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date: 18 October 2019

(p. ix) List of Figures

(p. ix) List of Figures

  1. 3.1 Recovery Rate/Default Rate Association. 49

  2. 4.1 Optically inverted Z‐spread term structure. Ford Motor Credit, 31 December 2002. 77

  3. 4.2 Z‐spread, CDS, and BCDS term structures. Ford Motor Credit, 31 December 2002. 77

  4. 4.3 Implied default probability, average pricing error of the regression, and risk model specific risk, A‐rated industrials. 91

  5. 4.4 Survival probability term structures for different credit risk levels. 94

  6. 4.5 Hazard rates (forward ZZ‐spreads) for Ford and BBB Consumer Cyclicals, as of 31 December 2003. 96

  7. 4.6 Fitted par coupon for Ford and BBB Consumer Cyclicals, as of 31 December 2003. 98

  8. 4.7 Fitted Libor P‐spread for Ford and BBB Consumer Cyclicals, as of 31 December 2003. 99

  9. 4.8 CCP term structures and bond prices, Georgia Pacific, as of 31 December 2003. 100

  10. 4.9 Static hedging of a premium bond by CDS. 111

  11. 4.10 Static hedging of a discount bond by CDS. 112

  12. 4.11 Static hedging of a near‐par bond by CDS. 112

  13. 4.12 Coarse‐grained hedging of a premium bond by CDS. 115

  14. 4.13 Exponential spline factors. 118

  15. 7.1 5y CDS spread in basis points of five well‐known issuers. The spreads are plotted on a different scale before and after 20 June 2007. 198

  16. 7.2 ITX and CDX historical on‐the‐run index spreads. 205

  17. 7.3 CDX IG on‐the‐run index basis versus 5y index quote in basis points. 205

  18. 7.4 FTD spread ŝ1, STD spread ŝ2, and CDS spread ŝ in basis points (right axis) and Default Correlation (left axis) as a function of Jump Intensity. 208

  19. 7.5 On‐the‐run ITX Tranche Quote History (0–3% maps to right axis and is % up‐front. All other tranche prices are quoted as running (p. x) spreads and map to the left axis. Since Q1 2009 the 3–6% and 6–9% have also been quoted as % up‐front with 500bp running but for the purposes of this chart they have been converted to running spreads for the entire time series). Prior to Series 9 which was effective 21 Sept. 2007 and scheduled to roll on 20 Mar. 2008 the on‐the‐run index rolled every six months, subsequently, index tranche traders have continued to quote for Series 9. 212

  20. 7.6 Base Correlation Calibration to ITX Tranches (June 2009, calibrated to data in Table 7.2). 220

  21. 7.7 Cumulative loss probability implied from ITX tranche quotes. 220

  22. 7.8 Scatter plot of single‐name spread risk of ITX tranches against underlying single‐name spreads (June 2009). For clarity, risk for tranches with a detachment point of 100% are plotted on the right hand axis, all other tranches correspond to the left‐hand axis. 224

  23. 7.9 LHP cumulative loss distribution for P = 10% and various correlation assumptions. 243

  24. 7.10 LHP loss probability density for P = 10% compared to 100 name portfolio. 243

  25. 7.11 Base correlation calibration to CDX tranches (June 2009) GC = Gaussian copula, SR = Gaussian copula with stochastic recovery: showing that without SR, calibration of the 30% point fails even with an implied correlation of 100%. 252

  26. 8.1 Default distribution for a portfolio of 100 credits: λi = 2%, λw = 0.05%, λB = 5%,λSj=2.5%, pi,B = 0.24 andpi,Sj=0.16. 272

  27. 8.2 Default distribution for a portfolio of 100 credits: λi = 2%, λW = 0.05%, λB = 5%,λSj=2.5%, pi,B = 0 andpi,Sj=0. 273

  28. 8.3 Comparison of the default distributions for the calibrated Marshall‐Olkin, Gaussian and t‐copula. 274

  29. 8.4 Tail of the portfolio default distribution for the Marshall‐Olkin, Gaussian, and t‐copula. 275

  30. 8.5 Default correlation as a function of the time horizon for the Gaussian and Marshall‐Olkin copulas. 276

  31. 8.6 Base correlation skew. 282

  32. 9.1 Hazard rate for τmin = min(τ1,τ2). 287

  33. 9.2 State transitions for portfolio of size 5. 293

  34. 9.3 State space for rating transition model. 293

  35. 9.4 Distributions in the Infectious Defaults model. 295

  36. 9.5 Diamond Default model, version 1. 300

  37. (p. xi) 9.6 Calibrated parameters for Diamond Default model. 302

  38. 9.7 Diamond Default model, version II. 303

  39. 9.8 Inhomogeneous model: ordered states, n = 3. 304

  40. 9.9 Inhomogeneous model: unordered states, n = 3. 304

  41. 9.10 State space of enhanced risk model. 306

  42. 9.11 CDO premium reduction function for [a,b] tranche. 307

  43. 9.12 Aggregated four‐state Markov chain. 310

  44. 9.13 PDP representation of aggregated process. 311

  45. 9.14 State space of projected model. 319

  46. 9.15 (Xt, t) state space. 321

  47. 9.16 PDP sample function. 322

  48. 10.1 Illustration of construction for m = 3. Arrows indicate possible transitions, and the transition intensities are given on top of the arrows. 358

  49. 10.2 The next‐to‐default intensities, i.e. Qk k+1, in the three calibrated portfolios with parameters given by Table 10.3. The upper plot is for 0 ≤ k ≤ 125, while the lower displays Qk, k+1 when 0 ≤ k ≤ 26. 370

  50. 10.3 The five year implied loss distributions ℙ [L5 = x%] (in %) for the 2004‐08‐04, 2006‐11‐28 and 2008‐03‐07 portfolios, where 0 ≤ x ≤ 12 (upper) and 0 ≤ x < 22 (lower). The lower graph is in log‐scale. 371

  51. 10.4 The five‐year (upper) and fifteen‐year (lower) implied loss distributions (in %) for the 2004‐08‐04, 2006‐11‐28, and 2008‐03‐07 portfolios, where 0 ≤ x ≤ 60. Both graphs are in log‐scale. 372

  52. 10.5 The implied loss distributions ℙ [Lt = x%] (in %) for the 2004‐08‐04, 2006‐11‐28, and 2008‐03‐07 portfolios at the time points t = 1,5, 10, 15 and where the loss x ranges from 0% to 24%. 373

  53. 10.6 The implied loss surface for iTraxx Europe 28 November 2006, where 0% < x < 14% and 0 ≤ t < 10. 374

  54. 10.7 The implied portfolio losses in % of nominal, for the 2004‐08‐04, 2006‐11‐28, and 2008‐03‐07 portfolios. 375

  55. 10.8 The implied tranche losses in % of tranche nominal for the 2006‐11‐28 (left) and 2008‐03‐07 (right) portfolios. 375

  56. 10.9 The implied expected ordered default times 𝔼 [Tk] for the 2004‐08‐04, 2006‐11‐28, and 2008‐03‐07 portfolios where k = 1,…, 125 (left) and k = 26,…, 125 (right, for 2006‐11‐28 ). 376

  57. 10.10 The implied default correlation ρ(t) = Corr(1{τit}, 1{τjt }), ij as function of time for the 2004‐08‐04, 2006‐11‐28, and 2008‐03‐07 portfolios. 377

  58. (p. xii) 10.11 The structure of the non‐zero elements in the sparse matrix Q constructed in subsection 4.1.1 (see Equation (51)) with m = 10. 379

  59. 11.1 Illustration of the asymmetry of counterparty risk for a CDS. When buying protection, the maximum loss is 100% (reference entity default with zero recovery) but when selling protection it is smaller since it is related only to a tightening of the reference entity CDS premium. We have used ratings as a proxy for credit quality changes and have assumed a 5‐year maturity and CDS premiums of 25, 50, 100, 200, 400, 600, and 1000 bp for AAA, AA, A, BBB, BB, B, and CCC respectively. 386

  60. 11.2 Illustration of counterparty risk scenarios for a CDS contract. 387

  61. 11.3 Expected long protection CDS MtM value (replacement cost) at the counterparty default time computed using analytical and Monte Carlo approaches for an assumed correlation of 50% between the counterparty and reference entity. The Monte Carlo results use 1,000,000 simulations with the calculations bucketed with a width of 0.05 years. 392

  62. 11.4 Expected long protection CDS MtM value (replacement cost) at the counterparty default time as a function of correlation computed analytically. 392

  63. 11.5 Upper and lower bounds for the fair CDS premium when buying protection subject to counterparty risk compared to the standard (risk‐free) premium. 393

  64. 11.6 As Figure 11.5 but with the hazard rates of the reference entity and counterparty swapped. 393

  65. 11.7 Upper and lower bounds for the fair CDS premium when selling protection subject to counterparty risk compared to the standard (risk‐free) premium. 394

  66. 11.8 As previous figure but with the hazard rates of the reference entity and counterparty swapped. 394

  67. 11.9 Illustration of the index tranches corresponding to the DJ iTraxx Europe and DJ CDX North American credit indices. All tranches are shown to scale except the [22–100%] and [30–100%]. 397

  68. 11.10 Upper and lower bounds for the fair CDS premium when buying protection on a CDS index subject to counterparty risk compared to the standard (risk‐free) premium. 400

  69. 11.11 Upper and lower bounds for the fair premium when buying protection on the [0–3%] equity tranche (assuming the premium is paid on a running basis) as a function of correlation with the parameters giveninthe text. 400

  70. (p. xiii) 11.12 Upper and lower bounds for the fair premium when buying protection on the [6–9%] tranche as a function of correlation with the parameters giveninthe text. 401

  71. 11.13 Upper and lower bounds for the fair premium when buying protection on the [22–100%] super senior tranche as a function of correlation with the parameters given in the text. The fair premium based on a recovery only assumption is shown—this assumes the counterparty will never settle any losses before defaulting. 402

  72. 11.14 Impact of counterparty risk across the capital structure. Fair risky tranche premium divided by the risk‐free premium for all tranches in the capital structure and compared to the index ([0–100%] tranche). 403

  73. 11.15 As Figure 11.14 but for a less risky counterparty with hC = 1.5%. 404

  74. 12.1 The model implied survival probabilities for 1yT ≤ 10y (top lhs); CDS spreads for 1yT ≤ 10y (top rhs); volatility skew for CDSO (bottom lhs); volatility skew for put options with T = 1m, 6m, 12m(bottom rhs). 437

  75. 12.2 Asymptotic and numerical Green's functions for DNJs (lhs), and analytical, asymptotic and numerical Green's functions for ENJs (rhs). The relevant parameters are the same as in Figure 1, T = 10y. 448

  76. 12.3 Calibrated intensity rates for JPM (lhs) and MS (rhs) in the models with ENJs and DNJs, respectively. 456

  77. 12.4 PDF of the driver x(T) for JPM in the model with DNJs (top lhs) and ENJs (top rhs) and for MS in the model with DNJs (bottom lhs) and ENJs (bottom rhs). T = 1y, 5y, and 10y. 457

  78. 12.5 Lognormal credit default swaption volatility implied from model with Te = 1y, Tt = 5y as a function of the inverse moneyness K/c̄ (Te, Te + Tt). 457

  79. 12.6 Lognormal equity volatility implied by the model as a function of inverse moneyness K /s for put options with T = 0.5y. 458

  80. 12.7 Equilibrium spread for a PB (top lhs) and PS (top rhs) of a CDS on MS with JPM as the counterparty; same for a CDS on JPM with MS as the counterparty (bottom rhs) and (bottom lhs). In both cases input spread is the fair spread for non‐risky counterparties. 459

  81. 14.1 Realization of Bn, the number of exceedances in X1, X2, …, Xn above the threshold un. 509

  82. 14.2 Google equity data: opening daily prices for the period 19/8/2004–25/3/2009 (top) with the negative log‐returns below. 512

  83. 14.3 Hill‐plot for the Google‐data using the POT method. 513

  84. 14.4 POT analysis of Google‐data. The negative return data (black dots) on a log‐log scale, above the threshold u = 0.024. The solid line is (p. xiv) the POT fitted model to the tail. The parabolic type (dashed) curves are the profile likelihoods around VaR99% and E S99% with corresponding confidence intervals cut off at the 95% (dashed) line. 513

  85. 14.5 Hill‐estimator (19) for 1 (x) = x−1 and 2 (x) = (x log x)−1 (top for n = 1000, bottom for n = 10000). Each plot contains the Hill‐plot for the model 1, bottom curve, 2, top curve. 518

  86. 14.6 Simulated LN(0,1) data (top) with the Hill‐plot bottom; note that LN‐data correspond to ξ = 0, whereas the (wrongly used!) Hill estimator ( 19) yields a value of around 0.1, say. 519

  87. 14.7 Hill‐plots for the AR(1)‐model Xt = φXt−1 + Zt with dashed line corresponding to the Hill estimator from the Z‐data, whereas the full line corresponds to the Hill estimator from the X‐data, and this for three different values of φ : 0.9, 0.5, 0.2, top to bottom; n = 1000. 520

  88. 15.1 Real and imaginary parts of the inversion integrand M(s)esy where M(s) = (1‐ βs)α (the Gamma distribution), with α = 1, β = 0.5, y = 1. There is a branch point at s = 1/β and the plane is cut from there to +∞. The integrand is oscillatory for contours parallel to the imaginary axis. 541

  89. 15.2 Absolute value of the same inversion integrand M(s)esy as Figure 15.1, and path of steepest descent. The contour runs in a horseshoe from ∞ − πi, through 1 (the saddlepoint), round to ∞ + πi. 541

  90. 15.3 Equal exposures, very small portfolio (10 assets). •=Saddlepoint, ∘=CLT. Exact result (which is ‘steppy’) shown by unmarked line. 545

  91. 15.4 As Figure 15.3 but for larger portfolio (100 assets). 546

  92. 15.5 Inhomogeneous portfolio (100 assets). The largest exposure is 10 × median. 546

  93. 15.6 An unlikely case: extreme heterogeneity in which largest exposure is 150× median. The true distribution has a step in it at loss=150, which the saddlepoint approximation attempts to smooth out, thereby overestimating risk at lower levels. 546

  94. 15.7 Correlated test portfolio in Example 2: β =0.3. 550

  95. 15.8 As above but with β =0.5. 551

  96. 15.9 As above but with β =0.7. 551

  97. 15.10 As above but with β =0.9. 551

  98. 15.11 VaR and shortfall contributions, as % of portfolio risk, compared in a default/no‐default model. ‘A’ and ‘B’ are tail risks (large exposures to high‐grade assets) and ‘C’ is the opposite: a smallish exposure to a very low‐grade credit. In a standard deviation‐based optimization, (p. xv) C generates most risk, with A being of little importance; in a tail‐based one,Acontributes most risk. 554

  99. 15.12 VaR and shortfall contributions of one particular asset as a function of VaR. Dots show the exact result; the curve is the saddlepoint result. 557

  100. 15.13 VaR and shortfall in 10‐asset portfolio. For each, the dotted line shows the approximation, and the solid one shows the exact result. 557

  101. 15.14 VaR and shortfall in 50‐asset portfolio. For each, the dotted line shows the approximation, and the solid one shows the result obtainedbyMonte Carlo. 566

  102. 15.15 For all the assets in the portfolio, this plot shows their shortfall contribution divided into systematic and unsystematic parts. Correlated assets sit on the bottom right, large single‐name exposures on the top left. Assets in the bottom left contribute little risk. 566

  103. 15.16 [Left] Shortfall (solid line) vs quadratic approximation (dotted), as asset allocation is varied. The approximation requires only the shortfall, delta, and gamma in the ‘base case’, whereas the exact computation requires a full calculation at each point. [Right] Again for varying that particular asset allocation, this shows the delta of the asset and the systematic delta. The systematic delta remains roughly constant so the inference is that more unsystematic risk is being addedtothe portfolio. 567

  104. 16.1 USand European ABCP volume outstanding. 578

  105. 16.2 Securitization issuance:US and Europe. 580

  106. 16.3 European jumbo covered bond new issuance. 581

  107. 16.4 AAA SF Spreads incomparison with corporate spread. 586

  108. 16.5 CMBX—AAA. 586

  109. 16.6 CMBX—AA. 587

  110. 16.7 CMBX—A. 587

  111. 16.8 CMBX—BBB. 588

  112. 16.9 ABX—AAA. 588

  113. 16.10 ABX—A. 589

  114. 16.11 ABX—BBB. 589

  115. 17.1 The FHFAUSHPA rate versus the MBS rate since 1975. 607

  116. 17.2 Empirical versus theoretical unconditional statistics of the US HPA. 608

  117. 17.3 Cross‐correlation function between HPA and log‐payment. 611

  118. 17.4 Theflowchartofthe HPA model. 613

  119. 17.5 Composition of the 25‐MSA HPA (results of Kalman filtering, 1989–2008). 617

  120. (p. xvi) 17.6 FHFA‐25MSA HPI forecast from 2008–end. 618

  121. 17.7 FHFA‐25MSA HPI forecastvsactual from 2005–end. 619

  122. 17.8 FHFA‐25MSA HPI forecast versus actual from 2002–end. 619

  123. 17.9 Average annualized HPI volatility. 626

  124. 17.10 Empirical unconditional volatility: indices and geography. 628

  125. 18.1 Graph showing two levels of the collateral dependence for Biltmore CDO 2007–1 (represented in the centre of the graph). The collateral of this CDO is currently composed of 136 ABS deals (cash and synthetic) and 16 ABS CDOs (‘inner circle’ in the graph). These 16 CDOs in turn depend on 256 CDOs (‘outer circle’) and 2025 ABS deals. The number of ABS deals directly below each CDO is indicated between parenthesis. For obvious reasons, only two levels of collateral are depicted. 632

  126. 18.2 Historical prices of ABX indices. Each index references a pool of 20 sub‐prime ABS deals. There are 4 different vintages corresponding to the first and second halves of 2006 and 2007. For each vintage there are 6 indices referencing different tranches of the same securitization. 634

  127. 18.3 Plot of the S&P Composite‐10 Case‐Shiller Home Price Index. The peakofthe index occurredatJuly 2006. 639

  128. 18.4 Markit quotes and model quotes for ABS deals with sub‐prime collateral. The y‐axis is the price given as percentage of current notional and the x‐axis is the cusip index (where cusips are ordered by model quote). Each Markit quote is the average of the quotes contributed by several dealers. Error bars are given by the standard deviation of those quotes and illustrate the market uncertainty. 644

  129. 18.5 Model and market quotes for the 852 cusips used for calibration. These cusips represent ABS deals spanning different collateral types and vintages. 645

  130. 18.6 Cumulative probability profiles for the calibrated model using entropy weights α = 1 × 10−3, α = 2 × 10−3, α = 4 × 10−3 and α = 1 × 10−2. Out of 3,000 scenarios and in order to account for 99% probability the model required 2,307 scenarios for α = 1 × 10−2, 1,249 scenarios for α = 4 × 10−3, 470 scenarios for α = 2 × 10−3, and only 135 scenarios for α = 1 × 10−3. 646

  131. 18.7 Behaviour of calibration (circles) and out‐of‐the‐sample (triangles) average errors as a function of the entropy weight α. The optimal entropy weight (α = 4 × 10−3) is indicated by the vertical line. 646

  132. 18.8 Model and market quotes for the 800 calibration cusips belonging to the collateral of Biltmore CDO 2007‐1. All the ABS deals present (p. xvii) in the first collateral layer (136 deals) are calibrated and 664 of the ones in the second layer are also included in the calibration set. 648

  133. 18.9 Plot showing the sensitivity of the A1 tranche of Biltmore CDO 2007‐1 to the prices of the calibration ABS deals aggregated by asset type and vintage. 649