Tums

l

/\ ,,r
I 1
LAUKI

\

This is to certify that the

thesis entitled

FORECASTING CORN BASIS AND CASH PRICE DISTRIBUTIONS

presented by

MIN ZHUANG

has been accepted towards fulfillment
of the requirements for

M.S. Agricultural Economics
degree in

 

 

 

 

0' i ' «j
Mrofcssor

Date 11/06/2000

 

0-7639 MS U is an Affirmative Action/Equal Opportunity Institution

 

 

 

LIBRARY

Michigan State

University

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 chlRC/DateDue.p65-p.15

 

FORECASTING CORN BASIS AND CASH PRICE DISTRIBUTIONS

By

Min Zhuang

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

Master of Science

Department of Agricultural Economics

2000

ABSTRACT
FORECASTING CORN BASIS AND CASH PRICE DISDRIBUTIONS
By

Min Zhuang

This research estimates the basis distribution of corn using time series techniques
and combines the resulting forecasted basis distribution with the market implied futures
price distribution to generate a forecast of the cash distribution for corn.

The data used in this study are the storage basis on corn at Saginaw, MI, relative
July Futures price in Chicago Board of Trade. This research studies the stochastic
properties of the basis and provides evidence that multi-year basis data could be
stationary. An autoregressive conditional heteroscedasticity model (ARCH) and a
generalized ARCH model are specified to estimate and forecast the basis distributions.
The out-of-sample forecast evaluation implies the ARCH model provides a good
description of distribution for the Saginaw corn storage basis.

Then black’s option pricing model is used to generate the implied volatility of
futures price. Using simulation techniques, we combine forecasts of the basis distribution
with the market implied futures price distributions to get a forecast of the cash price
distribution. This combination gives us a new approach to generate a forecast of
commodity cash price distributions. With the predictability features of basis distributions
and the available market information reflected in traded commodity option prices, this
combination may outperform forecasts of cash distributions either from structural models

or pure series models.

To my parents Sun Lihua, Zhuang Jiedun

and my husband Ziqiang Li

ACKNOWLEDGMENTS

I am grateful to Professor Steve Hanson, my major advisor, for his support,
encouragement, and efforts to provide me with a complete training in scientific research,
and especially for his patience. I have benefited tremendously from Dr. Hanson’s love of
research, his perseverance, and his extraordinary kindness. I also thank Dr. Robert Myers,
who taught me valuable knowledge of econometrics and helped me throughout the
research. Thanks Dr. James Hilker for giving me empirical knowledge about basis and
many important materials for the research.

Dr. Hanson, Dr. Myers, Dr. Hilker’s contribution and their spirits are reflected
through this work. Without their help, it is impossible for me to carry out this research.
Their help is very important and highly appreciated.

I would like to thank the entire department staff for being so pleasant and helpful,
and especially thank Nicole for all her help. I also want to thank the many students in the
department for their friendship. They make my study at Michigan State a very memorable
period of my life.

All I accomplished is given by my parents, who, even in the most difficult years,
taught me the importance of education and knowledge.

Finally, I want to thank my husband, Ziqiang Li. His love has always been my

mental support and the source of my happiness.

TABLE OF CONTENTS

LIST OF TABLES ................................................................................. vi
LIST OF FIGURES ............................................................................... vii
Introduction .......................................................................................... 1
Literature Review .................................................................................. 5
Previous Research on Modeling Basis .............................................. 5
Previous Research on Modeling Commodity Cash Price Distributions ..... 8
Empirical Models of Basis ....................................................................... 9
Data .......................................................................................... 9
Stochastic Properties of Basis Data ................................................ 11
Estimate Basis Generating Process ................................................ 14
Simulating the Corn Basis Distribution ............................................ 21
Evaluate Different Basis Models .................................................... 26
Forecasting Cash Price Distribution ......................................................... 27
Black’s Model and Implied Volatility ................................................ 27
Correlation between Basis and Futures Prices ................................ 3O
Simulating the Corn Cash Price Distributions .................................... 32
Conclusion ......................................................................................... 35
APPENDIX 1 Basis Definitions ............................................................... 39
APPENDIX 2 Unit Root Tests on Basis Data ............................................. 41
APPENDIX 3 Independent Test for Basis and Futures Price Distributions ....... 45
References ........................................................................................ 49

LIST OF TABLES

Table 1 ARCH-normal model specification ................................................ 17
Table 2 GARCH-M-normal model Specification ......................................... 20
Table 3 Statistics of the Simulated Basis Distributions ................................. 24

Table 4 Statistics of Simulated Futures and Cash Price Distribution at Option

Maturity Time .......................................................................... 35
Table A Estimation of ARMA model for Basis ........................................... 46
Table B Estimation of ARMA model for Futures Prices ............................... 46

vi

LIST OF FIGURES

Figure 1 Weekly Corn Basis Level from 1974 to 1997 ................................. 11
Figure 2 Simulated July Basis Distributions

2(a) Forecasted July Basis Distribution from ARCH model .................. 23
2(b) Forecasted July Basis Distribution from GARCH-M model ............ 23

Figure 3 Dynamic Forecasts of Weekly Corn Basis

3(a) ARCH model ....................................................................... 25
3(b) GARCH-M model ................................................................. 25
Figure 4 Kernel Fit Plot of Basis Condition on Futures Price ......................... 32

Figure 5 Simulated 1997 Corn Futures Price Distribution at Options
Maturity Time ......................................................................... 34

Figure 6 Simulated 1997 Corn Cash Price Distribution at Options Maturity Time
............................................................................................... 34

vii

Introduction

Commodity prices vary across time as a result of seasonal and daily supply and
demand events. These price movements generate price uncertainty for farmers and
agribusiness firms. A risk averse decision maker's response to price risk generally
focuses on reducing the likelihood of unfavorable outcomes through the use of one or
more risk management strategies. One strategy is to use pricing instruments such as
forward, futures, or option contracts to offset or “hedge” price risk. Other strategies
include such things as diversifying risk by producing multiple commodities, or
maintaining flexibility in the selection of inputs and/or input levels.

Futures and option markets are important pricing instruments for many of the
major commodities in the United States including com, soybean, wheat, hogs and cattle.
Firms producing or using these commodities may be able to reduce price risk by taking a
“hedge” position in one or more of these markets. The essence of hedging is to have a
position in a pricing instrument that is "opposite" to the position held in the cash market.
Because cash and futures prices for a commodity tend to move in parallel, losses (gains)
in the cash market tend to be offset with gains (losses) in the opposite futures position.

A key to effective hedging in these markets is to understand how closely cash and
futures price movements move together. Basis is a concise way to represent the
relationship between futures and cash prices and is commonly defined as the difference
between a cash market price at a specific location and time and the price for a given
futures contract at a specific time. Working (1949) defines basis as the futures price

minus the cash price at a particular point in time. More formally, basis can be defined as:

(1) b: =ft(T)-Pr

where b, is the basis at time t; p, is the cash price at a particular location at time t;fi(1‘) is

the futures price at t for delivery at time T.

There are many ways to measure basis. “Nearby basis”, “new crop basis”,
"harvest basis" and “storage basis” represent alternative measures of basis that each can
have important uses (see Appendix 1). However, the focus of this study is on the storage
basis. Storage plays an important function in many commodity markets. Agents store
commodities for a variety of purposes. Producers may store grain while waiting for a
better price; elevators may profit by providing storage facilities to producers or by storing
grain for themselves; and processors may store grain to ensure an adequate input supply.
The storage basis is a key determinant of the potential benefit from storing a commodity.

To illustrate the role of the storage basis, consider a storage operator who takes a

long position in the cash market and a short position in the futures market in period t and

reverses the position in period t+I . The operator buys the cash commodity at p, and sells
at p,+ 1, and sells an equal amount of the fiitures at ffl) and buys it back at f,+1(7). The
total return is (p,+)-p,)+(f}(T)-fi+1(7)); or in terms of the basis defined in (1), the return is

given by b, - b,+ ,. Thus, the change in the storage basis over the life of the hedge

determines the operator’s gross return from the storage hedge.

Because cash and futures prices tend to move up and down in the same direction,
the variation in basis is typically small relative to variation in prices themselves. For
commodities with continuous inventories, like corn, new market information that affects
cash prices tends to have similar impacts on futures prices. If traders anticipate a

decrease in demand for grain, this will tend to depress both distant and nearby futures

since the different months are linked through inventories (Tomek, 1981). Therefore,
even if cash and futures price levels change greatly over time, their difference may not
Change much. Thus, compared to cash and futures price levels, basis is reasonably
predictable, especially as delivery time approaches on a particular futures contract.
Theoretically, the basis narrows as the delivery month approaches reflecting the
decreasing cost of storage and the converging supply and demand conditions in cash and
futures markets. The relatively high level of predictability of basis results in hedgers
typically preferring the "basis risk" associated with offsetting positions in the cash and
futures markets to the risk associated with a cash market position alone.

Most previous studies on basis focus on measuring basis changes over time or
identifying factors that influence basis levels (Working, Tomek). Only a small number
deal with forecasting basis levels, and even less work has been done on modeling or
forecasting basis distributions (Tilley, 1988, Liu, 1994).

The predictability and relatively low variability of basis are a key to effective
hedging. In addition, these features may facilitate improvements in forecasting cash price
distributions. Commodity cash prices observed through time are the result of a complex
mixture of changes associated with supply and demand conditions. The time series of
cash price may include seasonal, cyclical, trend, and irregular factors (Ender 1995).
Economists have attempted to identify empirical regularities in cash prices using a variety
of techniques. Unfortunately, it is often difficult to forecast cash prices accurately from
historic observations on past prices and other market factors. To complicate matters, data
on other market factors influencing supply and demand are ofien only available after

significant time delays.

Futures prices provide a forecast of the expected cash price at the delivery
location during the period when the contract matures. In addition, options on futures
prices can be used to generate forecasts of the distribution of futures price at contract
maturity using implied volatility techniques (Hull 1993, Hilliard and Reis 1999). An
implied volatility is the volatility of the underlying futures price implied by market traded
option prices. It can be used to monitor the market’s opinion about the volatility of a
particular futures contract. Because options premiums presumably account for all
available market information on the day the options is valued, implied volatility forecasts
of futures price distributions may be more reliable and up to date than estimates using
only historical data. Furthermore it may be that combining forecasts of the relatively
predictable basis distribution using public data with the forecasts of the futures price
distribution using information reflected in traded commodity option premiums will result
in improved forecasts of the local cash price distribution relative to forecasts based solely
on publicly available data.

This study estimates the basis distribution of corn using time series techniques
and combines the resulting forecasted basis distribution with the market implied futures
price distribution to generate a forecast of the cash distribution for corn. Because the
resulting cash price distribution estimate uses information contained in commodity
futures option premiums, the resulting price distribution should be superior to forecasts
based solely on historical data.

More specifically, the objectives of this research are: I) generate alternative time
series models that characterize the corn basis distribution; 2) use the basis distribution

models to develop forecasts of basis the distribution; 3) evaluate the performance and

forecasting ability of the alternative models; 4) combine the forecasted basis distribution
with a market forecast of the futures price distribution to generate a forecast of the corn
cash price distribution. The data we used to estimate the price distribution in this study
are storage basis on corn at Saginaw, Michigan relative July Futures price in Chicago
Board of Trade (CBOT). Because we only generate a single cash price distribution at one
point in time, we do not evaluate the performance of the cash price distribution forecast
in this study. Testing whether the method developed and illustrated in this study improves

forecasts of cash price distributions is left to future research.

Literature Review

Previous Research on Modeling Basis

We begin by discussing the theory underlying storage basis behavior and then
discuss time series approaches that can be used to estimate corn price basis distributions.
There are two basic approaches used to model basis. One approach is to model basis
directly using a structural model. This has been the approach taken in many of the studies
on basis behavior which rely on the storage theory developed by Working (1949) and
Brennan (195 8). The implication is that the basis for a storable commodity represents the
market’s estimate of the price of storage where the price of storage is, in turn, determined
by the supply and demand for storage. The theory of storage argues that if firms operate
in competitive markets, then in long-run equilibrium their expected gains from storage
for a given period must equal the storage cost of the firm. Kaldor (1939) and Working
(1949) modify the theory by introducing the concept of “convenience yield ” which is the

value generated by having the commodity readily available to meet users needs and will

be positive but approaches zero at high stock levels. Convenience yield may offset what
appears as a loss from the storage function itself and provides an explanation why storage
occurs when the speculative return from storage is below observable costs.

Tomek (1996) applies these arguments to derive a typical structural model for
basis that incorporates both observable economic costs as well as convenience yield. The
main components of the basis model (price of storage) are the sum of three items:
opportunity cost of carrying inventory from one period to the next (usually represented as
interest cost); cost of other inputs involved in storage; and convenience yield. In this

framework basis can be modeled as:
(2) b: = C(11) =p: rfl. T) + s(t. T) -y(t. T)
where c(t, T) is the marginal storage cost from t to T; p, is spot price at time t; r(t, T) is an

opportunity cost rate of return from t to T; s(t, T) is physical storage cost fi'om t to T,
which could include warehousing charges, cost of insurance, the cost of any processing
or transformation services, as well as any anticipated losses from deterioration; and y(t, T)
is convenience yield. This general model of storage basis provides a basic insight into
the underlying concepts that drive basis behavior for storable commodities. The model
can easily be extended to account for a number of additional factors, such as inventory
levels, risk premium and transportation cost.

Because of the difficulty of making ancillary forecasts of explanatory variables in
structural models, basis forecasts ofien have been made from time series models.
Autoregressive Integrated Moving Average process (ARIMA) and autoregressive

conditional heteroscedasticity (ARCH) models, and Generalized ARCH (GARCH)

models are popular time series methods to model and forecast commodity prices
distributions.

Generally, ARIMA models perform well relative to alternative methods when
forecasting over short time horizons, but performance deteriorates over longer time
horizons. The existence of stochastic trends commonly found in time series of
commodity prices can easily be accommodated in ARIMA models. These models are
typically reduced form equations utilizing the information in past commodity prices, but
may also take account of other available information that might impact the price levels.

Commodity price distributions often exhibit time-varying volatility in commodity
prices that can be observed as autocorrelation patterns in the conditional variance of price
innovations (Baillie and Myers 1992, Yang and Brorsen 1992). Although volatility
clustering has a long history as an empirical regularity characterizing high-frequency
commodity prices, it was not until recently that applied researchers recognized the
importance of explicitly modeling time-varying second-order moments. Two common
ways of accounting for time-varying volatility in commodity prices are the ARCH model
introduced by Engle (1982) and the GARCH model developed by Bollerslev (1986).
ARCH and GARCH models are highly nonlinear and maximum likelihood is the usual
estimation approach. ARCH and GARCH models also can capture some of the observed
non-normalities in the empirical distributions of commodity prices.

While time varying volatility in commodity cash and futures prices have been
well documented, the behavior of basis volatility has received relatively little attention.

Since basis represents the relationship between futures and cash price, one can generate

basis distributions from the estimates of futures price fi(T) and cash price p, distributions.

This can be done in a variety of ways ranging for estimating supply and demand systems
for a commodity to estimating reduced-form equations for futures and cash prices.
Baillie and Myers (1991) use bivariate GARCH models to estimate joint cash and futures
price distributions for six commodities. While the study did not specifically examine
basis relationships, the authors’ approach provides an implicit model of the basis
distribution for each commodity and could also be used to generate forecasts of basis

distributions using simulation techniques.

Previous Research on Modeling Commodity Cash Price Distributions

Methods used to estimate commodity price distributions are very similar to those
used to estimate basis distribution models. Structural models for cash prices are specified
according to economic theory and market conditions. Since prices of agricultural
products are often related, multiple-market partial equilibrium models may be specified.
Forecasts can then be generated from either single equation or from the unrestricted or
restricted reduced-form models. Depending on the research problem and commodity, the
model may include endogenous variables, lagged endogenous variables and exogenous
variables. Streeter and Tomek (1992) find that the variance of futures price changes
depends on a variety of factors, including time-to-maturity, seasonality, economic
conditions and market structure. Structural models have the potential to provide useful
information. However, as discussed in the previous section, one important issue in
generating forecasts from structural models is the timing of when information is known.
The analyst must not incorporate variables into the model whose values will not be
known at the time that actual forecasts are made. If there is such a variable in the model,

then its value must be forecasted independently and, hence, treated as exogenous in the

model. Most of the research on commodity price distributions use time series and non-
parametric approaches, such as ARCH, GARCH, Poisson Jump, time-independent
mixture-of-normal distribution model (Kim and Kon 1994) and exponentially weighted
moving average models (Venkateswaran and Meenakshi 1993). These models do a good
job of capturing the time-varying volatility of commodity prices. However, all these
methods only involve historical price data and do not include all available market
information. Thus, the forecasted price distributions are backward rather than forward
looking. This study will incorporate available market information into forecasts of cash
price distributions by combining the forecasts of basis distribution using historical time
series data with forecasts of the corresponding futures price distribution using the

information contained in traded option premiums.

Empirical Models of Basis

Data

The primary focus in this study is on basis distributions for intra-year storage.
Basis is measured relative to the July futures contract because it is nearest the end of the
crop year (storage season). Basis is calculated using local cash prices for corn at Saginaw,
Michigan, and July Futures price at the Chicago Board of Trade (CBOT). Cash and
futures prices are each quoted as the closing price on Wednesday each week. So the basis
data are daily prices sampled at weekly intervals. The sample data contains 1,218 basis
observations starting in the first week of August 1974 and ending in the third week in

Dec. 1997. To construct the continuous series when the current year’s futures contract

matures and is no longer traded, basis is calculated by switching to the following year’s
futures price.

Typically, the bulk of Saginaw’s corn harvest is in November. After harvest there
is usually a large amount of corn available for storage. Although supply is at its peak in
the fall and declines as the crop year progresses, the demand for corn is relatively
constant across time. Theoretically, the cash prices will typically be at their lowest point
at the peak of harvest time and then rise gradually until crop year ends. Because the
futures price is a forecast of cash price at a future delivery time, we would not expect any
seasonal component in futures price levels. Thus we would expect basis levels to exhibit
a seasonal component, which reaches its highest point at November harvest and then
decreases gradually through the storage season. However, basis may not follow this
pattern in the last few weeks of the crop year since futures prices are often highly variable
close to maturity. Thus there may be a type of structural change each year in the data
series at the contract switching period because the two prices represent different supply
and demand conditions. The graph of the basis realizations are shown in figure 1:

By inspection, the graph of storage basis for corn at Saginaw shows a fairly
consistent pattern across each year, although some variations are notable. The expected
seasonality and contract jump can be detected in the graph. As expected, the basis level is
generally wide from August to November (harvest time) and then narrows during storage.
The obvious outliers at end of the 1995 crop year (July 1996) are due to relative high
futures prices at contract maturity time, reflecting the expectation for a short supply
situation for next crop year. Visual examination of the basis series also suggests a

possible shift in the average level of the series, implying a potential structural change

10

cents/bu
150

 

100 .

 

-150

 

 

JUF74 JuF77 JULBO JuL83 JuL86 JuF89 JuF92 JuL95

Figure] Weekly Corn Basis Levels from 1974 to 1997

occurred around the 1982 crop year. However, visual examination at the series shows
little evidence of any random drift. Visual examination of the data always is the first step
to identify the data generating process. It indicates that the basis series generating process
may include seasonal components, contract jump effects and possible structural shift. The
information is very helpful in studying stochastic properties of the data and estimating an

appropriate basis model specification.

Stochastic Properties of Basis Data

An important question when using time series approaches is whether or not the
data is characterized by a stochastic trend. There is empirical evidence suggesting most

high frequency (daily, weekly, even monthly) commodity prices are non-stationary. If the

11

time series of interest is non-stationary, then conventional econometric techniques and
statistical inferences are no longer valid.

There are numerous studies that have examined stochastic properties of
commodity prices. Baillie and Myers (1991) test for unit roots in the logarithm of daily
cash and futures price levels of six commodities and find a unit root for each commodity
price over a one-year period. Goodwin and Schroeder (1991) finds unit roots in weekly
beef prices for eleven U.S. regional markets; and Ardeni (1989) find unit roots for a
group of nonstorable and storable commodities using monthly data. Goodwin (1992) tests
monthly wheat prices covering the period from January 1978 through December 1989 for
five important international wheat markets and finds evidence of nonstationary.

Despite the detailed analysis of stochastic properties of commodity prices, there
has been relatively little attention given to analyzing stochastic properties of basis. Some
studies examine cointegration relationships between futures prices and cash prices. These
studies attempt to determine whether the two markets are influenced similarly by pricing
information by investigating the properties of the price differences between the two
markets. Cointegration tests of futures and cash price relationships are measures of the
extent to whether basis is “stationary” over time. Note that a necessary condition for
cointegration to exist is that each of the cointegrated variables must contain a unit root.

Baillie and Myers (1991) suggest that basis are non-stationary since basis is
function of the interest rate which has been found to be nonstationary. Thus, a unit root in
interest rate will indicate a unit root in basis. This suggests there may be good reasons for
expecting not to find cointegration between cash prices and the prices on a particular

futures contract with a fixed maturity. They conduct cointegration tests on daily futures

12

and cash prices over one year and find no evidence of cointegration for any commodity.
However, they do not preclude some other linear combination of cash and fixtures prices
being stationary.

Fortenbery and Zapata (1993) analyze the relationship between cash and nearby
futures prices for corn and soybeans. Unlike previous studies, they investigate a much
longer time series, employing daily data from crop year 1980 to 1990. They conducted
cointegration tests on the daily basis data year by year, as well as for the aggregate period
(multiple-year). Year-by-year tests revealed basis in some years is cointegrated while in
other years it is not. When years were aggregated, evidence of cointegration was detected
for all cash and fiitures market pairs considered, that is, the multiple-year basis is
stationary. Though these studies do not account for contract switching in multiple-year
basis, they give an indication that the one-year period basis may have different stochastic
properties than the multiple-year basis.

Understanding the stochastic properties of the basis data can lead to improved
model specification for estimating and forecasting basis. Based on the cash and futures
price properties, July corn storage basis tends to be large in November and gradually
decrease as the maturity date approaches, reflecting the decrease of storage costs within
the crop year. The basis on the delivery date approximates transportation costs between
the cash and futures markets (from Saginaw to Chicago).

For basis series over a one-year period (daily or weekly data), the intra-year basis
may be nonstationary based on the theory that basis is a function of the interest rate. For a
multiple-year basis series, the basis generally reverts to approximately the level of

transportation costs at the contract maturity date. Because shocks to a stationary time

13

series are temporary, the effects of the shocks will dissipate over time and the series will
revert to its long-run mean level. For multiple-year basis series, if transportation costs are
stationary, the basis data may converge to this unconditional mean after eliminating any
deterministic trend. Thus the multiple-year basis data may be a stationary process even if
the intra-year basis is non-stationary.

Since our basis data are multiple-year data, the observations on dates when
futures contracts switch may cause the unit root test results to be biased. To eliminate the
contract switching effects, we conduct the unit root tests on multiple-year basis data after
removing the observations immediately before and after the contract switch. So the first
differencing data don’t include the differencing basis between different contracts. The
Augmented Dickey-Fuller tests are used on the data series after removing deterministic
components, such as time trend and seasonal effects (see Appendix 2 for detail). The test
results reject the null hypothesis that the basis data contains a unit root.l The unit root test
results for the multi-year basis data suggest we can work with basis levels directly. It
provides the evidence that multi-year basis data can be stationary. So stochastic trend that
dominate in high frequency movement might be countered over by structural breaks

(contract switching effects).

Estimate Basis Generating Process

In this section we present the results of the estimated basis generation process. We
begin by constructing a general ARMA model for the basis level. The correlogram plot

of basis data including autocorrelation function (AC) and partial autocorrelation function

 

' We also conduct the Dickey-Fuller unit root test for intra-year basis data. The results of crop year 74, 75,
76, 77, 78, 79, 96 show the null hypothesis of unit root can be rejected at 5% significant level.

14

(PAC) can help us decide the proper lags for AR and MA process. The correlogram plot
of basis shows that PAC cut off after lag2 and get higher around lag 7, AC declines
geometrically. The correlogram also shows distinct seasonal spikes in AC. The sine and
cosine functions are used to capture seasonal effects. So we try ARMA(2, 7) model with
seasonal variables and time trend first. However, the Q-tests of the residuals show that
there are autocorrelations in the residuals at 5% significance level. After adjust, we get
ARMA [(1,6), (1, 5, 6, 8)] and don’t find autocorrelation in the residuals.

Using Ljung-Box Q—statistics to test autocorrelation in the squared residuals of the
ARMA model, we find ARCH effects in the squared residuals suggesting the volatility of
the series is not constant and that there is serial correlation between periods of large (small)
volatility. Therefore univariate ARCH and GARCH models are used to represent the basis
generating process for corn. ARCH (GARCH) models can be used to produce both point

forecasts and volatility forecasts. The volatility forecasts can be measured by the variance

of the forecast error conditional on the current information 0, used to construct the
forecast, var(e,,m|9,). When ARCH (GARCH) is present, var(e,,m|Q,) will depend on

the elements of Q, and will, in general, be time varying. Again, the correlogram plot of

squared residuals can help us identify the proper conditional variance equation. The AC
and PAC show there are autocorrelation in lagl, 3, 5 and 6 in squared residuals suggesting
an ARCH (6) model for the conditional variance.

The t-statistics of the estimated parameters suggest that both basis mean and
variance have seasonality. Based on the visual examination of the data and the
preliminary diagnostic tests, the following simple univariate ARMA-ARCH

specification is first used to describe corn basis:

15

(3) bi = p+£ain~ 1' ifljet—j +51 +3:

i=1 j=l

h, =w+a1 312—1 +---+a,-e,2_,- +S,

 

where S,= éwj sin(25)+¢1 008(2flwt)152

(p=6. q=8)

The degree of frequency in seasonality variables can be determine by log-

likelihood ratio test, AIC and SIC criterions.

ARMA and ARCH parts are simultaneously estimated by numerically maximizing
the log likelihood using the Newton Remdt algorithm. We use the Ljung-Box Q statistics
to check the autocorrelation of the residuals from the ARMA part and the squared
standardized residuals from the ARCH part to get a "best-fitting" model specification. If
autocorrelation is present, we need to adjust the ARCH and ARMA models to eliminate
the autocorrelation in the residuals and squared standardized residuals. The estimated
results are shown in table 1.

The Q test results suggest there is no remaining autocorrelation in the residual
series or squared standardized residuals of the ARCH model at a 5% significance level.
Thus the model appears well specified in the sense that the models do a good job of
capturing all the autocorrelations. However, the kurtosis of this model is larger than the

normal distribution indicating a relative peak distribution. Many studies on empirical

16

Table 1. ARCH ~normal model specification

 

 

 

 

 

 

Coefficient Estimate t-value p—value
,1 1.171167 3.255497 0.001
91 1.570692 3.369696 0.001
4,1 1.712171 8.205744 0.000
(,2 1.226470 5.928530 0.000
(,2 0.422059 1.814145 0.070
(,1 0.908873 23.693460 0.000
as 0.060739 1.918030 0.055
Bl 0.153154 -2.868962 0.004
85 0.054035 0.632912 0.527
[36 0.112358 -2.870888 0.004
[,8 0.060424 -2.505826 0.012
Variance Level

0, 21.782927 0.000790 0.000
(,1 16.155443 0.000346 0.000
4,1 18.584877 0.000458 0.000
(,2 10.758560 6.41E-O6 0.000
4,2 1.223889 0.540180 0.589
(,1 0.321646 0.169086 0.000
(,2 0.092008 0.057418 0.395
0,} 0.169329 0.052172 0.085
(,4 0.059913 1.976094 0.048
95 0.013637 0.122843 0.902
as 0.014503 0.559427 0.576

Q(1) - 0-0002 Q2(1) - 0.0100

(2(5) - 1-8650 Q2(5) - 6.8321

Q(10) - 15-9750 Q2(10) - 9.2115

(205) - 22-4445 Q2(15) - 12.6287

Q(20) - 23-0954 Q2(20) - 13.9002

Number of Observation: 1218 Skewness 0.22895251

Log likelihood -3780.7238 Kurtosis 8.78171906

 

Note: Q (k) is a Q test for k degree autocorrelation in the residuals; Q2(k) is a Q test for k degree
autocorrelation in the squared standardized residuals.

17

commodity price distribution have shown prices are not independent identical normally
distributed. The tails of price distribution often appear to be much fatter than that of
normal, indicating excess kurtosis in commodity price distributions. ARCH and GARCH
models can lead to a partial solution of the excess kurtosis problem, but may not capture all
of the excess kurtosis in commodity price distributions. One solution is to assume the
conditional distribution of price innovations in ARCH (GARCH) models follow a t-
distribution with degree of freedom treated as a parameter to be estimated (Bollerslev 1987,
Baillie and Myers 1991, Yang and Brorsen 1992). Unfortunately, we were unable to fit
ARCH (or GARCH) -t model for this basis sample data The introduce of t—distribution in
the ARCH model causes autocorrelations in residuals and squared standardized residuals
which are difficult to eliminate.

The estimated model is a simple specification that does not include any specific
corrections for the potential structural change issues discussed earlier. Next we introduce
structural change variables into the basis model to see whether we can improve the
specification and forecasting performance of the basis model. To account for the annual
contract switches and apparent structural change during the 1982 crop year, we introduce
dummy variables to measure contract jumps and structural change. Including the dummy
variables may also change the specification of the trend, seasonality and volatility
components of the model. After restricting the model with the dummy variables for
structural change and contract switching we obtain a form of GARCH-in-Mean
(GARCH-M) model (Engle, Lilien, Robins, 1987) which contains the conditional

standard deviation in the mean equation. The GARCH-M basis model is:

18

P q
(4) b,=p+61d1+62d2+2a,-b,_,-+ ijeH +5, +r]‘/h:+e,
i=1 j=l

et lat—l ~ 0(0: hr)

 

2
2w 27M
where S = 2 -sin ——’ + -cos ’
t j=l[¢j ( 52 ) ¢J ( 52 )]

(p=6. q=8)
d1 represents contract jump dummy; d2 represents structural change dummy variable
where structural change happened in May 1983. The estimated model parameters are
shown in table 2.

The GARCH-M model is often used in financial applications where the expected
return on an asset is related to the expected asset risk. The estimated coefficient on the

expected risk is a measure of the risk-retum tradeoff. In this basis GARCH-M model, the

basis risk is measured by the conditional variance h,. Remember the storage basis

definition, b, — b,“ represents the return of a storage operator who takes a long position in
the cash market and a short position in the firtures market in period t and reverses the
position in period t+1. The negative GARCH-M term 77 indicates that larger basis variance
in period t+1 are associated with smaller basis level in period t+1. Thus the higher basis
risk at t+1 will lead to higher expected storage return. This is consistent with standard risk-

retum arguments.

19

Table 2. GARCH -M-norrnal model specification

 

 

 

 

 

 

Coefficient Estimate t-value p-value
,1 4.211778 4.570104 0.000
91 8.769474 5.117442 0.000
92 -1.497059 -4.230295 0.000
(p1 2.285101 5.103892 0.000
(1,1 1.187119 4.448481 0.000
(1)2 1.349246 5.632816 0.000
(1,2 -0. 144758 -0.508309 0.611
011 0.884268 32.110248 0.000
a 6 0.042354 1.970483 0.049
B 1 -0. 150849 -2.799777 0.005
BS -0.049928 -1.477248 0.140
P 6 -0.096190 -2.899947 0.004
P 8 -0.047532 -2.018285 0.044
11 -0.156100 4.916877 0.055
Variance Level

(0 7.833631 4.275425 0.000
(p1 5.589322 3.411699 0.001
(1, 1 7.242676 4.658292 0.000
(p2 4.323185 3.772534 0.000
(1,2 1.121909 2.114415 0.035
(11 0.332767 6.167886 0.000
P 1 0.492811 4.264462 0.000
132 0.251041 2.050258 0.041
133 -0.168414 -2.191059 0.029

Q(1)- 0-1109 Q2(1) - 0.0603

Q(5)-19655 Q2(5) - 7.7843

Q(10)- 14-763 Q2(10) - 11.218

Q(15) - 21-079 Q2(15) - 11.984

Q(20) - 21-720 Q2(20) - 13.600

Number of Observation: 1218 Skewness -0. 1 15

Log likelihood -3754.2458 Kurtosis 9.04

 

The t-statistic of 01 and 62 indicate that both contract jump dummy and structural

change dummy variables are statistic significant. It is worth nothing that the kurtosis level

20

in this model is slightly higher than in the previous model in terms of the parameter

estimation of the new added variable (d! and d2) seem to provide us additional information

on the basis generating process. The estimate of the contract jump dummy 191 shows that the

first basis observation following each contract switch is average 8.8 cents/bushel higher

than the last observation of the previous contract. The estimate of structural dummy 0;

provides evidence that a structural change occurred sometime around the crop year 1982,
lowering the average corn basis level by 1.5 cents/bu from previous period, perhaps due to
the change of transportation costs and interest rate.

In general, both basis models appear well specified in terms of capturing all the
autocorrelations in residual series and squared standard residuals. The GARCH-M model
with the structural change and contract switching dummies appears to provide additional
information to explain the basis behavior. However, because the purpose of the models is
to forecast basis distributions, we evaluate these basis models by comparing their out-of-

sarnple forecasting performance.

Simulating the Corn Basis Distribution

We can use the basis generating process to forecast the basis distribution m
periods from the current period t to terminal period T. Let b, be the initial basis price on
Dec. 17, 1997, which is assumed to be the post-harvest time of corn. The terminal date is
the second week of July 1998, which is the assumed maturity time of July corn futures.
The simulation procedure for the basis distribution is as follows:

1. Simulate h by iterating the variance equation forward one period.
2 2

ht+l =w+a1 e, +"'+aiet—i Tfllht +”'+:6jht-j +St+l

21

3:41 = 21¢,- sin(— ’2’“ 1+ 45,- cost—5521)]

Calculate conditional variance h, based on previous conditional variance and realized

values of previous squared innovation e}_,- (B = O for ARCH model).

2. Generate random draw (v) from a normal distribution with zero mean and unit
variance so that («ihm v) are independently distribution normal vector with mean 0

and variance h,“ .

3. Calculate the error one step ahead forecast as:

e H] = \i ht+1 V
where the error e H1 is now sampled from a normal distribution with mean zero and

variance h,+1.
4. Calculate basis price at t+1 by using the errors calculate from 3.
bt+1 = l1 + fiaith—i + £15 jet-I-l— j + St+l + et+l
i=1 j=1
5. Let t=t+1 and repeat step 1 - 4 to calculate basis for next period until one realization
of by is obtained.
6. Repeat step 1-5 n times to generate an n-size sample of the basis price distribution at
maturity time, where n is 5000 in this study.
The histogram of the simulated distributions of basis prices at the maturity time
(m = 30) from ARCH and GARCH-M models are shown in figure 2. The first four

moments of the basis distribution are calculated and listed in table 3.

22

800

 

700 4-
600 4~
500 9-

400 J—

Frequency

300 -~

200 ~—

100 ~—

 

 

 

0 : ——
-60 -40 -20 0 20 40 60 80
cents/bu
2(a) Forecasted July basis distribution from ARCH model

 

 

1000
900 -F
800 4
700 -~
600 4
500 i
400 -4
300 ~F
200 -_
100 ~~

o -
-60 -40 -20 O 20 40 60 80
cents/bu
2(b) Forecasted July basis distribution from GARCH-M model

Frequency

 

 

Figure 2 Simulated July Basis Distributions

23

 

Table 3 Statistics of the Simulated Basis Distributions

 

 

Cents/bu Mean Standard Deviation Skewness Kurtosis
ARCH(6) 26.2763 9.1036 -0. 1447 4.9679
GARCH(1,3)-M 22.3223 13.9754 -O.3525 6.7353

 

For July 1998, the simulated basis distributions have mean level of 26.2763 and
22.3223 cents/bu from ARCH and GARCH-M models respectively. The realized basis
value is 24 cents/bushel. As expected from the estimated basis process, the simulated
basis distribution at maturity time is not normally distributed. Both simulated
distributions are skewed slightly to the left and have excess kurtosis.

The in-sample forecasts from both models appear to track the basis realizations
well. The true test of the models is to forecast beyond the period of fit, or out-of-sample.
Also, in-sample forecasts should necessarily perform well. Figure 3 shows out-of sample
forecasts of basis.

Note the forecasts of weekly basis from ARCH and GARCH-M models are very
similar. Both forecasts seem to do a good job capturing the trend of basis series except
during weeks 15 - 17. We can see that forecasted basis narrows from December to mid
June (weekl to 26) and then begin to widen after mid June. This is because corn harvest
reaches a peak at early November and cash prices usually fall to their lowest point of the

season during November and then rise gradually during the storage season.

24

50

 

basis (centslbu)

 

 

 

 

0 fl T i I I I I

1357911131517192123252729
forecast forward week

_._forecast basis +true basis cumulative STD - - - conditional STD

3(a) ARCH Model

 

so
45 -
40
35
30
25
20
15 .
10 -

5 -

 

 

basis (cents/bu)

 

 

 

0 I I T I I I I

13 5 7 91113151719 21 23 25 27 29
forecast forward week

_._ forecast basis + true basis .. ....... cumulative STD
3(b) GARCH-M model

Figure 3 Dynamic Forecasts of weekly Corn Basis

25

Figure 3(a) also shows the conditional standard deviation of the basis forecasts
using a dotted line. The conditional standard deviation measures the basis risk for the
corresponding forecasted basis levels and it varies over time. The relatively larger
conditional standard deviation at the last few weeks of forecasts implies larger volatility
or larger basis risk at delivery, partially reflecting that futures prices usually are quite
variable at contract maturity time. From week 27 to 30, the forecasts of basis level from
both models widen slightly, also reflecting the variability of the storage basis at maturity
time.

Unlike the conditional standard deviation, the cumulative standard deviations for
the basis forecasts increase as we forecast further into the future reflecting an increase in
the size of the confidence intervals of the forecasted mean as we forecast further into the

fiiture.

Evaluate Difi'erent Basis Models

Although the out-of-sample forecasts are similar for these two basis models, we can use
statistics to help evaluate which does a better job of forecasting during the period under
consideration. We use the root mean squared forecast errors (RMSFE) to evaluate the two

basis models. The function of RMSFE is calculated as follows:

 

_ _1_m f_ r 2
RMSFE— 2(1), 6,)
’"1=1

where bf are the forecasted basis and b,’ are the realized basis; m is the number of

forecasted observations. The bf are derived from the two alternative models using only

information available before the out-of-sample forecast period (December 17, 1997).

26

RMSFE for the ARCH model is 2.4747 cents and RMSFE for the GARCH-M model is
2.6967 cents.

The RMSFE results show that ARCH basis model has smaller forecast error, thus
performs better in terms of forecasting basis levels out-of-sample. It’s important to note
that this evaluation only represents one storage season realization. The results could
differ if a more extensive performance evaluation was conducted. However based on

these results, we choose the ARCH model to forecast the basis distribution.

Forecasting Cash Price Distribution

Black's Model and Implied Volatility

As mentioned earlier, commodity Option models can be used to generate implied
volatility of futures price. Implied volatility estimation is based on the fact that options
are forward looking assets and therefore contain information on futures distributions. The
implied volatility reflects the expectation of market participants from the estimation day
to the options expiration day. Conditional on an option model, the volatility of the futures
distribution is implied from market traded option prices instead of historical time-series
data of the futures prices. Thus with the predictability and relatively low variability
features of basis, combining implied futures price distributions with our forecasted basis
distribution may outperform the forecasts of cash distributions either from structural
models or historical time series models.

Black uses arbitrage arguments to obtain an exact formula for pricing options on
futures contracts. One property of this Black's differential equation is that the equation

does not involve any variables that are affected by the risk preference of investors. The

27

variables that appear in the equation are the current futures price, strike price of options,
time to maturity, futures price volatility at maturity, and the risk-free rate of interest. All
are independent of risk preferences (J. C. Hull, 1993). If we know these variables, we can
compute the corresponding call option price and put option price. The Black’s formulas

for both European call and put options are presented below:

Pc = e‘” *[S* N(dr)-Xc * Md»)

(5) T
pp =_e' *[S*N(—d])-Xp *N(-d2)]

Where: d1=[ln(S/X,-)+(r+0.5*0'2)*T]/(a*~/T)
d2 = all — (a * ff)
P,- = value of the option; where PC for call options and Pp for put options
S = price of the current futures
X,- = strike price of the option; where XC for strike price of the call option and X p

for strike price of the put option

T = time to expiration in terms of proportion of a year

r = annual interest rate on low risk securities

0' = annualized standard deviation

N(d) = cumulative normal probability values of d

A useful insight for our purpose is that if we know the price of the option and we
can compute which value of a will produce that price, that is, implied volatility of the
futures price without adding more exogenous variables (Hull 199). We invert (3) to find a

formula that expresses the implied volatility as a function of the option price and

remaining needed parameters, that is 0(Pi, S, X 1': T, r). We can use maximum likelihood

28

method to calculate the implied volatility o by minimizing the sum of the squared pricing

errors of all options in the sample given by:

min[(PC — PC(S,XC,T,r;o))2 + (P. — 13,. (S,X,.,T,r;a))’]

where PC(0) and 1315(0) are Black option pricing formulas in (5). Pricing errors are
the difference between the actual option prices (Pi) and the option prices estimated from

the implied volatility. The data we used are July corn futures option price and option
strike price with largest trading volume on Dec. 17, 1997 and interest rate for 52 week-

term treasury bill. The estimate result of variance for July corn futures are shown below:
0(PC,P,,,S,X(.,X,,,T,r)= 0.3165

where PC =15 cents, Pp =13.25cents, S =286 cents/bu, XC =290 cents/bu, Xp=280cents/bu,
T =180/365, r = 5.468%.

A fundamental assumption in Black's model is that the volatility of a futures price
is exactly the same at each instant. That is, the variance of the futures price is constant

over time. Therefore, the volatility of July’s futures would always be 0‘2 from harvest

time to maturity time. In addition, Black's model assumes that futures distribution is
lognormal.

In reality, however, if we compute volatilities for futures price, we find that
volatilities change across time. Why do we use Black's model to estimate the futures
price distribution? One compensating factor is that the volatility that makes the Black’s
value agrees with the market price include all available market information. So if the
market participants have recognized the true distribution of the future price, the market

traded value of the options will reflect of some correction for the model misspecification.

29

Hopefully Black's model is a close representation of the true volatility of futures price in
the market (J. C. Hull, 1993).

There are other pricing models to generate the implied volatility of futures price.
For example, Hilliard and Reis (1999) suggest that a jump-diffusion option pricing model
may be more appropriate by allowing non-normal distribution. However, Black's model
is still a widely used option pricing model and we will use it to generate forecasts of the

futures price distributions.

Correlation between Basis and Futures Prices

A critical issue when we combine the distribution of basis with distribution of
futures is the possibility of correlation between the basis distribution and the futures
distribution. If they are independent, then these two distributions can simply be linearly

summed to generate the cash price.

To begin, let bm, fm and cm represent the basis, futures and cash price at futures
contract maturity time, respectively. Then the covariance between basis and futures
prices is:

COV(bm,fm) = COV(fm TCm9fm) = var(fm) —COV(fm,Cm)

At futures contract maturity time, if f m = cm , then

cov(bm,fm) = var(fm)—cov(fm,fm) = O

Theoretically, if futures price and cash prices converge at contract maturity time
to a constant (e.g. transportation costs) then the basis distribution and futures distribution
may be independent at maturity. To investigate the independence between futures

distribution and cash distribution further, we conduct several statistical tests.

30

First, a formal independence test for two time series based on cross correlation
theory was conducted. According to the theorem, cross correlation of two stationary time
series equal to 0 indicate the independence of the two series. Based on the method
developed by Brockwell and Davis (1996), we can test independence by testing cross
correlation. The test result shows the futures price does not have much influence on basis
series (see Appendix 3).

Second, we use Kernel function to estimate conditional mean of basis condition
on futures price. Kernel is the function used to weight the observations in each regression
of the first series Y on second series X in the group. Figure 4 shows scatter plot of basis
and futures price data and Kernel fit plot. The Kernel fit are non-parametric regressions
for basis, at each value of futures, by choosing the parameters to minimized the weighted
sum-of-squared residuals. The plot of kernel fit represents the conditional mean of basis
condition on futures prices. From Figure 4, we can see the conditional mean line (solid
curve) is close to the unconditional mean (dotted line), which means the basis prices are
not influenced by futures prices.

These test results indicate that the basis distribution and the futures distribution
have no strong impact on each other. The assumption of independence between basis and
futures price makes the simulating procedure of cash distribution relatively simple and

based on the above analysis should not have a serious impact on forecasting results.

31

cents/bu

 

 

 

 

 

Basis150
100 _ °
50 _.
0 _.
(32:): OOCDO
-50 T I I
100 200 300 400 500
Futures Price
Unconditional mean of basis — Kernel fit plot of conditional mean of basis

Figure 4 Kernel Fit Plot of Basis Condition on Futures Price

Simulating the Corn Cash Price Distributions

With the implied volatility of futures price generated from Black's option model,
we can forecast the futures distribution using available market information. Then we
combine forecast of basis distribution with futures price distribution to get a forecast of

the cash price distribution.
Again, let b, and f, be the initial basis and futures price respectively on Dec. 17,

1997. The terminal date is the third week of June 1998, which is the assumed maturity
time of option for July corn futures.

Since basis and futures prices are assumed to be independent, we can simulate the
cash price distribution as follow:

1. Generate log futures price at t+m simply by:

32

lgfmn =lgf, +lga\/T*u
where fl is current futures price; 0' is the annualized standard deviation that get
from Black's model; T is time to expiration in terms of proportion of a year

(number of days to maturity/365); 0' * 77"— can adjust standard deviation to the

period of forecasting; u are random draws from independent standard normal

distribution, so that (a * JT * u ) are independently distribution normal vector
with mean 0 and variance 0' 2T.

11. Take the anti log to obtain one futures price realization at period m.

III. Follow steps 1 — 5 to generate a basis realization (b,+,,,).
IV. Calculate one realization of the cash price at t+m as:

CH-m = fr+m _bt+m
V. Repeat step I to IV n times to generate an n-size sample of the cash price

distribution at maturity time, where n is 5000 in this study.

Using the above simulation approach, we can obtain a terminal distribution of
cash price conditional on information available at the time of the decision problem by
combine basis distribution with futures distribution.

The histograms of the simulated distributions of futures and corn cash prices for
the maturity time (m = 26) are shown in figure 5 and figure 6, respectively. The first four

moments of the futures and corn cash distribution are calculated and listed in table 4.

33

700

 

 

 

0 : —— .
20 100 180 260 340 420 500 580
cents/bu
Figure 5 Simulated 1997 Com Futures Price Distribution at Options
Maturity Time

 

700

 

 

600 ..

500 ..

b
O
0

Frequency
OD
O
o

200 -.

100 40

 

 

 

20 100 180 260 340 420 500 580
cents/bu

Figure 6 Simulated 1997 Corn Cash Price Distribution at Options
Maturity Time

 

Table 4 Statistics of Simulated Futures and Cash Price Distribution at Option

 

 

Maturity Time
Cents/bu Mean Standard Deviation Skewness Kurtosis
Futures 286.2972 65.9286 0.7085 3.9368
Cash 266.9008 67.3788 0.5973 3.6559

 

The mean level of futures distribution at options maturity time is 286.3 cents/bu,
consistent with the firtures price that we used in the Black's function (286 cents/bu). As
expected, the futures price distribution follows a lognormal distribution.

Combining the forecast basis distribution from basis model and forecast futures
distribution from Black's option model, the simulated cash distribution at maturity time
has a mean of 266.9 cents/bu, a larger cumulative standard deviation, less skewness and
less kurtosis than the forecasted futures price distribution. Because this combined cash
price distribution includes historic price information from the basis model and available
market information from Black option model, it could outperform the cash distribution

forecasted from a structural model or a pure time series model.

Conclusion

Basis is a key element of effective hedging strategies in futures markets. The
relatively predictability and stability of basis also can contribute to forecasting cash price
distributions. This study has developed time series models to estimate and forecast the

storage basis distribution for com. A forecast of the cash distribution is then generated by

35

combining the forecast of the basis distribution with a market implied forecast of the
futures price distribution. The data used to parameterize the model are representative of a
corn farm in Saginaw, Michigan from 1974 to 1997 crop year.

Previous studies on basis primarily concentrate on the basis behavior; a few
studies focus on forecasting basis level, but little work has been done on modeling and
forecasting basis distribution. This research studies the stochastic properties of the basis
and provides evidence that multiple-year basis data is stationary. Based on the stochastic
properties of the basis data, a type of autoregressive moving average processes with an
autoregressive conditional heteroscedasticity model (ARCH) is specified to estimate and
forecast the basis distributions. The forecasted basis distribution can be used to improve
farmers' hedging decisions.

Compared to cash prices, basis is reasonably predictable and has smaller
volatility. Most previous studies on modeling and forecasting commodity cash price
distributions use time series methods. These studies typically use historical price to
estimate these models. However, implied futures prices volatilities generated from option
pricing models contain available market information at the time the option is priced.
Using simulation techniques, we can combine forecasts of the basis distribution with the
implied futures price distributions to get a forecast of the cash price distribution.
Combining basis distributions with market implied futures price distributions results in a
new approach to generate a forecast of the commodity cash price distribution. With the
predictability features of basis distributions and the available market information
reflected in traded commodity option prices, this combination may outperform the

forecasts of cash price distributions either from structural models or pure time series

36

models. Thus this study develops and illustrates a method that may improve the precision
of forecasted cash price distributions.

This research considers only one crop; however it provides some basic results on
storage basis behavior and modeling which can be generalized to other commodities.
This approach to forecasting cash price distributions includes available market
information could outperform alternative structural or time series models. Future research
could focus on testing the performance of the approach improving the implied volatility

forecast of firtures price by using alternative option pricing models.

37

APPENDICES

38

APPENDIX 1

Basis Definitions

The “nearby basis” (or continuation basis) is simply the cash price on a given
day subtracted from that day’s futures price for the next available maturity contract
month. In term of (1), T corresponds to the nearest futures maturity after t and
continually roles forward to the next contract month whenever t = T. Continuation basis
provides a continuous measure of the “nearby” basis across time. It is a popular
measurement in technical analysis. However, continuation basis is somewhat difficult to
interpret across time because the underlying futures contract changes (i.e. the underlying
asset being priced is different).

The “new crop basis” is the futures price nearest to and after harvest less the cash

forward contract price, that is, fifl) - E,(pT). Here, E,@T) is expected harvest cash price at

time T conditional on information available at t, often represented by the forward
contract price for the new crop. T corresponds to the contract month nearest and after
harvest. New crop basis provides a market estimate of the expected basis and provides
useful information to aid in making pricing decisions by allowing users to calculate the
local expected cash price for a commodity. Grain buyers can adjust the size of the new
crop basis to control grain flows. Buyers who wish to encourage delivery of additional
grain reduce the size of the new crop basis by increasing the relative forward price.
Producers (grain buyers) who wish to hedge future production sell (buy) futures
contracts before harvest and buy (sell) them back at harvest time in effort to offset price

changes in their long (short) position in cash market. In this case, the “harvest basis”

39

f7{T) - pr plays a key role in pre-harvest hedging decision. Here the futures contract
month T is at harvest. This is an important basis measure for farmers hedging their
production or buyers hedging input costs at harvest.

The “storage basis” is the current cash price subtracted from the futures price for
some month during the storage season. In (1), T can correspond to the any contract month

during the storage season. The storage basis represents the return to storing the

commodity from t to T and transporting it from the local region where cash price p, is
offered to the delivery point on the futures where fifl’) is offered. When the contract

matures (t = T), there will be no additional storage return and b, represents transportation

costs. For storage operators hedging in the futures market, knowledge of the storage basis
can help them design effective hedging strategies. The harvest basis and the storage basis

are most important concepts for hedgers.

40

APPENDIX 2

Unit Root Tests on Basis Data

Dickey and Fuller (1981) have developed unit root tests involving the OLS
regression technique using standard t and F statistics that follow non-standard
distributions to test for the present of a stochastic trend (unit root). Augmented Dickey-
Fuller (ADF) tests improved original Dickey-Fuller tests by including lagged differences
of the dependent variables as additional explanatory variables to account for
autocorrelation in the residuals from Dickey-Fuller regressions (Dickey and Fuller,
1981)

The Augmented Dickey-Fuller unit root test is conducted on the multi-year basis
sample. Since a trend can have deterministic and stochastic components, the testing
procedure is confounded by the presence of the deterministic regressors. We begin by
removing the deterministic component from the data. The characteristics of storage
commodity suggest the presence of a seasonal pattern in the storage basis. We use sine
and cosine function to capture the seasonal effects. With the following regression:

J

(la) b, = a) + at + 2w,- sin(27ziw/ 52) + w,- cos(27ziw/ 52)] + e,

j=l

where b, represents basis at time t, J is the degree of fiequency in seasonality variables,

and w is week, e, is the residual at time t. The order of J is determined by using

likelihood ratio (LR) test.

41

The log-likelihood ratio test states that the twice of the difference between the

log-likelihood of the unrestricted model and the restricted model is X2 distributed with
the number of restrictions as the degree of freedom. The test hypothesis is:
H03 ¢i = W: =0
H1: ¢,~ ¢Oor w,- #0
Calculate the likelihood Ratio of the two model specifications:
LR = 2(LU— LR) ~31; (2)
where LU and L R are the log-likelihood of the unrestricted and restricted model,

respectively. The log-likelihood and LR tests are listed in table below:

LR tests for degree of frequencies in seasonality

 

 

J Log-likelihood LR Ho

5 501.1062 1.2996 ¢5 -.= W5 :0
4 500.4564 7.5686 (1),, = W4 =0
3 496.6721 14.561 4,3 = W3 =0
2 489.3916 42.6166 4,2 = W2 :0
1 468.0833

 

Critical value for X’(2)5%: 5.99147; 1%: 6.6349

The null hypothesis of no seasonality versus the alternative of 4 degree of
seasonality can rejected at 1% significant level.

The test results provide some evidence that the seasonal factor may be
characterized by fourth order frequency of sine and cosine variables. So we get J = 4.

Using the residuals from (1a), ADF test can be conducted using the following

regression:

42

k
(1b) Ae, = ,BeH + ZYiAer—i + residual,

1
Ho: ,6=0 unit root process
H1: fl<0 stationary process
Since we use the residuals after regressing the deterministic seasonality from the
basis data, ADF test doesn’t include constant variable again.

In order to eliminate the contract switch effects, we conduct the unit root tests on
multiple-year basis data by taking out the difference (Ae,) occurred between the two

contracts from the data process. This technique only allows the data differencing in the
contract and thus eliminates the contract switches effects. However, it causes 23
observations lost each time we include one more lag of the dependent variable.

An additional difficulty is that the true order of the autoregressive process is
usually unknown, so that the problem is to select the appropriate lag length. We can start
with a relatively long lag length and pare down the model by the usual t-test and/or F-
tests. Repeat the process until the lag is significant different from zero. Moreover, the
correlogram of the residuals should appear to be white noise. We start with 53. The
Ljung-Box Q-test of first 100 lags shows no autocorrelation in the residuals at 5%
significant level. Then we decrease lag one by one. The Q-tests show autocorrelations in
residuals at 1% significant level until we decrease to lag 14. So we include lag 15 in the
regression.

Then use OLS to estimate value of B and it’s associated t-statistic. Comparing the
resulting t-statistic with the critical value reported by Dickey and Fuller allows us to

determine whether to accept or reject the null hypothesis fl = 0. Based on the unit root

43

regression equation, we use 1, critical value at 5% significant level (-3.41) for regression

(1b). The t-statistic for B in the test is —6.8747, less than -3.41 (r,). We reject the null

hypothesis that the process is unit root. Previous studies find one may fail to reject the
null hypothesis of a unit root because of a mis-specification concerning the deterministic
part of the unit root regression. If the null hypothesis of a unit root is rejected, there is no

need to proceed (Ender 1995). Thus, price level will be evaluated throughout this study.

44

APPENDIX 3

Independent Test for Basis and Futures Price Distributions

Ho: b, and f, are independent series;

H]: b, and fi are not independent series.

According to the theorem 7.3.1 in Brockwell and Davis (1996), cross correlation
of two stationary time series equal to 0 indicate the independence of the two series. If one

of the two processes is white noise, then the cross correlation between the two processes

is approximately normally distributed with mean 0 and variance % , that is cross

correlation p12 ~ N (0, %). This make the independence test straightforward to test the

hypothesis of p12: 0. Since the basis and futures prices are both stationary series by
previous unit root tests, we can develop a method to test for independence of them by
testing cross correlation.

Any test for independence of the two component series cannot be based solely on
estimated values of cross correlation without taking into account the nature of the two
component series. In practice, the difficulty can be solved by transforming the two series
to white noise by application of ARMA method. The estimations for basis and futures

prices are shown in Table A and B:

45

Table A: Estimation of ARMA model for Basis

 

 

 

 

 

Variable Coefficient Std. Error t-Statistic Prob.
C 0.316979 0.064708 4.898580 0.0000
SINl 0.110637 0.013186 8.390291 0.0000
0051 0.073650 0.011493 -6.408183 0.0000
SINZ 0.013617 0.010696 1.273139 0.2032
C052 0.041254 0.009520 -4.333391 0.0000
AR(l) 0.920142 0.049804 18.47513 0.0000
AR(6) 0.065791 0.049334 1.333583 0.1826
MA(l) 0.200366 0.092397 -2.168548 0.0303
MA(S) 0.100803 0.070443 -1.430987 0.1527
MA(6) 0.245944 0.098230 -2.503760 0.0124
MA(8) 0.090963 0.041591 -2.187081 0.0289
Table B: Estimation of ARMA model for Futures Prices
Variable Coefficient Std. Error t-Statistic Prob.
C 2.738024 0.141283 19.37966 0.0000
AR(l) 1.011541 0.016388 61.72516 0.0000
AR(6) 0.030709 0.015820 -1.941176 0.0525
MA(3) 0.113554 0.033199 -3.420431 0.0006
MA(S) 0.050144 0.033635 -1.490823 0.1363
MA(7) 0.100556 0.029205 3.443086 0.0006
MA(8) 0.076515 0.029407 2.601954 0.0094

 

Check the residuals from the ARMA models for basis and futures, respectively

(Wu and wtz). There is no autocorrelation at residuals. They are white noise sequences.

Then an approximately test of independence between basis and futures series is

independence test of w" and Wm

Ho: w” and we are independent series;

H]: w” and we are not independent series.

46

This test equal to:

Ho: Cross correlations between w” and we (p12) = 0;

H1: Cross correlations between w” and Wu ( p12 ) at 0.

At 5% significant level, compare the absolute values of cross correlation between

I
w” and we with 1.96n 7- (n =1218). Under Hg, for any fixed p12(h),h = 0,i1,---, falls

between the bound i 0.0566 with a probability less than 95%. When we calculate

p,2 (h),h = 0,i1,---i150 , all except eleven of the correlations are between the bounds of

i0.0566. 1%01= 0.0365 < 5%, suggest that the two residual series are uncorrelated.

None hypothesis of independence is fail to reject. Hence we can conclude that the basis

and futures price series are independent.

47

REFERENCES

48

References

Ardeni, Pier G. “Does the Law of One Price Really Hold for Commodity Prices?”
American Journal of Agricultural Economics, v71(August 1989), p661-669

Baillie, R. T. and R. J. Myers, “Bivariate GARCH Estimation of The Optimal
Commodity Futures Hedge” Journal of Applied Econometrics, v6(l991): p109-124

Bollerslev, T. “Generalized Autoregressive Conditional Heteroskedasticity ” Journal of
Econometrics v3 1(1986): p3 07-327

Bollerslev, T “A conditional Heteroskedastic time series model for speculative prices and
rates of return”, Review of Economics and Statistics, v69(1987): p542-547

Bollerslev, T., R. Y.Chou and K. F. Kronoer “ARCH modeling in finance -A review of
the theory and empirical evidence” Journal of Econometrics v2(1992): p5-59

Brennan, M. J. “The Supply of Storage” American Economics Review v48(1958): 195-
206

Brockwell, P. J. and R. A. Davis Introduction of Time Series and Forecasting 1996
Springer-Verlag New York, Inc. p230-232

Chriss, N. A. BIack-Scholes and Beyond: Option Pricing Models 1997 McGraw-Hill,
p327-360

Dickey, DA. and W. A. Fuller (1981), “Likelihood Ratio Statistics for Autoregressive
Time Series With A Unit Root”, Econometrica v49, n5(1981): p1057-1072

Ender, W. Time Series Model, New York: Wiley 1995

Engle, R. F. “Autoregressive Conditional Heteroscedasticity with Estimates of the
Variance of United Kingdom Inflation” Econometrica, v50, n4 (July, 1982)

Fama, E. F. and K. R. French “Commodity Futures Prices: Some Evidence on Forecast
Power, Premiums, and the Theory of Storage” Journal of Business, v60, nl(1987)

Ferris, J. N. Lecture Material for Commodity and Futures Marketing, Department of
Agricultural Economic, Michigan State University, 1997, p17-91

French, K. R. “Detecting Spot Price Forecasts in Futures Price” Journal of Business,
v59, n2,(1986)

Fortenbery, T. R., and H. O. Zapata , “An Examination of Cointegration between Futures
and Local Grain Markets” The Journal of Futures Markets, v1 3(1993): p921-932

49

Galloway, T. M. and R. Kolb “Futures Price and the Maturity Effect " The Journal of
Futures Market, v16(1996): p809-828

Garcia, P., and R. M. Leuthold and M. E. Sarhan “Basis Risk: Measurement and Analysis
of Basis Fluctuations for Selected Livestock Markets” American Journal of Agricultural
Economics v66 (1984): p499-504

Garcia, P. and D. R. Sanders “Ex Ante Basis Risk in the Live Hog Futures Contract: Has
Hedges’ Risk Increased?” The Journal of Futures Market v16, N4 (1996), p421-440

Goodwin, B.K. and T. C. Schroeder, “Cointegration Tests and Spatial Price Linkages in
Regional Cattle Markets”, American Journal of Agricultural Economics v73, n2(1991),
452-464

Goodwin, B.K. (1992), “Multivariate Cointegration Tests and the Law of One Price in
International Wheat Markets”, Review if Agricultural Economics v14, n1(1992), 117-124

Gordon, J. D. “The Distribution of Daily Changes in Commodity Futures Pieces”
Technical bulletin No.1 702 ERS USDA (1985)

Gujarati, D. N. (1995), Basis Econometrics, p709-729, McGraw-Hill, Inc. Third Edition

Hilliard, J. E. and J. A Reis "Jump Processes in Commodity Futures Prices and Options
Pricing" Aamerican Journal of Agricultural Economics v81, n2(1999): p273-286

Hull, J. C. Options, Futures, and other Derivative Securities, Second edition 1993
Prentice-Hall, Inc. p224-237

Kaldor, N. “Speculation and Economic Statility” Review of Economic Studies, v7(1939):
p1-27

Kahl, K. H. and C. E. Curtis “ A Comparative Analysis of the Corn Basis in Feed Grain
Deficit and Surplus Areas” Review of Research in Futures Markets v5, n3(1986): p220-
232

Kamara, A. “Issue in Futures Markets: A Survey” Journal of Futures Market v2(l982):
p261-294

Kenyon, D., E. Jones and A. McGuirk “Forecasting Performance of Corn and soybean
Harvest Futures Contracts” American Journal of Agricultural Economics v75(1993):
p399-407

Khoury, N. T. and J. Martel “A Supply of Storage Theory with Asymmetric Information”
Journal of Futures Market v9, n6 (1989): p573-581

50

Kim, D. and S. J. Kon "Alternative models for the conditional heteroscedasticity of stock
returns" The Journal of Business v67, n4(1994): p536-563

Lence S. H., M. L. Hayenga and MD. Patterson “Storage Perfotability and Hedge Ratio
Estimation” The Journal of Futures Markets v16, n6(1996): p655-676

Liu S., B. W. Brorsen, C. M. Oellermann and P. L. Farris “ Forecasting the Nearby Basis
of Live Cattle” The Journal of Futures Markets v.14, n3(1994): 259-273

Mann, J. S. and R. G. Heifner “ The Distribution of Short-run Commodity Price
Movement” Technical Bulletin N0. 1536 ERS USDA (1976)

Martin, L., J. L. Grotnewegen and E. Pidgeon “Factors Affecting Corn Basis in
Southwestern Ontario” American Journal of Agricultural Economics v62(1980): p107-
112

Myers, R. J. “Time Series Econometrics and Commodity Price analysis: A Review”
Review of Marketing and Agricultural Economics v62, n2 (1994)

Myers, R. J. and S. D. Hanson “Optimal Dynamic Hedging in Unbiased Futures
Markets” American Journal of Agricultural Economics, v78, n1(1996): p1 3-20

Oxley, L., D. A.R. George, C. J. Roberts and S. Sayer Surveys in Econometrics Blackwell
Publishers 1995 p215-273

Streeter, D. H. and W. Tomek "Variability in Soybean Futures Prices: An Integrted
Framewor " Journal of Futures Markets v12 (1992): p705-728

Tilley, Daniel and S. K. Campbell “Performance of the Weekly Gulf-Kanasas City Hard-
Red Winter Wheat Basis” American Journal of Agricultural Economics v70 (1988):
p929-935

Tomek, W. G. & Gray R.W. “Temporal Relations among Prices on Commodity Futures
Market: Their Allocative and Stabilizing Roles.” American Journal of Agricultural
Economics, v152 (1970): p372-3 80

Tomek, W. G. and K. L. Robinson Agricultural Product Prices Cornell University Press
(1984) 224-230

Tomek, W. G. and R. J. Myers “ Empirical Analysis of Agricultural Commodity Prices:
A Viewpoint” Review of Agricultural Economics, v15, n1 (Jan. 1993): p181-202

Tomek, W. G. “Commodity Futures Price as Forecasts”, Review of Agricultural
Economics v19, n1(1996): p23-44

51

Venkateswaran, M. and B. W. Brorsen and J. A. Hall “The Distribution of Standardized
Futures Price Changes” The journal of Futures Markets, v13(1993): 279-298

Wang, H. “Farmer Risk Management Behavior and Welfare under Alternative Portfolios
of Risk Instruments” Ph.D. Dissertation, Department of Agricultural Economics,
Michigan State University, 1996, p56-108

Working, H. “The Theory of Price of Storage” American Journal of Agricultural
Economics v3 9(1 949): p1254-1262

Yang, S. and B. W. Brorsen “Nonlinear Dynamics of Daily Cash Prices”, American
Journal of Agricultural Economics, v74(1992), p706-715

52

M IIIIIIIIIIIIIIIIIIIIIIIIIIIIII

IIII1111311111111116111211111