3GBINDEX 11/18/2013 16:4:36 Page 315

Index

Albanese, C., 285 about, 251–252 Altman E., 61 Basel I, 23–24, 252 Andersen, L., 134 Basel I, reason for, 23 Anderson, M., 65 Basel II, 23, 153, 252 Anderson Darling test, 51 Basel II, credit ARCH model, 50, 161, 177 (CVaR) approach, 252–258 Artificial intelligence and financial Basel II, default probability- modeling, 287–299 default correlation relationship, Bayesian probabilities, 295–297 259–260 chaos theory, 291–295 Basel II, reason for, 23 chaos theory and finance, Basel II, required capital (RC) for 293–295 credit risk, 258–259 fuzzy logic, 290 Basel III, 23, 153, 252, 262 genetic algorithms, 290–291 Basel III, credit value at risk genetic fuzzy neural algorithms, (CVaR) approach, 262 291 Basel III, reason for, 23 neural networks, 287–289 and double default treatment, Artificial neural network (ANN), 269–274 288 double-default approach, Asset modeling, 174–176 270–274 Asset value, 159 substitution approach, Asymptotic single risk factor (ASRF) 269–270 approach, and correlation, Basel Committee for Banking 274 Supervision, 16 Attractor, 292 Baxter, N., 134 Autocorrelation (AC), 50 Bayes, Thomas, 295 Bayesian methods, 297, 299 Backhaus, J., 134 Bayesian probabilities, 295–297 Backpropagation, 288, 298 Bayesian statistics, 299 Backwardation techniques, 289 Bayesian theorem, 295–296 Bahar, R., 74 Bellaj, T., 285 Basel, multiplier, 268–269 Binomial approach, 213 Basel accords. See also Correlation Binomial correlation approach, and Basel II and III 212

315 Correlation Risk Modeling and Management: An Applied Guide Including the Basel III Correlation Framework—with Interactive Correlation Models in Excel/VBA, First Edition. Gunter Meissner. © 2014 John Wiley & Sons Singapore Pte. Ltd. Published 2014 by John Wiley & Sons Singapore Pte. Ltd. 3GBINDEX 11/18/2013 16:4:37 Page 316

316 INDEX

Binomial correlation measure Capital asset pricing model. See about, 72–73 Black-Scholes-Merton (BSM) binomial correlation measure pricing model application, 73–74 Capital charge, 17, 273–274 vs. Pearson correlation model, 90 Capponi, A., 215 Binomial correlation measure Carr, P., 148, 159 application, 73–74 Cash CDOs, 103 Bivariate Gaussian copula, 75 n1, CDO’s (collateralized debt 132 obligations). See Collateralized Bivariate normal distribution, 80, debt obligations (CDOs) 270 CDS (credit default ). See Black, F., 108, 176 (CDS) Black-Scholes-Merton (BSM) 1973 Chang, E., 148, 159 option pricing model, 9, 58, Chaos theory, 291–295, 299 106, 167 and finance, 293–294 Black-Scholes-Merton (BSM) option and finance application, 294–295 pricing model, 6 Chaotic system criteria, 291 Bollerslev, T., 160–161, 176 Cherubini, U., 65 Bond correlation Chi-squared test, 51 and default probability Cholesky decomposition, 81, 91–93, correlations, 53–54 116, 132 distribution, 54 Circularity, 89, 151 Bounded Jacobi process, 165–167 Clark, P., 148 Brace, A., 176 Clustering of , 161 Brigo, D., 72, 215, 221, 225 Collateralized debt obligations Briys, E., 109 (CDO’s) Brooks, B., 61 about, 101–102 Brown, R., 163 advantages, 102–103 Brun, Marie-France, 169 basics, 101–105 Bubbles, 18–19 binomial correlation valuation, 73 Buraschi, A., 169, 171–172, 176 CDO complexity, 114 Buraschi, Porchia, and Trojani correlation, 222 model (2010), 168–169, global financial crisis, 18–19 171–172, 176 market price risk, 183 Burtchell, X., 65 model limitations in valuation, Buying correlation, 11 115 problems with Gaussian copula Calibration valuation, 113–116 financial models, 59 recovery rate, 113 Gaussian Copula limitations, 87 types of, 103–105, 115 Call options, 237 valuing, 105–113 buying and selling, 13 Commodity market, 181 3GBINDEX 11/18/2013 16:4:37 Page 317

Index 317

Commodity risk, 14 definition, 182 Computational finance, 284 financial practice examples, Concentration ratio, 33 184–187 Concentration risk, definition, 30 financial practices, 182–184 Conditional correlation, 162 and Gora in investments, Conditional defaults, 121 187–189 Conditional VaR risk, 24 measures of, 198–199 Conditionally independent default of risk/return ratio, 187 (CID) Cora financial practice examples application of, 149 in market risk management, contagion correlation models, 90 189–197 correlation approach, 134 option Cora and Gora, 185–187 correlation modeling, 122 option Vanna, 184–185 OFGC model, 120, 138 Cora in market risk management Constant asset correlation, vs. about, 189–195 stochastic asset correlation, 163 Gap-Cora, 196–197 Cont, R., 148 Correlated default distribution, Contagion correlation, 134 111 Contagion correlation models, Correlated default risk, 33 88–91, 148–150 Correlated default time Contagion default modeling, derivation of, 124 150–153 for multiple assets, 81–82 Copula applications, 85 using survival principles, 82–84 Copula correlation model, 18, 60 Correlated market risk, 33 Copula correlations Correlating Brownian motions, 90. about, 74–75 See also Geometric Brownian advantages and limitations, 90–91 motion (GBM) copula applications, 85 Correlation correlated default time for definition, 34–35 multiple assets, 81–82 definition by usage, 36–37 correlated default time using Correlation, basics, properties and survival principles, 82–84 terminology Gaussian copula, 76–81 about, 1–2 Gaussian copula limitations, correlation risk, 2–5 85–88 correlation risk as part of finance Copula functions, 75 risk, 24–33 Copula model, 27, 58 finance usage, 6–24 Copulas, limitations of, for finance, Correlation, empirical properties 90–91 bond correlations and default Cora probability correlations, 53–54 of a CDO, 229–230 equity correlation and for CDSs, 223–225 autocorrelation, 50–51 3GBINDEX 11/18/2013 16:4:37 Page 318

318 INDEX

Correlation, empirical properties Correlation modeling future, (Continued) numerical finance, 283–287 equity correlation distribution, Correlation models, 34 51–52 Correlation risk equity correlation volatility as about, 2–5 indicator, 52–53 in Basel framework, 278 in expansion through recession in a collateralized debt obligation periods, 43–46 (CDO), 227 mean reversion and, 46–49 and concentration risk, 30–33 summary, 54–55 Cora for different tranches, 230 Correlation and Basel II and III Cora parameter, 182 Basel accords, about, 251–252 and credit risk, 25–27 Basel accords and double default hedging correlation risk, 238 treatment, 269–274 and market risk, 24 Basel II and III credit value at as part of finance risk, 24–33 risk (CVaR) approach, quantification of, 195 252–258 and systemic risk, 27–30 Basel II required capital (RC) for types of, 34–35 credit risk, 258–260 wrong-way risk (WWR), 222 credit value adjustment (CVA) Correlation risk as part of finance with wrong-way risk (WWR), risk, 24–33 264–269 correlation risk and concentration credit value adjustment (CVA) risk, 30–33 without wrong-way risk correlation risk and credit risk, (WWR), 261–264 25–27 debt value adjustment (DVA), correlation risk and market risk, 274–276 24 funding value adjustment (FVA), correlation risk and systemic risk, 276–278 27–30 wrong-way risk quantification, terminology, 33–34 268–269 summary, 34–35 summary, 278–279 Correlation risk in CDO Correlation coefficient, 33–34 Cora of a CDO, 229–230 Correlation concept Gora of a CDO, 230–231 about, 128–129 types of risk, 227–228 loss distribution of, 129–130 Correlation risk parameters Cora tranche spread-correlation and Gora, 182–184 relationship, 130–131 Correlation smile, 87, 136 Correlation desks, 8, 33 Correlation swap, 11–13, 244–245 Correlation , 189 payoff of, 12 Correlation matrix, 75 n1 Correlation swap hedge, 246 Correlation modeling, 144 Correlation trading, 8, 33 3GBINDEX 11/18/2013 16:4:37 Page 319

Index 319

Correlation via stochastic time indexes, 104 change, 148–150 market price risk, 183 Correlation via transition rate payoff tree, 206–208, 216–217 volatilities, 146–147 payoff tree and CDS spread Correlation volatility, 45, 53 payment tree, 219–220 Correlation-dependent option, payoff tree and CDS spread tree, 239–244 210–211 Counter-party default risk, 211 premium tree, 206 Counter-party risk, 102 vs. , 247 Counting process, 152 spread, 205 Covariance, 36 spread impact testing, 211–213 Covered put buying, 236 spread payment tree, 217–219 Cox, J., 108, 184 spread tree, 207–210 Cox-Ingersoll-Ross (CIR) process, in synthetic CDO, 103 70, 173, 221 Credit exposure, 266, 268 Credit collection risk Credit products, 183 definition, 202 Credit risk positive vs. negative, 204 collateralized debt obligation Credit correlation, 34 (CDO), 101–102, 227 Credit correlation risk, 231 and correlation risk, 24 Credit correlation risk quantification credit exposure, 266 about, 201–203 credit value at risk (CVaR) in a CDS, 203–205 calculation, 252, 263–264 correlation risk in CDO, 227–231 types of, 201 pricing CDSs including credit Credit triangle, 112 correlations, 215–227 Credit value adjustment (CVA) pricing CDSs with entity- types of correlations, 23, 251 counterparty credit correlation, without wrong-way risk (WWR), 205–215 261–264 summary, 231–232 with wrong-way risk (WWR), Credit counterparty risk, 251 264–269 Credit default swap (CDS). See also Credit value at risk (CVaR) Pricing CDSs Basel accord, 251–253, 271 bond risk, 208–210 copula model, 58 counterparty risk with correlated default risk, 33 correlation-dependent option, default probability PD(T), 257 239–244 definition, 252 in a credit correlation risk valuation, 253 quantification, 203–205 Crossover method, 290 defined, 13 n3 Cumulative default probability, definition, 203 145 example of, 4 Currency risk, 14 3GBINDEX 11/18/2013 16:4:37 Page 320

320 INDEX

CVA liability. See Debt value and zero default correlation, adjustment (DVA) 257–258 Default risk, 25, 143, 201, 231 Da Fonseca, Grasselli, and Ielpo Default term structure, 27 model (2008), 168–169, Default threshold, 110 171–172, 176 Default time copula, 81 Da Fonseca, J., 169, 171–172, 176 Delta hedge, 237 Das, S., 61, 85, 160 Dependence, 35–36 Davis, M., 88, 91 Dependence and correlation De Varenne, F., 109 correlation, 36–37 Debt value adjustment (DVA), dependence, 35–36 274–276 independence and Debt/equity ratio, 223 uncorrelatedness, 37–38 Debt-GDP ratio, 18 Dependence measures, 35 Default contagion, 151 134, 151 Dependency coefficient, 34 Default correlation Dependency models, 34 and asset price correlation, 72 Derivative, 236–237 binomial correlation, 74 Derivative transaction, 261 of CDO assets, 110–113 Derived base correlation, 136 impact of, 21 Derman, E., 176 impact on creditors, 27 Deterministic nature, 293 between industries, 25 Deterministic process, 157 OFGC model measurement, 120 Ding, X., 144, 150–151 properties of, 54 Directing process, 148 terminology usage, 34 Distance to default (DD), 108 top-down correlation model Diversification, 102 approach, 153 Donnelly, C., 87 Default distribution, 111, 144, Dorsey, R., 289 146 Double default approach, 272–274, Default intensity, 81, 120, 206, 279 224 n7 Double t copula, 213 Default intensity contagion, 89 Double-default approach, 270–274 Default intensity correlation, 226 Dow correlation levels and Default probability PD(T) volatility, 44 assumptions regarding, 119 Dow Jones Industrial Average (the binomial correlation measure, 73 Dow), 13, 29 in CDO’s assets, 106–109 Duellmann, K., 176 correlation of default vs. systemic Duffie, D., 84–85, 109, 112, 135, risk, 260 277 correlation relationship, 259–260 Düllman, K., 163 and credit risk, 267 Dynamic conditional correlations example, 77 (DCCs), 160–162, 177 3GBINDEX 11/18/2013 16:4:37 Page 321

Index 321

Dynamic financial correlations, 2, Finance usage 34 call options, buying and selling, Dynamic theory, 296 13 correlation swap, 11–13 Eckner, A., 85, 135 global financial crisis of 2007/ Econophysics, 284 2008 and correlation, 18–23 Embrechts, P., 87 investments and correlation, 6–8 Emmerich, C., 167 multi-asset options, 9–10 Enders, W., 161 quanto option, 10–11 Engle, R., 160, 162, 176–177 regulation and correlation, 23–24 Enterprise risk management (ERM), risk management and correlation, 14 14–17 Equity correlation trading and correlation, 8–14 and autocorrelation, 50–51 swap, 14 autocorrelation (AC), 50–51 Financial Chaos Theory (company), behavior in economic periods, 43 294 bonds vs. equities, 53–54 Financial correlation modeling distribution, 51–52 about, 69–72 mean reversion, 46–47 binomial correlation measure, volatility as indicator, 52–53 72–74 Equity market, 181 Cholesky decomposition, 91–93 Equity risk, 14 contagion correlation models, Ertuk, E., 74 88–90 European debt crisis, 18 copula correlations, 74–88 Exchange option price, 10 Gaussian default time copula, 93 Expansion through recession summary, 90 periods, 43–46 Financial correlation models Expansionary period, 44 about, 143–144 Expected default frequency (EDF), contagion default modeling in 108 top-down models, 150–153 (ES) models, 146–150 market risk measures of portfolio, One-Factor Gaussian Copula 14, 24 Model (OFGC) revisited, vs. VaR, 17 144–146 Exposure at default (EAD), 113 top-down approaches, 143–154 Extensions, 132–135 summary, 153–154 Financial correlations, 43 Fadlalla, A., 289 Financial modeling, 66 Fair tranche spread, 127, 129 Financial models Fan, K., 72 about, 57–59 Finance risk, correlation risk as part calibration, 59 of, 24–33 limitations, 60 3GBINDEX 11/18/2013 16:4:38 Page 322

322 INDEX

Financial products, 58 Gençay, R., 289 Financial risk management, 14 General wrong-way risk (WWR), Finger, C., 87 251, 264–265, 279 Finite-state Markov chain, 146 Generator matrix, 146, 148 Fitting tests, 51 Genetic algorithms, 290–291, 299 market, 181 Genetic fuzzy neural algorithms, Foreign exchange market, 181 291 Forward default probability, 83 Genetic fuzzy neural networks Forwards, 236 (GFNNs), 290–291, 299 Fractal finance software, 294 Geng, G., 61, 160 Fractals, 292 Geometric Brownian motion Frailty variables, 121 (GBM), 70, 158, 163–164, 174, Freed, L., 61, 160 177 Freisleben, B., 289 Giesecke, K., 134–135, 144, Frequentist probability theory, 150–152 295–296 Gimonet, G., 285 Frey, R., 134 Girsanov theorem, 149 Funding cost, 277 Global financial crisis of 2007/2008, Funding value adjustment (FVA), and correlation, 18–23 274, 276–279 Global financial crisis of 2007/2009, Future trades, 236 74 , 247 causes of, 22 Fuzzy logic, 290, 298 Goldberg, L., 135, 144, 150–151 Gora. See also Cora Gamma process, 148 of a CDO, 230–231 Gap-Cora, 183, 196–197 for CDSs, 225–227 GARCH model, 50, 161, 177 definition, 183 Gatarek, D., 176 in market risk management, Gaussian copula 197–198 Basel II use of, 253 Gordy, M., 274 bivariate case of, 132 Gourieroux, C., 169 calibration, 87 GPUs (graphical processing units). category of, 75 See Graphical processing units default time copula, 93 (GPUs) derivation, 76 Gradojevic, N., 289 example of, 76–81 Graphical processing units (GPUs) limitations, 85–88, 115–116 benefits, 285–287 risk management, 87–88 limitations, 287 tail dependence, 85–87 model for valuing portfolio valuation problems and CDO’s, counterparty risk, 285 113–116 technology, 284, 298 Geman, H., 148 Grasselli. M., 169, 176 3GBINDEX 11/18/2013 16:4:38 Page 323

Index 323

Great Recession of 2007 to 2009, correlating Brownian motions, 21, 54, 251 69–72, 221 Gregory, J., 65 extensions of, 168–172 Gumbel copula, 87 with stochastic correlation, 168–172 Hacker, R., 161 Ho-Lee model (1986), 176 Hagan, P., 72, 184 Horel, G., 85, 135 Hatemi-J, A., 161 Housing market, 18 Hatfield, G., 289 Huang, H., 72 Heath, D., 176 Hudson, R., 292 Heath-Jarrow-Morton (HJM) Hull, J., 73, 115, 159–160, model, 146 176–177, 195, 236, 277 Hedge funds, 19–20 Hurd, T., 144, 148–149, 152–153 Hedges/hedging Hybrid-CID-contagion modeling, correlation risk, 235–248 134–135 definition, 235 with future or swap, 247 Ielpo, F., 169, 176 with an option, 248 Implied correlation, 136 Hedging correlation risk , 13 n2 about, 235–238 Independence and uncorrelatedness, challenges of, 238–239 37–38 options vs. futures use, 247–248 Indexes, 13 summary, 248 Ingenhousz, J., 163 n2 Hedging correlation risk examples Initial conditions, 291–293 CDS counterparty risk with Interest rate risk, 14 correlation-dependent option, Interest rates, 46 239–244 International Swaps and Derivative with correlation-dependent Association (ISDA), 278 option, 239–244 Internet bubble, 19 VaR correlation risk with Investments and correlation, 6–8 correlation swap, 244–247 Hedging financial correlation risk, Jaeckel, P., 72 239 Jamshidian, F., 176 Hedging strategy, 22 Jarrow, R., 88–89, 91, 109, 146 Henry-Labordere, P., 72 Joe, H., 85 Heston, S., 69, 90 Johnson SB distribution, 51, 54 Heston correlation approach, Joint default correlation, 4 71 Jorion, P., 195 (1993), 212 applications of, 72 Kahl, C., 72 Buaschi (2010) and Da Francesca Kapadia, N., 61, 85, 160 models (2000), 171–172 Kawano, N., 288 3GBINDEX 11/18/2013 16:4:38 Page 324

324 INDEX

Kendall t (tau), 64–65 Longstaff, F., 108 Kherraz, A., 87 Lorenz, E., 291 Kim, J., 108 Loss distribution, 129–130 Kolmogorov-Smirnov test, 51 Lu, X., 172, 175–177 Kuell, J., 176 Lucas, D., 72, 212 Kukolj, D., 289 Luciano, E., 65 Küll, J., 163 Kumar, D., 72, 184 Ma, J., 167 Kunisch, M., 163, 176 Madan, D., 148, 159 Kuznetsov, A., 144, 148–149, Mandelbrot, Benoit, 292 152–153 Marked-to-market (MtM) value, 263 Lando, D., 84, 109, 112, 146 Market correlation risk, 198 Langnau, A., 72 Market correlation risk Laplace, Pierre, 295 quantification Large homogeneous portfolio (LHP) about, 181 assumptions regarding, 138, 144 Cora and Gora in investments, individual default probability, 187–189 254 Cora financial practice, 182–184 One-factor Gaussian Copula Cora financial practice examples, (OFGC) model, 119–120 184–187 test results of, 87 Cora in market risk management, top-down correlation model, 153 189–198 Latent variables, 121 correlation risk parameters Cora Laurent, J-P, 65 and Gora, 182–184 Lehman Brothers, 22 summary, 198–199 Lesniewski, A. S., 72, 184 Market risk, 14, 24, 252, 264 Leverage, 171 definition, 181 Leveraged super-senior (LSS) factors, 265 tranches, 21 measures of portfolio, 199 Li, David, 60, 74, 76, 87, 90 Market value at risk (VaR), 252 LIBOR market model (LMM) Markets, types of, 181 process, 72, 208, 211–212, Markov chain models 214, 222 contagion correlation models, Lin, C., 289 148–150 Linear dependence, 38 correlation via stochastic time Linear relationships, 67 change, 148–150 Lo, V., 88, 91 correlation via transition rate Local correlation model (LCM), 72, volatilities, 146–147 134 Markov property, 158 model of Dupire, 72 Married put, 236 Long correlation, 229 Mathematical inconsistencies, 59 3GBINDEX 11/18/2013 16:4:38 Page 325

Index 325

Maturity adjustment, 259 Numerical finance Mean reversion artificial intelligence and financial equity correlations, 46 modeling, 287–299 of financial correlations, 164–165 correlation modeling future, low levels of, 166, 173 283–287 quantification of, 47–49 definition, 284 stochastic process modeling, 176 GPU benefits, 285–287 Mean reversion level, 53 GPU limitations, 287 Measures GPU model for valuing portfolio Kendall t (tau), 64–65 counterparty risk, 285 Pearson Correlation, approach GPU technology, 284 and limts, 60–62 summary, 298–299 Spearman’s Rank Correlation, 62–64 One-factor copulas, 115, 213 Measures of association, 35 One-Factor Gaussian Copula Meissner, G., 72, 74, 172, 175–177, (OFGC) 236, 288–289 Basel II use of, 253, 256 Merton, R, 107, 208 correlation risk parameters Cora Merton 1974 model, 108 and Gora, 227 Migration risk, 25, 201, 231 credit risk methodology to pricing Milne. F., 148 LHP, 119 Miltersen, K., 176 dynamic version of, 177 Model input parameters, 221 version testing, 212 Model limitations in valuation, 115 One-Factor Gaussian Copula Model pricing CDSs (OFGC) Model with entity-counterparty credit about, 119–120 correlation, 206–215 benefits of, 135–136 including credit correlations, correlation concept in, 128–131 216–223 extensions of, 132–135 Moral hazard, 23 hybrid-CID-contagion modeling, Multi-asset options, 9–10, 167 134–135 Multilayer perception (MLP) limitations of, 136–138 network, 288 original model, 121–122 Musiela, M., 176 randomness in, 127–128 Mutation, 290 revisited, 144–146 and standard copula, 131–132 Nagpal, K., 74 valuing tranches of CDO with, Negative correlation, 222 122–128 Neural networks, 287–289, 298 summary, 135–139 NINJA loans, 21 One-factor Student’s t copula, 133 Nonadditivity, 195 Operational risk, 14, 24, 252 Normal economic period, 44 Option Cora and Gora, 185–187 3GBINDEX 11/18/2013 16:4:38 Page 326

326 INDEX

Option premium, 248 Posterior probability, 299 Option Vanna, 184–185 Predictability, 293 Options “Predictability: Does the Flap of a call options, 237 Butterfly’s Wings in Brazil Set call options, buying and selling, 13 Off a Tornado in Texas?” vs. futures use, 247–248 (Lorenz), 291 with an hedges/hedging, 248 Presdescu, M., 115 multi-asset options, 9–10, 167 Pricing CDSs including credit options vs. futures use, 247–248 correlations put options, 236 about, 215–216 put options vs. credit default swap CDS payoff tree, 216–217 (CDS), 247 CDS payoff tree and CDS spread quanto options, 10–11, 167 payment tree, 219–220 Ordinal correlation measures, CDS spread payment tree, 217–219 62–63, 67 Cora for CDSs, 223–225 Ordinal rank correlation measures, Gora for CDSs, 225–227 65 model, 216–223 Ornstein, L., 165 model input parameters, 221 Ornstein-Uhlenbeck process, 47 results, 221–223 Outliers, 66–67 Pricing CDSs with entity- Outstanding notional (ON), 126 counterparty credit correlation Overcollateralization, 102 about, 205–206 CDS payoff tree, 207–208 Pairs trading, 8 CDS payoff tree and CDS spread Pallavinci, A., 72, 221, 225 tree, 210–211 Partial truth, 298–299 CDS spread impact testing, Pearson coefficient, 34 211–213 Pearson correlation approach, 8, CDS spread tree, 208–210 60–62, 73 models, 206–215 Pearson correlation coefficient, 7, results, 213–215 37, 61, 160 Prior probability, 299 Pearson correlation model, 10, 34, Probability space, 31–32 36, 60, 90 Put options, 236 Pearson covariance, 36 Percentage and logarithmic changes, Quanto, 10 38–39 Quanto options, 10–11, 167 Pietronero, G., 285 Poisson process, 152 Ramaswamy, K., 108 Porchia, P., 169, 171–172, 176 Random factor loading (RFL) Portfolio risk, 189 model, 134 Portfolio variance, 253 Randomness, 127–128 Positive correlation, 223 Rating agencies, 23, 73, 252 3GBINDEX 11/18/2013 16:4:38 Page 327

Index 327

Recession, 44 Sandmann, K., 176 “Recipe for Disaster: The Formula Schönbucher, P., 134–135, 144, That Killed Wall Street” 146, 153 (Wired), 101 Schubert, D., 134 Recovery rate, 113, 120 Schwartz,, E., 108 Reduced form approach, 109 Schweizer, B., 74 Regime changes, 292 Selection methods, 290 Regimes, 293 Self-similarity, 292–293 Regulation and correlation, 1, 5, Selling correlation, 11 23–24 Seneta, E., 148 Renault, O., 160 Servigny, A., 160 Repo, 22 n13 Sidenius, J., 134 Required capital (RC) Singleton, K., 84, 109 Basel II, 258–260 Sironi, A., 61 for credit risk, 259, 278 Sklar, A., 74, 90 CVaR for, 273 Slim-tails, 195 Resti, A., 61 Sondermann, D., 176 Return/risk ratio, 6–7 Spearman’s Rank Correlation, Ripper. K., 289 62–66 Risk. See also specific risk, types of, Special purpose vehicle (SPV), 14, 227–228 102 Risk management Specific wrong-way risk (WWR) Cora financial practice examples, Basel accords definition, 264, 189–197 266–268, 279 Cora in market risk management, credit value adjustment (CVA) 189–197 correlations, 251 Cora market correlation risk , 19, 239 quantification, 189–198 Standard copula, 131–132 and correlation, 14–17 Standard deviation, 7 Gaussian copula, 87–88 Static correlations, 1, 34 Gora in market risk management, Statistical correlation models 197–198 application Risk neutral framework, 186 financial models, 57–60 Risk-adjusted return of a portfolio, measures, 60–65 187 Spearman’s Rank Correlation and Rooder, S., 72 Kendall t(tau), 65–66 Roulette wheel selection, 290 summary, 66–67 Rubinstein, M., 289 Statistical finance, 284 Stochastic alpha beta rho (SABR) Saita, L., 85, 135 model, 72, 90, 184 Sampling correlation from Stochastic asset correlation vs. distribution, 159–160 constant asset correlation, 163 3GBINDEX 11/18/2013 16:4:39 Page 328

328 INDEX

Stochastic correlation and volatility Subordination, 102 about, 172–174 Subordinator process, 148 asset modeling, 174–176 Substitution approach, 269–270, usage of, 176 279 summary, 177 Sufana. R., 169 Stochastic correlation models Sundaresan, S., 108 dynamic conditional correlations Survival probability, 83–84, 123 (DCCs), 160–162 Swap, 236. See also Credit default Heston model with stochastic swap (CDS) correlation, 168–172 correlation swap, 11–13, sampling correlation from 244–247 distribution, 159–160 correlation swap hedge, 246 stochastic correlation and hedges/hedging with, 247 volatility, 172–176 , 14 stochastic process, 157–159 Swap spread payment tree, 209 summary, 177 Synthetic CDOs, 103 Stochastic correlation standard Systematic risk, 260, 270, 278. See models also Market risk bounded Jacobi Process, Systemic factors, 25 165–167 Systemic market factor, 253, 255 geometric Brownian motion Systemic risk (GBM), 163–164 and correlation risk, 29 Vasicek 1997 model, 165 definition, 28 Stochastic , 177 Stochastic differential equations Tail dependence, 85–87 (SQEs), 69 Tail risk, 24 Stochastic model, 158 Tan, C., 289 Stochastic process, 157–159 Tankov, P., 148 , 173 Term structure model, 208 Strange attractors, 292 Terminology, 33–34 Stress tests/testing Tetrahex (company), 294 of Cora and Gora, 196–197, 199 The (Mis)Behavior of Markets: A crisis scenarios, 115 Fractal View of Risk, Ruin, and for different time horizons, 88 Reward (Mandelbrot and GPU technology use in, 286 Hudson), 292 magnitude of, 188 “The Formula That Felled Wall of models, 59 Street” (Financial Times), requirements for, 116 101 Structural approach, 109 Thinning process, 150–151 Structured investment products, Time period, 291 18 Time value, 107 Student t copula, 212–213 Tomecek, P., 1, 152 3GBINDEX 11/18/2013 16:4:39 Page 329

Index 329

Top-down approaches, 143–154 Variance-covariance matrix, 170 Top-down contagion, 150–153 Variance-covariance VaR approach, Top-down correlation model, 144 14 n4 Toxic assets myth, 22 Vasicek, O., 74, 87, 90, 119, 138, Toy, W., 176 144, 153, 165, 167 Trading and correlation, 8–14 Vasicek model (1997), 47, 144, 165, Trading book, 16 177, 212 Tranches VIX, 172, 175 in CDOs, 19 Volatility. See also Stochastic spread, 125–126 correlation and volatility spread-correlation relationship, clustering, 161 130–131 correlation, 45, 53 valuing CDOs, 122–128 Dow correlation levels and, 44 Transition matrix, 150 implied, 13 n2 Treasury inflation-protected security as indicator, 52–53 (TIPS) futures, 72 as standard deviation of returns, 6 Trojani, F., 169, 171–172, 176 stochastic, 173 Turnbull, S., 109, 146 Volatility skew, 58 Two-parameter copulas, 75 Volatility smile, 58, 87, 136, 289 Volatility-asset return correlation, Uhlenbeck, G., 165 34 Unexpected loss, 29, 34 U.S. debt, 18 “Wall Street Wizards Forgot a Few Variables” (New York Times), Value at risk (VaR) 101 concept, 3 Weber, S., 134–135 correlation risk, 244–245 West, G., 72 correlation risk with correlation White, A., 73, 115, 159–160, swap, 244–247 176–177, 277, 289 defined, 14 Wiener process, 147 limitations of, 195 Willeman, S., 134 market risk measurement, 24, 58 Wilmott, P., 167 model, 1 Wishart affine stochastic correlation portfolio risk, 189 (WASC) model, 169, 177 Valuing CDOs, 105–113 Wolff, E., 74 about, 105–106 Woodward, D. E., 72, 184 default correlation of CDO assets, Wrong-way risk (WWR) 110–113 Basel III, 23–24 default probability in CDO’s CDS default intensity correlation, assets, 106–109 222 Valuing tranches of CDOs, 122–128 correlation factor for, 271 Variance swap, 14 credit value adjustment, 262 3GBINDEX 11/18/2013 16:4:39 Page 330

330 INDEX

Wrong-way risk (WWR) Yildirum, Y., 72 (Continued) Yili, L., 289 positive default correlation, 4 Yor, M., 148 quantification, 268–269 Yu, F., 88–89, 91, types of, 264 134

Yao, J., 289 Zhou, C., 72