Can one hear the shape of a target zone?
Jean-Louis Arcand, Max-Olivier Hongler, Shekhar Hari Kumar, Daniele Rinaldo
CCan one hear the shape of a target zone?
Jean-Louis Arcand Shekhar Hari Kumar Max-Olivier Hongler Daniele Rinaldo ∗ The Graduate Institute of International and Development Studies, Geneva École Polytechnique Fédérale, Lausanne Faculty of Economics, University of Cambridge
Abstract
We develop an exchange rate target zone model with finite exit time and non-Gaussiantails. We show how the tails are a consequence of time-varying investor risk aversion, whichgenerates mean-preserving spreads in the fundamental distribution. We solve explicitly forstationary and non-stationary exchange rate paths, and show how both depend continuouslyon the distance to the exit time and the target zone bands. This enables us to show howcentral bank intervention is endogenous to both the distance of the fundamental to the bandand the underlying risk. We discuss how the feasibility of the target zone is shaped by theset horizon and the degree of underlying risk, and we determine a minimum time at whichthe required parity can be reached. We prove that increases in risk after a certain thresholdcan yield endogenous regime shifts where the “honeymoon effects” vanish and the targetzone cannot be feasibly maintained. None of these results can be obtained by means ofthe standard Gaussian or affine models. Numerical simulations allow us to recover all theexchange rate densities established in the target zone literature.
JEL Classification : F31, F33, F45, C63
Keywords : Target zones; Non-stationary exchange rate dynamics; Dynamic mean-preserving spreads;Target zone feasibility; Spectral gap; Endogenous regime shifts.
The exchange rate target zone literature pioneered by Krugman (1991) is based on a stochasticflexible price monetary model in continuous time. This literature highlights the role of marketexpectations concerning fundamentals in shaping exchange rate movements. Given its assump-tions of perfect credibility, it implies that central bankers need only intervene marginally at thebounds of the target zone or allow honeymoon effects to automatically stabilize the exchangerate. The European Monetary System (EMS) and the Exchange Rate Mechanism (ERM), whichexisted from 1979 to 1999 (until participating countries adopted the Euro), provided a naturaltest bed for this theory. ∗ Corresponding author. [email protected]
The authors thank Didier Sornette, Ugo Panizza and Cédric Tille for their precious insights. a r X i v : . [ ec on . GN ] N ov he target zone model is both well accepted theoretically and has provided the intellectual jus-tification for a nominal anchor for monetary policy. However, there is scant empirical supportfor the validity of the framework. The U-shaped distribution within the target band and thenegative correlation between the exchange rate and the interest rate differential implied by theKrugman model have found little counterpart in the data. In spite of this, the practice of usingtarget zones continued through the 2000’s with new member states joining the ERM-II targetband and slowly adopting the Euro. It is conceivable that future new member states will gothrough the ERM process, making target zone modeling of current relevance. Our purpose inthis paper is to unpack target zone feasibility, while incorporating non-stationary dynamics anda rigorous measure of risk that captures the presence of non-Gaussian tails.We make three main contributions. First, we show how such tails can emerge in the the funda-mental dynamics of the exchange rate as a consequence of time-varying investor risk aversion.In our setting, the risk aversion of agents is subject to risk-on and risk-off shocks generating atime-varying coefficient of risk-aversion. These dynamics alter the idiosyncratic country-risk pre-mium of small open economy, which destabilises the fundamental process via a sudden bonanzaor sudden stop of capital flows during the target zone process. These tails are fully described bymeans of a definition of risk which corresponds to the dynamic equivalent of a mean-preservingspread. As is well known, risk and variance are not necessarily equivalent: variance is oftenused as a proxy for risk, but by construction it cannot capture tail risk in a random variablegenerated by a potentially non-Gaussian process.Second, we explicitly consider non-stationary dynamics for a currency to exit a target zone, andshow how the feasibility of the latter is shaped both by the finite time horizon and the degreeof underlying risk. Solving for explicitly time-dependent dynamics also allows us to show howthe exchange rate is continuously determined by the distance to the time horizon as well as itsdistance to the target bands. It turns out that the underlying dynamics are similar to the phe-nomena famously described by Kac (1966), where he asked whether one could “hear the shape ofa drum.”
In the case of exchange rates, in certain situations this can indeed happen, especiallywhen the exchange rate is pushed to the sides of the target band by an additional external force:intuitively, this corresponds to the acoustic difference between striking a tense membrane (largeshifts in risk aversion) versus a loose one. This is what we describe in our paper. This allowsus to show how the central bank determines its intervention strategies by the degree to which it“feels” the presence of the target zone bounds, and depends critically on the degree of underlyingrisk and the band size.Third, we show how large shocks to the investor risk aversion, leading to proportional increasesin risk in the fundamental distribution, can potentially yield a regime shift once a certain riskthreshold is crossed. This shift does not allow for honeymoon effects to happen anymore aroundthe target zone bands, since the increase in risk destabilizes the exchange rate dynamics to thepoint that the target band is hit with excessive force, smooth-fitting procedures cannot be ap-plied by central bank interventions and the target zone becomes untenable.The standard case of exchange rate dynamics in a finite target zone with Gaussian-driven funda-mentals is a simplified, limiting case of our model for which risk and variance are the same, andwhich fails to provide a palatable explanation for well-known exits such as ERM-I. Correctlyspecifying risk aversion shocks implies dynamics in which the exchange rate fundamental has a As of writing this paper, Croatia, Bulgaria and Denmark are in the ERM-II target-zone. Croatia and Bulgariaintend to adopt the Euro whereas Denmark has a special opt-out clause from Euro adoption.
The seminal paper by Krugman (1991) hinges on the assumption of perfect credibility of thetarget zone, which gives rise to a U-shaped distribution of the exchange rate. This implies thatthe exchange rate spends most of its time near the bands of the zone, as well as a negativerelationship between the interest rate differential and exchange rate volatility. Given this “hon-eymoon effect”, the central bank only has to intervene marginally at the bands. The only sourceof risk in this model is the volatility of the Gaussian distribution. The theoretical predictions ofthe model have been shown not to hold empirically by Mathieson et al. (1991), Meese and Rose(1991) and Svensson (1991). This led to the development of so-called second-generation models,which relax Krugman’s assumptions across two dimensions, to allow for imperfect credibility ofthe target zone and for intramarginal intervention. The first dimension is studied by Bertola andCaballero (1992) and Bertola and Svensson (1993), who relax the notion of credibility and allowfor time-varying credibility or realignment risk. They show that honeymoon effects disappearwhen there is a high probability of exchange rate revaluation. Furthermore, Tristani (1994)and Werner (1995) study endogenous realignment risk, and include mean-reverting fundamentaldynamics.Allowing for the possibility of realignment is a way of characterizing a riskier fundamentalprocess, motivated by speculative attacks and constant realignment of the ERM currencies.This is achieved by using a diffusion process with jumps, as an ad-hoc way of thickening thetails of the distribution in order to better fit the data. The second dimension explored by second-generation models focuses on allowing the fundamental process to be controlled intramarginally,thus generating a hump-shaped distribution where the exchange rate spends most of its timearound central parity. Dumas and Delgado (1992) and Bessec (2003), using controlled diffusionprocesses, show that the honeymoon effects are considerably weakened, putting into question the3ecessity of a target zone when central banks intervene intramarginally. Serrat (2000) generalizesthe target zone framework to a multilateral setting, and shows how spillovers from third-countryinterventions can increase conditional volatilities compared to free-float regimes. Moreover, thisimplies that exchange rate volatility does not need to be monotonically related to the distanceto the target zone bands, which in turn can reduce honeymoon effects. Bekaert and Gray (1998)and Lundbergh and Teräsvirta (2006) test the implications of the second-generation models, andfind mixed evidence with a slight tendency towards the intramarginal interventions hypothesis.Lin (2008) proposes a framework with an interesting analogy to our model, where the spotrate can be stabilized by imposing a target zone on the forward rate. This framework requiresthe setting a sequence of terminal maturity dates for the forward contracts, which generateforward-looking expectations that effectively endogenize the bands. This is similar to our modeldynamics, which imply that the finite exit time from the target zone is akin to the maturitydate of a forward contract, at which the spot rate is required to converge.Ajevskis (2011) extended the basic target zone model to a finite termination time setting whilemaintaining the assumptions of the original model: it is the closest to our approach. Ajevskis(2015) extends his earlier contribution by allowing the exchange rate to follow a mean-revertingOrnstein-Uhlenbeck (OU) process and compares the difference in exchange rate-fundamental-target zone dynamics between the OU process and Brownian motion. He solves the stationaryproblem for the OU process but is unable to explicitly solve the non-stationary part of theprocess. Recently, Studer-Suter and Janssen (2017) and Lera and Sornette (2016, 2018 and2019) find empirical evidence for the target zone model for the EUR/CHF floor target zone setby the Swiss National Bank between 2011 and 2015, the latter mapping the Krugman model tothe option chain.In particular, Lera and Sornette (2015) show how the standard Krugman model can hold in spe-cific cases, such as the EUR/CHF target zone, because of a sustained pressure that continuouslypushes the exchange rate closer to the bounds of the target zone, which the central bank tries tocounteract. In this particular case, the sustained pressure stemmed from the Swiss Franc beingused as a safe asset in the middle of the European crisis. This implies that there is a sourceof additional risk which is radically different from the diffusive nature of Gaussian noise. Thisrisk destabilises the exchange rate fundamentals and creates an extra tendency to escape fromits mean and move towards the boundary. Rey (2015) famously argued that the global financialcycles stemming from the United States generate additional risks for central banks targeting anominal anchor. Additionally, Gopinath and Stein (2019) and Kalemli-Özcan (2019) show howUS monetary policy shocks can affect the exchange rate of a country with minimal USD exposurebecause of the dominant nature of the USD as a trade currency. All of these examples representpossible sources of external risk that need to be included in the modeling of the fundamentalprocess. Lastly, Bauer et al. (2009) shows how a model with heterogeneous agents and perfectcredibility can create hump-shaped exchange rate distributions because of the contrasting forcesbetween informed and uninformed traders. All pre-existing attempts at modeling fundamentalrisk involve either the variance of Gaussian noise or the addition of ad-hoc jumps, or by assumingdeviations from rational expectations. In our paper, we show how all these resulting exchangerate densities can be recovered by a rigorous characterization of fundamental risk.The paper is organised as follows. In Section 3 we define possible interpretations and sourcesof external risk, in Section 4 we extend the traditional stationary framework in order to includenon-stationary dynamics, modeling the risk of the fundamental process by means of dynamicmean-preserving spreads. Section 5 discusses the connection between risk, target zone widthand feasibility. In Section 6 we show the emergence of regime shifts once a critical thresholdof risk is reached. Section 7 explains the numerical methods employed in the simulations, andpresents the results. Section 8 presents the policy implications of our model, while 9 concludesand discusses an agenda for future research. 4
Risk aversion shocks, external sources of risk and mean-preservingspreads
In this paper we want to characterize a modern target zone mechanism in which the fundamentalprocess can be destabilized by external risk factors, generating thick non-Gaussian tails in itsdistribution. Inclusion of these characteristics in the analysis is made necessary by the presenceof risk-averse investors who have time varying risk-aversion modulated by the global financialcycle. Entering a target zone increases the capital market integration of the country in questionwhich exposes countries’ fundamentals to an increased share of global and regional risk factors. Evidence from New Member States suggests that the magnitude of capital flows received may bevery high even if the member state does not enter the target zone process for adopting the theEuro (Mitra, 2011). In short, this framework allows us to consider additional fundamental riskarising from time-varying risk aversion generated by the global financial cycle when a currencyenters a target zone. Let us start with the standard flexible-price monetary model of exchange rate as in Ajevskis(2011). The money demand function is given as m t − p t = θ y y t − θi t + (cid:15) (1)where m is log of the domestic money supply, p is log of the domestic price level, y is the logof domestic output and i is the nominal interest rate. θ y is the semi-elasticity of the moneydemand with respect to output whereas θ i is the absolute value of the semi-elasticity of moneydemand with respect to the domestic nominal interest rate and (cid:15) is a money demand shock. Thesecond block is given by the expression for the real exchange rate q which is defined as q t = X t + p ∗ t − p t where p ∗ is the log of the foreign price level. The third block of this model is the uncoveredinterest rate parity condition which in a linearised form is given by E dX t = ( i t − i ∗ t ) − η t (2)where E dX t is the is expectation of the exchange rate conditional on information available tilltime t and i ∗ t is the foreign interest rate. The target zone framework depends critically onthe uncovered interest rate parity condition, with the currency in the target zone convergingto the target nominal interest rate at time of exit to the currency union. The UIP conditionrequires risk-neutral preferences to hold. This is usually not the case when we are consideringreal world situations, as investors are generally risk-averse. η t is a time-varying risk premiumand is a consequence of risk-averse foreign investors who demand a higher compensation forholding home bonds. η t is widely accepted to be dependent on investors’ risk aversion. Risk Fornaro (2020) finds that entering a currency union increases financial integration between member states.This is due to reduction of currency risk and the associated easing of external borrowing constraints, driven inpart by loss of national monetary and fiscal autonomy. A target zone setting is a quasi-currency union with thechosen target zone band representing the range of expected fluctuations. This may be considered analogous to the index effects documented by Hau et al. (2010) for emerging marketcurrencies. Destabilization of country fundamentals is also possible via shocks to dominant global currencies such as theDollar, which may affect the price of risk for both the non-Dollar target currency and the target zone currency.See Rey (2015); Avdjiev et al. (2019) for more details. U ( c t ) discounted at γ . B ht is the holding ofhome (small open economy) bonds B ft is the holding of foreign bonds by a representative agent.Consumption and bond holdings in period t and t + 1 are given by the problem max c t +1 ,B ht ,B ft ∞ (cid:88) t =0 γ t U ( c t ) c t = B ht + X t B ft E [ c t +1 ] = (1 + i t ) B ht + E [ X t +1 ](1 + i ∗ t ) B ft At any time t , if at the future time t + 1 the agent’s coefficient of relative risk aversion − cU (cid:48)(cid:48) /U (cid:48) were to be incremented by an amount ± ( λ ∈ R + ) which can be either negative (risk-on) orpositive (risk-off) with equal probability, yielding a new utility function ¯ U . Using the first-order conditions of the problem, this implies that the asset pricing kernel (the stochastic discountfactor) will be given by γ ¯ U (cid:48) ( c t +1 ) U (cid:48) ( c t ) = γ U (cid:48) ( c t +1 ) U (cid:48) ( c t ) ∆ U (cid:48) ( c t +1 ) = M t ∆ U (cid:48) ( c t +1 ) , where M t is the pricing kernel without the change in risk aversion and ∆ U (cid:48) ( c t +1 ) is the change incurvature of the utility function due to the change in risk aversion. Note that this last term is alsoa random variable. As an example, if we assume CARA utility and a log-normal consumptionprocess with mean µ and variance s , this extra term is equal to e ± λ ( µ − c t − λs / , noting thata realized − λ in the utility functional implies an increase in risk aversion. More generally, ifconsumption of bonds is at two discrete time points but their evolution is continuous, this extraterm is equivalent to the Radon-Nikodým derivative for the change of measure between thedensities generated by the differently curved utility functions. The investors’ pricing kernel istherefore γ ¯ U (cid:48) ( c t +1 ) U (cid:48) ( c t ) = d Q d P d ˜ Q d Q where Q is the foreign martingale measure of the home bond under the original measure P , and ˜ Q is the foreign martingale measure under the new utility function. The modified UIP conditionis then given by E { dX t } (1 + i ∗ t )(1 + i t ) = d Q d ˜ Q , (3)where the excess returns required to complete the no-arbitrage condition decreases with theinvestors’ risk aversion, since d Q d ˜ Q increases with a realization of + λ (decreased risk aversion) andvice versa. This is equivalent to the modified UIP condition in (2), where the time-varying riskpremium is dependent on the change in investor risk aversion. If we assume again log-normalityof the foreign bond, since the change in risk aversion is equally likely on each side (each ± λ isrealized with probability 0.5), it’s easily shown that that the new measure after the change inrisk aversion is given by a Gaussian density identical to the pricing kernel without the curvature This framework is equivalent to assuming heterogeneous investors, identical in everything except in riskaversion, where between t and t + 1 each changes her own risk aversion to a specific amount, and the resulting ± λ is the aggregate overall change in the representative utility function. ± λ with equal probability, represented by aBernoulli variable, that makes the overall process non-Gaussian. We note that the overall newmeasure d ˜ Q /d P is still a martingale but is not Gaussian, even assuming an underlying Gaussianprocess: the oscillation of the change in curvature of the utility function generates an extra term d ˜ Q d Q = 12 (cid:16) e − ( x + λ ) / + e − ( x − λ ) / (cid:17) , (4)up to a normalization constant, which is exactly the perturbation of a Gaussian process bymeans of a Bernoulli variable in the drift. We therefore have a risk premium that is dependenton the oscillation of investors’ risk aversion, ± λ with equal probability. Using equations 1-4, werecover the standard flexible-price monetary model of the exchange rate as θ E dX t − θ y y t + q t + p ∗ t + θi ∗ t + θη t − (cid:15) (cid:124) (cid:123)(cid:122) (cid:125) v t + m t (cid:124) (cid:123)(cid:122) (cid:125) f t = X t X t = θ E t { dX t } + v t + m t = θ E t { dX t } + f t , (5)where v t is a money demand shock (velocity) m t is money supply, usually assumed to be con-trolled by the central bank. The fundamental process for the exchange rate evolves accordingto df t = dv t + dm t . The velocity includes all money demand variables, and therefore includesthe varying risk premia from the modified UIP condition. It is commonly modeled as a driftlessBrownian motion, but in order to include from (2) the perturbations caused by time-varyingrisk aversion given by (3) and (4) we augment it with the Bernoulli variable λ B , where B takesvalues ± with probability 0.5. This allows us to represent the non-Gaussian dynamics requiredby the modified UIP condition, and allows us to write the fundamental process in absence ofcentral bank interventions as df t = λ B dt + dW t , f t =0 = f , (6)where dW t is the standard Brownian motion and B is a Bernoulli random variable obtainingvalues {− , } each with probability 0.5 and λ ∈ R + . This non-Gaussian diffusion process, calledthe dynamic mean-preserving spread (DMPS) process, has been studied by Arcand et al. (2020).The oscillation of the Bernoulli random term generates a probability spread - an increase in risk- around the mean of the fundamental. For any f ∈ R , the fundamental dynamics do not affectthe systematic average of f t , but the extra random term B introduces an extra tendency to shiftaway from f . The random term λ B is not a drift that pushes systematically the fundamentalaway from the mean, but is rather a destabilizing force that pushes probability from the cen-ter to the tails of the distribution. As such, the process f t admits a stationary measure suchthat lim t →∞ P ( x, t | x , t ) = P ( x ) which is the sum of two Gaussian distributions, each centeredaround ± λ . By modulating λ , the resulting fundamental processes can be ranked unambigu-ously in terms of their risk, as the dynamic equivalent of the two Rothschild and Stiglitz (1970)integral conditions for increasing risk are satisfied. For all results concerning the process we referto Arcand et al. (2020). The term λ B is indeed a force, being the derivative of the probabilistic potential of the process f t . λ in the velocity, therefore, cause an increase in risk in thefundamental that push probability away from the mean and generate non-Gaussian tails, whilstleaving the systematic average unchanged. What generates risk in the fundamental process isthe size of the change in risk version, rather than the direction per se. As seen in 3, suchshocks cannot be represented by Gaussian fluctuations. Uncertainty in the fundamental processis thus comprised of two parts: Gaussian fluctuations, as in the standard framework, and thedestabilizing force that shifts probability to the tails and allows the fundamental process toescape normality. We can allow for a rescaling of the log-fundamental process by a sensitivityparameter σ < , and equation (6) can be written as: df t = β B dt + σdW t , (7)where the Bernoulli variable B now takes values ± , where β = λσ is the rescaled risk parameter.We have identified the source of external exchange rate risk as exogenous changes in investor riskaversion: we note, however, the process (6) can be used to represent a variety of other sourcesof risk and destabilizing forces that cannot be represented reliably by Gaussian fluctuations, aswell as to investigate the implications of such a modeling in a tractable way. In Appendix Awe report alternative reduced form interpretations of λ as a source of destabilizing risk. Fur-thermore, in this paper we choose to focus on a fundamental process that remains stationary indistribution around its long-run level, here normalized to 0 without loss of generality. This isthe case for most target zone cases. However, if the fundamental was substantially misalignedfrom its long-run level, then the choice of a mean-reverting process could be more appropri-ate. The analysis of this case is presented in Appendices E and F, where we fully solve bothOrnstein-Uhlenbeck (O-U) and non-Gaussian, softly attractive dynamics. The latter can be ofinterest for researchers as an alternative to the O-U process, since it allows one to again escapeGaussianity and to model an ergodic process with light attraction towards its long-run level,whilst maintaining analytical tractability.This also allows us to precisely characterize the interplay of diffusive fluctuations (variance) anddestabilizing forces (risk, via changes in investor risk aversion): the tendency of external risk toshift the exchange rate away from the mean and towards the bounds of the zone is counteractedby the central bank’s efforts to maintain the fundamental fluctuating around its mean. This isprecisely what is argued by Lera and Sornette (2015). The standard Gaussian case is a limitingcase for which the risk parameter is zero and there is no change in investor risk aversion. This isa more realistic characterization of fundamental risk, especially considering the influx of externalrisk given by global financial cycles. We show how the solution to the model is made up of twoparts. The first is the time-independent stationary part, which corresponds to the behavior ofthe exchange rate at the time of entry in the target zone. The second is the transient part,which describes the sensitivity of the exchange rate to the distance to the bands, as a functionof risk, band size and time to exit. We study the exchange rate equation derived from (5), which we write as X t = f t + 1 α E { dX t } . (8)We allow explicitly time-dependent dynamics X t = X ( t, f t ) , and therefore study non-stationarybehavior. At a fixed time T the spot exchange rate is set to exit the target zone and match8he target fundamentals. The absolute value of the semi-elasticity of money with the nominalinterest rate, θ , is always greater than unity: we rewrite it in the form /α , with < α < ,thus interpreting it with the dimension of a frequency (i.e. / [time unit] ) which modulates thesize of the forward-looking time window.Using Itô calculus, Eq.(8) can be rewritten as: ∂ t X ( t, f ) + σ ∂ ff X ( t, f ) + β B ∂ f X ( t, f ) − αX ( t, f ) = − αf. (9)Note the presence of the additional term ∂ t X ( f ) in Eq.(22) which does not appear when onefocuses only on stationary situations. The solution of (22) can be written as the sum of thetime-independent stationary solution and the transient solution: X ( τ, f ) = X ∗ ( τ, f ) + X S ( f ) . (10)Appendix B shows how the stationary solution of (10) is given by: X S ( f ) = 1cosh( βf ) { A Y ( f ) + B Y ( f ) + Y P ( f ) } , (11)where we have: Y ( f ) = exp (cid:110) + (cid:113)(cid:2) β + ασ (cid:3) f (cid:111) , Y ( f ) = exp (cid:110) − (cid:113)(cid:2) β + ασ (cid:3) f (cid:111) ,Y P ( f ) = α ( f ( α + β ( − σ )) cosh( βf )+2 βσ sinh( βf ) ) (2 α + β (1 − σ )) (12)In Eqs. (11) and (12) the pair of constants A and B can be determined by smooth fitting at thebounds f = − f : ∂ f X S ( f ) | f = f = ∂ f X S ( f ) | f = f = 0 . (13)The two constants of integration A and B can be obtained in closed form but their expressionis lengthy and is therefore omitted. An illustration of the stationary solution (11) is presentedin Figure 1, which also shows how an increase in the riskiness β of the fundamental prompts the(stationary) exchange rate to behave more independently of the dynamics of the fundamental.At high levels of β , the exchange rate dynamics are driven mostly by the risk and depend lesson fundamentals, especially around the bounds, as represented by the steepening of the centralslope. In this figure, f = 10% and we assume a quasi-daily time step for the expectation α = 0 . .Our parametrization of α = 0 . corresponds to a case of fast agent updating, which is similar tothe case studied by Ferreira et al. (2019) and Coibion and Gorodnichenko (2015). Changing the α to a lower fundamental updating frequency will reduce the sensitivity of the exchange rate tothe fundamentals. For simplicity we focus our attention to targets zones symmetric with respect to f = 0 , although the resultshold for general bounds. To see this, notice that the general solution is unaffected by the bounds, which enterthe particular solution only via the scalar quantities A and B . We continue using a symmetric band [ − f, f ] forclarity of exposition. igure 1 Effect of varying β on stationary exchange rate dynamicsWe now turn to the transient dynamics. At a given time horizon t = T , we fix the predeterminedexchange rate X ( T, f ) = 0 . In terms of the backward time τ = T − t , we write the transformation X ∗ ( τ, f ) = Y ∗ ( τ, f ) / cosh( βf ) . The time-dependent partial differential equation we need to solveis therefore given by: ∂ τ Y ∗ ( τ, f ) − σ ∂ ff Y ∗ ( τ, f ) + (cid:20) β α (cid:21) Y ∗ ( τ, f ) = 0 . (14)with boundary conditions given by: (cid:2) ∂ f Y ∗ ( τ, f ) − β tanh( βf ) Y ∗ ( τ, f ) (cid:3) f = f = 0 , (cid:2) ∂ f Y ∗ ( τ, f ) − β tanh( βf ) Y ∗ ( τ, f ) (cid:3) f = f = 0 . (15)We express the solution Y ( τ, f ) as Y ∗ ( τ, f ) = φ ( τ ) ψ ( f ) , and proceed to solve this equationby separation of variables and expansion over the basis of a complete set of orthogonal eigen-functions. Sturm-Liouville theory allows us to state that on the interval [ − f , + f ] , one has acomplete set of orthogonal eigenfunctions ψ k ( f ) satisfying Eq.(13), namely: ψ k ( f ) = sin (cid:32) √ k σ f (cid:33) ∈ [ f , f ] , k = N + , (16)where each eigenvalue Ω k solves the transcendental equation: √ k σ cot (cid:32) √ k σ f (cid:33) = β tanh( βf ) . (17)Furthermore, the eigenvalues are real and span a discrete spectrum: { Ω k } := (cid:8) Ω k ( β, f ) (cid:9) , k ∈ N + . igure 2 Target band and spectrum(a) ¯ f = 10% (b) ¯ f = 20% Graphical illustration of the solution of equation (17), showing the effect of varying ¯ f on the spectrum Ω k . and can therefore be ordered as: Ω ( β, f ) < Ω ( β, f ) < · · · . For any k ∈ N + , the corresponding Ω k ( β, f ) solves the transcendental equation (17), and has tobe calculated numerically. For a general β > , one observes that the successive eigenvalues arenot evenly spaced, and display a distance which decreases in k . The spectrum is controlled bythe width of the target zone ¯ f : the wider the band, the smaller the separation. The spectrumand its relationship with the target band size are illustrated in Figure 5. Observe also that inthe limit β = 0 , one straightforwardly verifies that from Eq.(16) one obtains the evenly spacedset Ω k (0 , f ) = (2 k + 1) π f .The development of the non-stationary solution X ∗ ( τ, f ) over the complete set { ψ k ( f ) } enablesone to finally write the full expansion as: X ∗ ( τ, f ) = X ∗ ( T − t, f ) , t ∈ [0 , T ]= 1cosh( βf ) ∞ (cid:88) k =1 c k exp (cid:2) − (Ω k + ρ )( T − t ) (cid:3) sin (cid:34) √ k σ f (cid:35) (18) c k = − f (cid:90) + f − f X S ( f ) sin (cid:34) √ k σ f (cid:35) dfρ = (cid:20) β α (cid:21) . The full derivation is reported in Appendix B. When t = T , from Eq.(19), by construction ofthe Fourier coefficients c k , we have X ∗ (0 , f ) = − X S ( f ) and so X ( T, f ) = X ∗ (0 , f ) + X S ( f ) = 0 thus reaching the required fixed parity. An illustration of the non-stationary exchange ratedynamics, as well as the overall transition dynamics throughout the time interval [0 , T ] , ispresented in Figure 3. This solution allows one to express the movements of the exchange ratevia a weighted sum of its stationary behavior, its distance to the exit time and the distancebetween its value at any time t and the target band. The eigenvalues modulate the frequencyof both fundamental and exchange rate movements within the band. The Fourier coefficients c k igure 3 Non-stationary dynamics(A)(B)
This figure shows the evolution of X ( T − t, f ) of the non-stationary dynamics in the target zone. Panel (A) showsthe behavior of the time-dependent part: we assume a target zone which has been set to T = 3 years, with β = 1 for a given set of fundamentals. For the sake of brevity we truncate the figure towards the end of the targetzone to effectively illustrate the non-stationary dynamics. Panel (B) shows the full dynamics for an increase inrisk. Here we have assumed a target band symmetric around zero, i.e. ¯ f = 10% = − f . We also assume α = 0 . .We truncate the eigenfunction expansion at 50. The second panel illustrates the change in dynamics from β = 0 (Gaussian) to β = 5 . “if one had perfect pitch” , one would be able to “hear” theshape of the target zone. This formulation of the solution allows us to uncover the unique nature of the smooth-pastingconditions: the exchange rate process is not reflected at the bounds in the probabilistic sense,since this would have been modeled as a zero derivative condition on the transition probabilitydensity function. We are in the presence of “soft” boundaries, where the central bank interven-tions are determined by the interplay of the distance of the exchange rate to the bounds as wellas the tendency of the fundamental to hit them (the risk): this is what is implied by the eigen-function expansion of the solution. This allows us to “endogenize” the bands: because of thepresence of expectations in the exchange rate equation (22), we have a second-order term whichallows us to solve the equation in its Sturm-Liouville form and eigenfunction expansion. TheFourier coefficients modulate the sensitivity of the exchange rate to the distance to the band,allowing for the central bank to intervene whenever the fundamental is “felt” to be approachingthe bounds. This “feeling” is in fact a direct translation of how much the fundamental tendsto escape and how much the central bank needs to intervene marginally or intramarginally:it is a direct consequence of the presence of expectations in the exchange rate equation. Inother words, the higher the tendency to hit the bounds, the greater is the likelihood that thecentral bank will actually intervene intramarginally, with increasingly less weight placed on theactual position of the fundamental within the band. One can therefore see that the higher isthe risk (the fundamental’s tendency to escape from its central position), the more the centralbank intervenes intramarginally. The same applies when the target band shrinks. The standardKrugman framework applies when the fundamental is a pure Brownian motion and the centralbank only intervenes marginally. Note that this phenomenon is directly a consequence of ourrigorous characterization of fundamental risk. In Section 7 we show how the model can replicatethe different exchange rate densities under different assumptions of feasibility and intervention. It is also worth noting that this framework potentially allows for the existence of de jure and defacto bands, as noted by Lundbergh and Teräsvirta (2006): if the de jure band is large, expecta-tions over the magnitude of risk may react to a narrower de facto band. This is a phenomenoncommonly observed in most ERM countries.
Here we discuss the interplay between the risk parameter β , the size of the target band [ − f , + f ]and the feasibility of the time horizon T at which to reach the target zone. We first notethat at the initial time t = 0 , from Eq.(19) we have X ∗ ( T, f ) ≈ and therefore X (0 , f ) = X ∗ ( T, f ) + X S ( f ) ≈ X S ( f ) . Since Ω ( β, f ) < Ω ( β, f ) < · · · , one can approximately write: X ( T, f ) (cid:39) X S ( f ) + O (cid:16) e − (Ω + ρ ) T (cid:17) . Note that the time-independent part of the problem is a one-dimensional Neumann problem on the boundary ∂D = [ f, f ] (cid:40) ∆ f + Ω f = 0 ∇ f | ∂D = 0 , which is exactly the problem of finding the overtones on a vibrating surface. For additional information, see Figure 2 in Crespo-Cuaresma et al. (2005) X ( T, f ) = X S ( f ) , one sees immediately that X ( T − t, f ) = X S ( f ) + X ∗ ( T − t, f ) with X ∗ ( T − t, f ) given by Eq.(19) nearly matches theexact solution, provided we have an horizon interval T (cid:38) t relax where t relax := (cid:0) Ω + ρ (cid:1) − is thecharacteristic relaxation time of the exchange rate process. This provides a validity range forthe non-stationary dynamics given by the expansion Eq.(19).Hence, at time t = 0 , the required initial probability X S ( f ) law is reached only for a largeenough time horizon T (cid:38) t relax . This now enables us to link the non-stationary dynamics of X ∗ ( t, f ) to the feasibility of the target zone: the relaxation time τ relax determines the minimumtime interval for which a feasible target zone may be maintained. The larger β (the risk of thefundamental, stemming from larger shifts in agents’ risk aversion), the greater is the tendency ofthe fundamental to escape from its mean; the authorities need therefore to maintain the targetzone for a longer minimum duration. An increase in risk, for a given ¯ f , implies that the targetzone would have to be set for a longer horizon T to be feasible. Alternatively, for a given risk β ,an increase of the target zone width f , requires a longer minimal T implementation to ensurethe overall feasibility of the policy. In other words, the central bank has to impose that the timehorizon T is at least as large as the relaxation time t relax .An intuitive interpretation of the relaxation time in this framework is to understand t relax as thecharacteristic elapsed time required to “feel” the first effects of the home central bank’s actionsaimed at reducing fluctuations of the exchange rate, compared to a free float. The bank’sactions may be then viewed as a de facto reduction of the target zone band over time, whilstthe de jure band remains unchanged. A possible implication would be that t relax would be theminimum time for agents to update their priors accurately, generating self-fulfilling expectationsthat create the honeymoon effect.The inverse of the relaxation time is determined by the spectral gap , which is the distancebetween 0 and the smallest eigenvalue. We therefore have the relationship ( t relax ) − = (Ω + ρ ) .The spectral gap controls the asymptotic time behaviour of the expansion given by (19), and itis continuously dependent on risk β and band ¯ f . This relationship is illustrated in Figure 4. Figure 4
Interaction between β and f Note:
This figure shows the interaction of varying risk ( β ) and varying the band size ( f ). An increase in risk, fora given f , implies that the lowest eigenvalue Ω falls (Panel (A)). The inverse of this value controls the t relax . Let us now study analytically the behaviour of the solution Ω of the transcendental Eq.(17).Writing z = √ f , Eq.(17) implies that the product βf is the determinant of the amplitude of14 . An elementary graphical analysis enables one to conclude that two limiting situations canbe reached: βf << ⇒ z (cid:46) π ⇒ Ω (cid:46) π f ⇒ t − (cid:46) (cid:104) π f (cid:105) + β + α,βf >> ⇒ z (cid:38) π ⇒ Ω (cid:38) πf ⇒ t − (cid:38) (cid:104) πf (cid:105) + β + α. and therefore: (cid:104) πf (cid:105) + β + α ≤ t relax ≤ (cid:104) π f (cid:105) + β + α . (19)Eq.(19), together with Figure 4 shows how an increase in risk β affects t relax more strongly whenthe exchange rate is allowed to float in a wider band width ¯ f . Figure 5
Risk, target band and regime shifts(a) ¯ f = 2% (b) ¯ f = 15% Regime shift and eigenvalue jump as a function of risk, for different target bands
An unique phenomenon that emerges when considering non-converging drifts, and in particularour specification, is the emergence of a regime shift . Figure 5(b) shows that for a large enoughtarget band, after a threshold level in β , the relaxation time suddenly jumps to a much lowervalue and remains almost constant (though very slowly increasing) for further increases in risk.This effect happens because when the tendency β of the noise source driving the fundamentalreaches and surpasses a certain level, the destabilizing risk component in the noise source over-comes the diffusion. The force β B in the mean-preserving spread becomes the main driver of thestochastic process driving the fundamental, and therefore f t becomes a process with a tendencyto escape from its mean that is stronger than the tendency to diffuse around its central value.While this may look like a sudden emergence of supercredibility, it is in fact the opposite: thetarget zone cannot be feasibly held. This implies that the fundamental process escapes its initialposition with such force that it hits the band at every dt , and interventions become almost con-tinuous. Furthermore, smooth-pasting conditions cannot be applied anymore. The central bankwill have to either increase the size of the band or to allow the spot rate to float freely. Thishas a direct implication for honeymoon effects: Appendix C shows how, after a threshold level15f risk has been surpassed, the smooth fitting procedure at the boundaries cannot be applied,and hence honeymoon effects when the fundamental approaches the band become unobtainable.This implies that a high level of risk denies a central bank monetary autonomy up until themoment of entering the currency zone. This phenomenon is illustrated in Figure 6.Consider now the smooth-pasting conditions (15): one can separate the contribution of theeigenfunction to the one given by the probability spread and obtain: ∂ f sin (cid:16) √ k σ f t (cid:17) sin (cid:16) √ k σ f t (cid:17) − β tanh( βf t ) = 0 (cid:109) ∂ f EIG (Ω k , f t ) EIG (Ω k , f t ) − MPS ( β, f t ) = 0 . (20)The first term is a total sensitivity term, closely related to the elasticity of the eigenfunctionwith respect to the fundamental, and it represents the overall variation of the exchange ratewith the fundamental. The second term represents the increase in risk, as well as the desta-bilizing component that represents the tendency of the fundamental to hit the target bands.The solution of this equation yields the spectrum { Ω k } , for k = N + . The difference of the twoterms represents the residual tendency of the home country fundamental to avoid converging tothe target fundamental. The spectral gap, therefore, represents the intensity of the probabilityspread. The regime shift will happen at a threshold value β e , only obtainable numerically, forwhich the spectral gap will suddenly jump upwards: the destabilizing force has dominated overthe diffusive part and the first eigenvalue jumps higher. The oscillating part of the expansionincreases in frequency, and the time-dependent exponential decay increases in speed. A graphi-cal illustration is shown in Figure 7: one can easily show that the lower bound for the threshold β e is given by / ¯ f . This allows one to uncover the close relationship between the regime shiftand the size of the target band. This regime shift cannot occur with a Gaussian process orwith mean-reverting dynamics. Figure 6
Risk threshold, distance to the bands and honeymoon effects(a) β > β e . No honeymoon effects (b) β < β e . Large risk shocks vs. diffusion-driven regimes
In the diffusion-driven regime (characterised by a relatively low β < /f ), one observes that anincrease of risk implies a decrease in sensitivity, since t relax is increasing. This may seem coun-16erintuitive: but it must be remembered that at time t = 0 , the initial condition is the stationarysolution of the central bank-controlled diffusion for the given risk. Increasing β , therefore, islikely to load the stationary probability mass accumulated in the vicinity of the target zoneboundaries. Escape from this stationary state by bank action becomes more difficult, ultimatelyleading to an increase of t relax . Conversely, in high risk regimes where β > /f and where thedestabilizing dynamics dominate, the boundaries of the target zone are systematically hit by thefundamental. In this situation, the central bank will intervene almost entirely intramarginallyregardless of whether the fundamental is actually close to the bands, since honeymoon effectscannot exist anymore. This allows, in Eq.(12), for a sudden reduction of the probability masslocated at the bounds, and this generates the sharp drop of t relax . In other words, the bandimplicitly ceases to exist and the central bank operates effectively in an infinitesimally narrowband. This provides new insight into target zone feasibility: if risk is too high, exchange rateexpectations are no longer anchored to the band and the effectiveness of central bank interven-tion is greatly reduced. What the central bank could do is therefore either (i) to reduce risk,which in practice is often infeasible, or (ii) to increase the size of the target zone which itself isbounded by the free-float exchange rate volatility. The new size of the band would have to belarge enough for this new target zone to be “heard”. Figure 7
Risk and eigenvalue jump
Note:
Regime change For β = 15 . The force β tanh( βf ) (black curve) overcomes the diffusion component andgenerates the first eigenvalue jump. For β = 6 (red curve), the regime has not yet shifted. Here ¯ f = 0 . , σ = 1 , α = 0 . . We can therefore also connect the threshold β e at which the regime shift occurs to completefactor market integration: for lower levels of β , the home fundamental exhibits an idiosyncraticcomponent anchored to its original dynamics that is stronger than its tendency to convergeto the target fundamental. Once this component is overcome, the target zone ceases to existand the currency starts floating. This may also help explain why countries with a high level ofcapital integration with the target currency may have higher costs in maintaining a target zone.One implication of the suddenness of the regime shift is that the relationship between capitalintegration and the duration of the target zone is non-monotonic. This is precisely what Leraand Sornette (2015) illustrate with the case of the Swiss Franc floor between 2011-2015. We simulate central bank intervention by means of a symmetrized Euler scheme for stochasticdifferential equations. Since the original problem is a one-dimensional Neumann problem on theboundary ∂D = [ − ¯ f , ¯ f ] , the regulated SDE can be written as:17 t = (cid:90) t b ( f s ) ds + σ (cid:90) t dW s + (cid:90) t γ ( f s ) ds, where b ( f s ) is the nonlinear drift and γ ( . ) is the oblique reflection of the process on the boundary ∂D . This is the equivalent of the interventions, and we assume that for the unit vector field γ there exists a constant c so that γ ( x ) · (cid:126)n ( x ) ≥ c for all points x on the boundary D . This can beinterpreted as assuming bounded interventions. We use a regular mesh [0 , T ] for the numericalsimulation, for which the weak error is of order 0.5 when the reflection is normal (i.e. γ = (cid:126)n ),which is our case. We choose this method in order to obtain consistent Monte Carlo simulationof the resulting densities. The algorithm starts with f = 0 and for any time t i for which f t i ∈ D we have for t ∈ ∆ t = t i +1 − t i that: F N,it = f Nt i + ˆ b ( f Nt i )( t − t i ) + σ ( W t − W t i ) , as in the standard Euler-Maruyama scheme, and the nonlinear drift b ( . ) is approximated with asecond-order stochastic Runge-Kutta method. If F N,it +1 / ∈ ∂D , then we set: f Nt +1 = π γ∂D ( F N,it +1 ) − γ ( F N,it +1 ) , where π ∂D ( x ) is the projection of x on the boundary ∂D parallel to the intervention γ . If F N,it +1 ∈ ∂D , then obviously f Nt +1 = F N,it +1 . For more references, see Bossy et al. (2004). Theexchange rate path is then obtained simply by setting X Nt = X ∗ ( f Nt , T − t ) for every t ∈ [0 , T ] .It is of fundamental importance to set ∆ t equal to the update ratio given by /α in our model,so that the increment of the simulated exchange rate path has the same updating time frequencyas the central bank. We can now discuss two kinds of interventions: the kind that intervenes byreflecting the process so that it just stays within the band (sometimes called “leaning against thewind”), and the pure reflection variety, which projects the fundamental process by an amountequivalent to how much the process would have surpassed the boundary. This distinction canalso be understood as the amount of reserves the central bank has at its disposal in orderto stabilize the fundamental process: the greater this quantity, the more likely it is that theintervention will be of the pure reflection type. We also assume that an intervention is effectiveinstantaneously. This distinction also has important implications in the resulting exchange ratedensity: as shown in Figure 6, given our characterization of risk, the greater the β, the earlierthe central bank will have to intervene, given the fundamental’s increased tendency to escapetowards the bands. We present five possible scenarios by estimating Monte Carlo densities of thesimulated exchange rate process: the first two correspond to the Gaussian case, where β = 0 witheach of the two intervention strategies. The densities are obtained by Monte Carlo simulationof N sample paths, binning the data and limiting the bin size to zero to obtain the convolutiondensity, then averaging over the N realizations and interpolating the resulting points. For allfigures N is set to , σ = 0 . , r = 0 . , α = 200 , T = 3 and the exchange rate target bandto ± . For more references on the method, see Asmussen and Glynn (2007). We obtaina realization path for each of the two and obtain both U-shaped (corresponding to the baseKrugman case) and hump-shaped densities, corresponding to the Dumas and Delgado (1992)framework. The realized densities are plotted in Figure 8. We then simulate the case in which β > but is not large enough to trigger the regime shift, each one with a different interventionstrategy: in the marginal intervention case we obtain the two-regime density ( β = 5 ), as in theBessec (2003) framework, and in the intramarginal one we obtain a hump-shaped distributionas for all intramarginal intervention frameworks. These results are shown in Figure 9. Notethat this is a consequence of our characterization of risk: the tendency β of the fundamentalto hit the boundary generates the two-regime shape, since even in a marginal framework thecentral bank will already intervene when at a distance from the bands. Furthermore, this isthe case in which de facto bands start to appear. Finally, we present a case in which β is large18nough ( β = 50 ) to trigger the regime shift, and the band in fact ceases to exist: the tendency toescape leads to the fundamental process constantly surpassing the boundary, honeymoon effectsare impossible and pure reflection intervention concentrates most of the realizations around theinitial level. This, as N → ∞ , generates a Dirac delta function around the initial value of thefundamental. This is displayed in Figure 10. Figure 8
Exchange rate densities, β = 0 (a) Marginal intervention, LAW (b) Intramarginal intervention, pure reflection Figure 9
Exchange rate densities, β > , β < β e (a) Intramarginal intervention, LAW (b) Two-regime intervention, pure reflection Figure 10
Exchange rate densities, β > β e ,Target band too narrow given the level of underlying risk. A target zone with a terminal exit time to another currency has two objectives. First, the centralbank wants to limit the volatility of its exchange rate ( X t ) versus the anchor currency below thefree float level of the anchor currency ( Z t ). This provides us with a natural limiting condition19o the size of the band that the central bank can set: | f | ≤ σ z . This essentially means that the band size of ERM-II of ±
15% will never be breached if thecentral bank of the target zone currency pegs to the Euro, as the Euro itself has an annualisedvolatility versus other major currencies in the range of 7-10%. Second, the central bank needsits target-zone to be considered feasible, in order to enjoy “honeymoon effects”, which in turnreduces the cost of intervention for achieving the set parity. In our setup, we propose theconcept of a characteristic relaxation time τ relax which determines the minimum time a targetzone must be maintained to “feel” the first effects of the home country central bank’s actionsaimed at reducing fluctuations of the exchange rate, compared to a free float. This allows usto interpret τ relax as the minimum time for agents to update their previously held exchange rateexpectations, generating self-fulfilling expectations that create the honeymoon effect.This does not mean that a central bank cannot adopt a target currency overnight with anarbitrary parity being the close of day value of the target exchange rate. In such a case, agentswould not have had time to update their expectations and this would force the central bank touse a larger proportion of its assets (in the target currency) defending the parity level. Thisopens up many different avenues of enquiry into the expectation generation process of agents inforeign exchange markets. If t relax is the minimum time for agents to update their previously heldexchange rate expectations, this means that greater shifts in higher degree of agent risk-aversion(higher β ) will increase t relax . As shown by Osler (1995) and Lin (2008), this effect would workthrough the feasibility of the target zone in time shifting speculators’ horizons towards shortterm speculation, where t speculation ≤ t relax . This is a natural outcome of “honeymoon effects”which make intervention cheaper for central banks and harder for speculators after t relax .We find that t relax is increasing with the magnitude of the risk aversion shifts, for β ≤ β e .Moreover, the target band size is also increasing in shifts size for β ≤ β e up to | f | ≤ σ z .New relevance for our framework has emerged in a recent development for the Economic Com-munity of West African States (ECOWAS) as well as for new entrants in to ERM-II target zone.ECOWAS is planning to replace the current West African CFA Franc with a common currency,named Eco. The goal is for the 15 states to transition to the Eco via a target zone mechanism,similar to the ERM-II. Currently the CFA Franc is pegged to the Euro, with operational man-agement shared between the Bank of France and the local central banks. After the reform, thesecountries would have to manage their own exchange rate targets without any outside support.The main political reason is understandable, and lies primarily in the severing of the ties withthe former colonial ruler, France. The issues with this process, however, are multifaceted, andone of the main concerns is the short time horizon proposed for the target zone mechanism (oneyear). Moreover, there may be additional risk stemming from not allowing the ECB to haveoperational risk-sharing in the process, as well as the inherent risk faced by individual WestAfrican central banks. This translates directly to our framework, where the risk factor β maygenerate a relaxation time t relax for individual states that may be larger than the proposed con-vergence time T . This could have potentially devastating consequences for the credibility of theparticipating central banks, and for the overall process of creating the new common currency.The inability of some ECOWAS countries to achieve the convergence criteria would make theadoption of the ECO impossible in the near future.On the other hand, Bulgaria and Croatia officially entered ERM-II to replace their nationalcurrencies with the Euro in July 2020. The minimum convergence time T to exit to the Euro The bank’s actions may be then viewed as a de facto reduction of the target zone band over time, whilst the de jure band remains unchanged.
20s set at two years. Both the Lev and the Kuna have successfully pegged their currencies to theEuro over the last decade and may be considered as low β countries at the time of their entryinto ERM-II. However, in with Fornaro (2020)’s predictions, the accession of these countries tothe Euro will be followed by an upgrade in the country ratings for foreign currency debt. Theseratings upgrade at the time of the Covid-19 pandemic generates a higher probability of a capitalflow surge and consequently a higher probability of a sudden stop into these countries. Whilecapital flows might help with financing additional debt during the pandemic for these countries,it also generates a risk of these countries not meeting their fiscal criteria as well as destabilisingthe inflation expectations convergence process. It is highly unlikely given ECB support to theERM process that these countries will not be able to successfully adopt the Euro. However,Lithuania’s experience in adopting the Euro provides a useful benchmark. These countries dorun the risk of missing their fiscal and inflation criteria given the combination of push and pullfactors, which are likely to generate high capital flows. If this risk materialises, this would implythat the Lev and the Kuna may have to wait in the ERM-II for a longer time than originallyexpected. This uncertainty is not as destabilising as in the case of the Eco, given ECB support.Nevertheless, there may be political consequences with support of the new currency if the ERM-II process goes on for longer than expected. An important future contribution of our workwould be the structural estimation of the model parameters and an explicit computation of therelaxation time, thus effectively providing a lower bound for the necessary time for each countryto reach the desired parity.
In this paper we have explored the implications of extending exchange rate target zone modelingto non-stationary dynamics and heavy, non-Gaussian tails stemming from time-varying investorrisk aversion, which lead to mean-preserving risk increases in the fundamental distribution. Ourframework leads to a natural interpretation of target zone feasibility, driven by the interplaybetween two contrasting forces: a destabilizing effect driven by risk which pushes the exchangerate towards the bands, and a stabilizing diffusive force.Our model does not deal with optimal choices: indeed, the only choice variable potentiallyavailable to the authorities is the time horizon T by which the required parity needs to obtain.As such, from the policy perspective our model poses what is essentially a screening problemin the informational sense: in a worst case scenario, it is likely that neither of the two centralbanks knows the true riskiness of the fundamental process. If one chooses an exit time whichis lower than the required minimum time at which parity can be reached (the relaxation time),the target zone exit time is not feasible. However, setting a T which is too high exposes one toincreased business cycle risks, the dampening of which were a likely reason for entering a targetzone in the first place. We show how our model effectively endogenizes the presence of the bandsby the exchange rate expectations, and how the interplay between risk and target band has keyimplications in the credibility of the zone itself, as well as the possibility of honeymoon effects.Intervention is shown to be both marginal and intramarginal, depending on how much the centralbank “hears” the distance to the target zone band. The potential emergence of regime shifts,furthermore, can further erode the target zone credibility. This allows the methods employed inthis paper to be applied to a wide range of situations. An important future contribution of ourwork would be the structural estimation of the model parameters and an explicit computation ofthe relaxation time, thus effectively providing a lower bound for the necessary time for a countryto reach the desired parity. See https://tinyurl.com/yyg4wp9p and https://tinyurl.com/yyuxm3tr for more details on the potentialfor ratings upgrades due to ERM-II eferences Ajevskis, V. (2011). A target zone model with the terminal condition of joining a currency area.
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Journal of International Economics 39 (3-4), 353–367. ppendices A Alternative interpretations of risk
Figure 11
Estimated densities of the fundamental process (inflation expectations) for ERM-IIcurrencies
Inflation expectations (fundamental process) D en s i t y label BulgariaCroatiaEstoniaLithuaniaLatviaSlovakiaSlovenia (Left panel) Centered difference between Euro area inflation expectations and target zone country inflationexpectations, for the time each currency was in the target zone with the Euro. Kolmogorov-Smirnov tests greatlyreject each hypothesis of Gaussianity. The data for inflation expectations comes from the Euro Commission’sJoint Harmonised Consumer Survey. For more details we refer to Arioli et al. (2017). Bulgaria and Croatia haveonly recently acceded to joining the Euro and the data for them is backward-looking to give the reader a senseof pre-target zone differences in inflation expectations. (Right panel) Transition densities of the fundamentalprocess with mean-preserving spreads at time t = 1 , each with risk increases in the direction of the arrow. Destabilization is intrinsically connected to risk in the fundamental process. Besides the structural in-terpretation of risk as stemming from time-varying investor risk aversion, one could think of a variety ofother interpretations for the parameter λ of increasing risk, which generates mean-preserving spreads inthe density of the exchange rate fundamentals. A quick glance at the left-hand panel of Figure 11 showsthat the difference in inflation expectations, one of the key fundamentals in the determination of exchangerate target zones, is undoubtedly non-Gaussian, exhibits substantially heavier tails and presents bimodaltendencies stemming from both inflationary and deflationary pressures shifting probability away fromthe center. Such risk dynamics cannot be represented by the variance of Gaussian fluctuations, as theycannot affect the distribution tails, but rather requires the presence of forces that increase the tendencyof the fundamental process to escape its long-run level. The right panel of Figure 11 shows the transitiondensity of the DMPS process at an arbitrary time for increasing risk. The DMPS density with λ parame-ter fit by maximum likelhood is a better fit for the empirical densities for each of the densities shown in 11.Another way of interpreting of the risk parameter of our framework could be via the presence of capitalflows, especially in how the magnitude and the drivers of capital flows matter in determining the stabili-sation effects. First, capital flows may be driven by push factors creating cycles of bonanzas and suddenstops seen with New Member States. Hansson and Randveer (2013) argue that capital flow dynamicswere the key driver for cyclical developments in the Baltic ERM economies. This is might be a issuefor a small target zone country if the capital flows generate excess appreciation or depreciation pressureweakening the feasibility of the targetrep zone. This is particularly problematic if there is a sudden stopwith reallocation of capital flows to more productive economies in the target zone as seen during theEurozone crisis (Ghosh et al., 2020). Furthermore, assuming absence of macro-prudential tools, capitalflow volatility may generate foreign exchange intervention volatility inside the target zone, as the use ofinterest rates as a monetary policy tool can generate further pro-cyclicality in capital flows. This nexusbetween capital flows and target zone management may destabilise the convergence in the inflation pro- ess of the target zone country. This is the key source of additional risk in our setting. Let’s considerthe real interest rate version of the UIP condition: E { dX t } = ( r t − r ∗ t ) dt + E { dπ t − dπ ∗ t } , where π ∗ is the target country’s inflation measure and π is home inflation. If there are high capitalinflows that need to be counteracted by (unsterilised) intervention, this would generate a lower realinterest rate of financing by putting downward pressure on r t . This additional supply of credit is likelyto increase the E { dπ t } This would require an interest rate response by the national central bank, inthe absence of macro-prudential tools. We can see that in this particular case, increasing interest ratesmay be pro-cyclical to capital flows as long as the inflation process responds positively to the interestrate hike, causing a loss of monetary autonomy if the process is self-reinforcing. A destabilizing out-come of this setting would be if the inflation process does not respond to the interest rate moves andcauses an outflow of capital flows. This would jeopardise the feasibility of the target zone and couldcause the gap between r t and r ∗ t to become larger than before entering the target zone. The standardapproach of modeling risk in the target zone does not consider the risk stemming from the currencyunion itself. If the target currency union has real interest rate changes through lower expected inflationsurprises, it will also affect the stability of target zone by the capital flow mechanism we have described. This shows that one cannot conflate all information pertaining to risk with the variance parameter ofthe Gaussian distribution. We therefore adopt a definition of risk which corresponds to the concept ofa mean-preserving increase in risk, in the second-order stochastic dominance sense, often referred to asa mean-preserving spread. Risk forces are by construction characterized by a second-order stochasticdominance criterion, and therefore must leave the long-run level unchanged. We therefore need tocharacterize such forces as mean-preserving increases in risk. This concept was introduced in a staticsetting in the seminal contributions of Rothschild and Stiglitz (1970 and 1971), who define two sufficientintegral conditions that allow one to unambiguously rank distributions in terms of their riskiness, andextended to a dynamic setting by Arcand et al. (2020). This rigorous characterisation of risk is notconsidered by the extant target zone literature. Lastly, we note that our characterization of risk asdestabilizations caused by capital flows can be further extended to any source of external risk, and ourmodel framework would still apply.
B Derivations of the stationary and transient equations
For the derivation of the stationary solution, we first introduce the following integral transformation: X ( t, f ) = (cid:90) f cosh( βζ ) Y ( t, ζ ) dζ ⇐⇒ ∂ f X ( t, f ) = cosh( βf ) Y ( t, f ) , (21)which is Darboux-type functional transformation. As shown in Arcand et al. (2020), Eq.(22) can bewritten equivalently as ∂ t X ( t, f ) + σ ∂ ff X ( t, f ) + β tanh( βf ) ∂ f X ( t, f ) − αX ( t, f ) = − αf. (22)which leads to: ∂ t Y ( t, f ) + σ ∂ ff Y ( t, f ) − (cid:20) β α (cid:21) Y ( t, f ) = − αf cosh( βf ) . (23)Setting ∂ t = 0 one obtains a nonlinear ODE in f which has the closed form solution as given by (12),which is the sum of the general solution (two opposite-sided exponentials) and a particular solution.Inverting the transformation back to X one obtains (11). For simplicity, we do not consider the currency union having positive inflation surprises, even though in alow real interest rate setting, it may lead to capital flows to the target zone currency. This mechanism canbe amplified by presence of multiple currencies in the target zone with cross-currency constraints on movementversus the target currency (Serrat, 2000). or the transient dynamics, we need to solve the following equation: ∂ τ X ( τ, f ) − σ ∂ ff X ( τ, f ) − β tanh( βf ) ∂ f X ( τ, f ) + αX ( τ, f ) = + αf,X (0 , f ) = 0 . (24)Writing X ( τ, f ) = X ∗ ( τ, f ) + X S ( f ) , Eq.(24) implies: − σ ∂ ff X S ( f ) − β tanh( βf ) ∂ f X S ( f ) + αX S ( f ) = + αf,∂ τ X ∗ ( τ, f ) − σ ∂ ff X ∗ ( τ, f ) − β tanh( βf ) ∂ f X ∗ ( τ, ( f ) + αX ∗ ( τ, f ) = 0 . (25)While the first line in Eq.(25) has already being solved in Eq.(11), the second line needs now to bediscussed. Writing again X ∗ ( τ, f ) cosh( βf ) := Y ∗ ( τ, f ) , we obtain: ∂ τ Y ∗ ( τ, f ) − σ ∂ ff Y ∗ ( τ, f ) + (cid:20) β α (cid:21) Y ∗ ( τ, f ) = 0 . (26)The smooth-pasting conditions given by Eq.(13) imposes: ∂ f X ∗ ( τ, f ) | f = f = 0 ⇒ { [ ∂ f Y ∗ ( τ, f )] − β tanh( βf ) Y ∗ ( τ, f ) } | f = f = 0 ,∂ f X ∗ ( τ, f ) | f = f = 0 ⇒ { [ ∂ f Y ∗ ( τ, f )] − β tanh( βf ) Y ∗ ( τ, f ) } | f = f = 0 . (27)We solve (26) by separation of variables and expansion over the basis of a complete set of orthogonaleigenfunctions. The solution can be expressed as Y ∗ ( τ, f ) = φ ( τ ) ψ ( f ) , and therefore we can write it as ˙ φ ( τ ) φ ( τ ) = λ k = σ ψ (cid:48)(cid:48) ( f ) ψ ( f ) − ρ where ρ = (cid:104) β + α (cid:105) .The time-dependent part solves to ψ ( τ ) = exp( τ λ k ) , and the fundamental-dependent part can be writtenas ψ (cid:48)(cid:48) ( f ) − λ k + ρ ) ψ ( f ) = ψ (cid:48)(cid:48) ( f ) + 2 Ω k σ ψ ( f ) = 0 . The rest of the derivations follow straightforwardly, solving for ψ and obtaining the eigenfunctions ψ k ( f ) = c cos (cid:18) √ k σ f t (cid:19) + c sin (cid:18) √ k σ f t (cid:19) . which form an orthogonal basis for the space of f well-behaving functions. Smooth-pasting conditionsimpose c = 0 , c = 1 and we obtain the form of the eigenfunctions as given by (16). The Fouriercoefficients follow in their standard form, using the stationary equation X S ( f t ) . C Risk, regime shifts and honeymoon effects
We now briefly discuss the connection between risk and the honeymoon effect, and how such effectscannot be be obtained when the destabilizing effects of risk shocks in the fundamental are too strong.For illustrative purposes, let us consider a baseline case of our model in a symmetric band [ − f , f ] aroundthe parity , and let us compare the DMPS-driven model with the standard Gaussian one. Omittingtime dependency, we have again the framework given by = f + 1 α E { dX } dt , which leads to the following couple of PDEs, depending on the form of the fundamental process. (cid:40) X = f + ∂ ff [ X ( f )] ( Gaussian ) ,X = f + ∂ ff [ X ( f )] + β tanh( βf ) ∂ f [ X ( f )] (DMPS) . We now focus on the stationary regime for which get the general solutions: (cid:40) X ( f ) = f + A sinh( ρ f ) , ( Gaussian ) ,X ( f ) = f + A β sinh( ρ β f )cosh( βf ) , (DMPS) , where ρ β = (cid:112) β + 4 α and A β is a yet undetermined amplitude. We now apply the smooth fittingprocedure at the target level + ¯ f .For the standard Gaussian framework we have X ( f ) (cid:55)→ X ( f ) = f + a sinh( ρ f ) , since β = 0 andconsequently ρ (cid:55)→ ρ := (cid:113) ασ . We therefore have : X ( f ) = a tanh( ρ f ) + f, ρ = (cid:114) ασ , which is the same result as in the standard Gaussian models. In particular, denote W the contact pointwith the target boundary ± ¯ f , we have ¯ f = X ( W ) ⇒ F = W + a tanh( ρ W ) , ρ a cosh( ρ W ) (smooth fitting at position W ) . (28)From the second line in Eq.(28), we conclude immediately that: a = − ρ cosh( ρ W ) . and accordingly, we end with: X ( f ) = f − sinh( ρ f ) ρ cosh( ρ W ) (29)Furthermore, we can verify that W ∈ R + for all values of the parameter ρ > . Eq.(29) implies that: W − F = tanh( ρ W ) ρ . It’s immediately seen that the last equation always possesses a single solution
W > . Let us nowexamine the paper’s main framework, the case where β > . In this case, for a target zone with bandsize F and a smooth contact point W , we have: F = W + a sinh( ρW )cosh( βW ) + ω tanh( βW )0 = 1 + a cosh( βW ) [ ρ cosh( ρW ) − β sinh( ρW ) tanh( βW )] + ωβ cosh ( βW ) . (30)The second line of the last equation implies: Due to the symmetry, we have here only one amplitude A to determine since only one boundary needs to beconsidered. = − cosh ( βW ) + βω cosh( βW ) cosh( ρW ) [ ρ tanh( ρW ) − β tanh( βW )] (cid:124) (cid:123)(cid:122) (cid:125) :=∆ . From the last line, let us consider the equation ∆ = 0 . First we remember from the very definition that ρ ≥ β and hence the equation: ρβ tanh( ρW ) = tan( βW ) ⇔ ∆ = 0 . Since ρβ > the las equation has necessary a solution which is denoted ± W c . Note in addition that fora couple of β such that β < β , we have: β > β ⇔ W c, < W c, (31)and for β → ∞ , we shall have W c → . Now from W solving the first line of Eq.(30), we may have thealternatives: a ) W c < W,b ) W c ≥ W. (32)For case a), standard boundary techniques cannot be applied as in the Gaussian case, and hence thelimit W = W c explains the regime transition observed in the spectrum. This is due to the fact that forlarge β the honeymoon effect range becomes effectively large enough to preclude the possible existenceof a target zone. D Noise sources driving the fundamental
Let us now assume that the fundamental is driven by a couple of noise sources, namely i) compositeshocks v t and ii) fluctuations in the money supply m t , given by Gaussian noise around a drift µ . Wetherefore add another source of noise, but we are not necessarily increasing the risk in the fundamentalprocess. We then have df t = σ dW ,t + dm t ,dm t = µdt + σ dW ,t , m t =0 = m . (33)where the noise sources dW ,t and dW ,t are two independent White Gaussian Noise (WGN) processes.We then obtain f t as a Gaussian process, since trivially df t = µdt + (cid:113) σ + σ dW t and we are exactly in the standard framework (in the literature usually µ = 0 ), only with a change invariance. If however we wish to incorporate a general increase in risk, and one that may represent theforce that was discussed in Section 2, we can write the following more general framework: df t = σ dW ,t + dm t ,dz β,t = ζ ( β ; z t ) dt + σ ( β ) dW ,t , z t =0 = 0 . where β ≥ is a control parameter and the repulsive drift ζ ( β ; z ) = − ζ ( β ; − z ) < models an extra risksource via a dynamic zero mean process. We parametrize risk with β , and therefore β = 0 simply implies σ ( β ) = ζ (0; z t ) = 0 implying that the process is Gaussian and driven entirely by the composite shockprocess. Our candidate for ζ is the DMPS process: f t = σ dW ,t + dz t = β tanh( βz t ) dt + σ dW ,t + σ ( β ) dW ,t ⇓ dz t = β tanh( βz t ) dt + (cid:104)(cid:112) σ + σ ( β ) (cid:105) dW t , z t =0 = 0 . where we used the fact that the difference between two independent WGN’s is again a WGN with varianceas given in the previous equation. Alternatively one may formally write: df t = σ dW ,t + β tanh β z t (cid:122) (cid:125)(cid:124) (cid:123) ( f t − σ W ,t ) dt + σ ( β ) dW ,t = β tanh β z t (cid:122) (cid:125)(cid:124) (cid:123) ( f t − σ W ,t ) dt + (cid:104)(cid:112) σ + σ ( β ) (cid:105) dW t , Using the initial equation (8) and the previous equation and applying Itô’s lemma to the functional X ( f t , t ) , we obtain: (1 − r ) α ∂ t X ( f, t ) + ∂ f X ( f, t ) E { β tanh [ β ( f t − σ W ,t ] } (cid:124) (cid:123)(cid:122) (cid:125) = β tanh[ β ( f )] + (cid:2) σ + σ ( β ) (cid:3) ∂ ff X ( f, t ) = X t − r f t (34)In the last Eq.(34), the under-brace equality follows since all odd moments in the expansion of thehyperbolic tangent vanish and the tanh( x ) is itself an odd function. Now, normalizing as to have (cid:2) σ + σ ( β ) (cid:3) = 1 , we are in the nominal setting of our paper. E Attracting drift: mean-reverting dynamics
A fully similar discussion can be done for mean-reverting fundamental dynamics (Ornstein-Uhlenbeckdynamics) reflected inside an interval [ f , f ] . In this section, the fundamental is driven by the mean-reverting dynamics: df = λ ( µ − f ) dt + σdW t , where µ is the “long-run” level of the fundamental, and λ is now the speed of convergence, to highlightthe mean-reverting equivalent of the DMPS process. Following the previous exposition, we can obtainthe full solution for the exchange rate X ∗ ( t, f ) as the solution of ∂ t X + σ ∂ ff X + λ ( µ − f ) ∂ f X − α − r X = − rα − r f. As before, we have the stationary solution for a vanishing ∂ t , and here it reads X S ( f ) = A F (cid:20) α λ (1 − r ) ,
12 ; λσ ( f − µ ) (cid:21) ++ B √ λσ ( f − µ ) F (cid:20) α λ (1 − r ) + 12 ,
32 ; λσ ( f − µ ) (cid:21) ++ (cid:20) λµ (1 − r ) f + rαλ (1 − r ) + α (cid:21) (35)where F [ a, b ; x ] is the confluent hypergeometric function. The integration constants A and B , as before,are determined via smooth pasting at the target zone boundaries, namely: ∂X S ( f ) | f = f = ∂X S ( f ) | f = ¯ f =0 . Note that if µ = 0 , then A = 0 . Figure 12 shows the stationary dynamics of the exchange rate as igure 12 Mean-Reverting Stationary Dynamics function of the fundamental, for different values of long-run level µ and noise variance σ . The band isassumed symmetric around 0, and ¯ f = 10% .The associated Sturm-Liouville equation is now given by σ ∂ ff X + λ ( µ − f ) ∂ f X + ρX = 0 , where ρ = α − r , and the spectrum of the process can be obtained explicitly by solving a transcendentalequation involving Weber parabolic cylinder functions. As before, the complete solution is given by anexpansion on a complete set of orthogonal functions on the target band, namely: X ∗ ( T − t, f ) = X S ( f ) + ∞ (cid:88) k =1 c k exp[ − (Ω k + ρ )( T − t )] ψ (Ω k , f ) , where the Fourier coefficients c k again impose the terminal condition X ∗ (0 , f ) = − X S ( f ) . As workedout by Linetsky (2005) explicit though lengthy closed form expressions are obtainable (see Eqs.(39) and(40). For the case of a symmetric target zone f = − f , an approximation valid for large eigenvalues Ω k ,(i.e. large k ’s) is given in [L] and reads: Ω k = k πσ f + λ c + O (cid:18) k (cid:19) c = λ σ (4 ¯ f − f µ + 3 µ ) . (36)The normalised eigenfunctions, also up to O (cid:0) k (cid:1) , read: ψ k ( f ) = ± σ √ f − / exp (cid:20) λ ( f − µ ) σ (cid:21) (cid:20) cos (cid:18) kπf f (cid:19) + 2 ¯ fkπσ φ ( f ) sin (cid:18) kπf f (cid:19)(cid:21) φ ( f ) = λ σ f − λ µ σ f − (cid:34) λ (cid:32) √ λσ µ + 1 + c (cid:33)(cid:35) f + θµ (37)While strictly speaking Eq.(36) furnishes very good estimates for large k values, a closer look in [L] showsthat even for low k ’s, ( k = 1 , , · · · ) , pretty good approximations are also obtainable. In particular, for = 1 , we approximately have: τ relax (cid:39) [Ω ] − = (cid:34) πσ f + λ c (cid:35) − . For this mean-reverting dynamics, the interplay between risk (here solely due to the noise source variance σ ) and the target band width f on t relax is opposite compared to the DMPS dynamics of section 2.The tendency of the fundamental f to revert to its long-run level µ , for a narrow target band, generatesan effect of an increase in risk (variance) that is opposite of the one generated by an increase of β in theDMPS setting, because of the latter’s tendency to escape from the mean. If the band is larger, lowerlevels of σ initially increase the relaxation time, to ultimately achieving a decreasing effect. In bothcases, an increase in the size of the target band requires a higher T in order for the target zone to befeasible.We lastly notice that for the O-U case, zero is always the first eigenvalue (not surprising, given that it’san ergodic process) and a regime shift cannot be possible. F Alternative to O-U dynamics: softly attractive drift
We now present the model where we model the fundamental as an ergodic process with a softly attractivedrift instead of the Ornstein-Uhlenbeck dynamics. This framework has the advantage of incorporatingmean-reverting dynamics while retaining analytical tractability. By “softly attractive” drift we mean theDMPS drift with opposite sign, i.e. − β tanh( βf ) . This model presents similar dynamics to the O-Uframework, and allows for a stationary time-independent probability measure. The marginal differencewith the O-U advantage is that the reversion of the fundamental to the mean is softer, and the advantageis that the full spectrum is available and the dynamics do not require an approximation. The equationfor the exchange rate after applying Itô’s lemma is now given by ∂ t X ( t, f ) + 12 ∂ ff X ( t, f ) − β tanh( βf ) ∂ f X ( t, f ) − αX ( t, f ) = − αf. (38)Using the equivalent transformation as in the DMPS case, we plug in Eq.(38) into Eq.(21) and obtain: (cid:82) f cosh( βζ ) ∂ t Y ( t, ζ )+ [ β sinh( βf ) Y ( t, f ) + cosh( βf ) ∂ f Y ( t, f )] − β sinh( βf ) Y ( t, f ) − α (cid:82) f cosh( βζ ) Y ( t, ζ ) dζ = − αf. (39)Now, taking once more the derivative of Eq.(39) with respect to f , one obtains: ∂ t Y ( t, f ) + 12 ∂ ff Y ( t, f ) − (cid:20) β α (cid:21) Y ( t, f ) = − α f cosh( βf ) . (40)Observe now that Eq.(40) is once again equivalent to the standard BM motion case and we can repeat thesame procedure we . The spectrum will now include the eigenvalue zero since we deal with a stationarycase.We now proceed as before and Eq.(40) reads: − ∂ τ Y ( τ, f ) + 12 ∂ ff Y ( τ, f ) − (cid:20) β α (cid:21) Y ( τ, f ) = − α f cosh( βf ) . (41)Consider now the homogenous part of Eq.(41), namely: ∂ τ Y ( τ, f ) + 12 ∂ ff Y ( τ, f ) − (cid:20) β α (cid:21) Y ( τ, f ) = 0 . As done before, the method of separation of variables leads us to introduce Y ( τ, f ) = φ ( τ ) ψ ( f ) and theprevious equation can be rewritten as: − ∂ τ ψ ( τ ) ψ ( τ ) + 12 ∂ ff ψ ( f ) ψ ( f ) − (cid:20) β α (cid:21) = 0 . and therefore we can write: − ∂ τ ψ ( τ ) ψ ( τ ) = λ k , ∂ ff ψ ( f ) ψ ( f ) − (cid:104) β + α (cid:105) = λ k Defining Ω k = (cid:104) β + α (cid:105) + λ k , the relevant eigenfunctions reads: ψ ( f ) = c sin( √ k f ) + c cos( √ k f ) . Going back to Eq.(21), the boundary conditions at the borders of the target zone f = − f reads: ∂ f (cid:34)(cid:90) f cosh( βζ ) ψ ( ζ ) dζ (cid:35) | f = f = 0 . which implies that: cosh( βf ) ψ ( f ) ⇒ c = 0 and Ω k = (2 k + 1) π √ f . (42)We note that Eq.(42) implies : λ k = (2 k + 1) π f − β − α ≥ . (43)Lastly, as expected, for the soft attractive case we are able to derive the exact spectrum analytically andunlike the DMPS case, there is no spectral gap.(43)Lastly, as expected, for the soft attractive case we are able to derive the exact spectrum analytically andunlike the DMPS case, there is no spectral gap.