1998, we assumed that the firm’s net income was USD 100.As per 15 January 1999, a hedge intensity of ten months (covering February 1999to November 1999) was suggested by the procedure described in Fig. 15.4. Ninemonths, i.e., February 1999 to October 1999, were assumed to be covered in thepast, each by a proportion of 1/3 forwards and 1/3 straddle options. The remaining1/3 of the following 9 months’ exposure, along with the exposures from month 11to month 24, were assumed to remain unhedged. In order to obtain a recommendation on the composition of the hedge for maturity November 1999, the simulation/optimization procedure was started. Upon determination of the search, the bestvalues for the decision variables as obtained by OptQuest were recorded. The costof the hedge was calculated (straddle premium + transaction cost of 3 pips) and deducted from last month’s net income. Next, the clock was advanced by one month,and the financial losses or gains made on the forward and straddle positions weredetermined. Thus, on 15th January 1999, the first day of the backtesting period,all cashflow maturities have decreased by one month (i.e., the exposure in month2 becomes an exposure in month 1, the exposure in month 3 becomes an exposurein month 2, etc.) and a new exposure in month 24 appears in the 24 months timewindow. In realigning the forward and straddle contracts to the current spot levelseither some profit on the currently held contracts is realized or the selective hedgehas led to a loss. Since the decision model uses data sets, which were updated as westepped through time, the scenario generator created a completely new set of scenarios in each month looking ahead over a time horizon of T = 24 months. Giventhat the decisions were made altogether 72 times, by stepping through the time linewe processed the corresponding simulation/optimization model 72 times using theCrystal Ball/Optquest system. To verify whether multiproduct hedging with variableweights has advantages over simpler approaches, the dynamic rolling procedure wasnot only applied to the mean-variance-skewness optimization model but also to fourbenchmark strategies:1. 100% spot strategy (no hedging), i.e., xS = 1 and xF = xO = 0, for t = 0, . . . , T2. 100% forward strategy, i.e.,1 : t = 0, . . . ,t ∗∗• xF (t ) =0 : t = t ∗, . . . , T• xO = 0 for t = 0, . . . , T0 : t = 0, . . . ,t ∗∗• xS (t ) =1 : t = t ∗, . . . , T
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3. 100% straddle strategy, i.e.,• xF = 0 fort = 0, . . . , T1 : t = 0, . . . ,t ∗• xO (t ∗ ) =0 : t = t ∗, . . . , T0 : t = 0, . . . ,t ∗• xS (t ∗ ) =1 : t = t ∗, . . . , T4. Mixed strategy consisting of 1/3 at-the-money forwards, 1/3 at-the-money straddle call options, and 1/3 spot, i.e.,0.33 : t = 0, . . . ,t ∗∗• xF (t ) =0 : t = t ∗, . . . , T0.33 : t = 0, . . . ,t ∗• xO (t ∗ ) =0 : t = t ∗, . . . , T0.34 : t = 0, . . . ,t ∗• xS (t ∗ ) =1 : t = t ∗, . . . , TStrategies 1–4 are nonpredictive beyond the given grid recommendation since theydo not allow the hedge to time-vary but are fixed-weight strategies and therefore donot seek to exploit the history of market information.
15.3 Results15.3.1 Ex Ante PerformanceFigure 15.10 plots the objective function values of the proposed mean-varianceskewness utility function against the four benchmark strategies over the 8 year outof-sample backtesting period from January 1999 to December 2006. Grey bars indicate times when decisions on the optimal combination of instruments were delayedin order to reduce hedge intensities.On 15 December 1998 – as mentioned above – all four strategies start withsame initial product mix, a hedge intensity of ten future months (covering February1999 to November 1999) with each of them being covered by a proportion of 1/3forwards and 1/3 straddle options. The remaining 1/3 of the following 9 months’exposure along with the exposures from M11 to M24 remained unhedged.We observe that from here on the utility of the spot strategy gradually declinesover the next 9 months, and in September 1999 jumps down to a utility value below zero. Since no further hedges are made, the strategy has the highest varianceconsistently over time which is punished by the objective function. In addition,since it does not represent a hedge, the payoff distribution from which the mean andskewness values are derived becomes irrelevant. This explains the smooth path ofthe spot strategy at a negative level of utility.Visual inspection of Fig. 15.10 further reveals that the forward strategy provides the most volatile objective function values taking on relatively high – albeit
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Fig. 15.10 Historical comparison of objective function values
volatile – utility values from January 1999 to May 2003, and a low level of utilityfrom May 2003 onwards. This can be explained as follows. According to PPP, theUSD is strong compared to the EUR from January 1999 to May 2003. Relativelyfew USD have to be given up in order to buy one EUR. Since the USD is overvalued, we expect that it will become weaker in the future which is expressed by thestochastic smooth transition PPP process incorporating a nonlinear positive trend inthe exchange rate scenarios towards equilibrium. This in turn implies that forwardhedges implemented in the past are in-the-money which results in a positive meanof the payoff-distribution. From May 2003 onwards, the exact opposite holds. TheUSD overvaluation cycle has ended and the spot rate enters into a phase of USDundervaluation. The mean of the probability distribution of payoffs becomes negative because it is expected that – due to gravitation – future spot is likely to go downat a pace γ that is linked to the degree of USD undervaluation and forward hedgesimplemented in the past are therefore out-of-the-money. To sum up, entering intoa forward contract always includes a directional bet on the future path of the spotexchange rate while the variance of the open position is symmetrically reduced andthird-moment effects do not exist.The straddle strategy clearly dominates both the spot and forward strategiesthroughout the whole period. The reason is twofold. First, like forwards, straddleseliminate exchange rate risk as measured by σ . Second, unlike forwards, theyprovide a profit/loss potential beyond the forward contract depending on whetherexchange rate volatility expectations were closer to reality ex post than those ofthe counterparty selling the straddle. If exchange rate volatility was expected to behigher than implied volatility in the capital market, this had a direct positive impacton the mean of the straddle contract’s payoff distribution, and vice versa. From
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January 1999 to May 2003, spot upward movements were predicted to be largeenough by the smooth transition PPP model to activate the call option of the straddle and offset the premium paid. On average, the contribution of the mean payoffon the objective function value was higher than the skewness contribution. FromMay 2003 onwards, EUR/USD downward movements were predicted to be largeenough to activate the put option of the straddle which kept the strategy profitable.Still, expected straddle profits were not expected to be as high as the ones in thefirst half of the backtesting period. One reason is that the estimated values for γcontinually decline within our estimation procedure as a result of averaging. Thisresults in a slowing adjustment and therefore implies smaller profit potential forstraddles. The other reason is that hedge intensities were reduced during this periodwhich implies smaller profit potential due to less hedges. Instead, the skewnesscontribution increased in the second half.A comparison of the strategies confirms our expectation that the variable-weightmean-variance-skewness strategy’s utility level (black line) dominates the traditional fixed weight allocation prescriptions over time. The optimized meanvariance-skewness strategy is closely mirrored by the straddle strategy. Then, fromAugust 2003 onwards the two strategies diverge and the proposed strategy is bettermirrored by the 1/3 strategy. This is a natural result of the scatter search optimization routine tending to select the best of both worlds: completely closed positions upto the recommended hedge intensity with an emphasis on straddles during the USDovervaluation period from January 1999 until April 2003, and just as many hedgesas necessary to suffice the given degree of risk-aversion during the USD undervaluation. In particular, it was the skewness criterion that drove decisions from April2003 – the turning point in the backtesting period when the USD overvaluation(EUR undervaluation) cycle turns into a USD undervaluation (EUR undervaluation) phase – onwards. Although in the aftermath of April 2003, the mean of thestraddle’s payoff distribution is consistently higher than the mean of the optimizedpayoff distribution, this comes at a tremendous cost of skewness. This seemed atfirst surprising since the optimized strategy picks – on average – less straddle contracts (−21%), but a higher proportion of forwards (+10%) and a noticeably higherspot proportion (+11%) instead. Hence, despite the strategy containing less options,it is more positively skewed since the probabilistic losses of the forward proportionoffset part of the straddle’s probabilistic profits (asymmetric distribution). Furthernote, that in the first half of the backtesting period there were months when theoptimized strategy fell below the straddle strategy, and in the second half of thebacktesting period there exist months when the optimized strategy fell below themixed strategy of 1/3 spot, 1/3 forwards, and 1/3 options. The reason is, that at timeswhen hedging intensities were reduced (“do nothing”), i.e., exposures were left uncovered and completely exposed to spot rate fluctuations, worse objective functionvalues were achieved due to lower skewness if the optimized strategy included lessstraddle contracts than the competitive strategies. Overall, the straddle, forward, andspot strategies perform significantly worse than the mixed strategies. The problemwith the straddle strategy is that option premia are expensive and reduce the mean,in particular, at times when volatility in the market was lower than expected. In
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contrast, the forward’s purpose to cheaply reduce or eliminate income volatilityturns out to be problematic if spot moves in the opposite direction. One can benefitfrom such moves by following a pure spot strategy and leave the exposure totallyopen, however the high variability of results is punished by the objective function.
15.3.2 Ex Post PerformanceAn ex post comparison of the different hedging strategies’ net income is given inFig. 15.11.There is no dominant strategy that outperforms the other ones constantlyover time. However, both the straddle strategy and the optimized mean-varianceskewness strategy seem to be clearly preferable over the other strategies. All fivestrategies started with a net income of USD 100 which was reduced to USD 99.82in January 1999 as a result of paying straddle option premia on 1/3 of the exposedamounts for the next 9 months. The simplest approach, presented by the red line isthat of no hedge at all. In this case, the portfolio simply comprises a long position inthe spot market having zero payoff from October 1999 on, when the initial positionswere settled.The forward strategy leads to the most volatile P&L as a result of the forwardbias, a phenomenon vastly described in both academic [114, 125] and practicalliterature [29]. What this literature implies is that historically, rather than the USDdepreciating as suggested by the forward rate, the USD was actually appreciatingfrom January 1999 until the beginning of 2002 causing large losses of the forward
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Fig. 15.11 Historical comparison of profit and loss accounts
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strategy at expiry. This led to a minimum net income of USD 96.3 in January2002. In contrast, positive returns were achieved by the forward strategy over thefollowing months due to a forward premium discount bias, i.e., the USD turnedout to be weaker as initially expected by the forward rate. Hence, the realized netincome recovered since January 2002 exceeding the initial USD 100 two years laterin January 2004, reaching the maximum of USD 101.95 in May 2005, and movingsideways until the end of the backtesting period. The 1/3 strategy is the least volatilein terms of profit/loss, mainly due to its large spot proportion which limits upside,as well as downside profit potential. The straddle strategy seems to provide verydesirable results over time. One possible explanation is that the capital markets underestimated future EUR/USD volatility when pricing the options. Santa-Clara andSaretto [354] examined various strategies, including naked and covered position inoptions, straddles, strangles, and calendar spreads, and find support for mispricingin the options markets which cannot be arbitraged away due to high trading costsand margin requirements.The mean-variance-skewness strategy generates a growth path with similar patterns to those of the 1/3 strategy until December 2002, then rises strongly andoutperforms the 1/3 strategy and even the straddle strategy. The latter fact comesat the expense of increasing volatility. Therefore, the firm implementing the multiproduct optimal hedging model will be significantly less than fully hedged. Onepossible interpretation of the better performance of the dynamically optimizedstrategy over the naive hedges is that it uses short-run information, while the naivestatic hedges are driven by long-run considerations and an assumption that therelationship between spot and forward price movements is 1:1. For instance, if theindication is in favor of a USD appreciation, then being less than fully hedged willlead to gains in the spot market conversion, and lower losses (or no losses) on theforward contract leading to much better financial results. On the other hand, if theindication is in favor of a depreciation of the foreign currency, then being hedged ispreferable to being unhedged and will lead to gains in the forward contracts.An alternative way of presenting ex post results of the rolling decision model isgiven by Fig. 15.12 which provides a histogram of the different strategies’ realizedprofit/loss. In addition to Fig. 15.12, Table 15.3 provides the associated summarystatistics of the backtested hedging strategies’ probability distributions.The forward strategy has the longest left tail and implies the highest risk of ashortfall in net income. With a standard deviation of 1.95, it is also the most volatilestrategy. From a risk-return perspective, the forward strategy has the lowest SharpeRatio and the weakest return per unit of earnings-at-risk (EaR) which was calculatedas the difference between the initial amount USD 100 and the 5% quantile of the respective probability distribution of realized profit/loss. The straddle strategy and the1/3 strategy reveal similar risk-return characteristics if compared by their Mean/EaRratios with the straddle strategy at 79.84 being slightly better. However, if comparedagainst traditional Sharpe Ratio, superior results are provided by the 1/3 strategywhich is due to the lowest standard standard deviation among all strategies. The optimized mean-variance-skewness strategy provides superior out-of-sample results ifprofit/loss risk is not perceived symmetrically but in terms of downside risk. In fact,
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100.4099.29100.3299.82
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102.47101.95101.94101.21
67.4550.9397.88110.95
91.2827.2879.8478.75
most corporations and investors would be foolish to not accept the good risk, butwould be negligent to not eliminate the bad risk.Still, the explanatory power of the results is limited, especially if we considerthat exchange rate adjustment is a long-term phenomenon. We can only get a vagueidea on whether the forecasts from a STAR model are always superior to those ofthe linear models out-of-sample. Evaluating forecast performance is therefore intrinsically problematic. As noted by [403], even though nonlinear time series oftencapture certain characteristics of the data better than do linear or random walk models, the forecast performance of the former is not always superior, and is sometimeseven worse. Hence, there is no guarantee that the estimated LSTAR models willproduce superior forecasts. A necessary condition for that to happen would seemto be that the forecasting period contains nonlinear features [180]. For instance, anonlinear model may be expected to be superior to a linear one when the forecasting period contains the aftermath of a large negative shock. If that is not the case, alinear autoregressive model is likely to perform as well as a nonlinear one. Anotherpossible reconciliation is offered by [89], who suggest that regime switching modelsmay forecast poorly owing to the difficulty of forecasting the regime that the series
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will be in. Thus, any gain from a good fit of the model within the regime will belost if the model forecasts the regime incorrectly. Estimated values of parametersμ , σ , and γ in the exchange rate process using MCS are therefore indicative butnot perfectly predictive of similar parameters for future exchange rate returns. Thiswill be true for the parameters of any parametric exchange rate evolution model.Therefore, there is an inherent uncertainty associated with mean, standard deviation, adjustment speed, etc. of future exchange rate returns. As a result, the apparentprecision of an exact solution is somewhat deceptive since the problem may itselfbe viewed as not exactly defined due to lack of precision in the problem parameters.Therefore, precision of the parametric solution is a problem only if the parametersfor exchange rate dynamics are known with greater precision. In case parameters areknown with higher precision, a different parameterization may be used for a betterapproximation. This will however, result in a larger problem size. Despite the manyproblems that economic forecasts from economic systems confront, these modelsoffer a vehicle for understanding and learning from failures, as well as consolidating our growing knowledge of economic behavior.
Chapter 16
Exchange Rate Forecasting with SupportVector Machines
This thesis is an interdisciplinary work that combines computer science, econometrics and financial engineering in order to support human decision-making in thecontext of trading and hedging the foreign exchange market. The first part of thedissertation provides an introduction to basic macroeconomic and financial theories on exchange rates. While this part does not advance any new ideas, it presentsan original synthesis of existing work from different research directions, stressingthe relevance of computational complexity theory to address financial market phenomena which cannot be explained by conventional financial and macroeconomictheories. The remainder of the dissertation makes the following contributions.Abstract – Purpose: Part II examines and analyzes the general ability of support vector machine (SVM) models to correctly predict and trade daily EUR/GBP,EUR/JPY, and EUR/USD exchange rate return directions. When computers are applied to solve a practical problem, it is usually the case that the method of derivingthe required output from a set of inputs can be described explicitly. As computersare applied to solve more complex problems, however, situations can arise in whichthere is no known method for computing the desired output from a set of inputs,or where that computation may be very expensive. Forecasting financial time seriesevents such as daily exchange rate directions, for instance, is a problem that is veryrelevant for the financial community and known to be very difficult in practice. Weformally represent this problem as a classification task which is described by thelinear separability problem. In the special case of finding whether two sets of points(namely exchange rate ups and downs) in general space can be separated, the linearseparability problem becomes the binary classification problem whose most generalform, the case of whether two sets of points in general space can be separated by khyperplanes, is known to be NP-complete. It is generally believed that NP-completeproblems cannot be solved efficiently.Design/Methodology/Approach: We approach the task of solving the BCP inthe context of predicting daily exchange rate directions with supervised learning,a learning methodology for the computer that attempts to learn the input/outputfunctionality from historical training examples. In particular, we opt for SVM as aC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 16,c Springer-Verlag Berlin Heidelberg 2009
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supervised learning algorithm. SVM has been found to work well for classificationtasks across a variety of scientific disciplines. The basic idea is to project the inputdata via kernel into a more expressive, high dimensional feature space where theSVM algorithm finds a linear decision plane that has maximum distance from thenearest training patterns. Using a kernel function guarantees that linear classification in feature space is equal to nonlinear classification in input space. Our experiment is to compare six SVM models with varying standard kernels along with oneexotic p-Gaussian SVM in order to investigate their quality in separating Grangercaused input data in high dimensional feature space. To ascertain their potentialvalue as out-of-sample forecasting and quantitative trading tools, all SVM modelsare benchmarked against traditional forecasting techniques.Practical Implications: Apart from pure statistical evaluation, the performanceof SVM was tested in a real-world environment. We set up a trading simulationwhere return predictions were translated into positions first. Next, a decision framework was established that indicates when the underlying asset was bought or sold,depending on the SVM output. Several trading metrics were imposed on the forecasting results to measure the models’ success.Results/Findings: It is found that hyperbolic SVMs perform consistently wellin terms of forecasting accuracy and trading performance via a simulated strategy.Moreover, p-Gaussian SVMs perform reasonably well in predicting EUR/GBP andEUR/USD return directions.Originality/Value: The results of Part III shed light on the existence of a particular kernel function which is able to represent properties of exchange rate returns,generally well, in high dimensional space. The fact that hyperbolic kernels are suchpromising candidates can be valuable for institutional investors, private investors,and risk managers of nonfinancial corporations.Future research: Future research direction will focus on SVM models and theirtechnical improvements. In light of this research, it would also be interesting tosee if the dominance of hyperbolic SVMs can be confirmed in further empirical investigations on financial market return prediction. We believe that modern methodsof the machine learning, pattern recognition, and empirical inference will play anessential role in coping with a complex world.
Chapter 17
Exchange Rate Hedgingin a Simulation/Optimization Framework
Abstract – Purpose: Part III considers another relevant practical problem involvinguncertainty about future exchange rate developments: the problem of finding a possibly optimal combination of linear and nonlinear financial instruments in order tohedge foreign exchange transaction risk over a specified planning period. Whilemuch research has been done in order to find answers on why multinational firms inreality hedge their exchange rate exposures, newspapers regularly suggest that moreresearch should be directed towards how hedgers can improve their decision makingprocedures. Hedges are trades designed with the motivation to reduce or eliminaterisk which distinguishes them from pure speculation. Still real-world hedging policies often involve a speculative component in terms of exchange rate expectationswhich may deviate from those implied by derivative prices. For instance, if a hedgerbelieves that the currency is going to move in an unfavorable direction, he may askabout the appropriate strategy in terms of instrument selection. In contrast, if thecurrency is expected to move favorably, but the hedger is not entirely sure, shouldhe use a different risk management strategy?Design/Methodology/Approach: We address the conflicting empirical findingthat firms do like to try to anticipate events, but that they also cannot base riskmanagement on second-guessing the market. Our analysis therefore argues that away to understand corporate hedging behavior is in the context of speculative motives that could arise from either overconfidence or informational asymmetries. Theproblem is addressed in a formal way, taking into account one single source of uncertainty (the exchange rate), a given set of instruments consisting of spot, forward,and European straddle option contracts that the hedger is allowed to choose from,and his particular attitudes towards risk and return. Regarding the latter, financialliterature has shown that traditional mean-variance or mean-quantile-based performance measures may be misleading if products with nonlinear payoff profiles, suchas options, are used. In addition, literature on bounded rationality suggests that individuals perceive risk in a nonlinear fashion which is expressed by a preference forpositive skewness in the probability distribution of future payoffs. For these reasons,a weighted mean-variance-skewness utility maximization framework with linearC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 17,c Springer-Verlag Berlin Heidelberg 2009
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constraints is embedded in a single-period stochastic combinatorial optimizationproblem (SCOP) formulation with the goal to maximize expected utility at the planning horizon. Stochastic combinatorial optimization is the process of finding thebest, or optimal solution for problems with a discrete set of feasible solutions ina stochastic system. The single-period approach suggests that hedge ratios cannot be altered once hedging decisions are made. While much progress has beenmade in finding exact (proveably optimal) solutions to some combinatorial optimization problems, using techniques such as dynamic programing, cutting planes,and branch-and-cut methods, reaching “optimal solutions” is in many cases meaningless, as in practice we are often dealing with models that are rough simplificationsof reality. The underlying SCOP is proveably NP-hard and therefore, too difficult tobe solved analytically in polynomial time which requires the use of good heuristicmethods. In addition, the optimization problem exhibits multiple local extrema anddiscontinuities. In the absence of a tractable mathematical structure, we study thebehavior of heuristics, “quick and dirty” algorithms, which return feasible solutionsthat are not necessarily optimal. In particular, we apply a methodology called Simulation/Optimization which is a general expression for solving problems where onehas to search for the settings of controllable decision variables that yield the maximum or minimum expected performance of a stochastic system as presented by asimulation model. For modeling the EUR/USD exchange rate, a smooth transitionnonlinear PPP reversion model is presented. The simulation model is very attractive in the present context and unique to our knowledge. It addresses both, the firstand the second PPP puzzle, and provides a theoretically valid and visually intuitiveview on the corridor of future EUR/USD spot development. The key feature is asmooth transition function which allows for smooth transition between exchangerate regimes, symmetric adjustment of the exchange rate for deviations above andbelow equilibrium, and the potential inclusion of a neutral corridor where the exchange rate does not mean revert but moves sideways. Another advantage is thatonly two parameters need to be estimated, the speed of mean reversion and exchange rate volatility, which makes the model straightforward to use in practice.For the task of optimization, we propose the use of a metaheuristic combinatorialsearch algorithm. A metaheuristic refers to an intelligent master strategy that guidesand modifies other heuristic methods to produce solutions beyond those that arenormally generated in a quest for local optimality. The specific metaheuristic weuse is a variant of scatter search, a generalized form of path relinking, which inmany practical problems has proven to be an effective and efficient approach due toits flexibility to accommodate variations in problem structure and in the objectivesconsidered for the evaluation of solutions.Practical Implications: In order to show our simulation/optimization model’sapplicability in a practical context, a situation is presented, where a manufacturingcompany, located in the EU, sells its goods via a US-based subsidiary to the endcustomer in the US. Since it is not clear what the EUR/USD spot exchange ratewill be on future transaction dates, the subsidiary is exposed to foreign exchangetransaction risk under the assumption that exposures are deterministic. We take theview that it is important to establish whether optimal risk management procedures
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offer a significant improvement over more ad hoc procedures. We demonstrate thatthe establishment of simple rules which adjust the recommended hedge dynamically(without rebalancing) based on easily observed current market factors can lead tobetter performance and risk management than various nave static hedges. For thepurpose of model validation, historical data backtesting was carried out and it wasassessed whether the optimized mean-variance-skewness approach is able to outperform passive strategies such as unitary spot, forward, and straddle, as well as amixed strategy over time. The passive strategies are nonpredictive since they do notallow the hedge to time-vary but are fixed-weight strategies and therefore do not usethe history of market information. We compare the alternative strategies in dynamicbacktesting simulations using market data on a rolling horizon basis. The strategieswere evaluated both in terms of their ex ante objective function values – as well asin terms of ex post development of net income.Findings: Financial markets are perfect examples of dynamic systems of complex human behavior. Unsurprisingly to either algorithm