<s> Lecture Notes in Economicsand Mathematical SystemsFounding Editors:M. BeckmannH.P. KünziManaging Editors:Prof. Dr. G. FandelFachbereich WirtschaftswissenschaftenFernuniversität HagenFeithstr. 140/AVZ II, 58084 Hagen, GermanyProf. Dr. W. TrockelInstitut für Mathematische Wirtschaftsforschung (IMW)Universität BielefeldUniversitätsstr. 25, 33615 Bielefeld, GermanyEditorial Board:A. Basile, H. Dawid, K. Inderfurth, W. Kürsten
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Christian Ullrich
Forecasting and Hedgingin the Foreign ExchangeMarkets
ABC
Dr. Christian UllrichBMW AGPetuelring 13080788 MünchenGermanychristian.ullrich@bmw.de
ISSN 0075-8442ISBN 978-3-642-00494-0e-ISBN 978-3-642-00495-7DOI 10.1007/978-3-642-00495-7Springer Dordrecht Heidelberg London New YorkLibrary of Congress Control Number: 2009926055c Springer-Verlag Berlin Heidelberg 2009This work is subject to copyright. All rights are reserved, whether the whole or part of the material isconcerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting,reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publicationor parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violationsare liable to prosecution under the German Copyright Law.The use of general descriptive names, registered names, trademarks, etc. in this publication does notimply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelaws and regulations and therefore free for general use.Cover design: SPi Publisher ServicesPrinted on acid-free paperSpringer is part of Springer Science+Business Media (www.springer.com)
For Robin.
Foreword
Historical and recent developments at international financial markets show that it iseasy to loose money, while it is difficult to predict future developments and optimize decision-making towards maximizing returns and minimizing risk. One of thereasons of our inability to make reliable predictions and to make optimal decisionsis the growing complexity of the global economy. This is especially true for the foreign exchange market (FX market) which is considered as one of the largest andmost liquid financial markets. Its grade of efficiency and its complexity is one of thestarting points of this volume.From the high complexity of the FX market, Christian Ullrich deduces the necessity to use tools from machine learning and artificial intelligence, e.g., supportvector machines, and to combine such methods with sophisticated financial modeling techniques. The suitability of this combination of ideas is demonstrated by anempirical study and by simulation. I am pleased to introduce this book to its audience, hoping that it will provide the reader with interesting ideas to support theunderstanding of FX markets and to help to improve risk management in difficulttimes.Moreover, I hope that its publication will stimulate further research to contributeto the solution of the many open questions in this area.Karlsruhe, January 2009
Detlef G. Seese
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Preface
The growing complexity of many real world problems is one of the biggest challenges of our time. The area of international finance is one prominent examplewhere decision making is often fraud to mistakes, and tasks such as forecasting,trading and hedging exchange rates seem to be too difficult to expect correct or atleast adequate decisions. Could it be that it is too complex for decision-makers toarrive at an optimal solution from a computational complexity perspective?As a first task we address the problem of forecasting daily exchange rate directions. We challenge the widely believed idea of efficient markets as offered bymodern finance theory and represent the problem formally as a classification taskaccording to the binary classification problem (BCP), which is known to be NPcomplete. The BCP is addressed with support vector machines (SVM), a state-ofthe-art supervised learning system that has emerged from the fields of computational geometry and computational statistics and has proven to successfully exploitpatterns in many nonfinancial applications. Six SVM models with varying standardkernels, along with one exotic p-Gaussian SVM are compared to investigate theseparability of Granger-caused input data in high-dimensional feature space. To ascertain their potential value as out-of-sample forecasting and quantitative tradingtool, all SVM models are benchmarked against traditional forecasting techniques.We find that hyperbolic SVMs perform well in terms of forecasting accuracy andtrading performance via a simulated strategy. Moreover, p-Gaussian SVMs performreasonably well in predicting EUR/GBP and EUR/USD directions.Second, we address the fact that as business has become global, firms must protect themselves from the potential losses arising from fluctuating exchange rates.Given a stochastic representation of the exchange rate, we investigate the problemof optimally hedging foreign exchange exposure with a combination of linear andnonlinear financial contracts such that the expected utility at the planning horizon ismaximized. A weighted mean–variance–skewness utility maximization frameworkwith linear constraints is embedded in a single-period stochastic combinatorial optimization problem (SCOP) formulation in order to find optimal combinations forspot, forward, and European style straddle option contracts. The problem is shownto be NP-complete. In order to derive near-optimal decisions within a reasonble
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amount of time, a simulation/optimization procedure is suggested. Exchange ratebehavior is modeled via a nonlinear smooth transition PPP reversion model. Optimization is carried out with a variant of the scatter search metaheuristic. Dynamicbacktesting with real-world financial data is conducted to validate the model andto show its applicability in a practical context. We find in our experiments thatscatter search is a search method that is both aggressive and robust. The simulation/optimization approach adds value in terms of reducing risk and enhancing income in comparison to several passive strategies.The dissertation was supervised by Prof. Dr. D. Seese, Institute AIFB, Universityof Karlsruhe, and has been written in English language.München, January 2009
Christian Ullrich
Acknowledgements
I am most grateful to the kind support, the trenchant critiques, and the probing questions during the completion of this thesis of my thesis supervisor Prof. Dr. DetlefSeese. Sincere thanks to my reviewers Prof. Dr. Marliese Uhrig-Homburg, and Prof.Dr. Svetlozar Rachev for their input and interest in this research. Their detailed comments were of great importance in refining the publications and the draft version ofthe thesis to its final form. I am very grateful for the fruitful discussions with myfellow doctoral students Thomas Stümpert, Andreas Mitschele, Tobias Dietrich, andAmir Safari from the Complexity Management Research Group at the AIFB Institute, as well as Dr. Stephan Chalup from University of Newcastle’s Robotic Laboratory.Since this dissertation was mainly written during my time as a doctoral candidate in BMW Group’s financial risk management department, I am not less thanindebted to my supervisors Norbert Mayer for his ingenuousness, constant curiosity,and interest in innovation and ongoing improvement; Joachim Herr for his patience,dedicated guidance, quick grasp, and enviable promotional talents; Ron Tonuzi forproviding a stimulating and fun environment in which to learn and grow. I wouldalso like to thank my colleagues Karl Ertl and Peter Studtrucker for sharing valuableinsights and experiences in practical currency trading and soccer betting. This dissertation also drew upon the knowledge, insights and experience of numerous BMWinternals as well as external business partners from the banking and consulting industries. Although too numerous to name, they are appreciated for their individualcontributions and inspiration.Thank you also to the musicianship and different sides of my band who enabled me to have a wonderful time along the way. I am particularly indebted toone guy whose creative endeavors, coolness, and wisdom allowed me to maintainmy (in)sanity throughout the writing of this thesis: Bob Dylan. I want to thank myparents, who have always supported my educational pursuits and whose foresightand values paved the way with unconditional love and support. Finally, I want tothank my girlfriend Robin for her love and patience.
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Part I Introduction1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Analytical Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1 Foreign Exchange Market Predictability . . . . . . . . . . . . . . . . . . . . . . . 72.2 Exchange Rate Forecasting with Support Vector Machines . . . . . . . . 82.3 Exchange Rate Hedging in a Simulation/OptimizationFramework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Part II Foreign Exchange Market Predictability3
Equilibrium Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1 Purchasing Power Parity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.1 Absolute PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.2 Relative PPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.3 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.1.4 Explanations for Deviations from PPP . . . . . . . . . . . . . . . . . . .3.2 Interest Rate Parity (IRP) Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.1 Covered Interest Rate Parity (CIP) . . . . . . . . . . . . . . . . . . . . . .3.2.2 Uncovered Covered Interest Rate Parity (UIP) . . . . . . . . . . . .3.2.3 Empirical Evidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.2.4 Explanations for Deviations from IRP . . . . . . . . . . . . . . . . . . .
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Market Efficiency Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Informational Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Speculative Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
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Views from Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2 Calculating Fixed Point Market Equilibrium . . . . . . . . . . . . . . . . . . . .5.2.1 Computational Complexity of Centralized Equilibrium . . . . .
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5.2.25.2.3
Computational Complexity of Decentralized Equilibrium . . .Adaptive/Inductive Learning of Rational ExpectationsEquilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3 Computational Difficulties with Efficiency . . . . . . . . . . . . . . . . . . . . .5.3.1 Information Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.3.2 Computational Complexity of Arbitrage . . . . . . . . . . . . . . . . .6
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Part III Exchange Rate Forecasting with Support Vector Machines7
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
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Statistical Analysis of Daily Exchange Rate Data . . . . . . . . . . . . . . . . . . .8.1 Time Series Predictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2 Empirical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.2 Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.3 Linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.4 Heteroskedasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.5 Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Support Vector Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.1 Binary Classification Problem (BCP) . . . . . . . . . . . . . . . . . . . . . . . . . .9.2 On the Computational Complexity of the BCP . . . . . . . . . . . . . . . . . .9.3 Supervised Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.4 Structural Risk Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5 Support Vector Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5.1 Learning in Feature Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5.2 Kernel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9.5.3 Optimal Separating Hyperplane . . . . . . . . . . . . . . . . . . . . . . . .9.5.4 Generalized Optimal Separating Hyperplane . . . . . . . . . . . . .9.5.5 Generalization in High Dimensional Feature Space . . . . . . . .
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10 Description of Empirical Study and Results . . . . . . . . . . . . . . . . . . . . . . .10.1 Explanatory Dataset . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.1.1 Phase One: Input Data Selection . . . . . . . . . . . . . . . . . . . . . . . .10.1.2 Phase Two: Dimensionality Reduction . . . . . . . . . . . . . . . . . .10.2 SVM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 Sequential Minimization Optimization (SMO) Algorithm . . . . . . . . .10.4 Kernel Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 Cross Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.6 Benchmark Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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10.7 Evaluation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7.1 Statistical Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.7.2 Operational Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.8 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Part IV Exchange Rate Hedging in a Simulation/Optimization Framework11 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10712 Preferences over Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . 11712.1 Currency Hedging Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11712.1.1 Forward . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11712.1.2 Plain Vanilla Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11912.1.3 Straddle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12112.2 Formal Relationship Between Firm and CapitalMarket Expectations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12212.3 Specification of Probability Distribution Function . . . . . . . . . . . . . . . 12312.4 Expected Utility Maximization and Three-Moments Ranking . . . . . . 12412.4.1 Preference Structures over Lotteries . . . . . . . . . . . . . . . . . . . . . 12412.4.2 Preference Structures over Utility Functions . . . . . . . . . . . . . . 12612.4.3 Expected Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . 12612.4.4 Increasing Wealth Preference . . . . . . . . . . . . . . . . . . . . . . . . . . 12712.4.5 Risk Aversion Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12712.4.6 Ruin Aversion Preference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12812.5 Specification of Utility Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12913 Problem Statement and Computational Complexity . . . . . . . . . . . . . . . . 13313.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13313.2 Computational Complexity Considerations . . . . . . . . . . . . . . . . . . . . . 13513.2.1 Complexity of Deterministic Combinatorial Optimization . . 13513.2.2 Complexity of Stochastic Combinatorial Optimization . . . . . 13713.2.3 Objective Function Characteristics . . . . . . . . . . . . . . . . . . . . . . 13914 Model Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14114.1 Simulation/Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14114.2 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14214.2.1 Datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14214.2.2 Component 1: Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14314.2.3 Component 2: Nonlinear Mean Reversion . . . . . . . . . . . . . . . . 14514.2.4 Component 3: Gaussian Random Walk . . . . . . . . . . . . . . . . . . 14814.2.5 Aggregation of Components . . . . . . . . . . . . . . . . . . . . . . . . . . . 15014.2.6 Calibration of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15114.3 Optimization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15214.3.1 Solution Construction Algorithm . . . . . . . . . . . . . . . . . . . . . . . 15214.3.2 Scatter Search and Path Relinking . . . . . . . . . . . . . . . . . . . . . . 154
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15 Simulation/Optimization Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.1 Practical Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.2 Model Backtesting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16315.2.2 Data Inputs and Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16415.2.3 Evaluation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17015.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17315.3.1 Ex Ante Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17315.3.2 Ex Post Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176Part V Contributions of the Dissertation16 Exchange Rate Forecasting with Support Vector Machines . . . . . . . . . . 18317 Exchange Rate Hedging in a Simulation/Optimization Framework . . 185Part VI ReferencesReferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
Acronyms
ACFADFAICANNARARCHAR(I)MABCPBDSCIACIRPCPIDCOPDWERMESTARFXGARCHIIDIRPJBLMKKTKPSSKULBLPLSTARMAMCSMEH
Autocorrelation functionAugmented Dickey–FullerAkaike’s information criterionArtificial neural networkAutoregressiveAutoregressive conditional heteroskedasticityAutoregressive (integrated) moving averageBinary classification problemBrock–Dechert–ScheinkmannCovered interest arbitrageCovered interest rate parityConsumer price indexDeterministic combinatorial optimization problemDurbin–WatsonEmpirical risk minimizationExponential smooth transition autoregressiveForeign exchangeGeneralized autoregressive conditional heteroskedasticityIndependent identically distributedInterest rate parityJarque–BeraLagrange-multiplierKarush–Kuhn–TuckerKwiatkowski–Phillips–Schmidt–ShinKurtosisLjung-BoxLinear programingLogistic smooth transition autoregressiveMoving averageMonte Carlo simulationMarket efficiency hypothesis
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NIDOLSPACFPCAPCOPPPPPPPTSPQPRBFSCOPSETARSIPSKSMOSRMSTARSVSVCSVMTARTSPUIAUIRPVARVC
Acronyms
Normally identically distributedOrdinary least squaresPartial autocorrelation functionPrincipal component analysisProbabilistic combinatorial optimization problemPhillips–PerronPurchasing power parityProbabilistic traveling salesman problemQuadratic programingRadial basis functionStochastic combinatorial optimization problemSelf-exciting threshold autoregressiveStochastic integer programingSkewnessSequential minimization problemStructural risk minimizationSmooth transition autoregressiveSupport vectorSupport vector classificationSupport vector machinesThreshold autoregressiveTraveling salesman problemUncovered interest arbitrageUncovered interest rate parityVector autoregressiveVapnik–Chervonenkis
Chapter 1
Motivation
The growing complexity of many real-world problems is one of the biggest challenges of our time. Complexity is also one of the factors influencing decision making in the area of international finance which has taken on great significance overthe last decade. As the result of structural shifts in the world economy and in theinternational financial system, the foreign exchange market has been profoundlytransformed, not only in size, but also in coverage, architecture, and mode of operation. Among the major developments that have occurred in the global financialenvironment are the following:1. A basic change in the international monetary system from the fixed exchange raterequirements of Bretton Woods that existed until the early 1970s to the floatingexchange rate system of today.2. A tidal wave of financial deregulation throughout the world, with massive elimination of government controls and restrictions in nearly all countries, resulting in greater freedom for national and international financial transactions, andin greatly increased competition among financial institutions, both within andacross national borders.3. A fundamental move towards institutionalization and internationalization of savings and investment, with funds managers and institutions having vastly largersums available, which they are investing and diversifying across borders and currencies in novel ways.4. A broadening and deepening trend towards international trade liberalization,within a framework of multilateral trade agreements, such as the Tokyo andthe Uruguay Rounds of the General Agreement on Tariffs and Trade, the NorthAmerican Free Trade Agreement and U.S. bilateral trade initiatives with China,Japan, India and the European Union.5. Major advances in technology, making possible instantaneous real-time transmission of vast amounts of market information worldwide, immediate and sophisticated manipulation of that information in order to identify and exploit marketopportunities, and rapid and reliable execution of financial transactions.6. Breakthroughs in the theory and practice of finance, resulting not only in thedevelopment of new financial instruments and derivative products, but also inC. Ullrich, Forecasting and Hedging in the Foreign Exchange Markets, Lecture Notesin Economics and Mathematical Systems 623, DOI: 10.1007/978-3-642-00495-7 1,c Springer-Verlag Berlin Heidelberg 2009
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advances in thinking that have changed market participants’ understanding ofthe financial system and their techniques for operating within it.