Hedging Mismatched Currencies with Options and Futures ∗
Donald LIEN �
University of Texas at San Antonio
Maurice K. S. TSE �
University of Hong Kong
Kit Pong WONG §
University of Hong Kong
November 2002
JEL classification: F23; F31; D81
Keywords: Options; Futures; Multiple currency risks; Hedging effectiveness
∗We would like to thank Ho Yin Yick for excellent research assistance.�Department of Economics, College of Business, University of Texas at San Antonio, 6900 North
Loop 1604 West, San Antonio, TX 78249-0631, U.S.A. Tel.: 210-458-7312, fax: 210-458-5837, e-mail:[email protected] (D. Lien).
�School of Economics and Finance, University of Hong Kong, Pokfulam Road, Hong Kong. Tel.: 852-2857-8636, fax: 852-2548-1152, e-mail: [email protected] (M. K. S. Tse).
§Corresponding author. School of Economics and Finance, University of Hong Kong, Pokfulam Road,Hong Kong. Tel.: 852-2859-1044, fax: 852-2548-1152, e-mail: [email protected] (K. P. Wong).
Hedging mismatched currencies 1
Hedging Mismatched Currencies with Options and Futures
Abstract
This paper develops an expected utility model of an exporting Þrm in a developing country wherecurrency derivative markets do not exist. The Þrm faces exchange rate risk exposure to a foreigncurrency cash ßow. To cross-hedge against this risk exposure, the Þrm uses currency futures andoptions between the foreign currency and a third currency. Since a triangular parity condition holdsamong the three given currencies, this available hedging opportunity, albeit incomplete, is provedto be useful in reducing the Þrm�s exchange rate risk exposure. We show that currency optionsare redundant under two rather restrictive conditions: (i) logarithmic utility functions and/or (ii)independent spot exchange rates. In a more realistic cross-hedging environment, we show thatcurrency options are optimally used by the Þrm to incompletely span the missing currency futuresbetween the domestic and foreign currencies. We also estimate the beneÞts of using currency optionsfor cross-hedging purposes. Hedging effectiveness is shown to be improved substantially.
JEL classification: F23; F31; D81
Keywords: Options; Futures; Multiple currency risks; Hedging effectiveness
1. Introduction
Since the collapse of the Bretton Woods Agreement in 1973, exchange rates have be-
come substantially volatile (Meese, 1990), making exchange rate risk management a fact of
Þnancial life (Rawls and Smithson, 1990). As documented by Bodnar, Hayt, and Marston
(1996, 1998) and Bodnar and Gebhardt (1999), the use of currency forwards, futures, and
options is the norm rather than the exception among non-Þnancial corporations.
While currency derivative markets are the hallmark of industrialized countries, they
are seldom readily available in less developed countries (LDCs) wherein capital markets are
embryonic and foreign exchange markets are heavily controlled.1 Also, in many of the newly
industrializing countries of Latin America and Asia PaciÞc, currency derivative markets are
1Even if currency forward contracts are available in some LDCs, they are deemed to be forward-coverinsurance schemes rather than Þnancial instruments whose prices are freely determined by market forces(Jacque, 1996).
Hedging mismatched currencies 2
just starting to develop in a rather slow pace.2 To indirectly hedge against their exchange
rate risk exposure, exporting Þrms in these countries are obliged to use derivative securities
on related currencies. Such an exchange rate risk management technique is referred to as
�cross-hedging� (Sercu and Uppal, 1985).
The purpose of this paper is to study the hedging decision of an exporting Þrm in a
developing country where currency derivatives markets do not exist. We are particularly
interested in examining the hedging role of currency options in the context of cross-hedging.3
To this end, we develop an expected utility model of the exporting Þrm facing exchange
rate uncertainty. There is a third country which has well-developed currency futures and
options markets. Since a triangular parity condition holds among the domestic, foreign,
and third currencies, this available hedging opportunity, albeit incomplete, remains useful
in reducing the Þrm�s exchange rate risk exposure. We show that currency options are
redundant under two rather restrictive conditions: (i) logarithmic utility functions and/or
(ii) independent spot exchange rates. In a more realistic cross-hedging environment, we
show that currency options are optimally used by the Þrm to incompletely span the missing
currency futures between the domestic and foreign currencies. The driving force is the
triangular parity condition which gives rise to an exchange rate risk that is multiplicative,
thereby non-linear, in nature. This source of non-linearity creates a hedging demand for
non-linear currency options, as distinct from that for linear currency futures.
To estimate the beneÞts of using currency options for cross-hedging purposes, we con-
sider a Taiwanese exporting Þrm which encounters a unit cash ßow denominated in one of
the Þve currencies: (i) the Australian dollar, (ii) the Indonesian Rupiah, (iii) the Japanese
yen, (iv) the Philippine peso, and (v) the Thai baht. The United States is taken as the
third country. We Þnd that including currency options in hedge positions always improves
the hedging effectiveness as compared to using currency futures alone for all of the Þve
2See Eiteman, Stonehill, and Moffett (1998) for a description of the currency regime and the status ofcurrency derivatives in many of these so-called �exotic currencies.�
3Indeed, as shown by Benninga and Blume (1985), investors in the Black-Scholes (1973) world wouldpurchase options only when their utility functions exhibit bizarre properties, which is highly unlikely. Seealso Leland (1980), Brennan and Solanki (1981), and Franke, Stapleton, and Subrahmanyam (1998) for thedemand for options in the setting of portfolio insurance.
Hedging mismatched currencies 3
currencies. The improvements in hedging effectiveness range from 3.89% (the Indonesia
rupiah) to 40.10% (the Australian dollar) and thus can be quite substantial.
This paper is closest in the spirit of Chang and Wong (2002) who also examine the
hedging role of currency options in the context of cross-hedging. The main difference is
on how market incompleteness is introduced. In Chang and Wong (2002), no derivative
securities exist for the foreign currency but there are currency futures and options between
the domestic and third currencies. The degree of incompleteness is less sever in their setting
because the hedging instruments are directly related to the domestic currency. The source of
non-linearity is shown to be quadratic in nature, which makes the analytical results elegant.
In our model, however, the hedging instruments do not involve the domestic currency and
the source of non-linearity is of a higher order, thereby making the analysis rather difficult.
Due to the more non-linear structure, the improvements in hedging effectiveness in our
setting are in general much higher than those reported by Chang and Wong (2002) when
currency options are included in hedge positions.4
The rest of the paper is organized as follows. In the next section, we develop an expected
utility model of an exporting Þrm facing multiple currency risks and cross-hedging oppor-
tunities. Section 3 derives sufficient conditions under which currency options are never
used by the Þrm. Section 4 shows how the Þrm can incompletely span the missing cur-
rency futures between the domestic and foreign currencies with currency options. Section
5 offers empirical evidence on the hedging effectiveness of including currency options for
cross-hedging purposes. The Þnal section concludes.
2. The model
Consider a one-period, two-date (0 and 1) model of an exporting Þrm domiciled in a
developing country where currency derivatives markets do not exist. At date 0, the Þrm
4In Chang and Wong (2002), the improvements in hedging effectiveness range from 0.6% to 5.34%.
Hedging mismatched currencies 4
concludes a transaction in a foreign country, which results in a net foreign currency cash
ßow, X , to be received at date 1. While the size of X is known with certainty ex ante, the
Þrm does not know the then prevailing spot exchange rate at date 1, denoted by �S, which
is expressed in units of the domestic currency per unit of the foreign currency.5 The Þrm
as such encounters exchange rate risk exposure of �SX.
To hedge against its exchange rate risk exposure, the Þrm has to rely on a related
currency of a third country which has well-developed currency derivative markets. DeÞne
�S1 as the spot exchange rate of the domestic currency against the third currency at date
1. Likewise, deÞne �S2 as the spot exchange rate of the third currency against the foreign
currency at date 1. Based on these two spot exchange rates, one can derive the cross-rate of
the domestic currency against the foreign currency at date 1 as �S1�S2. It follows immediately
from the Law of One Price that �S = �S1�S2. Such a triangular parity condition is depicted
in Figure 1.
�S = �S1�S2
�S1�S2
home country foreign country@@
@@
@@
@@
¡¡
¡¡
¡¡
¡¡
third country
Figure 1. Triangular parity condition
At date 0, the Þrm can trade inÞnitely divisible currency futures and options (calls and
puts) which call for delivery of the third currency per unit of the foreign currency at date
1. Without any loss of generality, we restrict the Þrm to use currency futures and put
options only.6 Furthermore, for the pure sake of simplicity, we consider only one strike5Throughout the paper, random variables have a tilde (∼) while their realizations do not.6Because payoffs of any combinations of futures, calls, and puts can be replicated by any two of these
three Þnancial instruments (Sercu and Uppal, 1995), one of them is redundant.
Hedging mismatched currencies 5
price, denoted by K, for the currency put options. The strike price, K, is exogenously
determined, thereby not a choice variable of the Þrm. Let F be the futures price at date
0 and P be the premium on the currency put options, where P is compounded to date 1.
To isolate the Þrm�s hedging motive from its speculative motive, it suffices to restrict our
attention to the case where the currency futures and put options are perceived as jointly
unbiased by the Þrm. As such, we assume throughout the paper that F equals the expected
value of �S2 and P equals that of max(K − �S2, 0).7
The Þrm�s date 1 proÞt denominated in the domestic currency is given by
�Π = �S1�S2X + �S1(F − �S2)H + �S1[P −max(K − �S2, 0)]Z, (1)
where H and Z are the numbers of the currency futures and put options sold (purchased
if negative) by the Þrm at date 0, respectively. The Þrm possesses a von Neumann-
Morgenstern utility function, U(Π), deÞned over its domestic currency proÞt, Π, with
U 0(Π) > 0 and U 00(Π) < 0, indicating the presence of risk aversion.8
The Þrm is an expected utility maximizer and has to solve the following ex ante decision
problem at date 0:
maxH,Z
E[U(�Π)], (2)
where E(·) is the expectation operator with respect to the Þrm�s subjective joint probabil-ity distribution function of �S1 and �S2, and �Π is deÞned in equation (1). The Þrst-order
conditions for program (2) are given by
E[U 0(�Π∗) �S1(F − �S2)] = 0, (3)
E{U 0(�Π∗) �S1[P −max(K − �S2, 0)]} = 0, (4)
7Our intention here is not to impose an ad hoc option pricing theory but to focus on the hedging role ofthe put currency options.
8For privately held, owner-managed Þrms, risk-averse behavior prevails. Even for publicly listed Þrms,managerial risk aversion (Stulz, 1984), corporate taxes (Smith and Stulz, 1985), costs of Þnancial distress(Smith and Stulz, 1985), and capital market imperfections (Stulz, 1990; and Froot, Scharfstein, and Stein,1993) all imply a concave objective function for Þrms, thereby justifying the use of risk aversion as anapproximation. See Tufano (1996) for evidence that managerial risk aversion is a rationale for corporate riskmanagement in the gold mining industry.
Hedging mismatched currencies 6
where an asterisk (∗) indicates an optimal level.9
3. Redundancy of options
Since F = E( �S2) and P = E[max(K − �S2, 0)], we can write equations (3) and (4) as10
Cov[U 0(�Π∗) �S1, �S2] = 0, (5)
Cov[U 0(�Π∗) �S1,max(K − �S2, 0)] = 0, (6)
where Cov(·, ·) is the covariance operator with respect to the Þrm�s subjective joint proba-bility distribution function of �S1 and �S2. Partially differentiating E[U
0(�Π∗) �S1|S2] yields
∂
∂S2E[U 0(�Π∗) �S1|S2] = E
½U 0(�Π∗)[1−R(�Π∗)]∂
�S1
∂S2
+U 00(�Π∗) �S21
·X −H∗ − ∂
∂S2max(K − S2, 0)Z
∗¸¯̄̄̄S2
¾, (7)
where R(Π) = −ΠU 00(Π)/U 0(Π) is the Arrow-Pratt measure of relative risk aversion. If∂ �S1/∂S2 ≡ 0 (i.e., �S1 and �S2 are independent), or if R(Π) ≡ 1 for all Π (i.e., U(Π) is
logarithmic), it is evident from equation (7) that E[U 0(�Π∗) �S1|S2] would be invariant to
different realizations of �S2 when H∗ = X and Z∗ = 0 and thus equations (5) and (6) hold
simultaneously.
To see the underlying intuition, consider Þrst the case that the Þrm�s utility function,
U(Π), is logarithmic. In this case, the objective function of program (2) becomes
E(ln �Π) = E(ln �S1) + E
½ln{ �S2X + (F − �S2)H + [P −max (K − �S2, 0)]Z}
¾.
It is evident that �S1 does not affect the Þrm�s optimal hedge position and thus the celebrated
full-hedging theorem applies. Now, consider the case where the two random spot exchange9The second-order conditions for a maximum are satisÞed given risk aversion.
10For any two random variables, �X and �Y , Cov( �X, �Y ) = E( �X �Y )− E( �X)E( �Y ).
Hedging mismatched currencies 7
rates, �S1 and �S2, are independent. When H = X and Z = 0, equation (1) implies that
�Π = �S1FX. Thus, it follows that equations (5) and (6) hold simultaneously at H∗ = X and
Z∗ = 0. In either case, the currency put options are not used by the Þrm for cross-hedging
purposes.
4. Hedging role of options
In the previous section, we have shown that currency options are redundant under
logarithmic utility functions and/or independent spot exchange rates. However, it is unduly
restrictive to assume either condition to hold. In other words, in a more realistic cross-
hedging environment, we would expect currency options to be an integral part of the optimal
hedge positions of exporting Þrms in developing countries.
To verify the above conjecture, let us consider the following simple example. Suppose
that �S1 = S1 + β�θ and �S2 = S2 + �θ, where Si is the expected value of �Si (i = 1, 2), β is
a scalar, and �θ is a zero-mean random variable. Thus, �S1 and �S2 are perfectly negatively
(positively) correlated if β < (>) 0. Let �θ take on three possible values: −Θ with probabilityp, 0 with probability 1− 2p, and Θ with probability p. Then, we have
�S = �S1�S2 = (S1 + β�θ)(S2 + �θ) = S1S2 + (S1 + βS2)�θ + β�θ
2. (8)
Given the assumed three-point probability distribution function of �θ, we have �θ2 = Θ[�θ +
2max(−�θ, 0)]. Thus, using this fact and equation (8), the exchange rate risk exposure facedby the Þrm is given by
�SX = S1S2X + (S1 + βS2 + βΘ)�θX + 2βΘmax(−�θ, 0)X. (9)
Let K = F . Since F equals the expected value of �S2, we have F = K = S2. Then, the
payoff of a hedge position, (H,Z), is given by
�V = (S1 + β�θ){�θH + [max(−�θ, 0)− P ]Z}. (10)
Hedging mismatched currencies 8
Given the assumed three-point probability distribution function of �θ, we have �θ2 = Θ[�θ +
2max(−�θ, 0)] and �θmax(−�θ, 0) = −Θmax(−�θ, 0). Thus, equation (10) can be written as
�V = [(S1 + βΘ)H − βPZ]�θ + [2βΘH + (S1 − βΘ)Z]max(−�θ, 0)− S1PZ. (11)
Inspection of equations (9) and (11) reveals that the Þrm�s exchange rate risk exposure can
be completely eliminated by the hedge position, (H∗, Z∗), which satisÞes
[S1 + βS2 + βΘ]X = (S1 + βΘ)H∗ − βPZ∗, (12)
2βΘX = 2βΘH∗ + (S1 − βΘ)Z∗. (13)
Solving equations (12) and (13) yields
H∗ = X − S1 − βΘ2βΘ
Z∗,
Z∗ = − 2β2S2ΘX
(S1 − βΘ)(S1 + βΘ) + 2β2PΘ.
As long as β 6= 0, we have Z∗ < 0. Thus, in order to synthesize a short position of the
missing currency futures between the domestic and foreign currencies, a long position of
the currency put options between the foreign and third currencies is called for irrespective
of whether �S1 and �S2 are negatively or positively correlated.
In general, complete spanning of the missing currency futures contracts between the
domestic and foreign currencies is not feasible. To facilitate the empirical study of hedging
effectiveness in the next section, we hereafter restrict our attention to the case where the
Þrm�s objective is to minimize the variance of its domestic currency proÞt:
minH,Z
Var(�Π) = E{[ �Π− E(�Π)]2}, (14)
where Var(·) is the variance operator with respect to the Þrm�s subjective joint probabilitydistribution function of �S1 and �S2, and �Π is deÞned in equation (1).
Hedging mismatched currencies 9
The Þrst-order conditions for program (14) are given by
Cov[�Π∗, �S1(F − �S2)] = 0, (15)
Cov{�Π∗, �S1[P −max(K − �S2, 0)]} = 0. (16)
where an asterisk (∗) indicates an optimal level.11 Solving equations (15) and (16) yields H∗
Z∗
= −A−1BX, (17)
where
A =
Var[ �S1(F − �S2)] Cov{ �S1(F − �S2), �S1[P −max(K − �S2, 0)]}
Cov{ �S1(F − �S2), �S1[P −max(K − �S2, 0)]} Var{ �S1[P −max(K − �S2, 0)]}
,
B =
Cov[ �S1�S2, �S1(F − �S2)]
Cov{ �S1�S2, �S1[P −max(K − �S2, 0)]}
.Inspection of equation (17) reveals that Z∗ is in general non-zero. However, without speci-
fying the underlying joint probability distribution function of �S1 and �S2, the sign of Z∗ is
a priori indeterminate.
When the currency put options are absent (i.e., Z ≡ 0), it is easily shown that the Þrm�soptimal futures position, H0, is given by
H0 = −Cov[�S1�S2, �S1(F − �S2)]
Var[ �S1(F − �S2)]. (18)
The sign of H0 is opposite to that of Cov[ �S1�S2, �S1(F − �S2)], which is also a priori indeter-
minate without knowing the functional form of the joint probability distribution function
of �S1 and �S2.
11The second-order conditions for a minimum are satisÞed.
Hedging mismatched currencies 10
5. Hedging effectiveness
To implement the empirical tests, we assume that �S1 and �S2 are jointly log-normally
distributed: ln �S1
ln �S2
∼ N µ1
µ2
, σ2
1 ρσ1σ2
ρσ1σ2 σ22
.
Alternatively, we can write �S1 = exp(µ1 + σ1�ε1) and �S2 = exp(µ2 + σ2�ε2), where �ε1 and �ε2
are both standard normal random variables with Cov(�ε1, �ε2) = ρ. In Appendix A, we prove
the following useful result:
E
µ�Sm1 �S
n2 I{S̃2<K}
¶= exp
µmµ1 + nµ2 +
m2σ21
2+mnρσ1σ2 +
n2σ22
2
¶
× ΦµlnK − µ2
σ2−mρσ1 − nσ2
¶, (19)
where I{S̃2<K} is the indicator function for the event that�S2 < K, and Φ(·) is the cumulative
standard normal distribution function.
Equation (19) can be used repeatedly to express all the moments in equation (17) as
functions of the underlying distribution parameters. For example, by setting m = 0, n = 1,
and K =∞ in equation (19), we have
F = E( �S2) = exp
µµ2 +
σ22
2
¶. (20)
Note that
P = E[max(K − �S2, 0)] = E
·(K − �S2)I{S̃2<K}
¸= KE
µI{S̃2<K}
¶− E
µ�S2I{S̃2<K}
¶.
The Þrst term on the right-hand side of the above equation corresponds to equation (19)
when m = n = 0 and the second term to m = 0 and n = 1. Thus, we have
P = KΦ
µlnK − µ2
σ2
¶− E( �S2)Φ
µlnK − µ2
σ2− σ2
¶.
Hedging mismatched currencies 11
When K = E( �S2), the above equation can be further reduced to
P = exp
µµ2 +
σ22
2
¶·1− 2Φ
µ− σ2
2
¶¸. (21)
In Appendix B, we report all the variances and covariances contained in equation (17)
in terms of the underlying distribution parameters. It is interesting to observe that µ1 and
µ2, albeit essential for all the relevant moments, do not appear in the ultimate optimal
hedge ratios. As such, no information on the means of ln �S1 and ln �S2 is needed for Þnding
the optimal hedge rations.
The empirical exercises are as follows. We always refer to Taiwan as the home country
and the United States as the third country. The foreign currency cash ßow, X, is normalized
to one unit of the foreign currency so that the Þrm�s hedge position is simply the hedge
ratio. The prices of the currency futures and put options with K = F are artiÞcially set
equal to equations (20) and (21), respectively. The data on daily spot exchange rates are
extracted from Datastream for the period from January 1, 1997 to December 31, 2001.
We Þrst consider Japan as the foreign country. In this case, �S1 is the random spot
exchange rate for the Taiwanese dollar against the US dollar (NTD/USD) and �S2 is that
for the US dollar against the Japanese yen (USD/JPY). Since we assume that �S1 and �S2
are jointly log-normally distributed, the logs of these two random variables provide us the
time series, lnS1,t and lnS2,t, for our empirical analysis.
Applying the Perron-Philips tests for unit roots (Philips and Perron, 1988) to the
NTD/USD series, the Perron-Philips statistic for lnS1,t is −1.712 and that for the Þrstdifference of lnS1,t is −15.55. The hypothesis of a unit root cannot be rejected in theformer but can be rejected in the latter. For the USD/JPY series, the two Perron-Philips
statistics are −1.649 and −35.70, respectively. Once again, we cannot reject the hypothesisof a unit root in lnS2,t but can reject that in the Þrst difference of lnS2,t. As such, both
lnSi,t and lnS2,t are integrated of order one, i.e., I(1). Using the Johansen (1988) method,
we Þnd that lnS1,t and lnS2,t are cointegrated. The cointegrating variable is given by
zt = lnS2,t + 0.9925 lnS1,t − 1.3413, (22)
Hedging mismatched currencies 12
where the t-statistic for the coefficient of lnS1,t is 2.0512.
Since the two I(1) series, lnS1,t and lnS2,t, are cointegrated, to estimate σ1, σ2, and
ρ we use the following vector autoregressive-error correction model (VECM) (Engle and
Granger, 1987):
∆ lnS1,t = α10 +kXj=1
β1j∆ lnS1,t−j +kXj=1
γ1j∆ lnS2,t−j + φ1zt−1 + ε1,t, (23)
∆ lnS2,t = α20 +kXj=1
β2j∆ lnS1,t−j +kXj=1
γ2j∆ lnS2,t−j + φ2zt−1 + ε2,t, (24)
where ∆ lnSi,t = lnSi,t − lnSi,t−1 is the Þrst difference of lnSi,t (i = 1 and 2), and zt is
the cointegrating variable (the error correction term) deÞned in equation (22). The order
of the vector autoregression (VAR) is chosen by the statistical signiÞcance of the highest
order terms. We begin with an order of the VAR equal to 5. If either the coefficient of
∆ lnS1,t−5 or that of ∆ lnS2,t−5 is statistically signiÞcant at the 5% level, we set k = 5 in
our VECM. Otherwise, we reduce the order by one and redo the same procedure until we
Þnd an order of the VAR such that at least one of the highest order terms is statistically
signiÞcant at the 5% level. Table 1 reports the estimation results.
(Insert Table 1 here)
Let e1,t and e2,t be the residual series of equations (23) and (24), respectively. The condi-
tional variance-covariance matrix of (ln �S1, ln �S2) can be estimated by the sample variance-
covariance matrix of (e1,t, e2,t). The estimated conditional variance of ln �S1 is 1.74× 10−5,
that of ln �S2 is 6.51×10−5, and the estimated conditional covariance between ln �S1 and ln �S1
is −3.29 × 10−6. Substituting these estimates into equations (17) and (18) and using the
results in Appendix B, we compute that H0 = 0.6822, H∗ = −0.7280, and Z∗ = −2.5756.The sample variance of �S1
�S2 is 7.5922× 10−5. If the optimal futures position, H0, is used,
the residual sample variance is 4.56 × 10−5, resulting in a variance reduction of 39.91%.
If the optimal futures and put option positions, H∗ and Z∗, are used, the residual sample
Hedging mismatched currencies 13
variance is 2.91×10−5, resulting in variance reductions of 61.68% compared to the no hedge
case and 36.22% compared to using the currency futures as the sole hedging instrument.
As a robustness check, we also separately consider four other currencies as the foreign
currency: (i) the Australian dollar, (ii) the Indonesian rupiah, (iii) the Philippine peso, and
(iv) the Thai baht. Tables 2 to 5 present the estimation results for the VECM of each of
these four currencies.12
(Insert Tables 2 to 5 here)
Table 6 reports the optimal hedge positions and the resulting hedging effectiveness for
each of the above currencies, including the Japanese yen. As shown in Table 6, using the
currency futures and put options for cross-hedging purposes results in variance reductions
ranging from 61.24% (the Philippine peso) to 94.86% (the Indonesian rupiah) as compared
to the no hedge case. When the hedging effectiveness with and without using the currency
put options are contrasted, we Þnd that including the currency options always contributes
positively to further variance reductions among all Þve currencies. The improvements in
hedging effectiveness range from 3.89% (the Indonesia rupiah) to 40.10% (the Australian
dollar) and thus can be quite substantial.
(Insert Table 6 here)
6. Conclusions
In the post-Bretton Woods era, exchange rates have been increasingly volatile, making
non-Þnancial Þrms take exchange rate risk management very seriously. This paper has stud-
ied how a risk-averse exporting Þrm, confronting a foreign currency cash ßow but possessing
no direct hedging opportunities, can employ derivative securities on a related currency to re-
12We consider a partial cointegrating system for each of these currencies. A complete cointegrating systemthat includes all these currencies is not examined.
Hedging mismatched currencies 14
duce its exchange rate risk exposure. Currency options play no role as a hedging instrument
under two rather restrictive conditions of logarithmic utility functions and/or independent
spot exchange rates. In a more realistic cross-hedging environment, we have shown how the
Þrm can use currency options to incompletely replicate the missing risk-sharing contract.
The driving force is a triangular parity condition which gives rise to an exchange rate risk
that is multiplicative, thereby non-linear, in nature. This source of non-linearity creates a
hedging demand for non-linear currency options, as distinct from that for linear currency
futures.
Cross-hedging is important because it expands the opportunity set of hedging alterna-
tives. Given the fact that currency derivative markets are not readily available in developing
countries and are just starting to develop in many of the newly industrializing countries
of Latin America and Asia PaciÞc, it is clear that, for many exporting Þrms exposed to
currencies of these countries, cross-hedging will continue to be a major risk management
technique for the reduction of their foreign exchange risk exposure.
Hedging mismatched currencies 15
Appendix
A. Derivation of equation (19)
Using the fact that �Si = exp(µi + σi�εi), i = 1 and 2, we have
E
µ�Sm1�Sn2 I{S̃2<K}
¶=
1
2πp1− ρ2
Z ∞
−∞
Z lnK−µ2σ2
−∞exp[m(µ1 + σ1ε1)] exp[n(µ2 + σ2ε2)]
× exp
·− ε
21 + ε
22 − 2ρε1ε2
2(1− ρ2)
¸dε1 dε2
=1
2πp1− ρ2
Z ∞
−∞exp
·m(µ1 + σ1ε1)− ε2
1
2(1− ρ2)
¸
×½Z lnK−µ2
σ2
−∞exp[n(µ2 + σ2ε2)] exp
·− ε
22 − 2ρε1ε2
2(1− ρ2)
¸dε2
¾dε1
=exp(mµ1 + nµ2)
2πp1− ρ2
Z ∞
−∞exp
½mσ1ε1 +
[ρε1 + (1− ρ2)nσ2]2 − ε2
1
2(1− ρ2)
¾
×½Z lnK−µ2
σ2
−∞exp
½− [ε2 − ρε1 − (1− ρ2)nσ2]
2
2(1− ρ2)
¾dε2
¾dε1
=1√2πexp
µmµ1 + nµ2 +
m2σ21
2+mnρσ1σ2 +
n2σ22
2
¶
×Z ∞
−∞exp
·− (ε1 −mσ1 − nρσ2)
2
2
¸Φ
· lnK−µ2σ2
− ρε1 − (1− ρ2)nσ2p1− ρ2
¸dε1
=1√2πexp
µmµ1 + nµ2 +
m2σ21
2+mnρσ1σ2 +
n2σ22
2
¶
×Z ∞
−∞exp
µ− z
2
2
¶Φ
µ lnK−µ2
σ2− ρz −mρσ1 − nσ2p
1− ρ2
¶dz
=1√2πexp
µmµ1 + nµ2 +
m2σ21
2+mnρσ1σ2 +
n2σ22
2
¶
Hedging mismatched currencies 16
× Ez·Φ
µ lnK−µ2σ2
− ρz −mρσ1 − nσ2p1− ρ2
¶¸, (A-1)
where z = ε1−mσ1−nρσ2 is a standard normal random variable and Ez(·) is the expectationoperator with respect to Φ(z). Lien (1985) shows that
Ez[Φ(α+ βz)] = Φ
µαp1 + β2
¶. (A-2)
Using equation (A-2) by setting α = lnK−µ2
σ2−mρσ1−nσ2 and β =
ρ√1−ρ2
, equation (A-1)
reduces to equation (19).
B. Derivation of variances and covariances
Note that Var[ �S1(F − �S2)] = E[ �S21(F − �S2)
2]−E[ �S1(F − �S2)]2, which can be written as
E( �S21)F
2 − 2E( �S21�S2)F + E( �S
21�S2
2)− E( �S1)2F 2 + 2E( �S1)E( �S1
�S2)F − E( �S1�S2)
2.
Using equation (19) repeatedly, we have
Var[ �S1(F − �S2)] =M × [exp(σ21)− 2 exp(σ2
1 + 2ρσ1σ2) + exp(σ21 + 4ρσ1σ2 + σ
22)
−1 + 2 exp(ρσ1σ2)− exp(2ρσ1σ2)],
where M = exp(2µ1 + 2µ2 + σ21 + σ
22). Similarly, we can show that
Cov[ �S1�S2, �S1(F − �S2)] =M × [exp(2σ2
1 + 2ρσ1σ2)− exp(σ21 + 4ρσ1σ2 + σ
22)
− exp(ρσ1σ2) + exp(2ρσ1σ2)],
Var{ �S1[P −max(K − �S2, 0)]} =M ×½exp(σ2
1)
·2Φ
µσ2
2
¶− 1
¸·2Φ
µσ2
2
¶− 1− 2Φ
µσ2
2− 2ρσ1
¶¸
+4 exp(σ21 + 2ρσ1σ2)
·Φ
µσ2
2
¶− 1
¸Φ
µ− σ2
2− 2ρσ1
¶
Hedging mismatched currencies 17
+ exp(σ21)Φ
µσ2
2− 2ρσ1
¶+ exp(σ2
1 + 4ρσ1σ2 + σ22)Φ
µ− 3σ2
2− 2ρσ1
¶
−·2Φ
µσ2
2
¶− 1− Φ
µσ2
2− ρσ1
¶+ exp(ρσ1σ2)Φ
µ− σ2
2− ρσ1
¶¸2¾,
Cov{ �S1(F − �S2), �S1[P −max(K − �S2, 0)]} =M ×½[exp(σ2
1)− exp(σ21 + 2ρσ1σ2)]
·2Φ
µσ2
2
¶− 1
¸
− exp(σ21)Φ
µσ2
2− 2ρσ1
¶+ 2exp(σ2
1 + 2ρσ1σ2)Φ
µ− σ2
2− 2ρσ1
¶
− exp(σ21 + 4ρσ1σ2 + σ
22)Φ
µ− 3σ2
2− 2ρσ1
¶
−[1− exp(ρσ1σ2)]
·1− Φ
µσ2
2− ρσ1
¶+ exp(ρσ1σ2)Φ
µ− σ2
2− ρσ1
¶¸¾,
Cov{ �S1�S2, �S1[P −max(K − �S2, 0)]} =M ×
½[exp(σ2
1 + 2ρσ1σ2)− exp(ρσ1σ2)]
·2Φ
µσ2
2
¶− 1
¸
− exp(σ21 + 2ρσ1σ2)Φ
µ− σ2
2− 2ρσ1
¶− exp(2ρσ1σ2)Φ
µ− σ2
2− ρσ1
¶¾
+2 exp(σ21 + 4ρσ1σ2 + σ
22)Φ
µ− 3σ2
2− 2ρσ1
¶+ exp(ρσ1σ2)Φ
µσ2
2− ρσ1
¶,
Var( �S1S2) =M × [exp(σ21 + 4ρσ1σ2 + σ
22)− exp(2ρσ1σ2)].
Hedging mismatched currencies 18
References
Benninga, S., and M. E. Blume, 1985, �On the Optimality of Portfolio Insurance.� Journal
of Finance 40, 1341�1352.
Black, F., and M. J. Scholes, 1973, �The Pricing of Options and Corporate Liabilities.�
Journal of Political Economy 81, 637�659.
Bodnar, G. M., and G. Gebhardt, 1999, �Derivatives Usage in Risk Management by US
and German Non-Financial Firms: A Comparative Survey.� Journal of International
Financial Management and Accounting 10, 153�187.
Bodnar, G. M., G. Hayt, and R. C. Marston, 1996, �1995 Wharton Survey of Derivative
Usage by US Non-Financial Firms.� Financial Management 25, 113�133.
Bodnar, G. M., G. Hayt, and R. C. Marston, 1998, �1998 Wharton Survey of Financial
Risk Management by US Non-Financial Firms.� Financial Management 27, 70�91.
Brennan, M. J., and R. Solanki, 1981, �Optimal Portfolio Insurance.� Journal of Financial
and Quantitative Analysis 16, 279�300.
Chang, E. C., and K. P. Wong, 2002, �Cross-Hedging with Currency Options and Futures.�
Journal of Financial and Quantitative Analysis, forthcoming.
Eiteman, D. K., A. I. Stonehill, and M. H. Moffett, 1998, Multinational Business Finance.
Massachusetts: Addison-Wesley.
Engle, R. F., and C. W. J. Granger, 1987, �Co-integration and Error Correction: Repre-
sentation, Estimation and Testing.� Econometrica 66, 251�276.
Franke, G., R. C. Stapleton, and M. G. Subrahmanyam, 1998, �Who Buys and Who Sells
Options: The Role of Options in an Economy with Background Risk.� Journal of
Economic Theory 82, 89�109.
Froot, K. A., D. S. Scharfstein, and J. C. Stein, 1993, �Risk Management: Coordinating
Corporate Investment and Financing Policies,� Journal of Finance 48, 1629�1658.
Hedging mismatched currencies 19
Jacque, L. L., 1996, Management and Control of Foreign Exchange Risk. Boston: Kluwer
Academic Publishers.
Johansen, S., 1988, �Statistical Analysis of Cointegration Vectors.� Journal of Economic
Dynamics and Control 12, 231�254.
Leland, H. E., 1980, �Who Should Buy Portfolio Insurance.� Journal of Finance 35, 581�
594.
Lien, D., 1985, �Moments of Truncated Bivariate Log-Normal Distributions.� Economics
Letters 19, 243�247.
Meese, R., 1990, �Currency Fluctuations in the Post-Bretton Woods Era.� Journal of
Economic Perspectives 4, 117�134.
Philips, P. C. B., and P. Perron, 1988, �Testing for a Unit Root in Time Series Regressions.�
Biometrika 75, 335�346.
Rawls, S. W., and C. W. Smithson, 1990, �Strategic Risk Management.� Journal of Applied
Corporate Finance 1, 6�18.
Sercu, P., and R. Uppal, 1995, International Financial Markets and the Firm. Cincinnati:
South-Western College Publishing.
Smith, C. W., and R. Stulz, 1985, �The Determinants of Firms� Hedging Policies.� Journal
of Financial and Quantitative Analysis 20, 391�405.
Stulz, R., 1984, �Optimal Hedging Policies.� Journal of Financial and Quantitative Analy-
sis 19, 127�140.
Stulz, R., 1990, �Managerial Discretion and Optimal Financial Policies.� Journal of Finan-
cial Economics 26, 3�27.
Tufano, P., 1996, �Who Manages Risk? An Empirical Examination of Risk Management
Practices in the Gold Mining Industry.� Journal of Finance 51, 1097�1137.
Hedging mismatched currencies 20
Table 1. Estimates of the VECM: The Japanese Yen
Variable ∆ lnS1,t ∆ lnS2,t
Constant 0.0002 −0.0001(0.0001) (0.0002)
zt−1 −0.0030 −0.0030(0.0014) (0.0027)
∆ lnS1,t−1 −0.0590 −0.0023(0.0279) (0.0539)
∆ lnS1,t−2 −0.0726 −0.0220(0.0278) (0.0539)
∆ lnS1,t−3 0.0600 −0.0504(0.0277) (0.0535)
∆ lnS2,t−1 −0.0628 0.0113(0.0144) (0.0280)
∆ lnS2,t−2 −0.0262 0.0278(0.0145) (0.0282)
∆ lnS2,t−3 0.0010 −0.0222(0.0146) (0.0282)
R2 0.0317 0.0031
F-statistic 6.0411 0.5683
This table shows the estimates of the VECM in equations (23) and (24). S1,t is the spot
exchange rate for the Taiwanese dollar against the US dollar and S2,t is that for the US
dollar against the Japanese yen. ∆ lnSi,t = lnSi,t − lnSi,t−1 is the Þrst difference of lnSi,t(i = 1 and 2), and zt is the the error correction term deÞned in equation (22). The order
of the VAR is 3. The numbers in parentheses are the standard errors.
Hedging mismatched currencies 21
Table 2. Estimates of the VECM: The Australian Dollar
Variable ∆ lnS1,t ∆ lnS2,t
Constant 0.0002 −0.0003(0.0001) (0.0002)
zt−1 −0.0018 0.0009(0.0008) (0.0015)
∆ lnS1,t−1 −0.0497 −0.1529(0.0278) (0.0491)
∆ lnS1,t−2 −0.0750 −0.0085(0.0278) (0.0490)
∆ lnS1,t−3 0.0586 −0.0085(0.0277) (0.0490)
∆ lnS2,t−1 −0.0480 −0.0554(0.0158) (0.0279)
∆ lnS2,t−2 −0.0438 −0.0054(0.0159) (0.0281)
∆ lnS2,t−3 −0.0050 −0.0307(0.0158) (0.0280)
R2 0.0271 0.0111
F-statistic 5.1351 2.0729
This table shows the estimates of the VECM in equations (23) and (24). S1,t is the spot
exchange rate for the Taiwanese dollar against the US dollar and S2,t is that for the US
dollar against the Australian dollar. ∆ lnSi,t = lnSi,t − lnSi,t−1 is the Þrst difference of
lnSi,t (i = 1 and 2), and zt = lnS2,t + 3.0838 lnS1,t − 10.1897 is the error correction term.The order of the VAR is 3. The numbers in parentheses are the standard errors.
Hedging mismatched currencies 22
Table 3. Estimates of the VECM: The Indonesian Rupiah
Variable ∆ lnS1,t ∆ lnS2,t
Constant 0.0002 −0.0011(0.0001) (0.0009)
zt−1 0.0001 −0.0119(0.0005) (0.0038)
∆ lnS1,t−1 −0.0650 −0.2770(0.0283) (0.2113)
∆ lnS1,t−2 −0.0875 0.3370(0.0281) (0.2100)
∆ lnS1,t−3 0.0596 −0.0328(0.0281) (0.2098)
∆ lnS2,t−1 −0.0125 0.0366(0.0038) (0.0281)
∆ lnS2,t−2 −0.0144 0.0130(0.0038) (0.0283)
∆ lnS2,t−3 −0.0011 −0.0447(0.0038) (0.0283)
R2 0.0298 0.0150
F-statistic 5.6743 2.8088
This table shows the estimates of the VECM in equations (23) and (24). S1,t is the spot
exchange rate for the Taiwanese dollar against the US dollar and S2,t is that for the US
dollar against the Indonesian rupiah. ∆ lnSi,t = lnSi,t − lnSi,t−1 is the Þrst difference of
lnSi,t (i = 1 and 2), and zt = lnS2,t + 6.0202 lnS1,t − 11.9543 is the error correction term.The order of the VAR is 3. The numbers in parentheses are the standard errors.
Hedging mismatched currencies 23
Table 4. Estimates of the VECM: The Philippine Peso
Variable ∆ lnS1,t ∆ lnS2,t
Constant 0.0002 −0.0005(0.0001) (0.0003)
zt−1 −0.0010 0.0008(0.0005) (0.0012)
∆ lnS1,t−1 −0.0626 −0.1654(0.0281) (0.0639)
∆ lnS1,t−2 −0.0814 0.0883(0.0281) (0.0638)
∆ lnS1,t−3 0.0660 −0.1110(0.0280) (0.0636)
∆ lnS2,t−1 −0.0506 0.0795(0.0123) (0.0281)
∆ lnS2,t−2 −0.0281 −0.0062(0.0124) (0.0282)
∆ lnS2,t−3 0.0006 −0.0671(0.0124) (0.0282)
R2 0.0317 0.0238
F-statistic 6.0329 4.5006
This table shows the estimates of the VECM in equations (23) and (24). S1,t is the spot
exchange rate for the Taiwanese dollar against the US dollar and S2,t is that for the US
dollar against the Philippine peso. ∆ lnSi,t = lnSi,t− lnSi,t−1 is the Þrst difference of lnSi,t(i = 1 and 2), and zt = lnS2,t + 5.3720 lnS1,t − 14.8923 is the error correction term. Theorder of the VAR is 3. The numbers in parentheses are the standard errors.
Hedging mismatched currencies 24
Table 5. Estimates of the VECM: The Thai Baht
Variable ∆ lnS1,t ∆ lnS2,t
Constant 0.0002 −0.0004(0.0001) (0.0003)
zt−1 −0.0024 0.0047(0.0009) (0.0023)
∆ lnS1,t−1 −0.0746 −0.1525(0.0281) (0.0724)
∆ lnS1,t−2 −0.0826 0.0866(0.0280) (0.0723)
∆ lnS1,t−3 0.0702 −0.2952(0.0277) (0.0716)
∆ lnS2,t−1 −0.0598 0.0254(0.0108) (0.0280)
∆ lnS2,t−2 −0.0301 −0.0055(0.0109) (0.0282)
∆ lnS2,t−3 0.0029 −0.1397(0.0109) (0.0283)
R2 0.0486 0.0375
F-statistic 9.4380 7.1851
This table shows the estimates of the VECM in equations (23) and (24). S1,t is the spot
exchange rate for the Taiwanese dollar against the US dollar and S2,t is that for the US
dollar against the Thai baht. ∆ lnSi,t = lnSi,t − lnSi,t−1 is the Þrst difference of lnSi,t(i = 1 and 2), and zt = lnS2,t + 3.5004 lnS1,t − 8.6343 is the error correction term. Theorder of the VAR is 3. The numbers in parentheses are the standard errors.
Hedging mismatched currencies 25
Table 6. Optimal Hedge Ratios and Hedging Effectiveness
Currency Australian Indonesian Japanese Philippine ThaiDollar Rupiah Yen Peso Baht
H0 0.6470 0.9606 0.6822 0.7436 0.7909
H∗ −0.6461 1.0467 −0.7280 −0.4039 0.2556
Z∗ −2.4354 0.1691 −2.5756 −2.0476 −0.9658ProÞt 6.8222 95.2910 7.5922 9.6244 11.7930Variance
Residual 4.5409 5.0983 4.5625 4.6257 4.6615Variancewith Futures
Residual 2.7200 4.9000 2.9100 3.7300 4.3400Variancewith Futuresand Options
Hedging 33.44% 94.65% 39.91% 51.94% 60.47%Effectivenesswith Futures
Hedging 60.13% 94.86% 61.68% 61.24% 63.20%Effectivenesswith Futuresand Options
This table shows the estimates of the optimal hedge ratios in equations (17) and (18).
Hedging effectiveness is gauged by the percentage decrease in variance relative to the no
hedge case.
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