Numerical Schemes for Variational Inequalities Arising in International Asset Pricing.
This paper introduces a  valuation model of international pricing in  the presence of political risk. Shipments between countries are  charged with shipping costs and the country specific production  processes are modelled as diffusion processes. The political risk is  modelled as a continous time jump process that affects the drift of the  returns in the politically unstable countries. The valuation model  gives rise to a singular stochastic control problem that is  analyzed numerically. The fundamental tools come from the theory of  viscosity solutions of the associated HamiltonJacobiBellman equation  which turns out to be a system of integral-differential  Variational  Inequalities with gradient constraints.
Introduction
We develop a continuous time model of international asset pricing in a two-country
framework with political risks. The structure of the model is similar to
Dumas (1992) except for the inclusion of political risk. Assets are homogeneous
except for location, and serve as production inputs as well as consumption goods.
International capital markets are fully integrated in the sense that individuals from
each country can freely buy and sell claims to assets located in both countries.
However, individuals can only consume assets located in their country of residence.
Assets can be shipped between countries at a cost and subsequently may be utilized
for either production or consumption purposes at their new location.
Political risk enters the model via uncertainty in the drift of the stochastic
production process associated with the politically risky country. Fundamentally,
political risk represents uncertainty about future government actions which may
impact the value of firms and/or the welfare of individuals. If we focus simply
on the value of firms, there are still a lot of government actions which can affect
firm profitability and the values of securities it issues. Changes in the tax code (or
its implementation), price ceilings, local content requirements, quotas on imported
inputs, labor law provisions, and numerous other areas of government regulation can
affect firm profitability and/or security values. One could argue that all governments
exhibit political risk in that there is some uncertainty about their future actions.
However, the degree of risk tends to vary dramatically with some governments
(countries) viewed as "politically stable" and others as quite risky.
For simplicity, we treat one of our countries as exhibiting no political risk. That
is, the drift is known for the production process of assets located in that country.
There is still uncertainty about the value of production assets located in that country
due to market forces, technology, weather, etc. We associate such uncertainty with
a (country specific) brownian motion; however, the drift of the process is known in
the politically stable country. In the politically risky (unstable) country, we assume
that the expected productivity of assets can take on two values. The lower state
can be interpreted as representing the local government's ability to impose a tax
or regulation on firms producing in that country which negatively impacts their
profitability. Symmetrically, the high state can be interpreted as either a lower tax
rate or even a subsidy for local production, perhaps in an indirect form via changing
a restrictive regulation.
Furthermore, a negative action can be followed by a positive one and vice
versa. Recently, we have seen asset prices and exchange rates (a special type of
asset price) yoyoing up and down in response to sequences of government actions,
as well as conjectures about future actions. In a rather simplified manner, we are
attempting to capture this sort of phenomenon by having the drift in the risky
country determined by a continuous time Markov chain which can take one of two
possible values. Consequently, the extent of political risk in our model is determined
by both the spread between the two drift parameters from the Markov chain and
by the transition probabilities.
We formulate this model as a singular stochastic control problem whose states
describe the production technology processes in both countries. The collective utility
is the value function of this optimization problem and it is characterized as the
unique (weak) solution of the associated Hamilton-Jacobi-Bellman (HJB) equation.
Because of the presence of the shipping costs and the effects of the Markov chain
process, the HJB equation actually turns out to be a system of Variational In-
equalities, coupled through the zeroth order terms, with gradient constraints. Such
problems typically result in a depletion of the state space into regions of idleness
and regions where singular controls are exercised. In the context of the model we
are developing herein, the singular policies correspond to "lump-sum" shipments
from one country to the other.
Similar problems with singular policies arise in a wide range of models in the
areas of asset and derivative pricing. They are essentially linked to the fundamental
issue of irreversibility of financial decisions in markets with frictions such as trans-action
or shipping costs or an irreversible loss of an investment opportunity related
to unhedgeable risks. Unfortunately, such problems do not have in general smooth
solutions, let alone closed form ones. It is therefore imperative to analyze these
problems numerically by building accurate schemes for the value function as well
as the free boundaries which characterize the singular investment policies.
We undertake this task and we construct a family of numerical schemes for the
collective utilities and the equilibrium prices. These schemes have all the desired
properties for convergence, namely, stability, monotonicity and consistency. They
belong to the class of the so-called "time-splitting" schemes which approximate separately
- in each half-time iteration - the first- and the second-order derivatives.
These schemes are known to be very suitable for the approximation of the solutions
of a certain type of second-order nonlinear partial differential equations similar to
the ones arising in our model.
Although it is highly simplified, the proposed model captures some of the flavor
of an international environment where assets may be exposed to substantially
different risks because they are located within the jurisdictions of different govern-
ments. In effect, they are different assets and will generally exhibit different prices
because of their location. As we shall see, political risk not only influences asset
values but also consumption patterns. Furthermore, if we interpret the ratio of the
output prices in the two economies as a real exchange rate, then that exchange rate
will exhibit sustained deviations from its Purchasing Power Parity (PPP) value.
The paper is organized as follows. In section 2, we describe the basic model
and we provide analytic results for the value function. In section 3, we construct the
numerical schemes for the value function and the trading policies. In section 4, we
interpret the numerical results, and we provide some conclusions and suggestions
for extensions of this work.
2. The model and the associated Variational Inequalities.
In this section we describe the international asset pricing model we are going
to use in order to study the effects of political risk on international asset prices,
consumption and investment behavior across countries.
We concentrate on a simplified two-country model where capital markets are
fully integrated in the sense that individuals from each country can own claims
to assets located in either country. These assets serve as production inputs and
consumption goods. The production technology is stochastic and differs across the
two countries; however there is a single technology in each country. One of the
two countries is considered "politically unstable" and exhibits political risk. We
model the political risk via a continuous time Markov chain which affects the rate
of return of the stochastic production process in the politically risky country. We
assume, for simplicity, that there are only two states for the Markov chain, a low
and a high state. As it was pointed out in the introduction, the different states can
be interpreted, among other things, as representing the local government's ability
to alter the tax rate on firms producing in that country or to alter its subsidizing
policy on local production. In this context, the low state corresponds to a high tax
rate, with the high state representing either a low tax or perhaps a subsidy.
We denote the goods of the two countries by X and Y and the production
technology processes in the countries X and Y by X t and Y t respectively. Country
Y is considered to be politically stable and its production process Y t is modelled as a
diffusion process with drift coefficient b and volatility parameter oe 2 . Country X has
a production technology process with similar diffusion structure - with volatility
parameter oe 1 - but its drift coefficient is affected by a two-state Markov chain, say
z t , which represents the effects of the political instability.
Consumption on country X is denoted by C x , which includes both consumption
of local output and of imports from country Y. Consumption in country Y is defined
in an analogous manner and is denoted by C y . Cumulative shipments, as of time
t, from country X to country Y are denoted by L t ; such shipments (exports from
country X ) incur proportional shipping costs at a rate -. In a similar manner,
cumulative shipments from country Y (imports by country X ), denoted by M t
incur proportional shipping costs at a rate -. Without loss of generality, we assume
that country X is charged with the shipping costs.
Using the above definitions, we can write the state processes for the capital
stocks in the two countries as
with
t and W 2
being brownian motions on a probability
correlation this we can take W 2
being
a brownian motion independent of W 1
t . The constants oe 1 ; oe 2 and b are assumed to
be positive.
The process z t is a continuous-time Markov chain with two states z 1 and z 2
such that
As it was discussed above, the low state z 1 is associated with an unfavorable
political state (from the perspective of the production process owners) as opposed
to the high state z 2 which represents the favorable political state in country X . We
denote by p i;j , the transition probabilities of z t for the above states.
The collective (or integrated) utility payoff for consumers of both countries
over their consumption rates is
Z +1e \Gammaaet U(C x
A policy (C x
it is F t -progressively measurable -
being nondecreasing
CADLAG   processes such that C x
Z te \Gammaaes (C x
s +C y
and the following state constraints are satisfied
The collective consumer function U : [0; +1) \Theta [0; +1) ! [0; +1) is assumed
to be increasing and concave in both arguments. Also, U(0;
for some constants M ? 0 and
We define the collective across-countries value function V (x;
Az
Z +1e \Gammaaet U(C x
The set A z is the set of admissible policies (C x
are assumed
to satisfy the above measurability and integrability conditions and the state constraints
ae is a positive constant which plays the role of a discount factor.
In order to guarantee that V is finite for all x - 0, y - 0 and
impose the following restrictions on the market parameters.
First, we define oe;
A process is CADLAG if it is right-continuous with left limits.
and if b ! z 2 , 8
We assume that at least one of the following sets of inequalities holds
together with the additional related conditions
or (2.6b)
We continue with some elementary properties of the value function.
Proposition 2.1: The value function V is increasing and jointly concave in the
spatial arguments (x; y). Moreover, for fixed z, V is uniformly continuous on
Proof: The monotonicity follows from the fact that for the point (x+ ffl; y) (respec-
tively (x; y+ ffl) the set of admissible policies A z;(x+ffl;y) satisfies A z;(x+ffl;y) oe A z;(x;y) ;
the latter follows from the monotonicity and concavity of the utility function, the
form of the state dynamics and the definition of the value function. These proper-
ties, together with the state constraints (2.4) are also used to establish the concavity
of the value function. Indeed, if (C x
are optimal
policies for the points 1), the policy
for z). For the uniform continuity of the value
function on [0; +1) \Theta [0; +1) we refer the reader to Proposition 2.2 in Tourin and
Proposition 2.2: Under the growth conditions (2.6a) and (2.6b) and the properties
of the utility function U , the value function is well defined on [0; +1) \Theta [0; +1) for
The proof appears in Appendix B.
Remark 2.1: Even though we look at the case of linear coefficients in the state
equations (2.1) and (2.2), this assumption is by no means restrictive. In fact, all
the arguments presented herein can be easily generalized to the case of general co-efficients
long as the functions oe
[0; +1) satisfy the conditions: they are Lipschitz concave functions of their argument
with oe 1 and at least one of the oe i 's,
The motivation behind the choice of linear
coefficients is only for the sake of simplicity; the methodology is easier to present
and also, the numerical schemes are validated for such coefficients.
The classical way to attack problems of stochastic control is to analyze the
relevant equation that the value function is expected to solve, namely the Hamilton-
Jacobi-Bellman equation. This (HJB) equation is the offspring of the Dynamic
Programming Principle and stochastic analysis. When singular policies are allowed,
the HJB equation becomes a Variational Inequality with gradient constraints. These
constraints are associated to the "optimal direction" of instantaneously moving the
optimally controlled state processes. In the context of optimal consumption and
investment problems, such situations arise when transaction fees are paid (see, for
example, Zariphopoulou (1991), Tourin and Zariphopoulou (1994)). In the problem
we study herein, the analysis is more complicated because the drift of the state
process X t is influenced by the fluctuations of the Markov chain z t . This feature,
together with the presence of singular policies results into an HJB equation which
For a similar problem with general coefficients, we refer the interested reader
to Scheinkman and Zariphopoulou (1997).
is actually a system of Variational Inequalities coupled through the zeroth order
term (see equations (2.7) and (2.8)).
It is a well established fact that if it is known a priori that the value function is
smooth, then standard verification results guarantee that it is the unique smooth solution
of the HJB equation. Moreover, if first order conditions for optimality apply,
then they are sufficient to determine the optimal policies in the so-called feedback
formula. (See, for example, Fleming and Soner (1993)). Unfortunately, this is rarely
the case. As in our problem, the value function might not be smooth and therefore
it is necessary to relax the notion of solutions of the (HJB) equation. These weak
solutions are the so-called (constrained) viscosity solutions and this is the class of
solutions we will be using throughout the paper. In models of optimal investment
and consumption with transaction costs, which are a special case of the model described
above, this class of solutions was first employed by Zariphopoulou (1992).
Subsequently this class of solutions was used among others by Davis, Panas and
Zariphopoulou (1993), Davis and Zariphopoulou (1994), Tourin and Zariphopoulou
(1994), Shreve and Soner (1994), Barles and Soner (1995), and Pichler (1996). The
characterization of V as a constrained solution is natural because of the presence
of state constraints given by (2.4).
The notion of viscosity solutions was introduced by Crandall and Lions (1983)
for first-order equations, and by Lions (1983) for second-order equations.
Constrained viscosity solutions were introduced by Soner (1986) and Capuzzo-
Dolcetta and Lions (1987) for first-order equations (see also Ishii and Lions (1990)).
For a general overview of the theory we refer to the User's Guide by Crandall, Ishii
and Lions (1994) and to the book by Fleming and Soner (1993). We provide the
definition of constrained viscosity solutions in Appendix A.
The following theorem provides a unique characterization of the value function.
Its proof is discused in Appendix B.
Theorem 2.1. The value function is the unique constrained viscosity solution on
+1), of the system of the variational inequalities
min
aeV (x;
and
min
aeV (x;
in the class of concave and increasing functions with respect to the spatial argument
Here L is the differential operator
and
As it was mentioned earlier, the presence of singular policies leads to a depletion
of the state space into regions of three types, namely the "Import to country Y "
regions. The choice
for the country Y to be used as the baseline for the description of the optimal trading
rules is arbitrary and does not change the nature of the results. The regions
are related to the optimal shipping rules as follows: if at time t the
production technology state (X t belongs to NS region, only the consumption
processes are used and no shipments take place from one country to the other.
If the state (X t belongs to the (I Y ) (respectively, EY ) region, it is beneficial
to the central planner to import (respectively, export) a shipment from country
X to country Y. In other words, a singular policy-which represents the "lump-
sum" shipment-is used to reduce (respectively, increase) the value of X t in order
to increase (respectively, decrease) the value of Y t and move to a new state, say
which belongs to the boundary of the (NS) and the
solutions exist up to date for the free boundaries of the afore-mentioned
regions. Therefore, it is highly desirable to analyze
these boundaries as well as other related quantities, numerically. This is the task
we undertake in the next section.
Remark 2.2: In the special case of a collective utility function of the CRRA type,
show that the value function
is homogeneous of degree fl. This fact provides valuable information about the free
boundaries which turn out to be straight lines passing through the origin.
We continue this section by presenting some results related to analytic bounds
of the value function as well as alternative characterizations of it in terms of a class
of "pseudo-collective" value functions. The latter results are expected to enhance
our intuition for the economic significance of the proposed pricing model. We only
present the main steps of the proofs of these results; the underlying idea is to use the
equations (2.7) and (2.8) and interpret them as HJB equations of new pseudo-
utility problems. The comparison between the new "pseudo-value functions" and
the original value function stems from the uniqueness result in Theorem 2.1 as well
as the fact that the pseudo-value functions are viscosity solutions of the associated
HJB equations.
To this end, consider the following pairs
x t and -
y t solve respectively
dy t
and
y
are the two states of the process z t and x
y. The
above dynamics correspond to the case of deterministic drifts with no effect from
the Markov chain.
We define for (2.11), (2.12) and (2.13), (2.14) the sets of admissible policies
A z1 and A z2 along the same lines as before.
The following result shows that the original value function V is bounded between
v and - v with v and - v being respectively the value functions of two international
asset pricing models with the original collective utility but with no political
risk. More precisely, v (respectively -
v) is the collective value function for countries
X and Y (respectively X and Y) with X (respectively X ) not exhibiting political
instability but with a modified mean rate of return in its capital stock. Models of
this type were studied by Dumas (1992) in the case of CRRA utilities.
Proposition 2.3. Consider the value functions v, -
Z +1e \Gammaaet U(C x
and
Z +1e \Gammaaet U( -
x
y
Then
for
We finish this section by discussing two collective-utility asset pricing problems
without political risk but with different discount factors and "enhanced" collective-
bequest functions. It turns out, as it is stated in Proposition 2.4 that their value
functions below by (2.19) and (2.20) - coincide with the original
value function for states z 1 and z 2 . To this end, we consider two collective-bequest
functions
and the discount factors
Also, we consider the controlled state processes
solving
(2.11) to (2.14) and the associated sets of admissible policies A z1 and A z2 .
Proposition 2.4. Define the value functions
by
with
y, and
with -
as in (2.18) and (2.17). Then
of the proof:. Observe that V (x; is the unique solution of the HJB
equation (2.7), which can be rewritten as
min
We easily get that the above equation can be interpreted as the HJB equation of
a new stochastic control problem with value function u 1 . The result then follows
from the uniqueness of (constrained) viscosity solutions of (2.7) and the fact that
u 1 is a (constrained) viscosity solution of (2.22). The same type of arguments yield
the result for the state z 2 .
Our ultimate goal, besides understanding the behavior of the optimal shipping
policies is to specify the equilibrium prices of goods X and Y. Note that we can
interpret the ratio of the partial derivatives of V (x; as the relative price of good
X in terms of good Y. Then, the equilibrium prices for the states z 1 and z 2 will be
respectively
If we further assume that U(C x ; C y then each of the value
functions of the two pseudo-problems (2.19) and (2.20) is homogeneous of degree
fl. For the state z 1 , there will be a cone with linear boundaries in the (x; y) plane
within which no shipping occurs. Similarly, there will be a "no-shipping cone" for
the state z 2 ; however, these cones will generally be different. One can observe
that for the "politically favorable" state z 2 the expected returns for production
in country X is relatively high as compared with the situation for the politically
"unfavorable" state z 1 . Consequently, a shift to z 1 makes production in country X
less attractive and may cause a costly shipment to take place between countries. In
other words, transitioning from z 2 to z 1 (or vice versa) can cause the no-shipping
cone to shift. Hence, jumps in the coefficients of the asset prices can occur when
z t switches values; this contrasts with the smoothing changing prices obtained in
Dumas (1992). On the other hand, as in Dumas (1992), the prices are expected to
deviate from a parity value of one for potentially substantial periods of time. Both
prices are bounded between the (1 which represent the prices
at the cone boundary where shipments take place. However, shifts in the cone as z t
transitions between z 1 and z 2 can result in cone rotation. These observations are
further developed in section 4 after we obtain the numerical results.
3. Numerical Schemes
This section is devoted to the construction of numerical schemes for the solution
of the Variational Inequalities (2.7) and (2.8). Besides computing the value
function V , we also compute the equilibrium prices P 1 (x;
the location of the free boundaries related to the optimal lump-sum shipments. Fi-
nally, we study how the presence of political risk influences the trading policies and
equilibrium prices by also examining the model in the absence of political uncertainty

The first goal in choosing the appropriate class of schemes is to find a scheme
with three key properties: consistency, monotonicity and stability. We define these
properties below; we use a generic notation for our equation in order to simplify the
presentation.
To this end, we consider a nonlinear equation F (w; u(w); Du(w); D 2
for respectively the gradient and the second
order derivative matrix of the solution u; F is continuous in all its arguments and
the equation is degenerate elliptic meaning that F (w;
Definition 3.1: We consider a sequence of approximations S
\Theta\Omega \Theta R \Theta
We say that S is:
monotone if
consistent if
lim sup
stable if
8' ? 0, there exists a solution u its (local)
bound is independent of '.
The motivation to use such schemes for our model comes from the fact that they
exhibit excellent convergence properties to the (viscosity) solution of fully nonlinear
degenerate elliptic partial differential equations as long as the latter have a unique
solution. This result was established by Barles and Souganidis (1991)   and it is
stated below for completeness.
Theorem 3.1 (Barles and Souganidis): Assume that the equation
admits the strong uniqueness property, i.e. if u (resp. v) is a
viscosity subsolution (resp. supersolution) of v. If the approximation
sequence fS ' g satisfies the monotonicity, consistency and stability properties
then the solution u ' of S('; w; u ' (w); locally uniformly to the
unique viscosity solution of F (w; u; Du;D 2
We continue with the description of our scheme. To this end, we first write
(2.7) and (2.8) in the concise form
The variational inequalities (2.7) and (2.8) belong to the class of equations
that Barles and Souganidis (1991) examined. Our problem though is not entirely
identical to theirs due to the presence of the state constraints (2.5). The convergence
of our scheme, in the presence of the state constraints is not presented here.
where for at (x;
with
The first step consists of approximating the equation in the whole space by an
equation set in a bounded domain proving the existence of
a solution VR of the Variational Inequalities in BR and the convergence of VR to V
as R tends to the infinity. As there is no natural condition satisfied at infinity by
we have to decide what kind of condition we impose on
@BR . Barles, Daher and Romano (1995) answered to this question and exhibited
an exponential rate of convergence for the heat equation complemented either with
Dirichlet or Neumann conditions. The generalization of their result to more general
parabolic equations is straightfoward (for more details, see Barles, Daher and
Romano (1995)). In the degenerate elliptic case, there is no natural choice for the
Dirichlet or Neumann boundary value.
We impose here a simple, arbitrary Neumann condition @VR
@n (x;
n is the outer unit vector and K is a preassigned positive constant. Note that this
condition must be taken in the viscosity sense and that the corners of BR require a
specific treatment.
The second step is the approximation to the solution of the equation set in the
above bounded domain. We denote by \Deltax and \Deltay, respectively, the mesh sizes
in the x and y directions. Moreover x \Deltay are the grid points and
i;j ) are the approximations for the value function V (x;
at the grid point We then propose an iterative algorithm to
i;j . For this purpose, we introduce a time step \Deltat and the
approximation for
i;j ) at step n will be denoted by V 1;n
If (V 1;n
i;j ) is known at step n, the monotone scheme which allows us to compute
at step n
may be ultimately written as
and
are consistent with (2.7),(2.8) as \Deltat; \Deltax; \Deltay converge to 0
and n\Deltat converges to +1, satisfy the monotonicity and stability property defined
in (3.1). Note that (\Deltat; \Deltax; \Deltay; n\Deltat) correspond to the variable ' in Definition
3.1, whereas
i;j in S 1 (resp. V 2;n+1
i;j in S 2 ) represents
(w). Finally, the role of the variable u ' is played here for S 1 by
We continue with the description of the scheme. First, we define the following
explicit approximation to the gradient operators
\Deltay
\Deltay
i;j stands for both V 1;n
i;j and V 2;n
i;j . It is easy to verify that these approximations
are monotone as long as \Deltat - min(\Deltax;\Deltay)
2+- .
For the elliptic operator L, we use a time-splitting method in order to approximate
separately the first-order derivatives in a first-half iteration and the second-order
ones in the second-half iteration.
For the first-half iteration, we consider the first-order operator ~
L obtained by
eliminating the second-order terms in L, i.e., for
~
The solution to the equation ae ~
~
can be characterized
(see for example Lions (1983)) as the value function of a stochastic control
problem
~
~
A
where the state trajectories ~
We apply the Dynamic Programming Principle to the above control problem
and discretize it, that is, for \Deltat positive and sufficiently small, we choose a constant
approximation to each consumption rate on the time interval [0; \Deltat].
We then compute the optimum in closed-form in order to obtain the following
numerical scheme for ~
which is monotone for \Deltat sufficiently small:
in the case of a CRRA utility with 0:5, is defined by:
\Deltax
and z 1 x i - \Deltax
\Deltax
and z 1 x
and z 1 x
and
and z 1 x i - \Deltax
Symmetrically h 2 is deduced from h 1 by replacing respectively \Deltax, z 1 x i ,V 1;n
and V 1;n
i+1;j by \Deltay; by
i;j+1 and the approximation V 2;n
i;j is obtained
similarly.
A simple sufficient condition for the monotonicity of the previous approximation
is provided by the following upper bound on the time-step
\Deltat - min
\Deltay
The second order degenerate elliptic term is then approximated by the well-known
Crank-Nicolson scheme. To simplify the presentation, we chose the following
approximation for the second-order derivatives which in fact is not monotone but
the replacement by a monotone approximation is routine and this modification does
not affect the convergence of the scheme. It is worth mentioning that this scheme is
unconditionally stable, independently of \Deltat, which may be chosen large. As before,
we use the notation V n
i;j for both V 1;n
i;j and V 2;n
i;j .
\Deltay 2
\Deltay 2
4\Deltax\Deltay
4\Deltax\Deltay
On the x-axis, we impose for
i;j the gradient condition in the following
\Deltat
\Deltax
\Deltay
Similarly, on the y-axis, we impose
\Deltay
At each iteration, we choose the following adaptative time-step which actually
is not far from being constant but may evolve a little during the convergence:
min
\Deltay
Given the approximations to the elliptic and the gradient operators, V 1;n
i;j is
then set to the maximal value over these three ones. Futhermore, we let the algorithm
converge until the conditions sup i;j jV 1;n
sup i;j jV 2;n
are reached, ffl being a preassigned small positive constant.
After the last iteration, we compute the equilibrium prices by using centered finite
differences and finally, the no-shipping region is defined as the set of the points
where the approximation to the value function at the last step comes from the
discretization of the elliptic operator.
We continue with a brief description of the numerical experiments: let
We chose the following parameters:

Figures

1-4 show the no-shipping regions and the equilibrium prices for the
states z 1 and z 2 in the absence of political uncertainty. More precisely, Figures
correspond to z
correspond to z
Then, in order to study the influence of the transition probabilities, in Figures
5-12 are represented the no-shipping regions for the states z
respectively in the following four cases
Case A:
Case B:
Case C:
Case D:
Finally, in Figures 13-16 we graph the equilibrium prices for the above four
cases.
The scheme does not behave in a perfectly stable way at least for the no-
shipping region. If one lets the scheme converge for a very long time, the cone
remains globally the same, except that a few points which oscillate around the
free boundaries, that is, they appear and disappear from iteration to iteration.
Apparently, this phenomenon might be caused by possibly over-estimated Neumann
conditions for large values of x and y.
4. Concluding remarks
In this paper, we have developed a model of international asset pricing in
the presence of political risk. Although the model is considerably simplified, it
represents a substantial step towards understanding how uncertainty about future
government actions can affect the prices of tradeable assets. The recent turmoil in
asset prices for several Southeast Asian countries serves to emphasize the importance
of gaining a better understanding of the effects of political risk on asset prices.
Our numerical experiments with the model provide several interesting results.
As in Dumas (1992), we obtain a cone in the state space within which no shipping
occurs. That is, individuals find it optimal not to incur the shipping costs entailed
with adjusting their relative holdings of the two assets whenever their asset positions
lie within this cone. In our model, the size and location of this cone depend on
the political state in the Country X (the politically risky country). Consider, for
example, the figures 5 and 6. In figure 5, Country X is in the poor political state with
relatively low expected returns on asset X. In figure 6, Country X has now switched
to the favorable political state. Asset X is now more valuable and individuals are
less inclined to export from Country X . They are also more inclined to pay the
shipping cost to import from Country Y and, in effect, convert some of their position
in asset Y into asset X. These two changes in their relative willingness to trade are
manifested in the downward (clockwise) rotation of the no-shipping cone between
figures 5 and 6.
We can also see that the transition probabilities influence the rotation and size
of the non-shipping cone. Compare, for example, the figures 3 and 5. For both
figures, Country X is in the poor political state. However, in figure 5 there is a 10%
probability of transitioning to the better state whereas in figure 3 that transition
probability is zero. Intuitively, the increased probability of moving to a better state
increases the value of asset X and alters individuals' willingness to trade. In this
case, the primary effect is a reduced willingness to export X which results in a
downward rotation in the lower boundary of the no-shipping cone.
We also provide graphical comparisons on the relative prices of goods X and
Y. Consider, for example, figure 13. In this figure, the quantity of X is fixed while
the quantity of Y is varied and the relative price of X (in terms of Y) is plotted in
each of the two political states. In state z 2 (the favorable political states) asset X is
relatively more valuable. However, it is interesting to note that for situations where
asset Y is either quite scarce or extremely plentiful, the political state seems to
have a negligible effect on relative asset pricing. Intuitively, when Y is very scarce,
its value becomes extremely high and the relative value of X becomes so small
that the effect of differing political states is not apparent. A symmetric argument
applies when Y is extremely plentiful. Comparing, for example, figures 13 and 14
we can see that the transition probabilities have a substantial effect on the extent
to which the political state alters the relative pricing of X and Y. In figure 16, the
transition probabilities have both become so great that the relative price difference
across political states all but disappears.
In conclusion, we view the implications of political risk for asset pricing as
both interesting and of considerable economic importance. This paper represents
a step towards better understanding some of those implications, and we hope that
the paper will stimulate further research on this important issue.



--R

Convergence of numerical schemes for problems arising in Finance theory

Bounds on prices of derivative securities in an intertemporal setting with proportional transaction costs and multiple securities
Bounds on prices of contingent claims in an intertemporal economy with proportional transaction costs and general preferences
User's guide to viscosity solutions of second order partial differential equations.
Viscosity solutions of Hamilton-Jacobi equa- tions
selection with transaction costs.
European option pricing with transaction costs.
American Options and Transaction Fees.
Dynamic Equilibrium and the Real Exchange Rate in a Spatially Separated World
Controlled Markov Processes and Viscosity Solutions.
Optimal replication of contingent claims under transaction costs.
Viscosity solutions of fully nonlinear second-order elliptic partial differential equations
Optimal control of diffusion processes and Hamilton- Jacobi-Bellman equations
On transaction costs and HJB equations.
Environmental models with irreversible decisions
Optimal investment and consumption with transaction costs.
Optimal control with state space constraints.
Numerical schemes for investment models with singular transactions.
selection with transaction costs

Investment consumption models with constraints
--TR
Optimal control with state-space constraint II
selection with transaction costs
Investment-consumption models with transaction fees and Markov-chain parameters
European option pricing with transaction costs
Consumption-Investment Models with Constraints
Numerical schemes for investment models with singular transactions
political risk;shipping costs;international asset pricing;gradient constraints;viscosity solutions;variational inequalities