A disutility-based drift control for exchange rates

In this article we propose an exchange rate model as a solution to the disutility-based drift control problem. Given that the exchange rate is a function of the fundamental, we assume that government authorities control the fundamental dynamics aimed at minimizing the discounted expected disutility caused by the distance between the fundamental and some specific target. The theoretical model is solved using the dynamic programming approach and introducing the concept of viscosity solution. We contribute to research on exchange rate control policies by deriving the optimal interventions aimed at stabilizing the exchange rate and preserving macroeconomic stability. We also show that, under particular conditions, it is possible to derive the optimal width of the currency band.


Introduction
Since the breakdown of the Bretton Woods system, the relevance of the exchange rate stabilization policies has been causing frequent and forceful interventions of the government authorities. The reactions to the Asian financial crisis or the European Monetary System (ERM) accession represent only some recent examples. The economies of East Asia have adopted a variety of foreign exchange rate policies, ranging from currency board system to 'independently floating' exchange rates. Most of the Asian economies have implemented 'managed floats' that allow their local currency to fluctuate over time within a limited range [17]. The recent enlargement of the European Union (EU) to 27 countries requires that the new Member States fulfil a period of managed floating regime (ERM II) before the adoption of the euro. In this context, together with monetary and fiscal challenges, exchange rate policies have become a key tool for the new EU members. Optimal exchange rate policies have to be set to manage the hardening against the euro. Interventions by government authorities are required to stabilize the exchange rate even before the participation in ERM II [7]. *Corresponding author. Email: rita.decclesia@uniroma1.it Another example is provided by the Chinese exchange rate system. In July 2005, the Chinese authorities announced that the Renmibi (RMB) would be managed 'with reference to a basket of currencies' rather than being pegged to the dollar. According to the Public Announcement of the People's Bank of China (PBOC) [16] on reforming the RMB Exchange Rate Regime, the Chinese Authorities 'make adjustment of the RMB exchange rate band when necessary according to market developments as well as the economic and financial situation' and maintain 'the RMB exchange rate basically stable at an adaptive and equilibrium level, so as to promote the basic equilibrium of the balance of payments and safeguard macroeconomic and financial stability' (http://www.pbc.gov.cn). Although the RMB exchange rate adjustments were initially too cautious, the announcement made possible transitional arrangements like those applied in other emerging countries showing the PBOC's awareness of the unsustainability of the US Dollar pegging. The managed floating exchange rate system, together with a more independent monetary policy, might help the Chinese economy to cope better with both the internal and external macroeconomic shocks to which a developing country may be exposed [10].
Exchange rate stabilization policies represent a crucial issue, and they have been largely analysed in the literature. Krugman [13] emphasized the role of official interventions at the margin of a currency band, when the fundamentals driving the exchange rate follow a random walk with constant variance. Most empirical results are controversial, leaving many questions unanswered, such as the issues of the optimal monetary policy and the optimal width of the currency band (if adopted). Improvements of Krugman's framework are obtained thank to the extensions of the basic model (amongst others [5,11,12,14,15,20]. Jeanblanc-Picque´ [12] applies impulse control methods to show that using a diffusion process with constant coefficients, it is possible to keep the exchange rate in a given target zone with discrete interventions. Miller et al. [14] find a subgame-perfect solution for a central bank aiming at stabilizing the exchange rate in a target zone, given proportional costs of intervention. Mundaca and Oksendal [15], using a jump diffusion process for the exchange rate dynamics, combine continuous and impulse controls to stabilize the exchange rate. Im [11] presents the central bank optimal intervention strategies to find the policy which minimizes the value of the loss function. Assuming that the economy randomly switches between different regimes, with time-invariant transition probabilities, Zampolli [20] examines the trade-offs deriving from deviations of the exchange rate from fundamentals and from extreme changes. Castellano and D'Ecclesia [5] solve a stochastic optimal control model to describe exchange rate dynamics in a managed floating regime assuming government authorities aim to keep the aggregate fundamental not too far from a predetermined target and within an optimal currency band. In this article, optimal exchange rate stabilization policies are taken into account. We assume that the exchange rate is a function of the aggregate fundamental whose dynamics is described by a stochastic differential equation (SDE) with a general functional shape for the state-dependent drift and variance. The drift of the fundamental is the control variable to maintain the fundamental level as close as possible to a time-varying target. We introduce a disutility function that depends: (1) on the difference between the aggregate fundamental and its target dynamics and (2) on the control variable. The implicit costs associated with the interventions are measured in terms of disutility. The stochastic control problem is solved using the dynamic programming approach. The optimal strategies are obtained in two steps: first, deriving the unique solution of the Hamilton-Jacobi-Bellman (HJB) in the viscosity sense [2]; second, formalizing the existence of the optimal strategies and their related paths according to the regularity properties of the value function. The optimal trajectory of the exchange rate is fully characterized. We also show that, under particular conditions, the optimal width of the currency band can be determined.
The main innovations of this article are represented by: (1) the choice of a general-shaped function for the stochastic dynamics of the aggregate fundamental; (2) the introduction of a disutility function which measures the implicit costs of the intervention and (3) the definition of the endogenous currency band.
This work is organized as follows. The next section describes the model and the related optimal control problem; Section 3 presents the properties of the value function and the optimal strategies; in Section 4 a particular case is discussed; some concluding remarks are presented in Section 5 and the mathematical derivations are reported in the Appendix.

The model
This section describes the model developed to study the interventions of government authorities in a managed floating regime. The building blocks of the model are given by the exchange rate dynamics depending on some random fundamental, the presence of a time-dependent target and the optimization problem.

The exchange rate dynamics
We assume that the exchange rate depends on both some current fundamentals and expectations of future values of the exchange rate. The (log) of the spot exchange rate at any time t, s t , is assumed to depend on an aggregate 'fundamental', f t , and a speculative term proportional to the expected change in the exchange rate. As stated in Svensson [18], the fundamental absorbs the driving forces of the exchange rate (i.e. monetary and fiscal policy variables, domestic output, price level, foreign interest rate, etc.).
Given a filtered probability space (, F , {F t } t!0 , P), a simple representation of the spot exchange rate dynamics is given by where . s t is the logarithm of the exchange rate defined as unit of domestic currency per unit of the reference currency; . f t denotes the logarithm of the aggregate fundamental; . is a constant positive parameter which can be interpreted as the semielasticity of the exchange rate with respect to the instantaneous rate of currency depreciation; . E t [ds t ] measures the expected change of the exchange rate with respect to time t.
The process for the fundamental, f t , is given by where . t 2 Â, represents the control variable, used by monetary authorities to manage the current fundamental dynamics, Â is the admissible region defined as ð3Þ . B t is a standard Brownian Motion.
We assume that the initial value of the fundamental, f 0 , is deterministic. The effective aggregate fundamental, f t , consists of exogenous and endogenous components. We further assume that government authorities, using monetary, economic and fiscal policies, monitor the exchange rate and may intervene in order to maintain the fundamental, f t , broadly in line with its target. In particular, Equation (2) states that government authorities intervene on the control variable, t , to manage the drift of the fundamental, f ( f t , t ).
We set a target for the fundamental,f t , which includes a set of variables affecting the exchange rate. For instance, some of the parameters set by the European Commission during the process of EU accession have to show some specific behaviour, or some macroeconomic variables have to perform according to given targets. In the case of China, the PBOC officially sets targets for money supply [4] and credit growth 'to maintain stability of the value of the currency and thereby promote economic growth' (http://www.pbc.gov.cn).
The evolution of the potential target is described by an ordinary differential equation: where is defined on [0, þ1) andf 0 is the deterministic initial value off t . We define the state-variable x t :¼ f t Àf t whose dynamics, given in (2) and (4), on the filtered probability space (, F , {F t } t!0 , P), is given by where . is a continuous real value bounded function with respect to the process ; . x 0 ¼ x, is the deterministic starting point of the dynamics x t . and satisfy the usual regularity conditions for the existence and uniqueness of the solution for (5).

The optimization problem
The decision maker aims to reduce its disutility intervening on its preferences as well as on the fundamental through the control variable . The expected disutility allows an assessment of the government authorities' policies and the total 'social' costs of the stabilization process.
The disutility function depends on the distance of the fundamental value from its target, and it is controlled by t . The larger the distance between the fundamental and the target, the lower the satisfaction and the higher the disutility. The control problem is solved finding the optimal control rule t , as a function of the state variable x t that minimizes the expected discounted disutility and the implicit costs of the control policies. We formalize the dynamic optimization problem in terms of the value function, V : R ! R, presented as with Jð, xÞ : where . e Àt , with > 0, is the discount factor; . u : [0, þ1) Â [ m , M ] ! R is the government authority's disutility function; . jx t j ¼ j f t À e f t j is the distance of the fundamental from its target; . E x is the expected value of the disutility, u, depending on the absolute value of x t , whose dynamics are given by (5), with initial position x.
Since V is symmetric with respect to the origin we do not lose any generality assuming x ! 0.
As stated above, u is increasing with respect to jx t j and continuous with respect to t . With no loss of generality, we assume that the disutility function is essentially bounded with respect to x and this implies, together with the continuity of u with respect to in [ m , M ], that also V is essentially bounded with respect to x.
To clarify the concept of disutility function, it may be useful to list some specific functions: . uðjx t j, t Þ ¼ 2 t Á e jx t j ; the disutility grows rapidly as the distance of the fundamental from its target, jx t j, increases; government authorities should intervene as soon as possible to avoid an explosion of the disutility; the control variable t should be pushed downward to compensate the growth of e jx t j .
. uðjx t j, t Þ ¼ jx t jexp½ 1 t ; the disutility grows linearly as the gap jx t j increases at a rate depending on t ; government authorities should intervene to reduce the growth of the disutility, by pushing upward the control variable t .
The introduction of the disutility in the objective functional and its dependence on the control variable guarantees that the optimal solution is also the one minimizing the costs of the controls. The smaller the distance between the observed fundamental and the target, the smaller the cost of the control measured in terms of disutility.

The optimal policies
In this section, following the dynamic programming approach, we study the properties of the value function and derive the implied government authority's optimal strategies. THEOREM 1 The value function V is the unique classical solution of the HJB equation The proof is reported in the Appendix. By Theorem 1, the optimal strategies in feedback form can be obtained.
(1) The closed-loop equation admits a unique solution.
(2) Assuming that " x t is the solution of the closed-loop equation, we obtain t depending on " x t ; so we set " t :¼ Ã ð " x t Þ and obtain d " Þ ¼ VðxÞ holds, " t is the optimal value for the control variable, with the related optimal trajectory, " x t .
The proof is reported in the Appendix.
The existence of the optimal strategy " x t implies the existence of an optimal trajectory for the fundamental, f Ã t :¼ " x t þf t , given the relationship between x t and f t . Starting from the optimal fundamental, a characterization of the optimal exchange rate dynamics, s Ã t , can be provided. Next result allows to define the optimal exchange rate dynamics. PROPOSITION 3 Given the optimal fundamental f Ã t , then the optimal exchange rate dynamics can be written as s Ã t ¼ hð f Ã t Þ, where the function h is the solution of the following second-order differential equation: Proof We look for solution of (1) introducing a function h: Applying Ito's lemma to (11), we have The conditional expectation of ds Ã t is given by Therefore, Equation (1) can be rewritten as Given (11) and (13), h can be found as a solution of (10). g

Some applications: derivation of the optimal currency band
To provide an explicit formalization of the optimal values for the control variable, , as given in Theorem 1, in this section some particular cases are discussed. For our purpose, we remove the assumption that x > 0 and consider x 2 R. First of all, we argue that and u are assumed to exhibit the same behaviour w.r.t. in [ m , M ], and this means that if jxj increases, then u and increase. Therefore, the intervention of the government authorities through the control should push downward simultaneously and u. In this particular example we assume the existence of A, According to Theorem 1, the optimization problem is solved by minimizing the function g x w.r.t. . By assuming the right regularity for the functions and u and applying the first-order condition we get Under particular conditions, we are able to derive some intervention bands for x, i.e. the regions where the optimal control rule is invariant. In particular, it is easy to choose and u such that two thresholds x 1 , A and C represent two intervention bands for x. The optimal strategies can be written as follows: When x 2 [x 1 , x 2 ], Ã (x) can freely fluctuate. If x 2 [x 1 , x 2 ], then f Ã 0 2 ½x 1 þf 0 , x 2 þf 0 . Furthermore, the behaviour of the optimal exchange rate dynamics is fully described by the function h in (11), which depends on f , f , . Thus, when h is strictly monotonic, for instance increasing, then f Ã 0 2 ½x 1 þf 0 , x 2 þf 0 implies that s Ã 0 2 ½hðx 1 þf 0 Þ, hðx 2 þf 0 Þ. The interval ½hðx 1 þf 0 Þ, hðx 2 þf 0 Þ represents the optimal currency band for the exchange rate dynamics, where no interventions occur.
To provide an intuitive understanding of the optimal strategies, we introduce the functions: u 1 , 1 , : R ! R and u 2 , 2 : [ m , M ] ! R, such that the drift and the disutility functions can be defined, respectively, as where (18) and (19) satisfy the regularity conditions given in Section 2 and 1 is increasing in [0, þ1) and decreasing in (À1, 0). In (18), the government authorities may apply a control , through 2 , in order to let (x, ) be close to (0, ) and drive the process of the fundamental, f t , closer to its target,f t . Equation (19) provides a general example of the specific disutility functions introduced in Section 2.2.
Given (18) and (19), the map g x becomes g x ðÞ ¼ u 1 ðjxjÞu 2 ðÞ þ 1 ðxÞ 2 ðÞV 0 ðxÞ þ ðjxjÞ, ð20Þ and applying the first-order condition we get from which, by assuming u 1 (jxj) 6 ¼ 0, then the function ðÞ ¼ u 0 2 ðÞ 0 2 ðÞ is invertible and the optimal control Ã (x) is given by It is possible to consider the limit case of two dominant optimal policies: one expansionary and the other restrictive (i.e. optimal policies of bang-bang type). In this case, the optimal currency band collapses to a single value that is necessarily 0. Now, assume 2 () ¼ u . if then (26) is satisfied for each 2 [ m , M ] and the value x represents a specific distance between the fundamental and its target for which government authorities may apply arbitrary decision rules. It is natural that it must be x ¼ 0: here, the target of the government authority is reached and no intervention is needed; then (26) cannot be satisfied assuming an increasing (decreasing) n, i.e. when n 0 () > 0(<0) for 2 [ m , M ]. However, the continuity of g x and Weierstrass' Theorem guarantee the existence of the optimal strategies, belonging to { m , M }. More precisely, a critical region À R can be defined as follows: We have Since u is an increasing function of jxj, then by (6) and (7), V increases in [0, þ1) and decreases in (À1, 0). As a consequence, further assumptions on u 1 and 1 allow to derive some intervention bands for x. As a particular example, we have that if 1 (x) Á u 1 (jxj) > 0, for each x 2 [0, þ1) and 1 (x) Á u 1 (jxj) < 0, for each x 2 (À1, 0), then V 0 þ u 1 1 4 0 in [0, þ1) and V 0 þ u 1 1 5 0 in (À1, 0). Hence, À ¼ R. By (30) and (31), when À ¼ ; or À ¼ R, then there exists a unique optimal strategy Ã 2 { m , M } that the government authority can apply. In particular: . for À ¼ ; and n increasing (decreasing), When À ¼ (À1, x), then À and (x, þ1) represent two optimal intervention bands for x.
In this particular case, the optimal currency band for the exchange rates collapses to a singleton. Given the optimal fundamental path f Ã t :¼ " By definition of the function h in (11), we have that s Ã 0 ¼ hðx þf 0 Þ is the threshold for the exchange rate where no intervention is applied by the government authority. The set fhðx þf 0 Þg is the degenerate currency band.

Interpretation of the results
The optimal control is defined as a function of the gap registered between the theoretical and the observed fundamental, (24).
To fix ideas, assume x > 0 (the case x < 0 is analogous). Given (23), À1 is an increasing function of its argument and the relationship between the optimal control, Ã , and the variable x can be derived. Ã (x) is directly related to the disutility function, u 1 (jxj), and inversely related to the drift 1 (x) of the state variable, x t , and to the growth rate of the expected disutility function V 0 (x).
In other words, if the government authority's disutility is large, need for strong interventions occur and a large value of Ã is chosen; on the other hand, if the deterministic trend of x t or the change in the disutility is large, then there may be a need for a small intervention of the government authority.
For instance, considering the current international context, one can observe that some countries are experiencing a much lower growth rate than expected, and this causes a large value of the state variable, i.e. the distance between theoretical and observed fundamental, and high level of disutility. According to our model, this would require incisive interventions represented by large value of , i.e strong measures of fiscal and monetary policy.
The optimal control, Ã , described by (24) is applied whenever 2 ( m , M ). This means that theoretical and observed fundamentals may differ from one another, but the difference is still within the optimal tolerance band: When x becomes too large (x > x 2 ), or too small (x < x 1 ), the optimal strategies are given by (16) and (17) -or (30) and (31) -which may represent the extreme interventions that government authority have to choose in order to bring the fundamental closer to its target value.
In the case of the current economic situation, the observed fundamental may result very far from the theoretical one, therefore strong interventions have to be adopted, meaning large values of have to be chosen. For instance, in terms of monetary policies this may be translated in monetary control exercised via manipulation of the monetary base or control of domestic credit or bank credit or net foreign assets. The monetary policies to be adopted are defined according to the each country's internal targets such as inflation and unemployment, and external targets such as competitiveness, the current accounts and reserves.

Conclusions
This article presents a disutility-based drift control model for exchange rate dynamics, in the framework of managed floating regimes. The dynamics of the exchange rate is described as a function of the aggregate fundamental at time t, f t , which follows a Brownian Motion with state-dependent drift and volatility. The process for the fundamental dynamics are obtained as the solution of a stochastic control problem describing the government authorities' aim to keep the value of the fundamental as close as possible to its target. An expected disutility function minimization problem is developed, and the related HJB equation is solved in viscosity sense.
We show that under particular conditions, it is possible to obtain the optimal width of the currency band. The model is realistic since it suggests a more adequate process to describe the exchange rate dynamics and provides an accurate analysis of the observed phenomenon with respect to simple diffusion processes which may lack in economic content. The model takes into account the time-varying features of the dynamics of the exchange rates and the optimal strategies that can be applied by government authorities to stabilize the exchange rate within a band. THEOREM 5 The value function V is continuous in (0, þ1) and can be extended continuously on [0, þ1). Moreover, V is the unique viscosity solution of the HJB equation (8) with the boundary condition V(0) ¼ 0 and V(þ1) ¼ M.
Proof The proof is a direct consequence of a result in [3]. g We now need to discuss the regularity properties of the value function to prove Theorem 1. In fact, if V is at least twice differentiable, then Theorem 5 guarantees that it is the unique classical solution of (8) with boundary condition V(0) ¼ 0 and V(þ1) ¼ M.
The previous result implies the following corollary.
In the following lemma we recall an important general result due to Alvarez et al. [1]. This result is useful to prove concavity. Assume thatH satisfies the following properties: . there holdsH .
Let v lower semi-continuous in " I be a viscosity supersolution of (34) and define the convex envelope v ÃÃ of v as v ÃÃ ðxÞ :¼ inf Then v ÃÃ is lower semi-continuous in " I and it is a viscosity supersolution of (34).
Proof In order to prove the theorem, it is sufficient to prove that u :¼ ÀV is a convex function. We use Corollary 7 and apply it to Equation (33 is concave for every p. Furthermore, since 6 ¼ 0, for each x 2 [0, þ1),H is an elliptic operator. Since the hypotheses of Lemma 8 hold, the convex envelope v ÃÃ of v is a viscosity supersolution of (35).
Given this result and (36), the convex envelope v ÃÃ of v is a viscosity subsolution of (35). By Theorem 5 and Corollary 7, v ÃÃ is the unique viscosity solution of (35) and, hence, the unique viscosity solution of (33). Therefore v ¼ ÀV is convex in [0, þ1) and the theorem is completely proved. g Next result guarantees that the viscosity solution of the HJB equation is a classical solution.
The result is proved. g Proof of Theorem 1 By Theorems 5 and 10, we have the thesis. g