conditions, I can use GMM estimation to estimate parameters. In the
remainder of this section I will first provide a detailed description of the relevant moment
conditions. Then I discuss nonparametric estimators for unobserved volatility variables. Finally,
the finite sample properties of the parameter estimators are established in simulation experiments.

3.3.1 Moment Conditions
The first moment condition for the IV has previously been derived by Bollerslev and
Zhou (2002). The conditional moment of the IV under the P measure satisfies
(3.11)

,

where the coefficients
parameters

and

are functions of the underlying

and . The equation establishes the link between the expectation of the IV in the P

measure and its lagged value.
The second moment condition links the expected IV with the MF implied volatility.
(3.12)

,
91

where

and

are functions of the underlying parameters

and

. This equation is

derived in the Appendix F.
The third moment condition relates to jumps in asset returns. Since the difference
between the QV and the IV offers a simple nonparametric estimator for the jump component in
total price variation (Barndorff-Nielsen and Shephard, 2006), I use the following moment to
estimate the average jump component:
(3.13)

.

3.3.2 Volatility Measures
The QV and its separate components, such as IV, are not directly observable. I resort to
recently popularized model-free nonparametric consistent estimators. As demonstrated in
Anderson and Bollerslev (1998), the volatility calculated from high-frequency return data
provides a good ex-post measure of the actual volatility and is a better measure than that
estimated from daily data. I first estimate the QV using the RV. The RV is quantified by summing
the squared high frequency returns over the
(3.14)

where

time interval:
,

denotes the corresponding discrete-time within-day

returns, and

refers to the number of return observations over the trading period. It is known

that the RV will converge uniformly in probability to the QV as the sampling frequency of the
underlying returns approaches infinity (Barndorff-Nielsen and Shephard, 2002):
(3.15)

.

92

Similarly, I employ a consistent, nonparametric measure of the sum over cross products
of frequently sampled returns for the IV. I rely on the realized bipower variation (hereafter BV)
measure developed by Barndorff-Nielsen and Shephard (2004) to estimate the IV:
(3.16)

.

Importantly, for increasingly finely sampled returns the BV measure becomes immune to jumps
and consistently estimates the IV:
(3.17)

.

Additionally, I also have empirical implementation issues for the MF implied volatility;
option prices required for calculating the right hand side of eq. 3.8 are not available for all strike
prices. In fact only a finite number of strike prices are traded in the market, so I have to
approximate it from these limited observed option prices. I empirically implement the MF
implied volatility using Jiang and Tian approach (2005):
(3.18)

,
,

where

increment

and

are upper and lower truncation points of strike prices, a strike price

,

for

, and

. The approximation method produces two
measurement errors. One is truncation errors from a limited range of strike prices; the other is
discretization errors due to numerical integration. Jiang and Tian (2005) showed that these two
errors can be negligible if the truncation points are more than two standard deviations (SDs)

93

from

and the strike price increment

0.35 SDs. For options with the truncation range

beyond the available maximum and minimum strike prices, I assume their implied volatilities are
the same as endpoint implied volatilities and extrapolate their option values by using the
endpoint implied volatilities. The endpoint implied volatilities are calculated by using the inverse
BS formula.

35

These implied volatilities are translated into call prices with any unavailable strike

prices by using the BS formula once more.
In the implementation, a further challenge is that even in a given limited range
, only a sparse set of discrete strike prices are listed for trading in the market. I
apply a curving-fitting method to implied volatilities to interpolate between available strike
prices. Likewise, the inverse BS formula is used to obtain estimates of implied volatilities of
listed calls. The detailed implementation can be found in Jiang and Tian (2005).

3.3.3 The GMM Framework
The parameter vector

is estimated utilizing the moment conditions (3.11)-(3.13) after

replacing unobservable variables with their respective estimates. Additionally, I employ the
lagged value of BV and the MF implied volatility as instrumental variables to impose overidentifying restrictions. Therefore, the system of equations is:

35

Note that this procedure does not assume that the BS model is the true model underlying
option price process. It is merely used as a tool to provide a one-to-one mapping between option
prices and implied volatilities (Jiang and Tian, 2005).
94

(3.19)

The population moment is

. I estimate
, where

conditions

via standard GMM estimation such that

refers to the sample means of the moment

and

denotes the weighting matrix. I use an efficient

two-step GMM estimator in which the first estimator

is constructed using the covariance

matrix of the instrument vector to weigh the moment conditions and the second estimator is
chosen to minimize

, where

is the inverse of the asymptotic

covariance-variance matrix of

Empirically, I employ a heteroskedasticity and

autocorrelation consistent robust covariance matrix estimator with a Bartlett-kernel in
implementing

(Newey and West, 1987). At parameter estimators the objective function

times the number of observations is asymptotically

distributed with degrees of freedom equal

to the number of over-identifying restrictions. The test statistic provides a straightforward
method for testing over-identifying restrictions.

95

3.3.4 Simulation Analysis
To assess the finite-sample properties of the GMM estimators in the model, I conduct a
Monte Carlo study for the specialized Bates (2000) version of the model. The dynamics of (3.2),
(3.3), (3.4) and (3.5) are simulated with the Euler method. The parameter configurations are:
and

The MF implied volatility is

calculated using the above Jiang and Tian approach. RV and BV are constructed from the 5minute returns according to the formula (3.14) and (3.16) to approximate QV and IV. Inside the
simulations I also partition each five-minute interval into 5 smaller segments for the continuoustime record in order to calculate the true QV and IV (however, they are not observable in
practice). I compare the GMM estimator using the ―five-minute‖ RV and BV with the
corresponding non-feasible estimator using the true QV and IV. The comparison assesses
whether RV and BV constructed under the sampling frequency affect the estimation. The
accuracy of the asymptotic approximations is illustrated by contrasting the results for sample
sizes of 150 and 600. The Monte Carlo replications are 500. The detailed experimental design
can be found in the Appendix G.

96

Table 3.1. Monte Carlo Simulation Results
Mean
Median
RMSE
Mean Median
RMSE
T=150
T=600
MF, IV, and QV
2.2300
2.2200
0.8030
2.1400
2.1100
0.3432
0.0254
0.0253
0.0020
0.0253
0.0252
0.0008
-1.1200
-1.1100
1.0300
-1.0300 -0.9100
0.4500
0.01530
0.01487
0.0050
0.01537 0.01501
0.0020
0.1511
0.1501
0.0870
0.01522
0.0146
0.0370
MF,BV, and RV
2.2600
2.2700
0.8950
2.1400
2.1200
0.4060
0.0256
0.0255
0.0020
0.0253
0.0255
0.0008
-1.1600
-1.1700
0.1305
-1.0200 -0.9100
0.6601
0.01478
0.01463
0.0040
0.01557 0.01551
0.0017
0.0143
0.0142
0.0870
0.0150
0.0146
0.0445
Note: the table reports the estimation results in the GMM framework (eq. 3.19). Monte Carlo
replications are 500. The parameter configurations are:
and
MF, IV, QV, BV, and RV refer to MF implied volatility, integrated volatility,
quadratic variation, bipower variation, and realized volatility, respectively.

The results are summarized in Table 3.1. In addition to means and medians I also report
the root-mean-square-errors (RMSEs). There are several points to be highlighted from Table 3.1.
First, most parameter estimates using RV and BV are quite close to their corresponding true
values, measured in terms of mean and median. RMSEs for the parameter estimates decrease
roughly at the rate of 2 as the sample size goes from 150 to 600, showing the estimates converge
at

speed. Second, the use of RV and BV achieves a similar RMSE as the true infeasible QV

and IV. The results indicate the feasible estimator fares as well as the continuous-record QV and
IV. In summary, the finite sample results indicate that the GMM method and nonparametric
estimators for volatilities can recover the parameters of interest with reasonable precision.

3.4

Empirical Application
Our empirical analysis is based on MF implied volatilities, BVs and RVs for futures prices

of corn, a major agricultural commodity. In recent years agricultural commodity markets have
97

witnessed extreme and unpredictable increases in price volatility. However, gauging the level of
volatility risk and developing volatility-based tools in the agricultural commodity market has not
received so much attention as their counterparts in stock and currency markets. Faced with
volatility risk and lack of an instrument for hedging volatility, stakeholders in agricultural
commodities have to urge regulators of futures market to consider regulations on position and
trading limits. Against this backdrop, this application has significant implications not only for
precisely forecasting corn futures volatility and testing the efficiency of the corn option markets,
but also for developing volatility-based products in agricultural commodity markets. This section
will detail how to construct the risk-adjusted implied volatility for the corn market based on a 23year sample of daily options and high-frequency futures. The predictive power of the model will
be evaluated in the next section.

3.4.1 Data
MF Implied volatilities are extracted from call options contracts traded from Feb 25,
1987 to June 30, 2010. The intervals for which volatilities are computed are based on the
structure of expiration for corn futures and options. Corn futures contracts expire five times per
year, that is, March, May, July, September, and December. The corresponding options contracts
mature about one month ahead of futures expiration. Therefore, the five volatility intervals in
each year are November to February, February to April, April to June, June to August, and
August to November.

36

To extract the non-overlapping sample, I choose options on the

36

Although corn futures and options contracts are listed every year, lengths of intervals of each
year have slight date changes because the options expiration rules change and the number of
days varies year to year.
98

Wednesday immediately following the expiration date of the previous options contract.

37

The

intervals are either two or three months, resulting in a total of 118 observations. Call options are
filtered to exclude options that violate the boundary conditions, for example, in-the-money
options with a premium less than payoff for immediate exercise. Strike price increment is 10
cents and only options with strike prices inside the range of three SDs from
calculate MF implied volatilities.

38

are used to

For comparison purposes I also calculate BS implied

volatilities. BS implied volatilities are computed as the average of volatilities derived from two
nearest-to-the-money call options by inverting the BS formula. The risk-free rate is calculated by
compounding the corresponding three-month T-bill rate obtained from the Federal Reserve. RVs
and BVs should be good approximations to the true continuous quadratic variation and integrated
volatility measures. These are calculated based on the 5-minute returns on logarithmic corn
futures for the period matching the maturity of the corresponding options in the MF implied
volatility. For a typical trading day, I have 45 five-minute returns covering trading hours from
9:30 am to 1:15 pm. I also analyze the jump measure by calculating the difference between RVs
and BVs. Finally, all volatility measures are annualized for comparisons across varying intervals
and across years.
The upper panel of Table 3.2 presents summary statistics of all five volatilities in the
form of standard deviations and shows that the MF is on average higher than the RV. The
difference between MFs and RVs is sometimes used by market participants as a raw measure of
37

I select the sample this way because option trading seems to be more active during the week
following the expiration date and Wednesday has the fewest holidays among all weekdays.
38
Since the trading volumes of corn options with a $0.05 strike price increment (serial options)
are too small, I discretize the range of integration into a grid of 10 points, which also meets the
requirement that
0.35 SDs. The standard deviation of price movement before 2005 in the
corm market is set at 20%, while it is set at 30% after 2005, because the market is more volatile
in the latter period.
99

the volatility risk premium. But the difference is rather noisy and it is hard to determine its exact
value. The RV is systematically higher than the BV, implying positive jumps across time.
Table 3.2: Summary Statistics for Volatilities, 1987-2010
RV
BV
MF
BS
JUMP
Mean
0.236
0.196
0.253
0.221
0.040
Std. Dev.
0.088
0.067
0.082
0.092
0.028
Skewness
1.034
1.083
0.869
0.911
1.552
Kurtosis
3.913
4.057
3.814
3.981
5.229
Min
0.107
0.092
0.120
0.081
0.005
Max
0.534
0.422
0.579
0.590
0.148
Seasonality-mean
Feb-Apr
0.200
0.173
0.215
0.187
0.027
Apr-Jun
0.248
0.209
0.269
0.235
0.040
Jun-Aug
0.311
0.241
0.335
0.310
0.071
Aug-Nov
0.224
0.190
0.238
0.200
0.034
Nov-Feb
0.194
0.164
0.204
0.170
0.030
Note: in total, there are 118 observations. All volatilities are annualized standard deviations. RV,
BV, MF, and BS stand for realized volatility, bipower variation, MF implied volatility, and BS
implied volatility. JUMP is computed as the difference between the RV and the BV.

Empirically, corn price volatility exhibits seasonality. It tends to increase during summer
time as weather in this critical period causes uncertainty. Volatility is also sensitive to inventory
levels which can be seasonal. Periods of systematically greater and smaller volatility are present
in corn futures as depicted in Figure 3.1 and described in the lower panel of Table 3.2. For
example, the June-August interval that covers the most critical growing season and where the
weather effect is most pronounced displays the largest mean volatility, while the NovemberFebruary interval during the non-growing period has the smallest mean volatility. The repeated
movement pattern is embodied simultaneously in implied volatilities since market participants
incorporate this information into option prices. In addition, jumps also occur more often in the
June-August interval due to abrupt changes in weather conditions and supply forecasts.

100

Figure 3.1. Realized Volatilities (RVs), Bipower Variations (BVs), and MF Implied Volatilities
(MF) in Annualized Standard Deviation
0.7

RV

BV

MF

0.6
0.5
0.4
0.3
0.2
0.1

1987
1988
1988
1989
1990
1991
1992
1992
1993
1994
1995
1996
1996
1997
1998
1999
2000
2000
2001
2002
2003
2004
2004
2005
2006
2007
2008
2008
2009
2010

0

3.4.2 Estimation Results
I accommodate the seasonality feature in the empirical estimation by using four seasonal
dummy variables to proxy seasonal components so that moment conditions 3.11 and 3.13 can be
modified as:
(3.20)

,

(3.21)

,

,

where

, and

are dummies that take value 1 if volatility observations lie in Feb-

Apr, Apr-Jun, Aug-Nov, and Nov-Feb intervals, respectively, and take value zero otherwise;
and

(

are the corresponding coefficients. I define observations in the Jun-Aug

interval as the default.

101

Table 3.3: Parameter Estimates using the RV, BV and MF
Parameters
Estimates
Std. Err.
p-value
2.2450
0.6820
0.0010
0.0909
0.0204
0.0000
-4.4905
1.6034
0.0051
0.0105
0.0062
0.0886
0.0446
0.0042
0.0000
0.0046
0.0008
0.0000
0.0027
0.0009
0.0027
0.0014
0.0010
0.1408
0.0056
0.0012
0.0000
0.0055
0.0008
0.0000
0.0054
0.0008
0.0000
0.0038
0.0009
0.0000
0.0053
0.0007
0.0000
6.7019(0.5691)
(d.o.f=8) (p-value)
Note: realized volatilities (RVs) and bipower variations (BVs) are computed based on 5-minute
returns. The lag length in the Newey-West weighting matrix employed in the estimation is set at
5.

I estimate parameters of interest with two-month volatilities, that is,
month volatilities are converted to two-month ones using the adjustment coefficient

. Three. The

parameter estimates along with the corresponding asymptotic standard errors and p-values are
reported in Table 3.3. As can be seen in the first row of the table, the estimate for the mean
reversion parameter

is relatively high, suggesting a strong mean reversion pattern in volatility

under the P measure. As expected, the long-run mean parameter ( =0.0909) is close to the
sample mean of BV. Also, the results show that most seasonal dummy estimates are positive and
highly significant (
parameter

, implying that the BV in Jun-Aug is highest. The jump-adjusted

is statistically significant at the 10% level, which corroborates the difference

between the MF implied volatility and the risk-neutral expected IV (i.e., the approximation error
from Jiang and Tian (2005) is not negligible). The expected jump component in RV, , is also

102

significant (at the 1% level). All seasonal dummy variables in jumps are positive and highly
significant, consistent with earlier statistical analysis.
The volatility risk premium estimate

is negative and statistically significant at the 1%

level. This finding is consistent with other studies that have found negative risk premiums on
stochastic volatility in equity, currency, and energy markets. The magnitude of the estimated risk
premium is so large that it makes the volatility process non mean-reverting under the Q measure.
According to Gordon and St-Amour (2004), the stochastic volatility risk premium reflects the
risk preferences of a representative investor. This negative stochastic volatility risk premium
implies risk-aversion, which means that investors are willing to pay a risk premium to hedge the
volatility risk. Finally, as can be seen in the last row of the table, the chi-squared test of
overidentifying restrictions suggests that the overall specification is not rejected at conventional
significance levels.

3.4.3 Robustness Analysis
It follows from equations 3.15 and 3.17 that higher frequency data should provide better
possible estimates for QV and IV if the semi-martingale assumption is not violated. However, it
should also be kept in mind that, along with the presence of market microstructure noise
including price discreteness and bid-ask spread, ultra-high frequency returns may render the RV
and BV measures inconsistent since the market microstructure noise will invalidate the semimartingale assumption underlying equations 3.2 and 3.3 (Andersen, Bollerslev, and Huang,
2010). This prevents us from sampling too frequently while maintaining the fundamental semimartingale assumption. In order to investigate the robustness of our findings based on the 5minute returns, I first consider alternative volatility estimators constructed from more coarsely

103

sampled returns. This is a simple way to alleviate these contaminating effects, while maintaining
most of the relevant information in the high-frequency data. I re-estimate the model with RV and
BV constructed from 10-minute returns. As can be seen in Table 3.4, compared to the original
results, the parameter estimates show little difference in signs and significance levels, although
the volatility risk premium estimate falls a little from -4.4905 to -3.6993.
Table 3.4: Parameter Estimates using the RV_10, BV_10 and MF
Parameters
Estimates
Std. Err.
1.9214
0.5847
0.1049
0.0233
-3.6993
1.3752
0.0166
0.0046
0.0384
0.0038
0.0051
0.0007
0.0027
0.0008
0.0015
0.0009
0.0061
0.0011
0.0049
0.0007
0.0044
0.0008
0.0034
0.0008
0.0044
0.0007
8.2051(0.4137)
(d.o.f=8) (p-value)

p-value
0.0010
0.0000
0.0071
0.0003
0.0000
0.0000
0.0012
0.0761
0.0000
0.0000
0.0000
0.0000
0.0000

Note: RV_10 and IV_10 refer to RVs and BVs constructed from 10-minute returns. The lag length
in the Newey-West weighting matrix employed in the estimation is set at 5.

A better way to deal with the market microstructure noise is to construct a robust
estimator. This has been the subject of intensive research efforts recently (Barndorff-Nielsen and
Shephard, 2006; Barndorff-Nielsen et al., 2008). Barndorff-Nielsen and Shephard (2006)
developed a new BV measure robust to certain types of market microstructure noise. The BV
measure is the sum of the product of absolute returns and themselves lagged by two periods:
(3.22)

,

104

The staggering relative to the measure originally considered in eq. 3.16 alleviates the
confounding influences of the market microstructure noise, resulting in empirically more
accurate finite sample approximations (Huang and Tauchen, 2005). It is also shown that the
measure converges in probability to the IV under the general assumption that the logarithmic
price process is a semi-martingale. The results in Table 3.5 are produced using the new measure.
The parameter estimates as well as their standard errors are generally close to the original results.
Overall, the results clearly confirm the robustness of our previous findings with respect to market
microstructure noise.
Table 3.5. Parameter Estimates using the RV, BV_R and MF
Parameters
Estimates
Std. Err.
p-value
2.5722
0.7214
0.0004
0.0826
0.0172
0.0000
-4.3618
1.6284
0.0074
0.0152
0.0054
0.0050
0.0487
0.0047
0.0000
0.0046
0.0007
0.0000
0.0026
0.0008
0.0010
0.0018
0.0008
0.0263
0.0052
0.0010
0.0000
0.0058
0.0008
0.0000
0.0059
0.0009
0.0000
0.0039
0.0011
0.0003
0.0057
0.0008
0.0000
7.8308(0.4502)
(d.o.f=8) (p-value)
Note: BV_R refer to bipower variations robust to microstructure market noise by taking into
account an additional stagger. The lag length in the Newey-West weighting matrix employed in
the estimation is set at 5.

3.5

Forecast Evaluation
After estimating parameters of interest, it is straightforward to convert the risk-neutral

MF implied volatility into an adjusted equivalent. Consistent with the literature (e.g., Christensen

105

and Prabhala, 1998; Egelkraut and Garcia, 2006), the predictive ability of the risk-adjusted MF
implied volatility is assessed using three criteria: a) forecast unbiasedness, b) informational
efficiency, and c) superior predictive power relative to alternative forecasts. Three alternative
forecasts are considered: the MF implied volatility, the BS implied volatility, and a HV realized
during the past year.

3.5.1

Testing the Unbiasedness Hypothesis
I examine unbiasedness of AD using a univariate regression:
(3.23)

where

and

,
are the annualized RV and AD. Panel A in Table 3.6 summarizes the OLS

regression results for AD and alternative volatilities. These parameters are estimated following a
procedure robust to the presence of heteroscedasticity. The Durbin-Watson statistics are close to
two in most regressions, indicating that the regression residuals exhibit little autocorrelation. In
the regression of the AD, the slope coefficient is positive, significantly different from zero,
insignificantly different from one, while the intercept is small and insignificantly different from
zero. The Wald test shows that the null hypothesis of

and

cannot be rejected,

implying that the new volatility estimate is an unbiased estimator for future realized volatility.
This confirms our prediction that forecast bias of the risk-neutral MF implied volatility can be
eliminated after making the risk premium adjustment. Results in other model specifications show
that the unbiasedness hypothesis is strongly rejected and all the slope coefficient estimates are
downward biased. This finding is consistent with previous research (e.g. Szakmary et al., 2003;
Jiang and Tian, 2005) and supports our argument that investors’ risk aversion brings forth a

106

negative volatility risk premium and, consequently, leads to higher volatilities implied in option
prices than their observed counterpart.
Table 3.6. Testing Unbiasedness and Informational Efficiency
Durbin-Watson
-Test Wald Test
Panel A
118 0.010
0.936
---1.874
0.92
0.710
(0.470) (0.000)
(0.340)
(0.495)
118 0.015
-0.876
--2.210
4.450
6.620
(0.279)
(0.000)
(0.037)
(0.002)
118 0.068
--0.760
-2.200
18.9
20.060
(0.000)
(0.000)
(0.000)
(0.000)
113 0.109
---0.548
1.305
36.94
21.78
(0.000)
(0.000)
(0.000)
(0.000)
Panel B
118 0.027
0.594
-0.296
-2.063
2.160
0.950
(0.142) (0.034)
(0.209)
(0.145)
(0.419)
113 0.007
0.952
--0.002
1.865
0.300
0.280
(0.631) (0.000)
(0.997)
(0.837)
(0.837)
113 0.025
0.583
-0.318
0.005
2.066
2.260
0.630
(0.190) (0.038)
(0.181) (0.946)
(0.136)
(0.641)
Panel C
118 0.002
-1.126 -0.228
-2.166
0.160
5.040
(0.926)
(0.001) (0.435)
(0.693)
(0.003)
113 0.011
-0.872
-0.025
2.205
2.450
3.850
(0.405)
(0.000)
(0.728)
(0.121)
(0.012)
113 0.000
-1.080 -0.190 0.025
2.177
0.070
3.240
(0.983)
(0.001) (0.522) (0.729)
(0.791)
(0.015)
Note: is the sample size. The numbers in parentheses below the parameter estimates are pvalues. Each regression is implemented with a robust procedure taking into account of
heteroscedasticity. The t-test in Panel A is for the hypothesis:

. The

Wald test in panel A is for the null hypothesis:

. The

t-test in Panel B is for the hypothesis:
hypothesis:

and

The Wald test in panel B is for the null

and

, or

. The t-test in Panel C is for the hypothesis:
C is for the null hypothesis:

and
.

107

The Wald test in panel
, or

3.5.2

Testing Informational Efficiency
If AD is informationally efficient relative to alternative forecasts, the forecast will

subsume all information contained in them. The hypothesis is assessed in encompassing
regressions including two or more volatility forecasts as explanatory variables:
39

(3.24)

If the slopes of alternative forecasts are equal to zero, it means they provide no incremental
predictive information. A total of three encompassing regressions involving AD are analyzed for
different choices of volatility measures. As shown in Panel B of Table 3.6, the t-statistics cannot
reject the hypothesis that the slope coefficient for BS or HV is zero at any conventional
significance level. Furthermore, the joint test does not reject that both slope coefficients for BS
and HV are zero. It implies that alternative volatilities are redundant and their information
content has been subsumed in AD, supporting that AD is an efficient forecast for future realized
volatility. In addition, I jointly tested unbiasedness and efficiency of AD by formulating a joint
hypothesis as

, and

(

). The null hypotheses in all

three specifications are not rejected. This provides further evidence that AD is unbiased and does
subsume all information contained in both BS and HV.
For comparison purposes, I investigated the informational efficiency of the MF implied
volatility relative to BS and HV. As can be seen in Panel C of Table 3.6, the slope(s) of BS and/or
HV is statistically insignificant in all three specifications, suggesting that the MF implied
volatility is also efficient. But the joint hypothesis of unbiasedness and efficiency is strongly
rejected in all regression specifications due to the forecast bias.

39

Regressions (3.24) are appropriately specified only if the variables are stationary. This has
been confirmed by Egelkraut and Garcia (2006).
108

3.5.3

Testing Predictive Power
An unbiased forecast could increase its predictive power after the bias is removed. To

further assess the difference in accuracy of volatility forecasts, we use the following two loss
functions: mean absolute percentage errors (MAPE) and mean absolute errors (MAE)
(3.25)

,

(3.26)
where

,

is a volatility measure (

AD, MF, BS, or HV), T is the number of forecasts. The

MAPE loss function measures the relative accuracy, while MAE measures the absolute one. In
addition, they are generally more robust to the possible presence of outliers than the mean square
errors, which is especially important for small samples. I assessed the statistical significance of
difference in pairs of competing models by the Diebold-Mariano (DM) Test and one of its
modifications. Such tests are based on the null hypothesis of no difference in accuracy of any
two competing forecasts. I define the loss differential between two forecasts as
where

is percentage forecast error or forecast error. The DM statistic is computed for

one-step-ahead forecast as:
(3.27)
where

,

is the sample mean loss differential.

is an estimate of the asymptotic variance of ,

taking into account the autocorrelation of series

. Under the null hypothesis of equal forecast

accuracy, the DM statistic has an asymptotic standard normal distribution. Since the DM test can
be over-sized in small samples, I also consider a Modified DM (MDM) test, where DM is
multiplied by the factor

. The MDM test is useful for determining the significance of
109

differences in our competing forecasts because it does not rely on the assumption of forecast
unbiasedness.
Table 3.7. Testing Predictive Power
MAPE DM
MDM
MAE
DM
MDM
AD 16.512
0.038
MF 17.840 -1.715(0.086) -1.740(0.092)
0.041
-1.120(0.263) -1.110(0.269)
BS 18.915 -2.063(0.039) -2.046(0.043)
0.045
-2.543(0.011) -2.522(0.013)
HV 24.529 -3.995(0.000) -3.962(0.000)
0.060
-4.334(0.000) -4.297(0.000)
Note: the numbers in parentheses beside the statistics are p-values. MAPE and MAE are the mean
absolute percentage errors and mean absolute errors. The DM and MDM tests are reported when
the benchmark is the AD implied volatility, compared to each one of other forecasts.

Table 3.7 reports the DM and MDM tests when the benchmark is the AD forecast. The
signs of the DM and MDM statistics are always negative, implying that the benchmark’s loss is
lower than any other forecast. The findings show that AD has the smallest forecast error among
all volatility forecasts whether in MAPE or in MAE. For MAPE and for all models, I reject the
null of equal predictive ability, suggesting that the AD benchmark fares the best. For MAE and
for BS and HV forecasts, I reject the null of equal forecast accuracy, indicating that AD has
superior predictive power over BS and HV. However, when the benchmark is compared to the
second best (the MF implied volatility), I fail to reject the null of equal forecast accuracy.
Overall, the AD implied volatility tends to have superior predictive power over alternative
forecasts.

3.6

Conclusion
The extent to which implied volatilities provide unbiased forecasts of corresponding

future realized volatilities has attracted a great deal of attention in recent years. In testing the
unbiasedness hypothesis, the typical finding is an upward-bias predictor. Prior studies have
argued that the volatility risk premium may be an important factor in explaining the documented
110

bias in volatility forecast. An intuitive method to eliminate the bias is to directly transform the
risk-neutral implied volatility into an observational equivalent. Following Becker, Clements, and
Coleman-Fenn (2009), I proposed a risk-adjusted implied volatility by converting a risk-neutral
implied volatility using risk premiums. However, my approach extends Becker, Clements, and
Coleman-Fenn by using jump-diffusion models, which are widely accepted as more plausible
descriptions of asset returns than general diffusion processes. I explicitly derive the link between
the MF implied volatility and the QV in the jump-diffusion process, which is the risk adjustment
model. The model suggests that the jump risk premiums also contribute to the forecast bias. The
GMM estimation framework of Bollerslev, Gibson, and Zhou (2010) was also extended to a
jump-diffusion model. A Monte Carlo simulation study confirms that the framework performs
well in recovering the parameters of interest.
I illustrated the procedure using volatilities implied in the two-month options on corn
futures. Motivated by the success of the VIX index based on the liquid S&P500 index options, I
calculated a MF implied volatility, ―corn VIX‖, using Jiang and Tian (2005) method. This is the
first time a MF measure has been used to gauge volatility in commodity markets. Such a MF
measure may help facilitate volatility contracts in corn futures markets and expand risk
management portfolio for corn market participants. In the empirical model specification, I
accommodate seasonality characteristics in corn volatilities into the moment conditions and find
statistically significant seasonality effects. I also find that a negative volatility risk premium
exists in the corn futures, consistent with substantial evidence documented in the equity index
and currency markets. After evaluating forecast performance of the AD implied volatility, I find
that the adjusted implied volatility accurately reflects the patterns of realized volatility and
provides an unbiased forecast. Although both the adjusted and unadjusted MF implied volatilities

111

are more efficient than BS and HV, the former tend to have superior predictive power with
respect to relative accuracy.
Our method could be extended to equity and currency markets. Furthermore, while our
specification for the futures price process is quite general, our method could be extended to many
other continuous time models capturing more empirically relevant features of price and volatility
dynamics, such as time-varying risk premiums and jumps in volatility. The recent study by
Bollerslev, Gibson, and Zhou (2010) found significant evidence for temporal variation in the
volatility risk premiums depending on macro-finance state variables. I leave further work along
these lines for future research.

112

APPENDIX F

113

APPENDIX F: DERIVATION OF EQUATION 3.10
First, I construct the link between the MF implied volatility and the expected IV under the
Q measure. Integrating eq. 3.4 over time and taking expectations, I have
.
Therefore,
.
Jiang and Tian (2005) have proven that
.
It is valid for a very general class of asset price processes including a jump-diffusion process
because its derivation does not require any knowledge of the asset return process. The detailed
derivation can be found in the appendix of Jiang and Tian (2005). Hence,
(F.1)

,

where

.

Second, following from the results in Bollerslev and Zhou (2006), I establish the link between
the risk-neutral expectation of the IV and the expectation of the IV in the observed measure:
(F.2)

.

where
,

.

114

Finally, I derive the expression for the expected value of quadratic variation:
(F.3)

.

115

APPENDIX G

116

APPENDIX G: MONTE CARLO EXPERIMENTAL DESIGN
I assume one year has 252 trading days, and I divide up a day into 45 artificial ―5-minute‖
intervals.

40

I further partition each five-minute interval into 5 smaller segments for the

continuous record. The horizon is fixed at two month. Without loss of the generality, I set the
drift of the price dynamics in (3.2) to zero. The initial asset price is at 250 cents, and the initial
latent stochastic volatility with a two-month maturity begins at 0.025. The parameter
configurations in the P measure are:
and

,

,

,

,

, while other parameter configurations in the Q measure are:

and the volatility risk premium
parameters

and

41

.

,

,
,

,

With such parameter configurations, the composite

. The MF implied volatility is calculated using the

above Jiang and Tian approach. I only use call options with strike prices ranging three SDs from
current asset price

. The strike price increment is 10 cents, which is close to the 0.35 SDs.

Options valuation is based on the stochastic-volatility jump-diffusion model under the riskneutral measure, eqs. (3.4) and (3.5). I resort to Monte Carlo simulation method for pricing
options. The antithetic variable technique is adopted to reduce the standard error of the options
value and improve the efficiency of the results. For a 2-month to expiration option, 9450
(42

random shocks from a normal distribution are drawn for the price process. This

procedure is replicated for the volatility process with drawing shocks from a Poisson process.

40

In the empirical session, I will apply my method to corn futures. Corn futures trading hours
are from 9:30 am to 1:15 pm at the trading floor. Therefore, it generates 45 5-minute returns each
day.
41
Several parameter values refer to parameter estimates of futures price dynamics in the existing
agricultural commodity literature (Koekebakker and Lien, 2004).
117

The simulation sample path for options pricing is set to 10000. The option value for a call is then
calculated as the average value across all paths.

118

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119

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