random walk without a systematic tendency to revert towards its PPPequilibrium. Figure 14.2 illustrates, however, that this is not the case. According tothe first PPP puzzle, the so-called disconnect puzzle (see Sect. 3.1.3), exchange ratesoften get too far out of line with macroeconomic fundamentals in the medium term.This is better illustrated by Fig. 14.3 which displays the nominal effective exchange rate as a percentage deviation from equilibrium.We make three observations.1. The EUR/USD exchange rate seems to be more persistent when it is in the proximity of the long-run mean, whereas reversion towards the mean happens morerapidly when the absolute size of PPP deviation is large. In fact, the further awaythe real exchange rate moves from PPP, the stronger the adjustment intensitybecomes.2. This behavior can be observed in both states of positive disequilibrium and negative disequilibria, which gives rise to a symmetric kind of adjustment behavior.3. Within more than 30 years of data, the mean is crossed only 18 times whichindicates a remarkable degree of persistence of the real exchange rate.The observations are consistent with a strand of literature focusing on the existence of nonlinear dynamics in the real exchange rate, implying that the speed ofmean reversion is state dependent [387, 403]. This literature suggests that the observed real exchange rates may be stationary around a trend, albeit persistent, andthat it is very persistent in the neighborhood of PPP, while being mean-revertingat a faster speed when the deviation from PPP gets larger. Such models are called
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0.200.100.00−0.10−0.20−0.30−0.40Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan- Jan75 77 79 81 83 85 87 89 91 93 95 97 99 01 03 05 07 09 11+ Disequilibrium
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Fig. 14.3 EUR/USD disequilibrium
regime-switching models or (smooth) transition autoregressive ((S)TAR) models.References [387, 388, 403] have suggested a data-based modeling cycle for estimating STAR models. While this is certainly a rigorous procedure, it may lack economic intuition. We therefore propose a modeling procedure that is based on ourobservations and economic reasoning.
14.2.3 Component 2: Nonlinear Mean Reversion14.2.3.1 Transition FunctionLet G[st ; γ , c] be a continuous function (transition function) that determines thedegree of mean reversion and has the following properties: G[st ; γ , c] is boundedbetween 0 and 1 and is itself governed by• The transition variable st which is assumed to be a lagged endogenous variable[387], that is st = rt−d for certain integer delay parameter d > 0.1• The parameter γ > 0 which effectively determines the speed of mean reversion.• The locality parameter c which determines where the transition occurs. In casethe exchange rate is located close to equilibrium, there is high uncertainty about1 The transition variable can also be represented by an exogenous variable, or a function of laggedendogenous variables, or a function of a linear time trend, as given by [264].
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its future short-term course. Thus, it might be useful to determine a (close) corridor of exchange rate bandwidths that the future exchange rate is likely to passthrough.The regime that occurs at time t can be determined by the observable variable stand the associated value of G[st ; γ , c]. Different choices for the transition functionG[st ; γ , c] give rise to different types of regime-switching behavior. One popularchoice includes the first-order logistic function G[st ; γ , c] = (1 + exp{−γ (st − c)})−1which has been used for modeling asymmetric cycles of the underlying, for example, business cycle asymmetry to distinguish expansions and recessions [389, 390].However, in the case of exchange rates, this may not make sense. Our visual inspections and economic reasoning rather imply the existence of a symmetric bandaround the equilibrium rate in which there is no tendency of the real exchange rateto revert to its equilibrium value (inner regime). Outside this band (outer regime),commodity arbitrage becomes profitable, which forces the real exchange rate backtowards the band. If regime-switching of this form has to be captured, it appearsmore appropriate to specify the transition function such that the regimes are associated with small and large absolute values of st . This can be achieved by usingthe exponential (E)STAR function G[st ; γ , c] = 1 − exp{−γ (st − c)2 } according to[174]. The ESTAR function has been successfully applied to real exchange rates by([292, 386]) and to real effective exchange rates by [355]. However, according to[403], a drawback of the exponential function is that for either γ → 0 or γ → ∞ thefunction collapses to a constant (equal to 0 and 1, respectively). Hence, the modelbecomes linear in both cases and the ESTAR model does not nest a self-exciting(SE)TAR model as a special case, as would be given by the second-order logisticfunctionG[st ; γ , c] = (1 + exp{−γ (st − c1 )(st − c2 )})−1 , c1 ≤ c2 , γ ≥ 0
(14.2)
where now c = (c1 , c2 ) , as proposed by [205]. The properties of the second-orderlogistic function are depicted in Fig. 14.4.In this case, if γ → 0, the model becomes linear, whereas if γ → ∞ and c1 = c2 ,the function G[st ; γ , c] is equal to 1 for st < c1 and st < c2 and equal to 0 in between.Hence, the STAR model with this particular transition function nests a restrictedthree-regime SETAR model, where the restriction is that the outer regimes are identical. Note that for moderate values of γ , the minimum value of the second-orderlogistic function, attained for st = (c1 + c2 )/2 remains between 0 and 0.5, unlessγ → ∞. In the latter case, the minimum value does equal zero. This has to be kept inmind when interpreting estimates from models with this particular transition function.
14.2.3.2 Illustration and Theoretical RationalLet us further illustrate the concept of mean reversion by translating the logistictransition function of second order into the classical price chart. The future spot rate
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1.00c1
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Fig. 14.4 LSTAR2 transition functions for varying degrees of γ and fixed c
describes the deterministic sum of the PPP level at t = 0 and the future real exchangerate r̂t+1ŝt+1 = PPPt=0 + r̂t+1(14.3)
wherer̂t+1 =
rt :rt × G[st=0 ; γ , c] :
c1 ≤ st ≤ c2else
(14.4)
Hence, the future real exchange rate is dependent on the value of the transitionfunction G[st ; γ , c] if st lies in the outer regime, i.e., st < c1 or st > c1 .Figure 14.5 illustrates the smooth adjustment of future spot to the level of PPPat March 2005 according to (14.3) with PPPt=0 = 1.1250 and G[st=0 = 1.3184, γ =50, (c1 , c2 ) = (0, 0)]. Equilibrium theory states that if exchange rates get too far outof line with macroeconomic fundamentals, “gravitational forces” must bring thesystem back to equilibrium. These forces lead to nonlinearities in the adjustmentof exchange rates which can be justified from both a goods and a financial market perspective. As international goods arbitrage involves transaction costs, such ascosts of transportation and storage of goods, arbitrage sets in once the real exchangerate protractedly moves outside certain limits. Consequently, commodity arbitragedoes not compensate for the costs involved in the necessary transactions for smalldeviations from the equilibrium real exchange rate. From an asset market perspective, differing trading strategies could play a role with chartists dominating marketdynamics when the exchange rate is close to some perceived equilibrium, whilefundamentalists being at play once the exchange rate is increasingly misaligned.
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1.301.201.101.000.900.800.700.60Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10+ disequilibrium
− disequilibrium
Spot
PPP
Smooth Trans
Fig. 14.5 Nonlinear mean reversion
14.2.4 Component 3: Gaussian Random Walk14.2.4.1 DefinitionAccording to the second PPP puzzle, the so called excess volatility puzzle (seeSect. 3.1.3), the enormous short-term volatility of real exchange rates is contradicting to the extremely slow rate of adjustment. As we observed, despite the meanreverting behavior, the real exchange rate also indicates a remarkable degree of persistence. We therefore assume that in the short term volatility becomes increasinglyimportant. In fact, the adjustment process can be overshadowed/dominated by market volatility. This phenomenon is modeled by a simple one-dimensional randomwalk which is given by the AR(1) processŝt+1 = st + ε̂t+1
(14.5)
where ε̂t+1 ∼ NID(0, σ 2 ) with constant mean and finite variance.The normal distribution is based on the central limit theorem, which describesa tendency as the number of observations becomes infinitely large. Typically, thisassumption is not made because it allows to model financial markets more accurately than competing frameworks, but because it is transparent and tractable. Forexample, the assumption of normally distributed returns need not imply that we believe that returns have been normal in the past, or expect them to be normal in the
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future. We may simply believe that forecasts based on the normal distribution havenot done significantly worse than those based on other more sophisticated models.Or, even if we believe that other models have done better in the past, we may beunable to predict that they will do better in the future. In addition, it is clear thatour way of attempting to model the market is first and foremost determined by thesampling frequency of interest. In contrast to many financial firms, industrial firmsnaturally view their financial market exposures over monthly, or quarterly periodsof time. As a consequence, there is no need to sample the market on a daily or evenhigh-frequency basis and thus there is no need for modeling excess kurtosis, i.e.,fat tails. As the news flow to markets causes less variation in returns when comparing longer periods, extreme values are averaged out and therefore, normality tendsto become more pronounced as the sampling frequency of price changes decreases([375], pp. 145–165).
14.2.4.2 Simulation of a One-Dimensional Random WalkExchange rate returns can be computer-simulated by randomly (independently)drawing from historically observed returns (historical simulation), or drawing randomly (independently) percentiles from a certain probability distribution (MonteCarlo simulation). “Independently” means that every simulated exchange rate return εt is independent of εt−1 . We opt for the simple Monte Carlo method [202],in which the range of probable values for an uncertain input parameter is dividedinto ordered segments of equal probability. The values are sampled in such a way asto generate samples from all the ranges of possible values, producing informationabout the extremes of the output probability distributions. Compared to variancereduction sampling techniques, such as the Latin Hypercube method, which reducesthe number of solutions (and may therefore neglect the likelihood of extremes) inorder to decrease computation time, the simple method is found to be more appropriate in our context, that is for financial problems involving the assessment of risk.Based on today’s value of the spot rate s0 , randomly generated future exchange ratereturns are linked to paths of length T .The exchange rate st is derived from the exchange rate at st−1 to which the simulated change εt was added, where εt can be either positive or negative. Accordingto this principle exchange rate paths of any length can be simulated. The procedurecan be summarized in four steps as follows:1. Determine s0 (which is the spot rate in our case), I (which is the number ofscenarios), T (which denotes the simulation horizon)2. Simulation: Set i = 1. Generate the ith scenario εi,t for every period t = 1, 2, . . . , Twithin the simulation horizon3. Path Linking: st = s0 + εi,1 + εi,2 + . . . + εi,T4. Iteration: set i = i + 1; εi,t , i = 1, 2, . . . , I; t = 1, 2, . . . , T until i = I.
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1.301.201.101.000.900.800.700.60Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10+ disequilibrium
− disequilibrium
Spot
GGK
−20%
20%
Fig. 14.6 Random walk without drift
Figure 14.6 illustrates 10 out of 10,000 randomly generated, driftless, exchangerate scenarios starting at a spot rate of EUR/USD = 1.3184 as per March 2005 andan estimated standard deviation of σ = 0.025. The dotted bold black lines representthe 5% and 95%, i.e., according to our assumptions there is a 90% probability thatthe exchange rate will move within this bandwidth.
14.2.5 Aggregation of Components14.2.5.1 Gaussian Random Walk with Nonlinear DriftThe aggregation of the three single components results in a random walk model withnonlinear drift. Let Δt+1 = rt+1 − rt be the difference of the real exchange rate fromthe next state to the state of today and let rt+1 be defined according to (14.4). Then(14.5) can be written asst+1 = PPPt=0 + rt+1 = st + Δt+1
(14.6)
From (14.6) and (14.5) we obtainst+1 = st + Δt+1 + εt+1 , t = 0, 1, 2 . . . , Twhere εt+1 ∼ NID(Δt+1 , σ ).
(14.7)
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1.301.201.101.000.900.800.700.60Jan-99 Jan-00 Jan-01 Jan-02 Jan-03 Jan-04 Jan-05 Jan-06 Jan-07 Jan-08 Jan-09 Jan-10+ disequilibrium
− disequilibrium
Spot
GGK
Smooth Trans
Fig. 14.7 Random walk with nonlinear drift
14.2.5.2 IllustrationThe behavior of the model is illustrated by Fig. 14.7 and illustrates 10 out of10,000 randomly generated, exchange rate scenarios starting at a spot rate ofEUR/USD = 1.3184 as per March 2005 and an estimated standard deviation ofσ = 0.025. The drift is modeled by smooth adjustment of future spot to the levelof PPP as per March 2005 with PPPt=0 = 1.1250 and G[st=0 = 1.3184, γ = 50,(c1 , c2 ) = (0, 0)]. Accordingly, while the Euro real exchange rate can be well approximated by a random walk if PPP deviations are small, in periods of significantdeviations, gravitational forces are set to take root and bring the exchange rate backtowards its long-term trend. Exchange rate scenarios are PPP-reverting as shown bythe 5% and 95% bandwidths. Still, there may be scenarios where PPP adjustmentis overshadowed by market volatility and no direct adjustment can be recognized.This behavior is realistic since it matches the second PPP puzzle.
14.2.6 Calibration of Parameters14.2.6.1 Estimation of γ and cWe recommend to estimate the slope parameter γ and the location parameter c of(14.4) in three steps as follows:
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1. Given the current market circumstances, identify c subjectively2. Estimate γ by OLS, conditional on c3. Undertake maximum likelihood estimation of the selected nonlinear model fromStep 2, including estimation of γ and cWe found this procedure to work well in practice. Recall that the locality parameter c determines where the transition occurs. From an asset market perspective,we recommend to estimate c subjectively, rather than determining it by a purelydata driven approach [387, 403]. If “chartists” dominate market dynamics, thenthere usually exists something like a “perceived equilibrium,” i.e., a critical shortterm threshold. This may be hard to determine based on historical exchange ratedata since it depends on market expectations. Instead, one might use short-terminformation about technical equilibria as obtained from banks and brokers. Alternatively, companies might also use the interest rate differential as given by the forwardrate, as an indicator for short-term equilibrium value. For instance, one could perceive technical equilibrium to be 5% above/(below) fundamental equilibrium, i.e.,c = (c1 , c2 ) = ((1 − δ )PPPt=0 , (1 + δ )PPPt=0 ) with δ = 0.05. This would implythat, in case the exchange rate approaches c, there will be no further adjustmenttowards PPP since the model collapses to a Gaussian random walk without drift.Given c, we estimated γ by OLS, i.e.,T
I
min Z = ∑ ∑ (ri − r̂i )2
(14.8)
t=1 i=1
with rˆi according to (14.4) and t = 1, 2, . . . , T describes the backtesting period. Theindex i = 1, 2, . . . , I describes the forecasting period for γ which was set to I =36 months. OLS was performed with the Microsoft Excel 2000’s standard solver([150]) in order to derive optimal values for γ .14.2.6.2 Estimation of σThe standard deviation of ε̂t+1 from (14.7) was estimated by taking the square rootof an unbiased estimate of the empirical variancet1σ̂t =(14.9)∑ (si − s̄)2t − 1 i=t−23where s̄ is the mean of EUR/USD first differences over the last 24 months.
14.3 Optimization Model14.3.1 Solution Construction AlgorithmWe propose the use of a metaheuristic combinatorial search algorithm. The motivation is twofold. First, we have shown that our problem is NP-hard and cannot
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be solved in polynomial time which directly suggests that solutions can only beapproximated. Second, the problem is nonconvex and nonsmooth which makes itdifficult to find an optimal solution since one cannot simply follow the gradient ofthe objective function.Metaheuristics are modern heuristics which have been first introduced in [161,162], and have proven to be a very exciting and practical development in approximate optimization techniques. They have had widespread successes in attacking avariety of difficult combinatorial optimization problems that arise in many practicalareas. Osman and Laporte [316] conducted a comprehensive list of 1,380 referenceson the theory and application of metaheuristics. A more recent survey on metaheuristics in stochastic combinatorial optimization has been conducted by [36]. Until today, there is no commonly accepted definition, although researchers have mademany attempts (see, for instance, [46, 316]). In short, a metaheuristic refers to amaster strategy that guides and modifies other heuristic methods to produce solutions beyond those that are normally generated in a quest for local optimality. Inorder to do so, metaheuristics incorporate a dynamic balance between intensification and diversification. The term intensification refers to the exploitation of theaccumulated search experience, whereas the term diversification, generally refers tothe exploration of the search space.2 Some researchers consider these two requirements to be naturally conflicting [28], others to be interplaying [163]. Regarding thelatter, the distinction between intensification and diversification is often interpretedwith respect to the temporal horizon of the search.In some instances we may conceive of intensification as having the function of an intermediate term strategy, while diversification applies to considerations that emerge in the longerrun. [163]
Short-term search strategies can therefore be seen as the iterative application oftactics with a strong intensification character. When the horizon is enlarged, usually strategies referring to some sort of diversification come into play. Either way,a balance between intensification and diversification must be found, on the onehand to quickly identify regions in the search space with high quality solutions,and on the other hand not to waste too much time in regions of the search spacewhich are either already explored or which do not provide high quality solutions.The balance between intensification and diversification might be quite different notonly across different metaheuristics, but also for different components, which aregenerally understood as operators, actions, or strategies of a specific metaheuristicalgorithm.
2 Note that diversification is not the same as randomization. Whereas the goal of diversification isto produce solutions that differ from each other in significant ways, and that yield productive alternatives in the context of the problem considered, the goal of randomization, is to produce solutionsthat may differ from each other in any way, as long as the differences are entirely “unsystematic.”
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14.3.2 Scatter Search and Path Relinking14.3.2.1 Pseudocode TemplateWe use a procedure that is based on the scatter search methodology, but incorporates innovative mechanisms to exploit the knowledge of the problem and to createa trade-off between intensification and diversification for an efficient search. Theseminal ideas of scatter search originated in the late 1960s. A first description ofscatter search appeared in [161]. The modern version of the method is described in[239]. Scatter search and its generalized form, called path relinking [166,167], haveproven to be highly successful for a variety of known problems, such as vehiclerouting [18, 337, 338], tree problems [68, 423], mixed integer programing [165], orfinancial product design [81].Scatter search is a search strategy that operates on a set of solutions, the reference set, by generating linear combinations of the reference solutions to create newones. The resulting solutions are called trial solutions. These trial solutions may beinfeasible solutions and are therefore modified by means of a repair procedure thattransforms them into feasible solutions. A heuristic improvement mechanism is thenapplied in order to try to improve the set of trial solutions. These improved solutionsform the set of dispersed solutions. The new set of reference solutions that will beused in the next iteration is selected from the current set of reference solutions andthe newly created set of dispersed solutions. The coupling of heuristic improvementwith solution combination strategies, as is the case in scatter search, has inauguratedthe term memetic algorithms (see, e.g., [300]). The Pseudocode template for scattersearch, according to [163], has served as the main reference for most of the scattersearch implementation and can be stated as according to Fig. 14.8. The template isdivided into two phases: an initial phase that generates a first set of dispersed solutions, and a scatter search or path relinking phase that seeks to improve the initialsolution by iteratively searching through the trajectory.The six components processes may be sketched in basic outline as follows:1: Initial Phase:2:Seed Generation()3: Repeat4:Diversification Generator()5:Improvement()6:ReferenceSetUpdate()7: Until the reference is of cardinality n8: Scatter Search/Path Relinking Phase:9: Repeat10:Subset Generation()11:Solution Combination() //Path Relinking12:Improvement()13:ReferenceSetUpdate //Tabu List14: Until termination criteria metFig. 14.8 Scatter Search pseudocode template
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1. Seed: create one or more seed solutions, i.e., arbitrary trial solutions used toinitiate the remainder of the method.2. Diversification Generation Method: generate a collection of diverse trial solutions, using one or more seed solutions as an input. Check feasibility of trialsolutions and repair solution if it is infeasible.3. Improvement Method: transform a trial solution into one or more enhanced trialsolutions.4. Reference Set Update Method: build and maintain a reference set consisting ofthe b best solutions found, organized to provide efficient accessing by other partsof the solution procedure. Several alternative criteria may be used to add solutions to the reference set and delete solutions from the reference set.5. Subset Generation Method: operate on the reference set to produce a subset of itssolutions as a basis for creating combined solutions. The most common subsetgeneration method is to generate all pairs of reference solutions.6. Solution Combination Method: transform a given subset of solutions producedby the Subset Generation Method into one or more combined solutions.The interconnection between above methods is described in Pseudocode form byFig. 14.9 [163] and will be explained in more detail in the next paragraphs. Thescatter search methodology is very flexible since each of its above methods canbe implemented in a variety of ways and degrees of sophistication. The advancedfeatures of scatter search are therefore related to the way these methods are implemented. That is, the sophistication comes from the implementation of she scattersearch methods instead of the decision to include or exclude certain elements (as inthe case of tabu search or other metaheuristics).
14.3.2.2 ImplementationDiversification Generation MethodThe diversification generation method is used only once to build a large set of PSizedifferent solutions P at the beginning of the search, where PSize is 10 times the sizeof RefSet, and is never employed again. An initial reference set of solutions (RefSet)is created which consists of b = PSize/10 distinct and maximally diverse solutions.Consider the midpoint of the initial populationxi = li +
(ui − li ), i = 1, . . . , n2
(14.10)
where L = {li : i = 1, . . . , n} is the set of lower bound values and U = {ui : i =1, . . . , n} is the set of upper bound values for all candidate solutions xi ∈ X. Additional points are generated with the goal of creating a diverse population. A population is considered diverse if its elements are significantly different from one another.An Euclidean distance measure d(x, y) is used to determine how “close” a potentialnew point x, i.e., a candidate solution, is from the points y already in the population
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1. Start with P = .••••
Use the diversification generation method to construct a solution yApply the improvement method to y. Let x be the final solutionIf x ∈/ P then add x to P (i.e., P = P ∪ x), otherwise, discard xRepeat step until |P| = PSize. Build RefSet = {x1 , . . ., xb } with b diverse solutions in P
2. Evaluate the solutions in RefSet and order them according to their objective function valuesuch that x1 is the best solution and xb the worst. Make NewSolutions =TRUE.while (NewSolutions) do3. Generate NewSubsets, which consists of all pairs of solutions in RefSet that include atleast one new solution. Make NewSolutions =FALSE.while (NewSubsets = ) do4. Select the next subset s (i.e., the next pair (x , x )) in NewSubsets.5. Apply the Solution Combination Method to s in order to obtain one or more newsolutions y.//Path RelinkingProduce the sequence x = x(1), x(2), . . . , x(r) = xfor i = 1 to i < r/NumImp do6. Apply the Improvement Method to x(NumImp × i).end for7. Apply the Relinking Method to produce the sequence x = y(1), y(2), . . . , y(s) = xfor i = 1 to i < s/NumImp do8. Apply the Improvement Method to y(NumImp × i) and obtain x.end forfor (each generated solution x)if (x is not in RefSet and f (x) < f (xb )) then9. Make xb = x and reorder RefSet.10. Make NewSolutions =TRUE.end ifend for11. Delete s = (x , x ) from NewSubsets.end whileend whileFig. 14.9 Scatter Search/Path Relinking pseudocode
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(in RefSet), in order to decide whether the point is included or discarded. That is,for each candidate solution x in P − RefSet and reference set solution y in RefSet, ameasure of distance or dissimilarity d(x, y) is calculated. The candidate solution isselected which maximizesdmin (x) = miny∈RefSet {d(x, y)}
(14.11)
Since linear constraints are imposed on a solution x, every reference point x is subject to a feasibility test before it is evaluated, i.e., before the simulation model is runto determine the value of f (x). The feasibility test consists of checking whether thelinear constraints imposed are satisfied. An infeasible point x is made feasible byformulating and solving a linear programing (LP) problem with the goal of findinga feasible x∗ that minimizes the absolute deviation between x and x∗ .min d − + d +subject to AX ∗ ≤ B∗
−
(14.12)
+
X −X +d +d = 0L ≤ X∗ ≤ Uwhere d − and d + are negative and positive deviations from the feasible point x∗ tothe infeasible reference point x.
Improvement MethodOnce the population is generated, the procedure iterates in search of improved outcomes. In each iteration two reference points are selected to create four offsprings.Let the parent-reference points be x1 and x2 , then the offspring x3 to x6 are found asfollows:x3 = x1 + dx4 = x1 − dx5 = x2 + d
(14.13)
x6 = x2 − dwhere d = (x1 − x2 )/3. The selection of x1 and x2 is dependent on the objectivefunction values f (x1 ) and f (x2 ),