CURRENCY ARBITRAGE
6.3 EMPIRICAL MODELS OF THE HEDGE RATIO
Several empirical models are used to estimate the hedge ratio. These models are described in turn.
The naïve model
The naïve model simply implies that the hedge ratio is always 1. This is exactly the assumption we used to demonstrate the hedging decision in Chapter 5.
Since the naïve model is naïve, we need to go no further in elaborating on it.
The implied model
The implied model allows the estimation of the conditional covariance by employing the implied volatilities derived from currency options. These volatilities may be readily available or they can be calculated from the Black–Scholes option pricing formula.
The random walk model
The random walk model assumes that the most appropriate forecast of future variance and covariance is the variance and covariance observed today. Again, the simplicity of this procedure requires us to go no further in our exposition.
The conventional model
Also called the simple model and the historical model, the conventional model amounts to estimating the hedge ratio from historical data by employing a linear OLS regression model. Let pU and pA be the logarithms of the prices of the unhedged position and the hedging instrument respectively, such that RU = DpU and RA = Dp A . The regression equation corresponding to equation (6.18) is
a D t
= +h pA ,t +e (6.21)
DpU,t
in which case h is the estimated hedge ratio and the R2 of the regression measures the hedging effectiveness. Sometimes, the regression is written in levels rather than in first differences to give
a
= +hpA ,t +e (6.22)
pU,t t
in which case the optimal hedge ratio is defined as s(pU, pA )
(6.23) h=
s2(pA)
The implication of equations (6.22) and (6.23) is that the objective of hedging is to minimise the variance of the price of (rather than the rate of return on) the hedged position.
The error correction model
One problem with the conventional OLS model is that equation (6.22) ignores short-run dynamics, whereas equation (6.21) ignores the long-run relation
ship as represented by (6.22). Specifically, if pU and pA are cointegrated such that et ~ ( ) I0 , then equation (6.21) is misspecified, and the correctly specified model is an error correction model of the form
n n
= +
å
bi Dp D +å
gi Dp (6.24)DpU,t a U,t-i +h pA ,t A ,t-i +qet-1 +xt
i=1 i =1
where qis the coefficient on the error correction term, which should be signifi
cantly negative for the model to be valid. This coefficient measures the speed of adjustment to the long-run value of pU, as implied by equation (6.22). In other words, it is a measure of the speed at which deviations from the long-run value are eliminated.
Lien (1996) argues that the estimation of the hedge ratio and hedging effec
tiveness may change sharply when the possibility of cointegration between prices is ignored. In Lien and Luo (1994) it is shown that although GARCH may characterise the price behaviour, the cointegration relationship is the only truly indispensable component when comparing the expostperformance of various hedging strategies. Ghosh (1993) concluded that a smaller than optimal futures position is undertaken when the cointegration relationship is unduly ignored. He attributed the under-hedge results to model misspecification.
Lien (1996) provides a theoretical analysis of this conjecture by assuming a cointegrating relationship of the form ft =p A ,t -pU,t . A simplified error correction model, which implies that prices adjust in response to disequilib
rium, can be written as
(6.25) DpU,t =aft-1 +x1,t
(6.26) DpA ,t = -bft-1 +x2 ,t
D
A hedge ratio that minimises s2(DpU,t -h pA ,t ) is calculated as s(DpU t, , DpA ,t|ft-1)
( ç
h= =r x1,t , x2 ,t ) æs
(
(x1,t ) ö÷ (6.27)
s2(Dp ) ) ÷
A ,t|ft-1 èçs x2 ,t ø
where r x( 1,t , x2 ,t ) is the correlation coefficient between x1,t and x2 ,t . Alterna
tively it can be calculated from the regression equation
= +h pA ,t +gf +zt (6.28)
DpU,t a D t-1
If the cointegrating relationship is ignored, then the hedge ratio is calculated as
s(DpU,t , DpA ,t )
(6.29) h =
s2(DpA ,t )
From equations (6.25) and (6.26), we have s(
s(DpU,t , DpA ,t ) = -bft-1 +x2 ,t , aft-1 +x1,t )
(6.30)
( ( (
= -abs2(ft-1) +r x1,t , x2 ,t )s x1,t )s x2 ,t ) and
2 2(ft-1) +s2(x2 ,t ) (6.31) s2(DpA ,t ) =s2(-bft-1 +x2 ,t ) =b s
- -
- -
- -
- -
- -
Hence the hedge ratio is measured as -abs2(ft-1) +r x( 1,t, x2 ,t)s x( 1,t)s x( 2 ,t)
h= (6.32)
b s2 2(ft-1) +s2(x2 ,t)
Obviously, there is a difference between the expressions in equation (6.27) and equation (6.32). On the basis of these two expressions, Lien (1996) concludes that an errant hedger who mistakenly omits the cointegrating rela
tionship always undertakes a smaller than optimal position on the hedging instrument.
By using a general specification of equations (6.25) and (6.26), we have
n n
=aft-1 +
å
a p - iD - (6.33)DpU,t iD U,t i+
å
b pA ,t i +x1,ti=1 i=0
n n
= -bft-1 +
å
c p - iD - (6.34)DpA ,t iD U,t i +
å
d pA ,t i +x2 ,ti=0 i=1
in which case the hedge ratio calculated on the basis of the correctly specified model is given by
s(DpU,t, DpA ,t|ft-1, DpU,t i, DpA ,t i) h=
s2(DpA ,t|ft-1, DpU,t i, DpA ,t i)
(6.35) s x( 1,t, x2 ,t)
( (
= ç
s2(x2 ,t) =r x1,t, x2 ,t) æs x1,t) ö÷
( ) ÷ çs x2 ,t ø è
whereas the errant hedger who does not take into account the cointegration relationship will choose a hedge ratio that is given by
s(DPU,t, DpA ,t|DpU,t i, DpA ,t i) h=
s2(DpA ,t|DpU,t i, DpA ,t i)
(6.36) r x( 1,t , x2 ,t) ( s x1,t) ( sx2 ,t) -abs2(ft-1|DpU,t-i, DpA ,t i- )
= s2(x2 ,t) +b2s2(ft-1|DpU,t i, DpA ,t i)
which means that the errant hedger will undertake a smaller than optimal position on the hedging instrument, incurring losses in hedging effectiveness.
BivariateARCH/GARCH errorcorrectionmodels
A bivariate ARCH/GARCH error correction model can be used to accommo
date the two problems discussed so far, that of allowing for the possibility of cointegration and for the time-varying nature of the second moments and the hedge ratio. Kroner and Sultan (1993) use a bivariate GARCH error correction model of the form
a ( ) +x1,t (6.37)
DpU,t = +b pU,t-1 -dpA ,t-1
c ( ) +x2 ,t (6.38)
DpA ,t = +d pU,t-1 -dpA ,t-1 éx1,t ù
~ (0, HN t) (6.39)
Wt-1 ú êx2 ,tû ë
é s2 t (D pU,t|Wt-1) st(D pU,t, D pA ,t|Wt-1)ù
(6.40) Ht =
êê
ëst(DpU,t, DpA,t|Wt-1) s2 t (D pA ,t|Wt-1) ú úû ) =a +b x2 1 t
s2 t (D pU,t|Wt-1 1 1 1,t-1 +d s2(D pU,t-1|Wt-2 ) (6.41) ) =a +b x2 1 t
st 2(D pA ,t|Wt-1 2 2 2 ,t-1 +d s2(D pA ,t-1|Wt-2 ) (6.42) ht =s(DpU,t, DpA ,t|Wt-1 )
(6.43) s2(DpA ,t|Wt-1 )
Again, the time subscripts on the hedge ratio and the second moments imply time-variation.
TheKalmanfilter
A time-varying hedge ratio can be estimated by applying the Kalman filter to equation (6.18), which can be written in a general form as
RU( ) t =RA( ) ( ) t H t +u t( ) (6.44) where H(t) is a vector of time-varying parameters (hedge ratios) and u(t) is normally distributed with E[u(t)] = 0 and s2[ (u t)] =V. A common specification of the parameter variation is
H t( ) =AH t( -1) +w t( ) (6.45)
where w(t) is a vector random variable with E[w(t)] = 0 and s2[ (w t)] =W. Ais a diagonal matrix given by
0 0
éa11 ù
0 a 0
ê 22 ú
A = ê an-1,n-1 0 ú (6.46)
êë 0 0 0 annúû
such that 0 < aii< 1, i= 1, 2, ..., n. This restriction on the elements of Ais neces
sary to guarantee the stability of the generating process. The random walk model can be obtained by setting A = I, i.e. aii = "i. The estimation of the 1 vector H(t) can be carried out recursively using the Kalman filter technique.
This is because equations (6.44) and (6.45) form the state space representation of the system, in which equation (6.44) is the measurement equation and equa
tion (6.45) is the transition equation, which allows for systematically varying
parameters. The state of the system H(t) is not directly observable, but can be observed through RU(t). Let
[ ( ( |
E H t)|RU(t-1)] =H t t-1) (6.47)
and
[ ( ( | ( ( | |
E H t) -H t t-1)][H t) -H t t-1]¢ =s2(t t-1) (6.48) The Kalman filter equations are given by
( | 1) (
H t t- =AH t-1|t-1) (6.49)
s2( | 1)t t- =As2(t-1|t-1)A¢ +W (6.50) ( ) =s2( | ¢A( )[ ( )s2( | ¢A(
G t t t-1)R t RA t t t-1)R t) +V]-1 (6.51) ( | ( | 1) ( )[ ) ( |
H t t) =H t t- +G t RU(t) -RA(t H t t-1)] (6.52) s2( | t t) =V( |t t- -G t R1) ( ) ¢A( ) t RA( )st 2( |t t-1) (6.53) The initial conditions are given by H( | ) 0 0 =H(0) and s2( |0 0) =s2 (0). This system of equations tells us that the optimal estimator of H(t) at time t, H t( | t)is
( | ) represented by a linear combination of the previous estimator, H t t-1 and the current observation, RU(t). Equation (6.52) shows the recursive nature of the computation. For more detail, see Cuthbertson et al.(1992).
Nonlinear models
Broll et al. (2001) suggest that the hedge ratio should be estimated from a nonlinear model of the form
a A ,t +et (6.54)
pU,t = +hpA ,t +gp2
in which case the error correction term (et =pU,t -a-hpA ,t -gp2 A ,t ) becomes nonlinear. By applying this model to the futures market they found (i) preva
lent nonlinearities in the relationship; (ii) that the relationship is likely to be convex (positive g) rather than concave (negative g); and (iii) that the order of magnitude of the nonlinear component is by and large relatively small. They concluded that the firm should export more (less) and adopt an over (under) hedge in an unbiased currency market if the spot-futures relationship is convex (concave).
The following is a more general treatment of nonlinearity, which we will undertake by referring to equation (6.22), that is pU,t = +hpA ,t +et. There are two approaches to nonlinearity in a relationship like equation (6.22): by postu
lating either a nonlinear attractor, or a nonlinear adjustment to a linear attractor. The first approach is discussed in Granger (1991). Let us define two nonlinear functions, f1(pU) and f2(pA), both of which represent long-memory series. If we define ztas a short-memory series given by
a
zt =f1(pU,t) -lf2(pA ,t) (6.55)
( /
( / ( / ( /
then the equation of the corresponding nonlinear attractor is
) =lf2(p (6.56)
f p1( U,t A ,t)
Hallman (1989) shows how the functions f1(pU) and f2(pA) are estimated.
Nonlinearity in the error correction model is discussed in Escribano (1987), and the procedure is applied to a model of the demand for money in Hendry and Ericcson (1991). Nonlinearity in this case is captured by a polynomial in the error correction term. Thus the error correction model corresponding to equation (6.22) is
=AL +BL k i t i
DpU,t ( )DpU,t-1 ( )DpA ,t +
å
g ei - +xt (6.57) i=1where A(L) and B(L) are lag polynomials. Hendry and Ericcson (1991) suggest that a polynomial of degree three in the error correction term is sufficient to capture the adjustment process.
Multicurrency hedge ratios
Consider a model in which the percentage change in the base currency value of a firm, Vx, is influenced by changes in the exchange rates of the base currency against other currencies. This model can be represented by the regression equation
n
Vx =
å
aiS xyi ) (6.58)i=1
In this case, the hedge ratio corresponding to each exchange rate is equal to the regression coefficient on that exchange rate, which means that hi = ai for all i. Suppose, for example, that Vxis the US dollar value of a firm and that this value is affected by changes in the exchange rates of the US dollar against three currencies: Canadian dollar, Japanese yen and pound. Equation (6.58) can be rewritten for this special case as
V
USD =h1 S USD CAD) +h2S USD JPY) +h3 S USD GBP) (6.59) where h1, h2 and h3 are the hedge ratios corresponding to the three currencies respectively. If, for example, h1 > 0, h2 < 0 and h3 > 0, then hedging one US dollar of value requires taking a short position of h1 Canadian dollars, a long position of h2 yens and a long position of h3 pounds.
Recall from Chapter 4 that if V
CAD = ¢h1 S CAD USD( / ) + ¢h2S CAD( / JPY) + ¢h3 S CAD GBP( / ) (6.60) then h¢ =2 h2, h¢ =3 h3 and h¢ = -1 1 (h2 +h3 ). The hedge ratios applicable to the Canadian dollar value of the firm are h¢1, h¢2 and h¢3 , which correspond to the US dollar, Japanese yen and pound respectively.