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In this paper, we propose a model of exchange rate target zone based on a specification of the economic fundamentals known as a Geometric Brownian Motion. The rationale behind this specification is that the fundamentals series is not necessarily normally distributed as commonly assumed, as indicated by its excess kurtosis and ARCH properties. Therefore, assuming a normal specification can be problematic. The main difficulty is that with such a specification finding a closed form solution for the model becomes somehow more involved. We present some results in which the exchange rate formula is explicitly derived. Then we look at several types of central bank interventions in the foreign exchange market such as Krugman’s marginal interventions, central bank interventions a la Caballero and central bank interventions a la Flood-Garber. In addition, we present some empirical investigations where it is found that, for the most part, these exchange rate models do not fit the data well and a case where the model performs satisfactorily. We believe that the sources of the problem may reside in the complexity of estimating the models efficiently given that the theoretical approach is quite sound.

A target zone model is an exchange rate model where the monetary authorities are committed to keeping the exchange rate within some specific bands commonly known as a target zone [

Equivalently, we can write

Here, we propose a model where the fundamentals follow a geometric Brownian Motion (GBM). We will also consider the predictions of the model for target zone modeling under Krugman interventions as well as under discrete interventions by the monetary authorities. First, we present some background about the behavior of the fundamentals assumed to drive exchange rate movements.

The model assumes that the fundamentals follow a Geometric Brownian Motion (GBM)

where W(t) is a standard brownian motion or Wiener process. Here,the drift parameter,

Now, we want to investigate the distribution of the GBM process. The equation above is a diffusion stochastic differential equation. If we can solve the SDE, probably that can shed some light on the distribution of the process. The fact is that this class of linear stochastic differential equations can be solved analytically. For completeness, we will present the details of the solution method. Also, we will calculate some moments. First, to find the mean, we rewrite the SDE in an equivalent integral form

The first integral is the integral of a random function with respect to a standard measure, the Riemann measure, and the second integral is an Ito stochastic integral. We have

The last equation results from the Ito isometry theorem. Taking expectations, we ob- tain

Setting

This equation is an integral equation of the first kind which occurs commonly in the theory of integral equations and can be easily solved. Differentiate both sides to obtain

The solution to this linear differential equation is given by

Now, we look at the second moment, the variance of the process. To do that, we first try to compute

Then

This implies that

Let

Therefore, we obtain the following ODE

The solution is given by

The variance is given as

Now, we would like to determine the distribution of the process f(t). The interesting property of such a formulation is that we can solve explicitly the SDE as it is known in the literature. The SDE

is a linear SDE with nonconstant coefficients,we can solve the SDE using the integrating factor technique. But, using this method leads to stochastic integral equation which is more difficult to solve than the SDE itself. Alternatively, we use the common solution technique, though a general method of solving linear SDEs. Let

That is

This solution does not tell us much about the distribution of the stochastic process governing the fundamentals. But, we can find out about the nature of the distribution by observing the following.

Rewrite (18) as

It is clear that

It follows that

That is,

tion function is very important in applied work. We can analyze its kurtosis and skewness by using a very powerful statistical technique, namely, the moment generating function technique without having to rely on some relatively tedious calculations. We briefly define the lognormal distribution and show how the moment generating function technique can be applied to calculate some of the moments.

Suppose that the random variable X has a lognormal distribution with μ and variance σ^{2}. Now, we apply the moment generating function technique to find the first and second moment. Define the random variable

Similarly, we obtain

Therefore, the variance of X is given by

We see that we could have used this result on lognormal probability distribution functions to calculate the first moments of the f(t) process. A final remark to be made about the stochastic process governing the fundamentals f(t) is that the SDE that defines the process has two unknown parameters,

In this section, we pay a special attention to the task of estimating the parameters

tion we showed that

are modeled as a geometric Brownian motion. More specifically,

In this specific case the parameters can be estimated by exact maximum likelihood. For the sample

However, for more general cases, in particular when the geometric Brownian is regulated, the exact likelihood function is unknown and the researcher has to rely on approximation methods. A large body of literature has been devoted to parameter estimation of SDEs such as Discrete Maximum Likelihood (DML) methods, Hermite Polynomial Approximations, Infinitesimal Operator methods advocated by [

For an SDE of the form

where

The transitional PDF satisfies the Fokker-Plank equation with initial and boundary conditions

where S is the state space,

where

Other methods use a Milstein approximation algorithm which, in most cases, give superior estimators compared to the Euler-Maruyama scheme. Our model specification for the fundamentals process states that

Letting

where

Now, take a sample

where we define

The log-likelihood function is given by

The first order conditions entail

As indicated above, in our formulation, no approximation is necessary for estimation purposes given that a closed form solution for the likelihood function is available. We present the exact maximum likelihood estimation results as well as the pseudo maximum likelihood results below where we use velocity as economic fundamentals, as in [

The estimates are presented below (

It is clear that the model is consistent with the data.

The target zone model considered here is driven by the behavior of the process governing the fundamentals process. We argued previously that a formulation of the series governing the fundamental determinant of exchange rate, f(t), is supposed to be guided by empirical considerations on the behavior of the series f(t) and therefore, models, whatever variables being considered as economic fundamentals, should not be agnostic about such behavior. In effect, the fundamental series f(t), as we have previously highlighted, depends on the model of exchange rate determination being considered. Again, the emphasis is put on the classical monetary model of exchange rate determination as in [

As before, we assume that the dynamic behavior of the fundamentals is governed by the stochastic differential equation

where

or equivalently,

where

The model’s parameters are

Proposition 1. Suppose that the fundamentals, f(t), satisfy the stochastic differential equation

Exact MLE | Pseudo MLE | |||||
---|---|---|---|---|---|---|

estimates | ste | p-value | estimates | ste | p-value | |

m | 0.010 | 0.004 | 0.021 | 0.010 | 0.005 | 0.038 |

s | 0.033 | 0.002 | 0.000 | 0.034 | 0.002 | 0.000 |

Log-likelihood | −183.333 | 0.000 | 0.000 | −177.237 | 0.000 | 0.000 |

Also, assume that the exchange rate is determined by the equation

Then a family of solutions for the exchange rate is given by

where

where A and B are arbitrary constants.

Proof. The fundamentals are assumed to satisfy the SDE or GBM

Guess that the exchange rate function is a time invariant function of the current fundamentals, that is, s(t) has the strong Markov property. Thus, we can set

Then we obtain

This leads to the following second order differential equation given by

This is a functional equation of the form

This belongs to a well known class of ordinary differential equations known as the Cauchy-Euler or Equidimensional equation. Closed form solutions for such a class of differential equations exist. There are several methods for solving this class of ODEs. One such methods is he to use the theory of Laplace transforms. But, finding the inverse Laplace transform may be not that simple. The easiest solution method is to set

As usual, we solve the second order homogeneous ODE

The characteristic equation is given by

Given the definition of a, b, c, we have

Therefore, the general solution to the homogeneous equation is given by

To find a particular solution, suppose it is given by

Assuming that

Finally, the general solution is given by

Moreover, since

Since

where

The next task is to definitize the coefficients A and B. To do that, we consider different types of central bank interventions. As before, we consider a target zone

As we have previously mentioned, Krugman infinitesimal marginal interventions assume that the monetary authorities intervene in the foreign exchange market so as to prevent the exchange rate from ever leaving the target zone band. At the time of interventions,

Corollary 2. Under Krugman interventions and with a geometric brownian motion for the fundamentals:

where A and B are given by

Furthermore, assuming symmetry, in the sense that

Proof. We have

“Smooth-pasting” conditions imply

We obtain

For the symmetric case, we target zone becomes

A similar expression is obtained for B

Also, in the symmetric case, we have

It is widely accepted in the literature that discrete interventions are more realistic than infinitesimal marginal interventions à la Krugman due to the fact that the central bank starts with a fixed amount of reserves which will eventually be exhausted [

where

The monetary authorities intervene in the foreign exchange market whenever f hits either the upper or lower bound of the target zone by placing the fundamental back in the middle of the band.

When the fundamentals hit the lower band

At time

ties will reset f at the middle of the band, that is, at

Hence,

UIP implies no jumps and therefore

That is,

Similarly, at the instant

we see that

At the next instant

That is,

Equivalently, we can write

We obtain the following system of linear equations

To solve the system, we set

The solution to the system is therefore obtained as

The complete solution to the model under Flood-Garber interventions can be summarized in the following proposition

Proposition 3. Under Flood-Garber interventions, the solution to the target zone model is given by

where

To complete the discussion under this type of interventions, we consider the symmetric case. This has only a theoretical value given that for our model, the fundamentals are either positive or negative.

Suppose that

We present below a possible graph (

This type of intervention is justified by the fact that the monetary authorities start with a fixed amount of reserves which will eventually be exhausted and hence the target zone cannot be completely credible. Therefore, the assumption of perfect credibility is relaxed here. When the exchange rate hits either band limits, the monetary authorities either realign or mount a defense of the zone. Let p be the probability of realignment and 1 − p the probability that the monetary authorities mount a target zone defense. Suppose further that the fundamentals hit the upper bound,

At the next instant

1) With realignment (with probability p), we have a new band

2) If a defense is mounted, then the monetary authorities place the fundamentals back to the middle of the band:

Uncovered interest parity implies

But, it is seen that

As before, UIP implies

where

Now we consider the case where f hits the lower bound at time

If realignment occurs, then at time

If a defense is mounted, then we obtain

This implies that

As before, UIP implies that exchange rate does not jump so that

We can rearrange this equation to obtain

where

Finally, we consider the symmetric case. Suppose that

These results are summarized in the following proposition.

Proposition 4. Under Flood-Garber interventions, the solution to the target zone model is given by

where

and as given above

and

Parameter | estimate | t-statistic | J-Statistic |
---|---|---|---|

0.0728 | 2.2338 | ||

0.3257 | 4.8284 | ||

6.253 | 3.7351 | ||

0.2594 | 2.1443 | ||

5.2138 | 2.8573 | ||

2.9744 |

In this section, we present some estimates of the model’s parameters for the Krugman’s and Flood-Garber’s types of interventions. We use Daily exchange rate data for Japan and Sweden from 1987 to 1990. Though the data are relatively old, we use it to illustrate the findings that the model gives some satisfactory results contrary the basic target zones models in the literature. The estimation technique being used is the Simulated Method of Moments (SMM) procedure [

As indicated by the table, the estimates are quite reasonable in magnitude and, as expected, have the correct signs. Moreover, the J-Statistic of the one overidentifying restriction is not rejected and therefore the model does a good job in fitting the data, at the most commonly used significance levels.

We also found that the model does not perform well in the case for the Krugman marginal interventions. The estimates are reasonable in signs and magnitudes. However, the high value of the J-Statistics indicates that the model is not supported by the data. In essence, this can be due to the fact that it is commonly not easy to estimate these models efficiently.

In this paper, we have derived analytical or closed form expressions for the exchange rate function under the assumption that the fundamentals follow a geometric Brownian motion within the target zone band. Preliminary estimates from Simulated Method of Moments (SMM) show that the data do not show great support for the Krugman infinitesimal marginal interventions in the foreign exchange market, as indicated by the high value of the J-statistic. However, we found strong evidence for the target zone model for Flood-Garber interventions for the case of Japan, as indicated by the low value of the J-statistic. This indicates that it is likely that Japan has used an unofficial exchange rate target zone band during the 1987-1990 period. This is not a surprise due to the fact that central banks do not often reveal their intervention strategies. No such evidence has been found for the case of Germany.

Cupidon, J.R. and Hyppolite, J. (2016) A Target Zone Model Where the Fundamentals Follow a Geometric Brownian Motion. Journal of Mathematical Finance, 6, 866-886. http://dx.doi.org/10.4236/jmf.2016.65058