Advances in risk management

Advances in Risk Management Edited by Greg N Gregoriou ADVANCES IN RISK MANAGEMENT Also edited by Greg N Gregoriou ASSET ALLOCATION AND INTERNATIONAL INVESTMENTS DIVERSIFICATION AND PORTFOLIO MANAGEMENT OF MUTUAL FUNDS PERFORMANCE OF MUTUAL FUNDS Advances in Risk Management Edited by GREG N GREGORIOU Selection and editorial matter © Greg N Gregoriou 2007 Individual chapters © contributors 2007 All rights reserved No reproduction, copy or transmission of this publication may be made without written permission No paragraph of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1T 4LP Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages The authors have asserted their rights to be identified as the authors of this work in accordance with the Copyright, Designs and Patents Act 1988 First published 2007 by PALGRAVE MACMILLAN Houndmills, Basingstoke, Hampshire RG21 6XS and 175 Fifth Avenue, New York, N.Y 10010 Companies and representatives throughout the world PALGRAVE MACMILLAN is the global academic imprint of the Palgrave Macmillan division of St Martin’s Press, LLC and of Palgrave Macmillan Ltd Macmillan® is a registered trademark in the United States, United Kingdom and other countries Palgrave is a registered trademark in the European Union and other countries ISBN-13: 978–0–230–01916–4 ISBN-10: 0–230–01916–1 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data Advances in risk management / edited by Greg N Gregoriou p cm — (Finance and capital markets series) Includes bibliographical references and index ISBN 0–230–01916–1 (cloth) Investment analysis Financial risk management I Gregoriou, Greg N., 1956– II Series: Finance and capital markets HG4529.A36 2006 332.1068’1—dc22 2006045747 10 16 15 14 13 12 11 10 09 08 07 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham and Eastbourne Contents Acknowledgements xi Notes on the Contributors xii Introduction xxi Impact of the Collection Threshold on the Determination of the Capital Charge for Operational Risk Yves Crama, Georges Hübner and Jean-Philippe Peters 1.1 1.2 1.3 1.4 1.5 Introduction Measuring operational risk The collection threshold Empirical analysis Conclusion 11 16 Incorporating Diversification into Risk Management 22 Amiyatosh Purnanandam, Mitch Warachka, Yonggan Zhao and William T Ziemba 2.1 Introduction 2.2 Risk measure with diversification 2.3 Numerical example 2.4 Implementation 2.5 Pricing portfolio insurance 2.6 Conclusion 22 24 31 33 37 43 v vi CONTENTS Sensitivity Analysis of Portfolio Volatility: Importance of Weights, Sectors and Impact of Trading Strategies 47 Emanuele Borgonovo and Marco Percoco 3.1 3.2 3.3 3.4 Introduction Sensitivity analysis background Effect of relative weight changes Importance of portfolio weights in GARCH volatility estimation models 3.5 Empirical results: trading strategies through sensitivity analysis 3.6 Conclusion Managing Interest Rate Risk under Non-Parallel Changes: An Application of a Two-Factor Model Manuel Moreno 4.1 Introduction 4.2 The model 4.3 Generalized duration and convexity 4.4 Hedging ratios 4.5 A proposal of a solution for the limitations of the conventional duration 4.6 Conclusion An Essay on Stochastic Volatility and the Yield Curve 47 50 51 53 56 64 69 69 70 72 74 75 83 86 Raymond Théoret, Pierre Rostan and Abdeljalil El-Moussadek 5.1 Introduction 5.2 Variations on stochastic volatility and conditional volatility 5.3 Interest rate term structure forecasting 5.4 Interest rate term structure models 5.5 Methodology 5.6 Data and calibration of the Fong and Vasicek model 5.7 Simulation 5.8 Empirical results 5.9 Conclusion 86 88 92 92 94 97 98 99 102 CONTENTS Idiosyncratic Risk, Systematic Risk and Stochastic Volatility: An Implementation of Merton’s Credit Risk Valuation vii 107 Hayette Gatfaoui 6.1 6.2 6.3 6.4 6.5 Introduction The general model A stochastic volatility model Simulation study Conclusion A Comparative Analysis of Dependence Levels in Intensity-Based and Merton-Style Credit Risk Models 107 110 114 118 126 132 Jean-David Fermanian and Mohammed Sbai 7.1 7.2 7.3 7.4 7.5 7.6 Introduction Merton-style models Intensity-based models Comparisons between some dependence indicators Extensions of the basic intensity-based model Conclusion The Modeling of Weather Derivative Portfolio Risk 132 133 136 139 143 150 156 Stephen Jewson 8.1 8.2 8.3 8.4 Introduction What are weather derivatives? Defining risk for weather derivative portfolios Basic methods for estimating the risk in weather derivative portfolios 8.5 The incorporation of sampling error in simulations 8.6 Accurate estimation of the correlation matrix 8.7 Dealing with non-normality 8.8 Estimating model error 8.9 Incorporating hedging constraints 8.10 Consistency between the valuation of single contracts and portfolios 8.11 Estimating sampling error 8.12 Estimating VaR 8.13 Conclusion 156 157 159 160 162 162 163 164 165 166 167 167 168 viii 10 CONTENTS Optimal Investment with Inflation-Linked Products 170 Taras Beletski and Ralf Korn 9.1 Introduction 9.2 Modeling the evolution of an inflation index 9.3 Optimal portfolios with inflation linked products 9.4 Hedging with inflation linked products 9.5 Conclusion 170 171 173 182 189 Model Risk and Financial Derivatives 191 Franỗois-Serge Lhabitant 10.1 10.2 10.3 10.4 10.5 Introduction From mathematical theory to financial practise An illustration of model risk The role of models for derivatives The model-building process and model risk-creation 10.6 What if the model is wrong? a case study 10.7 Eleven rules for managing model risk 10.8 Conclusion 11 Evaluating Value-at-Risk Estimates: A Cross-Section Approach 191 194 195 197 199 201 203 210 213 Raffaele Zenti, Massimiliano Pallotta and Claudio Marsala 11.1 11.2 11.3 11.4 11.5 11.6 12 Introduction Value-at-risk Review of existing methods for backtesting An extension: the cross-section approach Applications Conclusion 213 214 214 217 219 224 Correlation Breakdowns in Asset Management 226 Riccardo Bramante and Giampaolo Gabbi 12.1 12.2 12.3 12.4 12.5 Introduction Data and descriptive statistics Correlation jumps and volatility behavior Impact on portfolio optimization Conclusion 226 226 228 237 237 CONTENTS 13 Sequential Procedures for Monitoring Covariances of Asset Returns Olha Bodnar 13.1 Introduction 13.2 Covariance structure of asset returns and optimal portfolio weights 13.3 Multivariate statistical surveillance 13.4 Simultaneous statistical surveillance 13.5 A comparison of the multivariate and simultaneous control charts 13.6 Conclusion 14 An Empirical Study of Time-Varying Return Correlations and the Efficient Set of Portfolios ix 241 241 243 246 251 253 258 265 Thadavillil Jithendranathan 14.1 14.2 14.3 14.4 15 Introduction Empirical Methodology and Data Results Conclusion The Derivation of the NPV Probability Distribution of Risky Investments with Autoregressive Cash Flows 265 267 270 276 278 Jean-Paul Paquin, Annick Lambert and Alain Charbonneau 15.1 15.2 15.3 15.4 Introduction Systematic risk and the perfect economy Total risk and the real economy The NPV probability distribution and the CLT: theoretical results 15.5 The NPV probability distribution and the CLT: simulation models and statistical tests 15.6 The NPV probability distribution and the CLT: simulation results 15.7 Conclusion 16 Have Volatility Transmission Patterns between the USA and Spain Changed after September 11? 278 280 282 285 288 289 293 303 Helena Chuliá, Francisco J Climent, Pilar Soriano and Hipịlit Torró 16.1 Introduction 16.2 Data 303 305 362 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVES 45 Gamma error 40 Vanna error 35 Volga error 30 Total error 25 20 15 10 Ϫ20% Ϫ15% Ϫ10% Ϫ5% Ϫ5 0% 5% 10% 15% 20% Figure 18.2 The hedging error probability density function obtained by assuming that the additional claim has the same expiry (1 year) as the option to hedge but different strike (K = 1) The trader uses a stochastic volatility model All errors are expressed as a percentage of the initial cost of the option 30.00 Gamma error 25.00 Vanna error Volga error 20.00 Total error 15.00 10.00 5.00 Ϫ40% Ϫ30% Ϫ20% Ϫ10% 0% 10% 20% Ϫ5.00 Figure 18.3 The hedging error probability density function obtained by assuming that the additional claim has the same strike (K = 0.975) as the option to hedge but different expiry (2 years) The trader uses a stochastic volatility model All errors are expressed as percentage of the initial cost of the option We study two different cases First we assume that we choose for D a liquid at-the-money option that has the same expiry as the option to hedge (i.e a one-year call) In Figure 18.2 we show the hedging error coming from the Gamma, the Volga and the Vanna terms separately, as well as the total error GIUSEPPE DI GRAZIANO AND STEFANO GALLUCCIO 363 We immediately notice that large errors are now less likely when compared to the BS delta hedging strategy In addition, the largest contribution to the total error comes from the two Vanna terms since they are the most affected by the wrong assumption on correlation As a second example, we consider the situation where D, like C, is an option struck at K = 0.975 but it expires one year later, i.e T = Results are shown in Figure 18.3 We now see that the largest contribution to the total error comes from the Gamma terms since the two options have very different Gamma in this case However, all errors are still much smaller than in the simple BS case CONCLUSION In this chapter we have examined the errors arising from hedging a contingent claim written on a tradable asset that follows a generic stochastic volatility model when traders have a bad assessment of the real dynamics We provide a general formula for the total hedging error extending some known results to the case of state variables driven by stochastic volatility processes We have numerically shown that errors due to a bad representation of the whole dynamics are significantly larger than those arising from just a bad estimation of model parameters, in general However, even if the trader uses a model that is formally equivalent to the true one, errors due to parameters misspecification can still be quite large This in particular should generate some concern when hedging is performed with a model that assumes no correlation between Wiener noises, a framework that has recently gained some favor in the market due to its mathematical tractability NOTES It must be noticed, however, that in general one needs to artificially assume strong time-dependency of model parameters to achieve a good model “calibration”, i.e to ensure that market vs model errors are within the bid–ask spread (Galluccio and Le Cam, 2005) Despite this, at least theoretically it is possible to well-approximate any smile shape by properly adjusting the postulated dynamics if coefficients are allowed to take arbitrary values We will assume throughout the paper that interest rates are deterministic Extending the present approach to include the effect of stochastic rates is possible but results are essentially unaffected by this choice in the range of options expiry we consider We recall that we not address here the question of whether equation (18.1) is the correct representation of reality, thus we not need to calibrate our “benchmark” model to the S&P market Instead, we assume that equation (18.1) is the true model and study the replication error induced by taking equation (18.2) as a good approximation of the market, described by equation (18.1) 364 MODEL SELECTION AND HEDGING OF FINANCIAL DERIVATIVES REFERENCES Andersen, L and Andreasen, J (2000a) “Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for Option Pricing”, Review of Derivatives Research, 4(3): 231–62 Andersen, L and Andreasen, J (2000b) “Volatility Skews and Extensions of the LIBOR Market Model”, Applied Mathematical Finance, 7(1): 1–32 Bates, D (1996) “Jumps and Stochastic Volatility: Exchange Rates Processes Implicit in Deutsche Mark Options”, Review of Financial Studies, 9(1): 69–107 Carr, P and Madan, D (1997) “Towards a Theory of Volatility Trading”, in R Jarrow (ed.), Volatility, pp 417–27 Chernov, M., Gallant, R., Ghysels, E and Tauchen, G (2003) “Alternative Models for Stock Price Dynamics”, Journal of Econometrics, 116(1): 225–57 Cont, R (2005) “Model Uncertainty and Its Impact on the Pricing of 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(eds), Applied Stochastic Analysis, pp 389–414 (New York, NY: Gordon & Breach) Hagan, P., Lesniewski, A., Kumar, D and Woodward, D (2002) “Managing Smile Risk”, Wilmott Magazine, September: 84–108 Harrison, J.M and Pliska, S.R (1981) “Martingales and Stochastic Integrals in the Theory of Continuous Trading”, Stochastic Processes and Their Applications, 11(3): 215–60 Heston, S (1993) “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options”, Review of Financial Studies, 6(2): 327–43 Hull, J and White, A (1987) “The Pricing of Options on Assets with Stochastic Volatility”, Journal of Finance, 42(2): 281–300 Jacod, J and Shiryaev, A.N (2003) Limit Theorems for Stochastic Processes, 2nd edn (Berlin: Springer-Verlag) Pan, J (2002) “The Jump Risk Premia Implicit in Options: Evidence from an Integrated Time Series Study”, Journal of Financial Econometrics, 63(1): 3–50 Piterbarg, V (2003) “A Stochastic Volatility Forward LIBOR Model with a Term Structure of Volatility Smiles”, Working paper, Bank of America Schoutens, W., Simons, E and Tistaert, J (2004) “A Perfect Calibration! Now What?”, Wilmott Magazine, March: 85–98 Stein, E.M and Stein, J.C (1991) “Stock Price Distributions with Stochastic Volatility: An Analytic Approach”, Review of Financial Studies, 4(4): 727–52 Index acceptable portfolios 23, 25–6 accumulated value 77–83 advanced measurement approach (AMA) calibration 12–16 aggregate desirability of a portfolio 29–30 Agrawal, D 108 Akhavein, J 109 α-stable distributions 146–50 α-stable intensity-based model 148–50 properties of the family 146–7 simulation 147–8 American Depository Receipts (ADRs) 305 Andersen, L 354 Andersen, T.G 115, 246 Andersson, E 242 Andreasen, J 354 approximations, models as 198 Aragó, V 304 Aramov, D 108 arbitrage models 92 ARCH models 48, 86–7 conditional volatility 89–91 ARCH test 306–7 Artzner, P 22, 24–5, 27, 28 asymmetric covariance 328–9, 348, 349–50, 351 asymmetric dynamic covariance (ADM) model 329–51 asymmetric volatility 328–9, 348, 349–50, 351 asymmetric volatility impulse response function (AVIRF) 311–12, 317–21, 322, 323 asymmetrical information 283 asymmetries analysis 342–5 asymptotic MEWMA control charts 251, 253, 254–8 atomic simulation 165 augmented Dickey–Fuller (ADF) test 307 Australia 328 autocorrelation 339 autoregressive models 48 NPV probability distribution and autoregressive cash flows 286–7, 288, 289–93, 297–301 average run lengths (ARLs) 247, 254, 255–8 average stock risk 108–9 Bachelier, L 194 backtesting 200 VaR models 214–17 backward-looking models 200 Balkema, A.A bankruptcy, probability of 283–4 Basel Committee on Banking Supervision Basel II framework 1–2, basic indicator approach (BIA) basic multivariate normal (BMVN) method 161–8 accurate estimation of correlation matrix 162–3 consistency between valuation of single contracts and portfolios 166–7 dealing with non-normality 163–4 estimating model error 164 estimating sampling error 167 estimating VaR 167–8 365 366 INDEX basic multivariate normal (BMVN) method continued incorporation of hedging constraints 165–6 incorporation of sampling error 162 Bayesian model averaging 164 Bekaert, G 329 BEKK model 330 asymmetric VaR-BEKK model 309–11, 312–17, 322 Berkowitz, J 216 Bermudian options 210 beta volatility and debt 118–23 volatility transmission in Europe 342–4, 345, 351 binomial distribution negative 4–5 Black, F 109, 110, 117, 194 Black–Scholes option pricing model 109, 198, 201, 208, 353, 356 implied volatility 195–7 blank sheet syndrome 205 block maxima method Blume, L 48 Blume, M 280 Bock, B 242 Bodnar, O 243, 258 Bohn, J 109 Bollerslev, T 54, 115, 246, 266, 311, 330, 335 Bollinger bands 87, 99, 100, 101–2, 103 bond portfolios 69–85 bonds inflation-linked 172–3; optimal portfolios 176–82 inflation-linked products and hedging 182–9 zero-coupon see zero-coupon bonds Booth, G 305 Borgonovo, E 50, 51 bounds for credit spreads 125–6 Braun, P.A 329 British stock market 331–51 Britten-Jones, M 258 burn analysis 160–1 CAC40 index 331–51 calibration, model 200 call options 74–5, 158 Campbell, J.Y 87, 108, 111 Campbell, S.D 215 Campolongo, F 49, 59 Canadian government yield curve 97–102, 103 capital asset pricing model (CAPM) 278–9, 280–4 conditional see conditional CAPM ADC model decision rule 282–3 capital charge for operational risk 1–21 capital investment projects 278–302 Carr, P 357 Ceci, V 48 central limit theorem (CLT) 285–301 and the first-order autoregressive process 297–301 and the NPV probability distribution 285–97; simulation models and statistical tests 288–9; simulation results 289–93; theoretical results 285–7 certainty equivalent approach 282–3 CEV-ARCH models 90–1 Chambers, J.M 148 Chapelle, A Chatfield, C 216 Chen, L 70 Chen, R.-R 125 chi squared test Christie, A.A 329 Christoffersen, P 214, 215, 216 collection threshold 2, 8–16, 17 impact on capital charge for operational risk 9–11; empirical analysis 11–16, 17, 18, 19 selection 8–9 Collin-Dufresne, P.P 108, 124 comparative Bayesian analysis 221–2, 223 conditional CAPM ADC model 329–51 asymmetries analysis 342–5 model estimates 335–41 volatility spillovers 345–8, 349–50 conditional correlations 210 conditional volatility 86–7, 88–92 Conover, W 163 Conrad, J 328 constant correlation coefficient model 266, 330 Cont, R 360 convexity, generalized 74, 76–7 copulas 163–4 correlated frailty intensity-based models 143–6 INDEX correlation breakdowns 226–40 correlation jumps and volatility behaviour 228–36 data and descriptive statistics 226–8 impact on portfolio optimization 237, 238, 239 correlation matrix 161, 162–3 correlations between default events see default events correlations empirical study of time-varying return correlations and the efficient set of portfolios 265–77 model risk and 210 costless contracting 283–4 Courtadon, G 210 covariance asymmetric 328–9, 348, 349–50, 351 conditional 342–7 covariance structure of asset returns and optimal portfolio weights 243–6 monitoring changes in covariance matrix see sequential control procedures Cox model 136–7 Crama, Y Cramer–von Mises test crash-phobia 196 credit risk valuation 107–31 general model 110–14; basic setting 110–12; stochastic volatility and Merton’s pricing 112–14 simulation study 118–26; credit spread 123–6, 127; volatility and debt 118–23 stochastic volatility model 114–17 credit spreads 123–6, 127 crises 304 Crnkovic, C 216 Crocket, J 280 Crosier, R.B 249 cross-section approach to VaR backtesting 217–24 applications 219–24 CUSUM control charts 248–50 projected pursuit CUSUM 249–50, 252–3 vector valued CUSUM 249 daily level simulation 165–6 Danilov, D 103 data verification 208 Day, J 92 367 de Haan, L De Jong, F 103 debt, volatility and 118–23 debt financing 285 debt pricing see credit risk valuation decrease in slope of yield curve (flattening) 79–81 default events correlations 134, 136, 138–9, 150–1 and default probabilities in intensity-based models 139–41 large time horizons 139, 152–4 default probabilities intensity-based models 137; and default events correlations 139–41 Merton-style models 133, 134 default risk 108 deflation protection 178–81 Delbaen, F 22, 24–5, 27, 28 Delianedis, G 108 dependence levels 132–55 comparison between dependence indicators 139–43 extensions of basic intensity-based model 143–50 intensity-based models 136–9 Merton-style models 133–6 derivatives 26, 34, 35, 159 evolution of pricing models 194–5 model risk and 191–212 model selection and its impact on hedging 353–64 role of models for 197–9 see also under individual types of derivative Derman, E 192, 354 deterministic (local volatility) models 354 Di Graziano, G 357, 358, 360 Diebold, F.X 103, 115, 216, 246 differential importance measure 49, 50–1, 65–6 trading strategies and 58–65 DIPO discount bonds price 70–2 discount rates 286–7, 288, 290–3, 293–4 diversification-based risk measure 22–46 economic motivation 29–30 implementation 33–7 numerical example 31–3 pricing portfolio insurance 37–43 368 INDEX diversification-based risk measure continued properties of the measure 27–8, 44–5 dollar-denominated risk 26–7 insurance and 42–3 double exponential distribution 288, 291–3, 294 Dow Jones Industrial Average Index time-varying return correlations and efficient set of portfolios 269–76 trading strategies through sensitivity analysis 56–66 Drachman, J 216 Drudi, F 48 Dupire, B 354 duration 69–70, 84 generalized 72–4, 75 proposed solution for limitations of 75–83 Durbin-Watson statistic 229, 230, 239 dynamic conditional correlation (DCC) models 266, 268–76 Ebens, H 115, 246 Eber, J.M 22, 24–5, 27, 28 Eberlain, E 109 economic motivation 29–30 efficient market hypothesis 282 efficient set of portfolios 265–77 El Karoui, N 357 elasticity 48–9, 51, 53, 65 Elton, E.J 108 Embrechts, P 4, energy sector 57, 63, 64, 65 Engle, R.F 53–4, 86, 266, 268, 330, 342 Eom, J.H 109, 128 equilibrium models of interest rate term structure 92, 93 equity 121–2 equity financing 285 Eraker, B 354 Ericsson, J 127 Euro area 226–36 Europe, volatility transmission in 327–52 European call options 172 debt pricing 113–14, 116–17 EWMA control chart comparison of multivariate and simultaneous 254–8 multivariate 250–1, 254–8, 259 simultaneous 253, 254–8, 259 exchange traded contracts 158 exotic derivatives 209–10 expiry value 167 extended Kalman filter (EKF) 87, 94, 95 algorithm 96 application to Fong and Vasicek model 96–7 simulation of interest-rate term structure 99–103 extreme value theory 7–8, 12–16, 217 algorithm for finding the threshold 8, 12, 18–19 failures analysis 222–4 falsifiability 281 Fama, E.F 108, 280, 282 Fernández, A 304 Figlewski, S 192 financial crises 304 financial derivatives see derivatives financial distress, probability of 283–4 financial services 57, 63, 64 financing of a firm 285 firm preferences 35–7 first-order autoregressive process 286–7, 288, 289–93, 297–301 Fisher equation 171 Flannery, M.J 285 flattening of yield curve 79–81 Follmer, H 23, 28 Fong, H.G 87 Fong and Vasicek model 87, 93–4 application of extended Kalman filter to 96–7; discretization 96–7; linearization 97 calibration 98 data 97–8 simulation of interest rate term structure 99–103 Fornari, F 87, 90–1 forward contracts 26, 34 forward-looking models 200 Frachot, A 3, 8, frailty models 136–7 see also intensity-based models French, K.R 108, 280 French stock market 331–51 frequency distribution 4–5, 17 Frey, R 48 frictionless economy 282 Friedman, M 280 Friend, I 280 Frisen, M 242, 258 FTSE100 index 331–51 INDEX futures contracts 26, 34, 35 Galluccio, S 357, 358, 360 Gallus, C 203 Gamma 358–9, 362–3 GARCH models 48 conditional volatility 89, 90, 91 importance of portfolio weights in GARCH volatility estimation models 53–6, 66 multivariate see multivariate GARCH models Gatfaoui, H 109, 110, 111, 126 Gemmill, G 124 Generale, A 48 Generalized Pareto Distribution (GPD) generalized scenarios 27 geometric Brownian motion 171–2 Georges, P German stock market 331–51 Geske, R 108 Gibbons, M.R 258 Gibson, R 192 Glaser, M 304 global minimum variance portfolio (GMVP) 242, 243–6, 247 global risk models 222 Glosten, L.R 330, 344 Goetzmann, W.N 128 Goldstein, R.S 108, 124 goodness-of-fit tests 5–6, 9–10 Gourieroux, C 48 Goyal, A 108–9 Green, T.C 192 Greenspan, A 195 Gruber, M.J 108 Gultekin, M.N 328 Gumbel, E.J Gunther, T.A 216 Hagan, P 360 Hamao, Y 305 harmonized consumer price index (HCPI) 171 Healy, J.D 248 Heath, D 22, 24–5, 27, 28 hedging constraints 165–6 of derivatives and model selection 353–64 with inflation-linked products 182–9 managing interest rate risk 74–5 369 model risk and 202–3 super-hedging strategies 203 hedging error 355, 356–63 analytical expression of total hedging error 357–9 numerical results 359–63 Helwege, J 109, 128 Hendry, O.L 328 Hentschel, L 342 Hertz, D.B 278 Heston, S 354, 356 heteroskedasticity 339 Hicks, D 158, 160 Hillier, F 278, 285–6 Hirsa, A 210 hit function, tests based on 214–15 Hoeffding, W 286 Hofmann, N 112 Hon, M.T 304, 323 Hotelling, H 248 Houston, J.F 285 Huang, J.-Z 109, 125, 128 Hübner, G Hull, J 109, 354 hurdle rates 281 Iachine, I.A 143 IBEX35 index 305–8, 312–23 idiosyncratic risk 107–31 illiquidity 35–7 Iman, R 163 implied volatility 109, 195–7, 354 increase in slope of yield curve (steepening) 81–2, 83 independence property 215 inflation index modeling the evolution of 171–3 optimal portfolios 178, 179 inflation-linked products 170–90 hedging with 182–9; investment in bond and stock 186–7; investment in bond, stock and inflation 184–6; numerical examples 188–9 optimal portfolios with 173–82 information asymmetries 283 information and communication technologies (ICT) 57, 63, 64 insurance contracts 159 portfolio see portfolio insurance intensity-based models 132–55 comparisons between dependence indicators 139–43 370 INDEX intensity-based models continued default events correlations 138–9 extensions 143–50; α-stable distributions 146–50; multi-factor model 143–6 loss distribution 137–8, 142–3 interest rate risk see two-factor model for interest rates interest rate term structure forecasting 87, 92–103 data and calibration of Fong and Vasicek model 97–8 empirical results 99–102, 103 methodology 94–7 models 92–4 simulation 98–9 inventory of models in use 204–5 Ito, H 304 Jagannathan, R 281, 330 Japan 226–36 Jarque-Bera test 270, 271, 274, 306, 307 Jarrow, R 23, 189 Jeanblanc, M 357 Jensen, M.C 283 Jewson, S 157, 162, 163, 164, 167 Jobson, J.D 258, 265 Johannes, M 354 Jones, S 157 Jostova, G 108 jump-diffusion (JD) models 354 Kallsen, J 109 Kalman filter 94 extended see extended Kalman filter Kani, I 354 Kaul, G 328 Kealhofer, S 127 Kearney, C 310 Kocagil, A.E 109 Kolmogorov–Smirnov test 6, 9, 10, 11 Koopman, S.J 108 Korkie, B.M 258, 265 Korn, R 170, 172, 173, 174, 176–7, 183–5 Koutmos, G 305 Kraft, H 170, 173 Kristen, J 109 Kroner, K.F 266, 328, 330, 342, 344 Kruse, S 170, 172 Kumar, A 128 Kumar, D 360 Kupiec, P 215, 220 Kurbat, M 127 kurtosis 270, 271, 274, 306, 307 lambda 118–23 Lange, R 92 large capitalization stocks 327–52 Laughhunn, D.J 283 Laurent, J.P 48 Le Cam, Y 360 Ledoit, O 163, 241, 258 Lee, D 304 left tail risk 88–9 Leland, H.E 122 leptokurtic distributions 5, Lesniewski, A 360 Lettau, M 111 leverage effect hypothesis 328–9 Levy, H 280–1 Lhabitant, F.S 192 Li, C 103 likelihood weighting 164 Lin, W.L 317 Lintner, J 278 liquidity 127 illiquidity 35–7 liquidity premium 127 Ljung–Box test 307 Lo, A.W 327 local volatility (LV) models 354 location parameter 147 long time horizons 139, 152–4 Longin, F 305 Lopez, J.A 217 loss distributions α-stable intensity-based model 149–50 intensity-based models 137–8, 142–3 measuring operational risk 3–8; empirical analysis 12–16; frequency distribution 4–5, 17; modeling extreme losses 7–8; severity distribution 5–7, 12, 13, 16, 17 Merton-style models 134–5, 142–3 multi-factor intensity-based model 144 loss functions, backtesting VaR models based on 217 Lucas, A 108 Luenberger, D.G 34 Mackinlay, A.C 327 Madan, D.B 210, 357 Mahmoud, M.A 258 INDEX Majnoni, G 48 Malkiel, B.G 108, 111 Mallows, C.L 148 Mandal, K 103 Manganelli, S 48, 54, 57, 66, 67 Mann, C 108 manufacturing 57, 63, 64, 65 market frictions 33, 35–7 market indices 331–51 market risk 69, 83 market value 167–8 marking to market 206–7 marking to model 206–7 Markowitz, H 194, 241, 243, 265 Martens, M 305 Martin, J.S 108 Masulis, R.W 305 maximum likelihood estimation (MLE) techniques 3–4 MC1 control charts 248, 252, 254–8 McNeal, A.J 48 mean absolute error 234–5, 236 mean excess function (MEF) plot 7–8, 12, 14 mean square error (MSE) 12, 15 mean-variance optimization models 265–6, 276 Meckling, W.H 283 Meier, I 281 Mele, A 87, 90–1 Meneu, V 311, 317 Merrill Lynch 208 Merton, R.C 23, 109, 110, 124, 192, 194 credit pricing model and stochastic volatility 112–14, 127–8 Merton-style credit risk models 132–6, 142–3, 150 Michaud, R.O 265–6 MIDCAC index 331–51 Miller, M.H 280 Mills, T.C 88 minimal equivalent martingale measure 113–14, 126 misspecification indicators 344–5 model-building process 199–201 model calibration 200 model selection/creation 199–200 model usage 200–1 model error 164 model-implied calibration 354–5 model misspecification 359, 360–1 model risk 191–212, 355 case study 201–3 examples and consequences 193 371 illustration 195–7 model-building process and model risk creation 199–201 rules for managing 203–10; correlations 210; define a model-testing framework 205–6; define what should be a good model 204; exotic derivatives 209–10; keeping track of models in use 204–5; marking to market 206–7; regular revision of models 206; simplicity 207–8; stress testing of models 209; use a model for its purpose 209; verification of data 208 model selection 199–200 and its impact on hedging derivatives 353–64 model-testing framework 205–6 model usage 200–1, 204 model validation team 205 Modigliani–Miller (MM) paradigm 281, 282 monotonicity 28 Monte Carlo simulation 87 comparison of multivariate and simultaneous control charts 253–5 forecasting interest rate term structure 98–102; Bollinger bands 99; empirical results 99–102, 103 Moreno, M 70, 74 Mossin, J 278 motivation, economic 29–30 Moudoulaud, O 8, Moustakides, G.V 249 moving average specification 266 Muirhead, R.J 259 multifactor models intensity-based 143–6 interest rate term structure 93 multiple VaR levels 216–17 multivariate CUSUM (MCUSUM) control charts 249, 252, 254–8 multivariate EWMA (MEWMA) control charts 250–1, 254–8, 259 multivariate GARCH models time-varying return correlations and the efficient set of portfolios 265–77 volatility spillovers in Europe 330–1 volatility transmission between USA and Spain 303–26 372 INDEX multivariate normal distribution see basic multivariate normal (BMVN) method multivariate statistical surveillance 246–51 comparison of multivariate and statistical control charts 253–8 multivariate t-distribution 244–5 Myers, S.C 283 NatWest 206 negative binomial distribution 4–5 negative returns 273 Nelken, J 109 Nelson, D.B 86–7, 90 news impact surfaces 342–4 Ng, V.K 266, 305, 328, 330, 342, 344 Ngai, H.-M 248, 249, 250 no-arbitrage models 92, 93 no default risk 281 non-normality 163–4 non-synchronous trading problem 304–5 normal distribution 288, 289, 290, 294 NPV probability distribution 278–302 and the central limit theorem 285–97; simulation models and statistical tests 288–9; simulation results 289–93; theoretical results 285–7 systematic risk and the perfect economy 280–2 total risk and the real economy 282–5 one-factor models of interest rate term structure 92–3 operational risk 1–21 collection threshold 8–11; empirical analysis of impact 11–16, 17, 18, 19 measuring 3–8 optimal weight changes 57, 58–63, 65–6, 67 optimization, portfolio see portfolio optimization option pricing models 199 Black–Scholes model see Black–Scholes option pricing model model risk 201–3 option pricing theory 112–14, 116–17 options 158 call options 74–5, 158 European call options 113–14, 116–17, 172 exotic 209–10 out-of-the-money options 26, 34, 35 orthogonalization 309 ORX out-of-control states 247, 255–8 out of sample model efficiency 234–5, 236 out-of-the-money options 26, 34, 35 over the counter (OTC) contracts 158 parallel change in yield curve 77–9 parameters misspecification 359–60, 361–3 Parner, E 143, 144 partial derivatives (PDs) 48–9, 50–1, 52–3, 65 Patton, A.J 310 peak over threshold (POT) method Peccati, L 50, 51 Peña, J.I 304 Penzer, J 162 Perez, J.V 304 perfect economy 280–2 Peters, J.P Philipov, A 108 Philips, T 242 Philips and Perron test 307 Phoa, W 128 Pickands, J Pignatiello, J.J 248 Pistre, N 192 Platen, E 112 Poisson distribution Pollak, M 249 Polson, M 354 Poon, S.H 305 Popper, K 281 portfolio holdings-based risk measure see diversification-based risk measure portfolio insurance 23, 30 pricing 37–43; insurance and dollar-denominated risk 42–3; insurance with rebalancing 39–42; insurance without rebalancing 38–9 portfolio optimization covariance structure of asset returns and optimal portfolio weights 243–6 impact of correlation jumps 237, 238, 239 INDEX with inflation-linked products 173–82 time-varying return correlations and the efficient set of portfolios 265–77 portfolio rebalancing see rebalancing portfolio weights see weights, portfolio positive homogeneity 28 Poterba, J.M 281 Poteshman, A.M 304 preferences firms’ 35–7 model user’s 204 price of risk 335–7, 351 price risk 77 pricing error 202 probability density function 216–17 projected pursuit CUSUM (PPCUSUM) control charts 249–50, 252–3 proportional weight changes 53, 55, 58–63, 65–6 purpose, model’s 209 put options 158 QQ plots 12, 13 quantile regression 217 quasi-debt leverage ratio 124–5 Ramchand, L 305 ratchet options 210 real economy 282–5 real option theory 279 rebalancing 23, 27, 30, 31–2, 33–4 portfolio insurance with 39–42 record keeping 204–5 reinvestment risk 77 relative weight changes 51–3 relevance 28 Renault, O 127, 143 Revised Framework of the International Convergence of Capital Measurement and Capital Standards (Basel II) 1–2, revision of models 206 risk measurement see diversification-based risk measurement riskfree capital monotonicity 28 RiskMetrics 226–8 Robbins, H 286 robust conditional moment test 344–7 Rockafellar, R.T 23 Roll, R 281 rolling estimator 267–76 Roncalli, T 3, 8, root mean square error (RMSE) 103 Ross, S.A 258, 328 Rubinstein, M 196 Runger, G.C 248 Runkel, D.E 330 373 100–2, S&P500 index 305–8, 312–23 SABR model 360 Saltelli, A 48, 49, 59 sampling error 162, 167 Santa-Clara, P.P 108–9 Sarnat, M 280–1 savage score correlation coefficients (SSCC) 49, 59–65 Savickas, R 119 Scaillet, O 48 scale parameter 146–7 scenario tests 164 Schied, A 23, 88 Schipper, S 242, 247 Schmid, W 242, 244, 247, 258 Scholes, M 109, 110, 117, 194, 280 see also Black–Scholes option pricing model Schönbucher, P.J 139–40 Schoutens, W 359 Schwebach, R.G 108 Schweizer, M 112 Schwert, G.W 246, 254, 329 SDAX index 331–51 second-order autoregressive process 289, 293 sensitivity analysis (SA) background 50–1 impact of collection threshold on capital charge for operational risk 16, 17, 18, 19 portfolio volatility 47–68; effect of relative weight changes 51–3; importance of portfolio weights in GARCH volatility estimation models 53–6; trading strategies through SA 56–65 September 11 2001 terrorist attacks 272, 273, 303 impact on volatility transmission patterns between USA and Spain 303–26 sequential control procedures 241–64 comparison of multivariate and simultaneous control charts 253–8; 374 INDEX behavior in the out-of-control state 255–8; sequential control procedures, behavior continued structure of Monte Carlo study 253–5 covariance structure of asset returns and optimal portfolio weights 243–6 multivariate statistical surveillance 246–51 simultaneous statistical surveillance 251–3 Servigny, A de 143 severity distribution 5–7, 12, 13, 16, 17 Shanken, J 258 Sharma, J 328 Sharpe, W.F 107, 194, 278 Shaw, S 284 Shephard, N 54 short selling constraints 267 shortest path 28 shorthand, models as 197–8 Shreve, S 357 shrinkage 162–3 shrinking method 249 Simon, C.P 48 Simons, E 359 simplicity 207–8 simultaneous MEWMA control statistic 253, 254–8, 259 simultaneous statistical surveillance 251–3 comparison of multivariate and statistical control charts 253–8 single contracts 166–7 skewness 270, 271, 274 skewness parameter 146–7 SMALL CAP index 331–51 small capitalization stocks 327–52 smile 109, 196, 354 Solnik, B 305 Spahr, R.W 108 Spain–USA volatility transmission patterns 303–26 spreadsheet syndrome 205 Sprecher, C.R 283 stability, index of 146–7 standard portfolio analysis of risk (SPAN) risk management system 25, 27 standardized approach (SA) state–space representation 95–7 statistical surveillance 242 comparison of multivariate and simultaneous 253–8 multivariate 246–51 simultaneous 251–3 steepening in yield curve 81–2, 83 Stein, E.M 354, 356 Stein, J.C 354, 356 Stein, R.M 109 Steinand, D 242 stochastic volatility 48, 86–106, 354 and conditional volatility 86–7, 88–92 idiosyncratic risk, systematic risk and 107–31; simulation study 118–26; stochastic volatility and Merton’s pricing 112–14; stochastic volatility model 114–17 and interest rate term structure forecasting 87, 92–103 stocks hedging with inflation-linked products 182–9 large and small capitalization stocks in Europe 327–52 optimal portfolio and inflation-linked bonds 181–2 Strauss, J 304, 323 stress testing for models 209 Stuck, B.W 148 Stulz, R.M 283, 284, 285 subadditivity 28 Summers, L.H 281 Sunderman, M.A 108 super-hedging strategies 203 Susmel, R 305 swaps 26, 34, 158 systematic risk 279 idiosyncratic risk, stochastic volatility and 107–31 and the perfect economy 280–2 T control charts 248, 252, 254–8 Taksler, G.B 108 Talay, D 192 Tay, A.S 216 Taylor, S.J 89 telecommunication 57, 63, 64 term structure of interest rates forecasting see interest rate term structure forecasting terminal portfolio values 24–5 terrorism see September 11 2001 terrorist attacks Thakor, A.V 284 INDEX Theodossiou, P.T 242 three-stage least-squares (3SLS) method 98 time-varying return correlations 265–77 Tistaert, J 359 Torra, S 304 Torró, H 311, 317 total risk 279 and the real economy 282–5 trading hours, non-synchronous 304–5 trading/reallocation strategies 56–66 transaction costs 35–7 transparent economy 282–3 Trautmann, S 170, 173, 174, 176–7 truncation 9–11 two-factor model for interest rates 69–85 basic model 70–2 generalized duration and convexity 72–4 hedging ratios 74–5 proposed solution for limitations of the conventional duration 75–83 Tzotchev, D 242 unconditional correlations 268–76 unconditional coverage property 215 uniform probability distribution 288, 290–1, 294 uniform weight changes 52–3, 55, 58–63, 65–6 United States (USA) 328 correlation jumps with Euro area and Japan 226–36 volatility transmission patterns between Spain and 303–26 value-at-risk (VaR) 22, 48, 213–25 asymmetric VaR-BEKK model 309–11, 312–17, 322 cross-section approach 217–24; comparative Bayesian analysis of performance 221–2, 223; failures analysis 222–24; intuitive example 219–21 review of existing methods for backtesting 214–17; tests based on hit function 214–15; tests based on multiple VaR levels or entire probability density function 216–17 weather derivatives portfolio risk 167–8 375 Vanna 359, 362–3 variance large and small capitalization stocks 342–4, 348, 351 mean–variance optimization models 265–6, 276 Vasicek, O.A 87 Vecchiato, W 48 VECH model 330 vector valued CUSUM 249 Vega 359 Venkataraman, S 285 verification of data 208 volatility asymmetric 328–9, 348, 349–50, 351 conditional 86–7, 88–92 correlation jumps and 228–36, 237–9; portfolio optimization 237, 238, 239 GARCH models compared with rolling estimates for time-varying return correlations 271–2 implied 109, 195–7, 354 sensitivity analysis of portfolio volatility 47–68 spillovers between large and small firms 345–8, 349–50 stochastic see stochastic volatility transmission between large and small firms in Europe 327–52 transmission patterns between USA and Spain 303–26 volatility feedback hypothesis 328–9, 337, 348 volatility impulse response function (VIRF) 311 volatility smile 109, 196, 354 volatility surface 196–7 Volga 359, 362–3 Wagle, B 278, 286 Wald, A 248 Wang, S 163 weather derivative portfolios 156–69 accurate estimation of correlation matrix 162–3 consistency between valuation of single contracts and portfolios 166–7 dealing with non-normality 163–4 defining risk for 159–60 estimating model error 164 estimating sampling error 167 estimating VaR 167–8 376 INDEX weather derivative portfolios continued incorporation of hedging constraints 165–6 incorporation of sampling error 162 methods of estimating risk in 160–2 nature of weather derivatives 157–9 Weber, M 304 weights, portfolio covariance structure of asset returns and optimal portfolio weights 243–6 importance in GARCH volatility estimation models 53–6, 66 relative weight changes 51–3 White, A 109, 354 Wolf, M 163, 241, 258 Wongswan, J 305 Woodall, W.H 258 Woodward, D 360 Wooldridge, J.M 311, 335, 342, 344, 345 Wu, G Xu, Y 329 108, 111 Yashchin, E 242 Yashin, A.I 143 yield curve 69–70, 83–4 forecasting stochastic volatility and 86–106 shifts in 77–83 yield curve options 209–10 yields 76–83 Yildirim, Y 189 Yong, S 304, 323 zero-coupon bonds 72–4 inflation-linked 173, 177–81 Zhang, J 248, 249, 250 Ziemba, W.T 23 ... of risk control and risk management in an incomplete market setting where practical constraints exist Introduction Chapter examines the estimation of operational risk exposure of financial institutions,... programming solves for the portfolio η* in section 2.4 26 INCORPORATING DIVERSIFICATION INTO RISK MANAGEMENT Define a trivial acceptable portfolio ηc consisting of $1 invested only in riskfree... non-normality, incorporate hedging constraints, estimate sampling error, allow consistency between single contract pricing and portfolio modeling, and give quick estimates of VaR Chapter links nominal interest