Hedge
Ratio and Hedging Horizon: A Wavelet Based Study of Indian Agricultural
Commodity Markets
Dr.
Irfan ul Haq
Abstract
In India the history of
commodity derivatives market has a long history, though a structured and
exchange traded derivative trading is not more than a decade long. The
derivatives market is established for the main purpose of hedging the price
risk. Since the inception of derivatives, the concern of how much to hedge
technically called the hedge ratio is widely debated and discussed. In present
paper, we has empirically estimated the hedge ratio using three different
methodologies viz. OLS, ECM and WAVELET Approach for ten agricultural
commodities traded on NCDEX platform. The results witnessed reveal that wavelet
hedge ratio is comparatively larger than OLS and ECM, and as we go on
increasing the hedging horizon hedge ratio increases.
Keywords:
Derivatives, Price risk, Hedging horizon
1. Introduction
Futures market has an
important function of hedging the price risk faced by producers, traders and
other stake holders. The concept of Hedging strategy is based on compensating anticipated
spot market losses with the gains from the futures market. In order to execute
hedging strategy one has to take opposite position in futures market with
respect to the position in spot market.
Most
studies in hedging are related to hedge ratio. It is either estimation of
optimal hedge ratio or optimal hedge ratio derivation based on certain
objective functions. The derivation is mostly based on maximization of expected
utility or minimization of return variance. The empirical estimation of hedge
ratio varies in terms of different methodologies used. The present study
focuses on estimating the optimal hedge ratio based on tradition and new
wavelet methodology to find out the optimal hedge ratio for different time
horizons.
The
estimation of optimal hedge ratio generally differs in the methodologies used.
The most common methodologies vary from simple ordinary least square (e.g Ederington,
1979, ; Malliaris & Urrutia, 1991; Benet, 1992) to more complex ones like
ARCH and GARCH methods (Cecchetti, Cumby, & Figlewski, 1988; Baillie &
Myers, 1991; Sephton, 1993) , Cointegration Method (Ghosh, 1993; Lien &
Luo, 1993; Geppert, 1995; Chou, & Lee, 1996), the random coefficient method
(Grammatikos & Saunders, 1983) , cointegration  heteroscadestic method
(e.g., see Kroner & Sultan, 1993) and wavelet method (Lien and Shrestha,20007).
However,
the problem with most of the empirical studies is they ignore the hedging
horizon which is different for different participants in the market. With the
increase in hedging horizon the optimal hedge ratios tends to increase as
witnessed by many researchers.
In this article, we have decomposed the original time series of futures and
spot prices into different horizons using wavelet methodology. The advantage
with wavelets is that the sample size is not reduced while matching the hedging
horizon and data frequency.
2.Methodology
Hedge ratio is defined
as the ratio of value of futures contracts to the value of the underlying
asset. Optimal hedge ratio is the ratio that eliminates or minimizes the price
risk. While hedging through the identical asset as underlying, the number of
contracts that are booked to cover the long or short positions is equal to the
exposure in the underlying asset. In such a case it is implicitly set the hedge
ratio equal to one, which is regarded as optimum.
The optimum hedge ratio
(value of futures contracts to the value of the underlying asset) is dependent
upon the degree of correlation between the spot prices and the futures prices.
For a hedge through futures the optimum hedge ratio is that ratio which
minimizes the risk of the combined portfolio of the underlying and the futures.
If h is the
futures contracts booked then the risk of the combined portfolio of the
underlying asset and the futures is given by variance of the return from the
portfolio so constructed.
Return from the
portfolio = (1)
Variance of the
portfolio, V =
=
For minimization of the
variance of the portfolio, .
Therefore, (2)
Where = optimal
hedge ratio
= Correlation
coefficient of spot and futures price
= Standard
deviations of spot and futures prices respectively.
For the estimation of
optimal hedge ratio both traditional and new methodology is used. The tradition
or conventional methodologies include Ordinary Least Square (OLS) and Error
Correction Model (ECM). The new methodology for time varying hedge ratio is
wavelet method. All the three methodologies are discussed below;
2.1Ordinary Least Square
(OLS)
The traditional
approach to estimate minimum variance hedge ratio is Ordinary Least Square
(OLS) technique, where spot returns are regressed on futures returns. The
regression equation for spot and futures returns can be expressed as;
(3)
Where, is a spot
return and is futures
returns. β is the minimum variance hedge ratio.
The problem with OLS is
that it does not incorporate long run equilibrium, so another model, error
correction is estimated to get the optimal hedge ratio.
2.2 Error Correction Model
Error Correction Model
is one of the multiple time series model used to estimate the speed of
adjustment with which dependent and independent variables return to equilibrium.
Error Correction Models (ECMs) are also used to estimate the short and long run
relationships between the time series. The general structure of any ECM is:
(4)
Where EC is the error correction term.
In the present study the ECM model
extended by Lien and Shrestha (2005) is used. The model is as;
(5)
Where is a residual
of cointegrating equation;
(6)
is
Error Correction (EC) hedge ratio.
The problem with both
the conventional methods is that data frequency is to be matched with hedging
horizon which will lead to smaller sample. To overcome this problem another
methodology where matching of frequency data and hedging horizon is not needed
is used here. This methodology is based on Wavelet Approach.
2.3 Wavelet Approach
The origin of Wavelets
though recognized in 1980s was before put forth in 1909 by Alfred Haar and
nowadays his contribution is recognized as Haar wavelets. Wavelet Analysis is
groomed by the collaborative work of Ingrid
Daubechies (Mathematician) and Stephane Mallat (Signal Processing
Professional). Daubechies (1996) points out that wavelet is more than the sum
of different ideas from different fields merged together. The different fields
which contribute to Wavelet analysis include mathematics, physics, engineering,
and computer science. Daubechies (1992) presented the theoretical background of
wavelets but before this its application was first explored by Mallat (1989).
Wavelet analysis is of two types  continuous wavelet analysis and discrete
wavelet analysis. The first one assumes continuous time functions and second
one assumes sampling at discrete the points which are equally spaced.
In common parlance
wavelets are functions with specific properties. These functions satisfy
certain mathematical requirements which find applications in representing other
functions or data. Mathematically wavelets is a function
over the real axis such that as.
From definition it
follows that wavelet is
localized in space or time i.e. with time oscillations of damp quickly
to zero. This localization property of wavelets makes them interesting and
useful in handling the non stationary data which changes rapidly over a period
of time. A time series can be presented as wavelet functions by applying
Wavelet transformation, which means a time series, can be decomposed into
multiresolution components.
Wavelets are of two
types father and mother wavelets.
,
(7)
In time series, father wavelet
represents smooth and low frequency components and mother wavelet represents
detail and high frequency components. In short we can say that father wavelets represent
trend components and mother wavelets represent all deviations from trend Two
scale dilution equation is used to derive the wavelets.
The dilution equation for father wavelet is
(8)
And the dilution equation for mother
wavelet is
(9)
Where and are low pass
and high pass coefficients respectively and are defined as;
,
(10)
In practice we deal
with time series rather than continuous functions. For this we employ wavelet
filters which are short sequence of values and are denoted by , L represents
width of the filter.
The restrictions, , (j= non zero
integer)must be satisfied by (filter
coefficient).
The filter coefficients and are related
to each other as;
The time series can be
represented in terms of wavelet coefficients as;
(11)
Or (12)
Equation (11) is a
representation of time series with different time scales obtained from the
decomposition of.
This decomposition process is called multi resolution analysis (MRA). In case
of discrete time series, discrete wavelet transform is used to perform MRA
using digital filtering technique. Suppose represents a
discrete time series. Then MRA leads to decomposition of as;
(13)
Discrete wavelet
transforms are of two types; DWT (discrete wavelet transform) and MOWDT (maximal
overlap discrete wavelet transform). DWT decomposes the original series using
orthonormal transformation. Suppose represent
observations of a discrete time series and N is an integer multiple of. Then column vector
of discrete wavelet coefficients w under DWT is given by
(14)
W is a transformed N x N real real
valued matrix satisfying.
Coefficient vector w is also divided into sub vectors, j= 1, 2……, J
and as;
(15)
Vector
and Vector.
and
actually
represent wavelet coefficient vector and scaling coefficient vector
respectively.
Another transformation
i.e. MOWDT involves nonorthogonal transformation which leads to J transform
coefficients vectors each of length N and N is not necessarily multiple of 2.
For MOWDT the coefficient vector of length (J + 1)N is given by
(16)
Where is an (J + 1)N
X N matrix and
(17)
Note and are wavelet
coefficient vector and scaling vector respectively.
If the data is sampled at an interval of
Δt, then is
associated with changes on the time scale of length (Percival and
Mofjeld 1997). For example, in case of daily data, , is
associated with daily time scale and is
associated 2 days time scale and so on. Furthermore, is associated
with length of the time scale which is equal to or longer. The
limit of J is sample dependent, and is always less than log_{2}(N), N
is the sample size.
Based on equation (11)
spot and futures return time series can be decomposed into different time
scales as;
(18)
(19)
Where, is the spot
return series and is
the futures return series.
Now using J decompositions to estimate J
regressions;
(20)
where is the minimum
variance hedge ratio associated with j^{th} time scale. Both DWT as
well MODWT can be used to perform the above analysis. In this study we have
used MODWT, because of its advantages over DWT.
3. Data
The sample of the study
is ten agricultural commodities viz. Barley, Chana, Chilli, Guar Gum, Guar
Seeds, Jeera, Pepper, Soy oil, Soy Bean and Turmeric. The data used in the
present study is obtained from National Commodities & Derivatives Exchange
Ltd (NCDEX). The data is collected for different periods based on the
availability of trading cycles and contracts. A pooled series of prices for
each commodity is generated by the process of roll over from one maturity to
the next. Daily closing futures prices of current maturity and corresponding
closing spot prices are used in the study. The detailed summary of data period
for different commodities is presented in Table 1.
4. Results
The results of various optimal hedge
ratios are provided in Tables 2 to 6. The hedge ratios are estimated for
different hedging horizons viz. 2 days, 4 days, 8 days, 16 days and 32 days
using three different methodologies OLS, ECM and Wavelets. The wavelet ratio
exceeds both OLS and ECM ratios in almost all the cases except in the lowest
scale i.e. 2 days. ECM and OLS ratios are almost similar. The wavelet ratio
being highest among the three methodologies can be attributed to the fact that
wavelet approach does not reduce the sample size which is quiet prominent in
other two methodologies. As we go on increasing the time scale hedge ratio goes
on increasing irrespective of the methodology used.
5. Conclusion
In Indian commodity markets, the
agricultural futures trading have got momentum after the establishment of
national commodity exchanges and some regulatory measures. It is important to
assess the functions for which the futures markets are established. In this
context this study has evaluated the optimal hedge ratios and hedging
effectiveness of agricultural futures. Three different methodologies OLS, ECM
and Wavelets have been employed to estimate optimal hedge ratio. The hedge
ratios suggested by OLS and ECM are almost equal but wavelet hedge ratios are
higher than the former two. The hedging effectiveness measured by Ederington
Measure suggests good amount of hedging in Indian markets and more particularly
in commodity with high trading volumes.
Table 1: Sample and
Sample Size
Commodity

Data
Period

No of Observations

Commodity

Data
Period

No of Observations

Barley
Channa
Chilli
Guar
Gum
Guar
Seeds

Mar.2007to
Dec 2011
Jan
2006 to Dec 2011
Feb
2006 to Dec 2011
Jan
2006 to Dec 2011
Jan
2006 to Dec 2011

805
1085
812
1289
1312

Jeera
Pepper
Soy
Oil
Soy
Bean
Turmeric

Jan
2006 to Dec 2011
Feb
2006 to Dec 2011
Jan
2006 to Dec 2011
Jan
2006 to Dec 2011
March
2009 to Dec 2011

1447
1665
1314
1314
481

Table
2: Optimal Hedge Ratio (2 day Returns)
Commodity

OLS

ECM

Wavelet

Barley
Channa
Chilli
Guar Gum
Guar Seeds
Jeera
Pepper
Soy Oil
Soy Bean
Turmeric

0.59
0.50
0.41
0.76
0.67
0.267
0.342
0.683
0.676
0.573

0.52
0.47
0.45
0.78
0.673
0.267
0.329
0.685
0.663
0.657

0.49
0.32
0.40
0.63
0.568
0.185
0.179
0.560
0.543
0.503

Table
3: Optimal Hedge Ratio (4 day Returns)
Commodity

OLS

ECM

Wavelet

Barley
Channa
Chilli
Guar Gum
Guar Seeds
Jeera
Pepper
Soy Oil
Soy Bean
Turmeric

0.64
0.48
0.42
0.75
0.673
0.263
0.348
0.685
0.656
0.604

0.52
0.48
0.42
0.76
0.663
0.256
0.327
0.677
0.655
0.597

0.68
0.6
0.41
0.84
0.731
0.302
0.406
0.741
0.746
0.607

Table
4: Optimal Hedge Ratio (8 day Returns)
Commodity

OLS

ECM

Wavelet

Barley
Channa
Chilli
Guar Gum
Guar Seeds
Jeera
Pepper
Soy Oil
Soy Bean
Turmeric

0.66
0.49
0.20
0.74
0.673
0.287
0.370
0.668
0.239
0.590

0.29
0.50
0.20
0.72
0.655
0.280
0.358
0.665
0.238
0.585

0.70
0.73
0.44
0.88
0.816
0.367
0.552
0.799
0.727
0.684

Table
5: Optimal Hedge Ratio (16 day Returns)
Commodity

OLS

ECM

Wavelet

Barley
Channa
Chilli
Guar Gum
Guar Seeds
Jeera
Pepper
Soy Oil
Soy Bean
Turmeric

0.86
0.51
0.44
0.738
0.684
0.269
0.370
0.700
0.680
0.600

0.57
0.53
0.44
0.729
0.661
0.267
0.358
0.696
0.681
0.588

0.73
0.66
0.47
0.920
0.843
0.463
0.644
0.799
0.792
0.717

Table
6: Optimal Hedge Ratio (32 day returns)
Commodity

OLS

ECM

Wavelet

Barley
Channa
Chilli
Guar Gum
Guar Seeds
Jeera
Pepper
Soy oil
Soy Bean
Turmeric

0.92
0.46
0.45
0.74
0.915
0.291
0.363
0.682
0.652
0.262

0.57
0.47
0.45
0.74
0.913
0.282
0.349
0.676
0.657
0.374

0.94
0.83
0.69
0.98
0.897
0.531
0.754
0.963
1.005
0.866

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