Systematic jump risks in a small open economy: simultaneous equilibrium valuation of options on the market portfolio and the exchange rate

Journal of International Money and Finance
20 (2001) 191–218
www.elsevier.nl/locate/econbase
Systematic jump risks in a small open
economy: simultaneous equilibrium valuation of
options on the market portfolio and the
exchange rate
Melanie Cao a, b,*
a Department of Economics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
b Department of Finance, Schulich School of Business, York University, Toronto, Ontario,
Canada M3J 1P3
Abstract
The valuation of stock options and currency options has witnessed an explosion of new
development in the past 20 years. These models, set up either in a partial equilibrium or a
general equilibrium framework, have certainly enriched our understanding of option valuation
in one way or the other. However, the main drawback of these models is that stock options
and currency options are analyzed in separate contexts. The co-movement of the stock market
and the currency market is absent from the option valuation analysis. Such co-movement is
extremely important and is best illustrated by the Southeast Asian financial crisis.
To overcome this drawback, this paper uses an equilibrium model to investigate the joint
dynamics of the exchange rate and the market portfolio in a small open monetary economy
with jump-diffusion money supplies and aggregate dividends. It is shown that the exchange
rate and the market portfolio are strongly correlated since both are driven by the same econ-
omic fundamentals. Furthermore, options on the exchange rate and the market portfolio are
evaluated in the same equilibrium context. The analysis shows that parameters describing the
same economic fundamentals have very different effects on currency and stock options ©
2001 Elsevier Science Ltd. All rights reserved.
JEL classification: F31
Keywords: Open economy; General equilibrium; Market portfolio; Exchange rate; Options; Jumps;
Siegel’s paradox
* Correspondence address: Department of Finance, Schulich School of Business, York University,
Toronto, Ontario, Canada M3J 1P3. Tel.: +1-416-736-2100 ext 33801; fax: +1-416-736-5687.
E-mail address: mcao@ssb.yorku.ca (M. Cao).
0261-5606/01/$ - see front matter © 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0 2 6 1 - 5 6 0 6 ( 0 0 ) 0 0 0 5 3 - X

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
1. Introduction
Derivatives valuation has witnessed an explosion of new development in the past
20 years. Examples for stock option valuations include Black and Scholes (1973),
Merton (1976), Cox and Ross (1976), Hull and White (1987), Bailey and Stulz (1989)
and Naik and Lee (1990). Examples for currency option models include Biger and
Hull (1983), Garman and Kohlhagen (1983), Grabbe (1983), Chesney and Scott
(1989), Amin and Jarrow (1991), Heston (1993), Bates (1996) and Bakshi and Chen
(1997). The references listed here are by no means exhaustive. These models, set
up either in a partial equilibrium or a general equilibrium framework, have certainly
enriched our understanding of option valuation in one way or the other.
However, the main drawback of these models is that stock options and currency
options are analyzed in separate contexts. The co-movement of the stock market and
the currency market is absent from the option valuation analysis. Such co-movement
is extremely important and is best illustrated by the recent Southeast Asian financial
crisis, which has swamped small economies like Thailand, Indonesia, Malaysia
and Korea.
During the crisis, the dramatic currency devaluations were always accompanied
by sharp decreases in their corresponding stock markets. As shown in Table 1, the
1997 average return on Southeast Asia’s currency and the stock market is about -
45%. The 1998 drastic devaluation of the Russia ruble and Russia’s stock market
only adds more evidence to the co-movement. Such evidence suggests that the stock
market and the currency market are affected by the same fundamental economic
factors. Failure to incorporate such simultaneous reactions to changes in the same
fundamental economic factors would misguide the derivative valuations.
The second drawback of the existing models is best summarized by Jorion (1988,
pp. 427-428):
Many financial models rely heavily on the assumption of a particular stochastic
process, while relatively little attention is paid to the empirical fit of the postulated
distribution. As a result, models like option pricing models are applied indiscrimi-
nately to various markets such as the stock market and the foreign exchange mar-
ket when the underlying processes may be fundamentally different.
Table 1
Summary of currency and stock index performance
Country
1997 returns on currency
1997 returns on the stock index
(%)
(%)
Thailand
45
54
Indonesia
56
37
Malaysia
35
52
Korea
47
38
Average
46
45

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193
Obviously, the information arrival process in the foreign exchange market differs
from that in the stock market, since exchange rates are directly influenced by monet-
ary polices that do not have apparent counterparts in the stock market. It is important
to directly investigate the effect of monetary policy changes on exchange rates and
hence on currency options. Such analysis can only be carried out in a general equilib-
rium framework where the relation between exchange rates and monetary policies
can be endogenized. In fact, indiscriminately applying the Black and Scholes (1973)
formula to both stock options and currency options yields the opposite pricing bias
pattern. The Black-Scholes formula generally overprices out-of-the-money stock call
options and underprices in-the-money stock call options (MacBeth and Merville,
1979), but it usually underprices out-of-the-money currency calls (Bodurtha and
Courtadon, 1987).
Another problem of applying stock option models to currency options is that the
assumptions for stock option models may not be valid for currency options. For
example, a number of scholars, such as Bodurtha and Courtadon (1987), Jorion
(1988) and Dumas et al. (1995), suggest that currency options should be priced with
Merton’s (1976) mixed jump–diffusion stock option model since jumps have been
found in exchange rates.1 The key problem with this application is that the jump
risk in Merton’s model is assumed to be uncorrelated with the market. Such an
assumption of uncorrelated jump risk may be reasonable if the concern were firm
specific stocks, but is problematic for currency markets. Since the exchange rate
reflects one nation’s purchasing power relative to another nation, the exchange rate
is inherently correlated with aggregate fundamental forces that affect the market.
The main objective of this paper is to overcome these drawbacks, by simul-
taneously analyzing option valuations for the exchange rate and the market portfolio
in a small open economy with systematic and non-systematic jump risks. I employ
a continuous-time extension of the Lucas (1978) asset pricing model to a small open
monetary economy, where money has a non-trivial role in the agents’ utility function.
Based on utility maximization, the equilibrium analysis endogenizes the precise
relationship between the exchange rate and the market portfolio which are functions
of the same fundamental forces. The explicit modelling of the relationship between
the exchange rate and monetary policies also helps to uncover the distinct nature of
the exchange rate process that differs from the stock price process. Under the logar-
ithmic utility function, the equilibrium exchange rate, expressed as the relative price
of foreign currency in terms of home currency, is affected by the domestic money
supply, aggregate dividends and the level of investments in foreign assets. In contrast
to the exchange rate, the real price of the domestic stock is affected by aggregate
dividends and the level of investments in foreign assets. This equilibrium formulation
also enables me to price options on the exchange rate and stock accordingly. Com-
parative analysis shows that currency options and stock options are affected differ-
ently by the parameters underlying economic fundamentals. In addition, this paper
1 See Akgiray and Booth (1988), Jorion (1988), Tucker (1991) and Ball and Roma (1993) for empirical
evidence on the jumps in exchange rates.

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
also addresses the analog of the so-called Siegel’s paradox in currency option valu-
ation with systematic jump risks, which is illustrated by Dumas et al. (1995).2
The current model is obviously different from the existing partial equilibrium
option models in which the exchange rate or the stock price is exogenous. As pointed
out by Bailey and Stulz (1989), the arbitrary choice of the exogenous process for
any security price in the partial equilibrium models is unlikely to be consistent with
the equilibrium conditions or to provide important insights into how derivative prices
may respond to changes in any fundamental economic variables.
Though the current model shares the equilibrium approach with the existing equi-
librium option models, the key difference is that the current model simultaneously
analyzes currency option and stock option valuation in a consistent manner. More-
over, the focus here is on a small open economy, which is different from a closed
pure-exchange economy as in Naik and Lee (1990) for stock option valuation, or a
two-country setting as in Bakshi and Chen (1997) for currency option valuation.
The remainder of this paper is organized as follows. Section 2 describes the econ-
omy and presents the equilibrium results. Section 3 examines the endogenized
exchange rate and the price of the market portfolio, and derives equilibrium prices
for European currency and stock options from the view of the domestic risk-averse
agent. Section 4 identifies the adjustments on the risk-neutral process of the exchange
rate that help to solve the analog of Siegel’s paradox in currency options. Section 5
extends the model to allow for a correlation between the money supply and aggregate
dividends. Section 6 concludes the paper and the appendices provide the neces-
sary proofs.
2. A small open monetary economy
Consider a small open economy with perfect capital mobility between itself
(termed the domestic country) and the rest of the world (termed the foreign country).
This economy consists of a single risk-averse representative agent whose lifetime
horizon is infinite. I adopt the standard formulation of a small open economy used
in the existing literature with the following characteristics.3 First, the agent in the
small economy has perfect access to the international goods and assets markets.
Since the small economy has little influence on the foreign country, it takes the price
of any foreign asset as given. Second, the domestic currency and domestic assets
held by the foreign country are assumed to be negligible, implying that the supplies
of these assets are cleared by domestic demands. Third, domestic aggregate con-
sumption is financed through both domestic aggregate output (dividends) and the
2 The paradox (Siegel, 1972) originally illustrates the discrepancy between the forward exchange rate
and the expected future exchange rate. That is, when the exchange rate is expressed as the price of the
domestic currency in terms of the foreign currency, the forward exchange rate is always less than the
expected future rate.
3 For a reference to a deterministic model of a small open economy, see Obstfeld (1981). An example
in the stochastic environment is Grinols and Turnovsky (1994).

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
195
return to holding foreign assets (which is paid in consumption goods). When the
sum of aggregate dividend and the return to foreign assets exceeds aggregate con-
sumption, the goods market is cleared by an increased holding of foreign assets (i.e.,
a current account surplus); when the sum of aggregate dividends and the return to
foreign assets falls short of aggregate consumption, the residual is financed by a
reduction in the holding of foreign assets (i.e., a current account deficit). This feature
distinguishes a small open economy from a closed economy.
I will first describe the primitives of the economy and then solve the agent’s
maximization problem. Equilibrium asset prices, including the domestic nominal
interest rate and the exchange rate, are determined by requiring goods, money and
financial markets to clear, as in Lucas (1982).
2.1. Structure of the economy
There is a single good traded worldwide with no barriers, which can be used for
consumption and investment. The nominal price of the good at home at time t is
denoted p . Let P∗ be the foreign price level measured in the foreign currency.
t
According to the law of one price in the good market, p equals the spot exchange
rate times P∗. Since the home country is small, it takes P∗ as given and so we can
simplify the discussions by normalizing P∗=1.4 Then, p equals the spot exchange
t
rate expressed as the relative price of the foreign currency in terms of the home cur-
rency.
The home government controls the domestic money supply, which is taken as
given by each domestic agent. The real money balances held by the domestic agent
at time t are defined as m =M /p , where M is the nominal quantity of domestic
t
t
t
t
money demanded by home agents. To assign a non-trivial role to money, I follow
Sidrauski (1967) to assume that real money balances yield utility to agents in addition
to their purchasing power. In particular, the agent’s period utility function,
U(c , m , t), depends positively on the agent’s real money balances, m , as well as on
t
t
t
consumption, c . The rationale is that larger real money balances reduce the trans-
t
action time in the goods market and hence allow the agent to enjoy more leisure.
As long as leisure yields positive marginal utility to the agents, real money balances
yield utility.5
The government’s purchase of goods and services is assumed to be constant and
so the change in the money supply is injected into the economy as lump-sum monet-
4 Allowing P∗ to follow a stochastic process complicates the analysis without changing the qualitative
results, provided that the process for P∗ is independent of the processes for domestic dividends and
domestic money supply.
5 The money-in-the-utility approach is also technically convenient in a continuous-time setting. On the
other hand, the cash-in-advance approach depends crucially on the timing of events and hence on the
discrete-time structure, as stated in Sargent (1987, p. 157). For the cash-in-advance constraint to bind,
all financial markets must be temporarily shut down when consumption goods are purchased with money.
In a continuous-time setting where agents can instantaneously sell goods and assets for money, the cash-
in-advance constraint imposed by Bakshi and Chen (1997) is technically difficult to implement.

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
ary transfers. As in Lucas (1982), I assume that the agent is endowed with one unit
of a claim on these monetary transfers. Denote the real price of this equity claim at
time t as L . The money transfer measured in real terms, l, can be understood as the
t
“dividend” for this claim. Therefore, L is the present value of future real monetary
transfers. Note that monetary transfers are lump-sum and hence are taken as given
by individual agents. The dynamics of the domestic money supply are described in
the following assumption.
Assumption 1 The domestic money supply, Ms, is assumed to evolve according to
the following mixed diffusion-jump process:
dMs (m l k ) dt s dz (Y 1) dQ , ∀ t (0, ).
(2.1)
Ms
m
m m
m
1
m
m
Here, m is the instantaneous expected growth rate of the money supply; s2 is the
m
m
instantaneous variance of the growth rate, conditional on no arrivals of new important
shock and dz is a one-dimensional Gauss-Wiener process. The element dQ
is a
1
m
jump process with a jump intensity parameter l and Y -1 is the random variable
m
m
percentage change in the money supply if the Poisson event occurs. The logarithm
of Y
is normally distributed with mean q
and variance f2 . The expected jump
m
m
m
amplitude, k =E(Y
1), is equal to exp(q +f2 /2) 1. Also, k¯ =E((1/(Y )) 1) is
m
m
m
m
m
m
equal to exp( q +f2 /2) 1. The random variables {z , t 0}, {Q , t 0} and {Y ,
m
m
1t
mt
mj
j 1} are assumed to be mutually independent. Also, Y
is independent of Y
for
mj
mj
j j . The parameters (m , s , l , q , f ) are constant.
m
m
m
m
m
The above money supply process incorporates both frequent fluctuations in the
money supply, which correspond to the diffusion part dz , and infrequent large
1
shocks to the money supply, which correspond to the jump part dQ . Both capture
m
changes in government monetary policies.
There is only one domestic risky stock, which represents the ownership of the
home productive technology for the single good. The total supply of this risky stock
is normalized to one. Denote its real price at time t as S and the dividend as d . The
t
t
dividend stream {d } can be understood as aggregate dividends in this small econ-
t
omy, which are exogenously given as:6
dd m(d) dt s(d) dz (Y
d
2
d
1) dQd,
(2.2)
where dz is a one-dimensional Gauss-Wiener process and dQ
2
δ is an independent
jump process, described more precisely later.
The specification of the aggregate dividend process corresponds to an economy
which is infrequently subject to real shocks of unpredictable magnitude. The shocks
6 Although dividends are not continuously distributed in reality, one may be able to find reasonable
proxies for aggregate dividends used here. Aggregate output and dividends on stock indices are the
examples.

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
197
on dividends could result from output shocks or shocks due to technological inno-
vations. For most of the discussion, the dividend process and the money supply
process are assumed to be independent, measured with respect to a given probability
space ( , F, P). Section 5 will extend the discussion to allow for a correlation
between the two processes.
There are foreign pure discount bonds available for trading to the home agent at
any time. A foreign pure discount bond pays 1 unit of consumption goods at maturity
and 0 at all other times. The agent can internationally diversify his portfolio by
holding the foreign bonds and the domestic financial assets. That is, the net trading
in assets between this small economy and the foreign country is positive and time-
varying. Since the country is small, the real price of the foreign bond at time t, F ,t
is taken as exogenous by the home agent. The dynamics of F are assumed below:
t
Assumption 2 F evolves as dF=rF dt, where r is a positive constant.
t
The processes for the money supply, the foreign bond price and the aggregate divi-
dend are the primitives of the economy. Together with the specification of the utility
function described below, they induce equilibrium prices for other assets. Among
these other assets, there are domestic nominal pure discount bonds in zero net supply,
with nominal rate of return i. A domestic nominal discount bond pays 1 unit of
domestic currency at maturity and 0 at all other times. Denote B as the nominal
t
price of the discount bond at time t. Then, dB=iB dt, where i is endogenously determ-
ined in equilibrium. The real price of the domestic bond at time t, b , is given as
t
b =B /p . In addition, there are many other contingent claims on the risky domestic
t
t
t
stock and the spot exchange rate available for trading at any time in the economy.
These contingent claims are all in zero net supply. Denote the real prices of the
contingent claims at time t by a vector x and the corresponding vector of real divid-
t
ends by dxt.
2.2. The agent’s optimization problem
The representative agent’s information structure is given by the filtration
F s(Ms
, m ,
t
t, dt; 0
t t). As described earlier, the period utility at time t is U(ct
t
t), where U(·, ·, t): R2→
+
R is increasing and strictly concave and satisfies the follow-
ing properties:
lim U (x , x ) 0 and lim U (x , x )
, j 1, 2.
j
1
2
j
1
2
xj→
xj→0
The agent’s intertemporal utility is described by
V(c, m) E
U(c , m , t) dt.
0
t
t
0

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
Initially, the agent is endowed with NF units of the foreign bond, one share of the
0
domestic risky stock, money holdings M and one share of the equity claim for
0
domestic monetary transfer. His consumption over time is financed by a continuous
trading strategy M , N , ∀ t 0, where M is the money holding at time t and
t
t
t
N =(NL, NF, NS, Nb, Nx ) is a vector which represents the portfolio holdings con-
t
t
t
t
t
t
sisting of all the financial assets traded in financial markets at time t. For example,
NF is the quantity of foreign bonds held by the domestic agent at time t. Denote the
t
real prices of all financial assets at time t by a vector X =(L , F , S , b , x ) and the
t
t
t
t
t
t
corresponding vector of real dividends by q . The cumulative dividends up to t are
t
t
defined as D = q
=N
t
t dt. At any point t
0, the agent’s wealth is Wt
tXt+Mt/pt and
0
the flow budget constraint is
1
ct dt Mt d
NX
p
t (dDt
dXt) dWt.
(2.3)
t
This constraint intuitively states that the sum of the wealth increase (dWt) and con-
sumption flow (ct dt) is bounded by the dividend and capital gain from the portfolio
{Mt, Nt}.
With this flow budget constraint, one can use the technique of optimal control to
derive the partial differential equations that are satisfied by the assets prices. In the
presence of the jump components in the money supply process and the dividend
process, these partial differential equations turn out to be very complicated. In con-
trast, the Euler equation approach appears much simpler and is adopted here.7 To
do so, transform the flow budget constraint into an integrated one (see Duffie, 1992,
p. 110, for a similar formulation):
t
t
t
M
1
M
c
0
t
t dt
M
NXX
NX
X .
(2.4)
p
t d p
p
0
0
t (dDt
dXt) NXt t
0
t
t
0
0
0
The agent chooses an optimal portfolio trading strategy {M , N , ∀ t 0} so as to
t
t
maximize his expected lifetime utility. Precisely, he solves:
max E U(c , m , t) dt s.t. (2.4) holds.
t
t
{ct, Mt, Nt}
0
The expectation is taken with respect to the filtration specified earlier. The Euler
equations are:
7 The Euler equation approach has been used in Naik and Lee (1990) and the two approaches are
equivalent in the sense that they lead to the same asset prices.

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
199
1
1
1
E
U (c
dt ,
(2.5)
p
U (c , m , t) t
m
t, mt, t)p
t
c
t
t
t
t
1
X
E
U (c
t
U (c , m , t) t
c
t, mt, t) dDt .
(2.6)
c
t
t
t
That is, the reciprocal of the exchange rate equals the expected discounted sum of
the future real wealth of one dollar, with the state price deflator being the marginal
rate of substitution between consumption and the real money balance. The price of
any other asset equals the expected discounted sum of dividends, with the stochastic
state price deflator being the marginal rate of substitution between consumption at
different dates.
As is typical for a small open economy, the exogenous foreign interest rate, the rate
of time preference and the parameters describing consumption must satisfy certain
restrictions in order to ensure the existence of an equilibrium. Such a restriction can
be obtained by examining an agent’s trade-off between consuming at time t and
purchasing the foreign bond. The net utility gain from purchasing bond is
dF/F
dU /U
E
c
c ,
dt
t
dt
where [(dF/F)/dt]=r is the rate of return to holding the bond and E ((dU /U )/dt) is
t
c
c
the utility loss due to the delay in consumption. Since optimality requires the net
utility gain to be zero, the equilibrium restriction is r= E ((dU /U )/dt).8
t
c
c
2.3. Equilibrium exchange rate and asset prices under logarithmic utility
Market clearing conditions are described as follows. The domestic currency held
by the foreign country is assumed to be negligible, implying that the money market
is cleared by domestic money demand,9 i.e., M =M. Similarly, the demand for the
s
risky stock equals the supply of shares, which is one share, and the demand for the
claim on monetary transfers equals the supply, which is also one. Equilibrium prices
are such that the representative agent holds neither the domestic nominal bonds nor
any other contingent claims, because the net supply of each such asset is zero. Note
that the supply of a domestic asset (or money) equals the domestic demand for the
asset (or money). This equality holds here not because the economy is closed but
rather because the economy is small relative to the outside world and so the foreign
8 This restriction can be formally derived from the Euler equation (2.6). When there is no uncertainty,
this restriction becomes the well-known equality between the real interest rate (r) and the rate of time
preference (r).
9 One can easily introduce a stochastic percentage of the domestic currency held by the foreign country,
however including an additional stochastic percentage will not change the main results in the current
paper, but only complicates the analysis.

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M. Cao / Journal of International Money and Finance 20 (2001) 191–218
demand for its asset (its money) is considered to be negligible, as discussed at the
beginning of Section 2.
On the other hand, the goods market clearing condition is quite different here
from that in a closed economy. Since the country can have current account surplus
or deficit, as discussed in the introduction, aggregate consumption does not necessar-
ily equal the aggregate dividend generated from the domestic stock. Since the country
can export the goods to the foreign country to increase its holdings on foreign bonds,
the total expenditure on goods is c dt+df, where f =NFF is the value of foreign bonds.
t
t
t
The total supply of goods is the sum of domestic dividends, d dt, and return to
holding foreign bonds, rf dt. Thus, the goods market clearing condition is
df (d rf c) dt.
This goods market clearing condition also differs from that in Lucas’ (1982) two-
country assets pricing model and its application in currency options by Bakshi and
Chen (1997). In these models, the equilibrium portfolio of each country is identical
to its initial endowment, so that the net trading in assets between the two countries
is zero in equilibrium. In contrast, here the net trading in foreign bonds must be
non-zero in equilibrium as d and c vary over time. This difference not only makes
it more challenging to solve for the equilibrium portfolio but also leads to important
differences in the behavior of the exchange rate: since the exchange rate clears the
goods market, the net trading volume affects the exchange rate.
For analytical tractability, I assume that preferences are given by:
Assumption 3 The risk-averse agent’s period utility is described by
U(c , m , t) e−rt[a ln c
(1 a) ln m ], a (0, 1).
(2.7)
t
t
t
t
The goods market clearing condition implies that the real wealth, f +S , is equal
t
t
to the expected present value of future consumption stream, c /r. This condition,
t
together with (2.6), helps to determine the equilibrium price of the domestic risky
stock, S , and the equilibrium quantity of the foreign bonds held by the domestic
t
agent, f (see Appendix A for a proof).
t
Proposition 2.1 Under Assumptions 1–3, the equilibrium real price of the domestic
risky stock at time t, S , is S =S(d )=d /r, ∀ t (0, ) and the equilibrium value of
t
t
t
t
foreign bonds held by the domestic agent is f =NFF =f e(r−r)t.
t
t
t
0
Given the logarithmic utility function in Assumption 3, the real price of the risky
stock is only affected by aggregate dividend. Precisely, the stock price equals the
present value of future dividends discounted at the rate of time preference. The
quantity of foreign bonds held by the domestic agent in equilibrium evolves at a
constant rate of r r. Equivalently, the level of investment in foreign bonds at time
t in equilibrium is determined as NF=NFe−rt. Therefore, the market portfolio in this
t
0

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