Credit Derivatives: fundaments and ideas

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1 Credit Derivatives: fundaments and ideas Roberto Baviera, Rates & Derivatives Trader & Structurer, Abaxbank I.C.T.P., Dec

2 About? 2

3 lacked knowledge Source: WSJ 5 Dec 07 3

4 Outline 1. overview 2. bootstrap interbank curve 3. single-name models 4. multi-name models 4

5 Outline: overview 1. overview 2. bootstrap interbank curve 3. single-name models 4. multi-name models 5

6 Outline: overview Credit event credit event credit risk components absolute priority and an (over)simplified balance sheet basic definitions (survival probability & hazard rates) building blocks: initial conditions in a simplified model Credit products single-name - bonds: floater & fixed coupon - credit derivatives: ASW, CDS portfolio related - bonds: ABX, CDO - credit derivatives: First-to-default, Loss-layer Credit markets market description and structure volumes 6

7 credit event when a borrower does not pay i.e. in case of: bankruptcy failure to pay (> threshold, after grace period) obligation default obligation acceleration Definition of event now standardized by ISDA (see ISDA Master Agreement). 7

8 credit rating AAA AA+ AA AA- 8 A+ A BBB BB Spread Over Libor Interbank market

9 an (over) simplified balance sheet assets liabilities firm value: material & immaterial assets equity debt (loans/bonds/other debt vs suppliers) V = S + B default event: when firm value < total debt priority rule: in case of default, debt is served in order of seniority 9

10 credit risk components Risks Mathematical description single-name arrival risk timing risk recovery risk relation with market risks probability distribution of time of default probability distribution of recovery rate correlation with other risk factors multi-name default dependence timing risk/clustering joint default probability/default correlation clusters of events/ fat tail in loss distribution 10

11 basic notation : recovery value : time to default : survival indicator function : survival probability between t and T : stochastic discount : default free ZC bond (or initial discounts) : defaultable ZC with NO recovery : hazard rate 11 : value of 1 in if default in

12 building blocs in single name credits : default free ZC bond curve (interbank curve) : defaultable ZC bond curve with zero recovery : value of a payoff of 1 in if default occurs in 12

13 building blocs in a simple model a simple model with additional assumptions: indipendence rates & defaults constant recovery rate a set of reset dates: default in (tn, tn+1] payment in tn+1 no default risk in derivative s counterpart (ASW & CDS) In this case: building blocs: 13

14 fixed coupon bond c t 0 = 0 c c c 1 Today t 1 ti t i+ 1 t N 14

15 floater spol s t 0 = 0 spol s L 0 ( t 0 ) spol s L ( t 1) L i t ) i 1 i spol s ( i L N ( t 1) 1 N 1 Today t1 ti t i+ 1 tn 15

16 asset swap pull to par: 1 C start date A B obligor 3m + C at start date: with 16

17 Credit Default Swap (CDS) t 0 = 0 fee leg Today t 1 ti t i+ 1 contingent leg 1 π at start date: 17

18 CDS: main relations In the simple model: (falling angel) 18

19 CDS: mkt quotes Intesa Sanpaolo CDS (in bps*), 6 Dec bid ask maturity (years) *1 bp = 10 2 % 19

20 ABS & CDO: waterfall structure Notes issued by SPV low risk reference portfolio senior tranche risk } mezzanine tranches equity tranche high risk SPV notes have different names depending on the r.p.: ABS (morgage loans), CDO (bonds)... 20

21 ABS & CDO basic structuring: reference portfolio (r.p.) is transferred to a Special Purpose Vehicle (SPV) SPV issues notes divided in different tranches r.p. income goes toward paying tranches coupons according to their seniority at maturity r.p. is liquidated and proceeds distributed to the tranches according to their seniority if a default occurs, recovery payments are reinvested in default-free securities tranche basic characteristics: ( ) K d subordination : amount of losses a portfolio can suffer before tranche s notional is eroded tranche cumulative loss L ( t) = L ( t) tranche [ K,0] max[ L ( t) K,0] max r p. d r. p.. u 21

22 ABS & CDO: subordination example Fig: Standard mezzanine ABS CDO subordination Subordination (%) Sr. AAA Jr. AAA AA A BBB BB Equity Tranche 22

23 itraxx: subordination example Fig: itraxx subordination 25 Subordination (%) Sr. AAA Jr. AAA AA A BBB Equity Tranche not traded 23

24 ABS & CDO: pricing pricing obtaining portfolio loss distribution at each reset date ( building block ) Fig: loss distribution in the Vasicek model for two different correlation values (p =5%) 0-5 % 5-20 % % loss distribution % 0% % 5% 10% 15% 20% 25% 30% 1.000% 0.100% 0.010% 0.001% loss fraction 10% 1% 24

25 portfolio credit derivatives: FtD & Loss layer First to Default (FtD) on a portfolio of reference credits (generally with the same weight) t 0 = 0 fee leg Today t 1 ti t i+ 1 contingent leg: 1 π i th if the i credit defaults Loss Layer on a portfolio of reference credits with notionals lower and upper notional bounds r.p. loss cumulative layer loss K, K d u L layer ( t) = L ( t) [ K,0] max[ L ( t) K,0] max r p. d r. p.. u 25

26 credit market: cash vs derivatives 26

27 credit market: derivatives by product 27

28 credit market: total CDO issuance 28

29 credit market: CDO by reference portfolio 29

30 Outline: bootstrap 1. overview 2. bootstrap interbank curve 3. single-name models 4. multi-name models 30

31 Outline: bootstrap interbank curve bootstrap interbank market curve interbank market - deposits - sht futures - swaps methodology interest rates dynamics 31

32 Credits and bootstrap IR curve BB Spread Over Libor AA AA- A+ A BBB AAA AA+ Interbank market bootstrap (interbank) IR curve: how to get from market observables 32

33 ingredients interbank deposits: zero coupon rates up to 6m Eurodollar short futures: first seven contracts Forward rate = Market Today Settlement TARGET 30-Nov-07 4-Dec-07 Depos BID ASK 0 sn 5-Dec w 11-Dec m 4-Jan m 4-Feb m 4-Mar m 4-Jun Future BID ASK 1 DEC Dec MAR Mar JUN Jun SEP Sep DEC Dec MAR Mar JUN Jun SEP Sep IR swaps: idea: - choose always the most liquid product - other discounts are obtained interpolating 33 Swap BID ASK 1 4-Dec Dec Dec Dec Dec Dec Dec Dec Dec Dec (...)

34 deposits zero coupon rates up to 6m Act/360 Depos MID # days year frac discount 0 sn 5-Dec w 11-Dec m 4-Jan

35 futures Eurodollar short futures: forward rates (in %) : 100 future price interpolation ( t t ) L ) i+ 1 i i ( t 0 Futures Fixing Start Date End Date 1 17-Dec Dec Mar Mar Mar Jun Jun Jun Sep Sep Sep Dec Dec Dec Mar Mar Mar Jun Jun Jun Sep-09 1 t = 0 0 ti t i+ 1 t = 0 0 ti t i+ 1 Today Start End Today Start 1 End 35

36 swaps IR swaps: example (EURO mkt): annual fixed coupon vs 6m Euribor 1st year next years interpolation bootstrap: given we get... and one should consider the basis adjustment 36

37 interpolation rule The two most common approaches: with 37

38 curve May-04 Feb-07 Nov-09 Aug-12 May-15 Feb-18 Nov-20 Jul-23 Apr-26 initial discounts Jan-29 Oct-31 Jul-34 Apr-37 Jan-40 Zero rates (%)... and the dynamics a curve dynamics 30 Nov :15 C.E.T.

39 Interest rates dynamics: HJM the simplest way to model discount factors (HJM) discounts initial condition where and a Brownian motion in with When is a deterministic function of time in, the model is called Gaussian HJM. In particular, within this frame, the simplest example is Vasicek model: (see e.g. Musiela Rutkowsky 1997) 39

40 Outline: single-name models 1. overview 2. bootstrap Interbank Curve 3. single-name models 4. multi-name models 40

41 Outline: single-name models firm value (structural) models motivations Merton model Black Cox model implications on credit spreads intensity based (reduced form) models deterministic intensity models stochastic intensity models building blocks calibration & simulation 41

42 Firm value models motivations: link equity debt instruments pricing convertible bonds corporate finance questions e.g. capital structure optimization, strategic defaults firm value dynamics: default trigger: D V ( T ) < D V ( t) < K( t) maturity less than debt : (Merton model) safety covenants : (Black-Cox model) 42

43 Merton model In the model we can intervene only at the maturity T of the debt. If the value is less than the debt, there is a default. I.e. Condition at maturity is basic dynamics: solution: where is Black-Scholes call solution 43

44 Merton model: calibration Since it almost impossible to infer firm value V from balance sheet... set of equations for : where the second equation is obtained observing that 44

45 Black-Cox model In presence of safety covenants, creditors can liquidate the firm if firm value falls below a threshold. Default occurs as soon as the bond is equal to NO default occurs default occurs Firm value dynamics is Merton s one. The simplest case is a constant threshold 45

46 Black-Cox model: idea V () t NO Default scenario V ( 0) T t K ( t) τ Default 46

47 Black-Cox model: solution In the case and of constant interest rates r the solution is with where... similar solutions can be obtained in the cases: stochastic IR (Gaussian HJM) impact of correlations rates defaults 47

48 Black-Cox model: solution Spread in the zero recovery case 20% 15% 10% 5% 0% V=2 V=3 V=4 V=5 Remark: The bump shape is not in market spreads! 48

49 firm value models: summary Advantages pricing convertible bonds link equity debt instrument (e.g. default correlation equity correlation) analising corporate finance issues (e.g. capital structure optimization in a firm) Disadvantages mkt credit spreads have NOT a bump shape 49

50 deterministic intensity based models local probability of default over inhomogeneous Poisson process, with local probaility to jump = basic definitions: counting process with intensity time of the first jump of survival probability 50

51 deterministic intensity based models: building blocs main property: purely discontinuous process zero covariation with continuous martingales (e.g. rates) building blocks: 51

52 deterministic intensity based models: simulation & calibration simulation draw flat 1 time of default : 0 t calibration in a simplified example: (flat &, paid continuously) Jarrow & Turnbull (1995)... and in general credit bootstrap is straightforward 52

53 stochastic intensity based models Cox processes: s.t and, conditional on, is a Poisson proces with intensity basic properties: 53

54 stochastic intensity based models: Gaussian HJM example bond dynamics: with building blocs: correlation effect 54

55 intensity based models: summary Advantages direct calibration to credit mkt spreads simple simulation it is possible to see impact of market risks (e.g. effects of IR correlation) Disadvantages link equity debt instrument 55

56 Outline: multi-name models 1. overview 2. bootstrap Interbank Curve 3. single-name models 4. multi-name models 56

57 Outline: multi-name models Default dependancy and new questions Single time step model description firm s value model (Vasicek or factor model): - loss distribution - large homogeneous portfolio (LHP) generalizations: t-student, double t-student, archimedean copula copula approach Default time models calibration & simulation 57

58 Why relevant? losses in the subprime mortgage cash market (simplified example) if this many borrowers defaulted on their loan... and the lender resold* the property for*... the loss severity is... total "pool" losses would be... losses on $1.4 trillion base case 30% 60 cents/$ 40% 30% x 40% = 12% = $168 billion hypothetical stress case 40% 50 cents/$ 50% 40% x 50% = 20% = $280 billion ABX index (November 20, 2007) 29% = $406 billion *net of foreclosure and resale expenses Source: FT, 6 Dec 2007 US subprime exposure is just the starting point of a rolling snowball: - which are the implications on other non-corporate US credits (Alt-A & prime morgages, credit cards, auto loans,...)? - what is the contagion effect on other forms of lending/credit? 1 US$ trillion = $ 1 US$ billion = 10 9 $ 58

59 orders of magnitude outstanding ( trio $ ) US home mortgages 10.1 subprime 1.4 Alt-A 1.2 jumbo 1.8 prime (80% agencies) 5.7 commercial Real Estate 3.1 credit cards 0.9 auto loans 1.0 non-corporate US credit 15.1 structured securities 9.0 corporate bonds 5.4 US Treasuries 4.3 source: FT, Sole24Ore, DB, GS, estimates 59 1 US$ trillion = $ 1 US$ billion = 10 9 $

60 orders of magnitude: assessing the risk of contagion outstanding charge-off losses ( bio $ ) ( trio $ ) low high low high US home mortgages subprime % 29.0% Alt-A % 14.0% jumbo % 7.0% prime (80% agencies) % 1.5% 6 86 commercial Real Estate % 3.5% credit cards % 11.0% auto loans % 5.5% non-corporate US credit 15.1 total US$ trillion = $ 1 US$ billion = 10 9 $ 60

61 objectives objectives: reproduce default dependancy of realistic magnitude reproduce timing of defaults and clustering e.g. saving & loan US banks : 962 bankruptcies on 4000 existing in and the model should present: straight calibration joint default information over a fixed time horizon individual bond term structures (for FtD & CDO) simple implementation parsimony 61

62 one time step model Each obligor defaults in the lag T (time window of interest) with a default probability, i.e. a survival probability Homogeneous Portfolio (HP) assumption: loss distribution default distribution Large Homogeneous Portfolio (LHP) assumption: loss fraction factor models allow simple computation 62

63 factor model: Vasicek multi-name firm value model obligor i defaults iff with where std normal i.i.d. Remark: given are indipendent Idea: use conditional indipendence HP default probability given : HP loss distribution: with 63

64 factor model: LHP in Vasicek LHP loss distribution: loss distribution % 0% % 5% 10% 15% 20% 25% 30% 1.000% 0.100% 0.010% exponential tail 0.001% loss fraction 10% 1% with p =5% 64

65 generalizations: t-student obligor i defaults iff with where with degrees of freedom std. Normal i.i.d. Remark: given are indipendent HP default probability given : LHP loss distribution with numerically 65 (O Kane & Schloegl 2005)

66 generalizations: double t-student obligor i defaults iff with where i.i.d. t-student LHP loss distribution with numerical inversion (Hull & White 2004) 66

67 generalizations: archimedean copula obligor i survival iff with where a decreasing function Remark: given are indipendent HP survival probability given : HP default probability given : LHP loss distribution 67

68 generalizations: examples archimedean copula Clayton Gumbel 68

69 Comparison: implied correlation itraxx tranches 69

70 copula: definition A function, s.t. is a distribution function of Main property: Sklar s theorem multi-variate distribution s.t with the set of univariate marginal distribution functions Remark. We can separate the two modelling aspects: single obligor default dynamics dependence structure 70

71 limit copula in the 2d case: 71

72 example: Gaussian copula with the I-d cumulated normal 72

73 example: archimedean copula with decreasing function with Main property: Marshall and Olkin theorem drawing: have the archimedean copula function with generator 73

74 copula: a simple (one-time-step) default model : individual (marginal) survival probabilities (one time step) : copula which describes default dependency Remark: 1 q 2 S1D D D u 2 S 1 S 2 D 1 S 2 0 q 1 1 u 1 Probability of no default Probability of survival of the first k obligors 74

75 multi-name default time model : individual (marginal) survival probabilities : copula which describes default dependency Remark: for each fixed T the model is a (one-step) copula default model with survival probabilities simulation: draw and obtain the default times given the scenario, evaluate the payoff calibration: calibrate the marginal survival probability of each obligor... the copula... 75

76 conclusions 76

77 conclusions......in CREDITS which is the financial problem? collect the relevant data (for calibration) select a modeling framework simulate critical analysis of the approach: -orders of magnitude - sensitivities vs required precison (mkt bid/asks) -parsimony reality is always more complicated 77

78 bibliography sketch P.J. Schonbucher (2003), Credit Derivatives Pricing Models, Wiley M. Musiela and M. Rutkowsky (1997), Martingale Methods in Financial Modeling, Springer N. Patel (2002), The vanilla explosion, Risk Magazine 2, D. Li (2000), On default correlation: a copula function approach, J. Fixed Income 9, R. Jarrow and S. Turnbull (1995), Pricing derivatives on financial securities subject to credit risk, J. Finance 50, L. Schloegl and D. O Kane (2005), A note on the large homogeneous portfolio approximation with the Student-t copula, Finance and Stochastics 9, J. Hull and A. White (2004), Valuation of a CDO and an n-th to default CDS without a Monte carlo simulation, J. Derivatives 2,

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