Dynamic Modeling of Portfolio Credit Risk with Common Shocks

Transcription

1 Dynamic Modeling of Portfolio Credit Risk with Common Shocks ISFA, Université Lyon AFFI Spring 20 International Meeting Montpellier, 2 May 20

2 Introduction Tom Bielecki,, Stéphane Crépey and Alexander Herbertsson

3 Introduction Main issue: hedging of portfolio credit derivatives 00% CDX North America Main 25 CDS, Investment Grade Second Super Senior First Super Senior Senior Senior Mezzanine Junior Mezzanine Equity 30% 5% 0% 7% 3% 0% Spreads, level of subordination Cash-flows driven by the realized path of the aggregate loss process L t = n n ( R i)ht i where R i is the recovery rate and H i t is the default indicator of obligor i i=

4 Introduction Hedging using the one-factor Gaussian copula model? Advantages: Bottom-up model: account for dispersion of default risk among names in the portfolio Copula construction of default times: Calibration of CDS spreads and CDO tranche quotes can be made using two separate numerical procedures Factor model: fast algorithms to compute aggregate loss distribution Drawbacks: Static model Base correlation approach unable to describe consistently the dependence structure of default times

5 Dynamic model of portfolio credit risk Simultaneous default model Defaults are the consequence of triggering-events affecting simultaneously pre-specified groups of obligors Example: n = 5 and Y = {{}, {2}, {3}, {4}, {5}, {4, 5}, {2, 3, 4}, {, 2}} t

6 Dynamic model of portfolio credit risk {,..., n} set of credit references Y = {{},..., {n}, I,..., I m} pre-specified groups of obligors λ Y = λ Y (t) deterministic intensity function of the triggering-event associated with group Y Y H t = (H t,..., H n t ) defined as multivariate continuous-time Markov chain in {0, } n such that for k, m {0, } n : P (H t+dt = m H t = k) = Y Y λ Y (t) {k Y =m}dt where k Y is obtained from k = (k,..., k n) by replacing the components k j, j Y, by number one. ex: (0,, 0, 0) {,2,4} = (,, 0, ) F t = σ(h u, u t) natural filtration of H

7 Dynamic model of portfolio credit risk Example: n = 2, Y = {{}, {2}, {, 2}}. H t = (H t, H 2 t ) is a multivariate continuous-time Markov chain with space set {(0, 0), (, 0), (0, ), (, )} and generator matrix (0, 0) (,0) (0, 0) (,0) (0,) (,) (0,) (,) λ {} λ {2} λ {,2} 0 0 λ {2} + λ {,2} 0 0 λ {} + λ {,2} Obligor defaults with intensity λ {} + λ {,2} regardless of the state of the pool Obligor 2 defaults with intensity λ {2} + λ {,2} regardless of the state of the pool No contagion effect : Past defaults do not have any effect on intensities of surviving names

8 Dynamic model of portfolio credit risk General case: Obligor i defaults with intensity η i(t) = Y Y λ Y (t) {i Y } P(H i t+dt H i t = F t) = P(H i t+dt H i t = H i t) = ( H i t)η i(t)dt Each default indicator H i, i =,..., n is a Markov process with respect to F

9 Pricing and Hedging Separate calibration procedure of CDS-s and CDO tranches For any i =,..., n, the price at time t of a CDS referencing name i (European-type payoff): ] [ ] [ ] E [Φ(HT i ) F t = E Φ(HT i ) Ht,..., Ht n = E Φ(HT i ) Ht i Hedging CDO tranches with single-name CDS Derive price dynamics of CDO tranche and single-name CDS-s Computation of min-variance hedging strategies in this incomplete market model But: price of portfolio loss derivatives solves a large system of Kolmogorov backward equations that is numerically intractable at least for large portfolios (n > 20)

10 Common-Shock Model Interpretation For any pre-specified group Y Y = {{},..., {n}, I,..., I m}, we define { t } = inf t 0 λ Y (s)ds > E Y τ Y where E Y, Y Y are independent and exponentially distributed random variables with parameter. τ Y is the arrival time of shock Y that yields default of non-defaulted names in group Y Default time of name i =,..., n defined by: τ i = 0 min τ Y {Y Y; i Y } Common-Shock Model Interpretation For all t,..., t n 0, the following relation holds P ( τ > t,..., τ n > t n) = P (τ > t,..., τ n > t n) where τ i := inf { t 0 H i t = } is the default time of name i in the Markovian model

11 Common-Shock Model Interpretation Using fast recursion procedure for pricing and hedging CDO tranches Thanks to the common-shock model interpretation: L t = n n d ( R i) {τi t} = n i= n ( R i) { τi t} where { τ t},..., { τn t} are conditionally independent Bernoulli s given common-shock indicators {τi t},..., {τim t} For any state of the Markov process H t, there exists an equivalent common-shock model that matches joint distribution of default times for non-defaulted names. Computation of min-variance hedging strategies is also tractable i=

12 Common-Shock Model Interpretation Two-step calibration procedure First step: CDS spread of name i at time t = 0 can ( be expressed as a function of survival probabilities P(τ i > t) = exp ) t ηi(u)du where 0 η i(u) = λ {i} (u) + m λ Ik (u) {i Ik } k= Marginal default intensities η i, i =,..., n, can be calibrated on single-name CDS curves using a standard bootstrap procedure Second step: Common-shock intensities λ Ik, k =,..., m are calibrated on CDO tranche quotes using the recursion algorithm

13 Calibration on CDX index Data set: 5-year CDX North-America IG index on 20 December 2007 Quoted spreads at different pillars of the n = 25 index constituents Quoted spreads of standard tranches [0,3], [3,7], [7,0], [0,5], [5,30] Model specification: Names are labelled with respect to decreasing level of spreads I 5 I 2 n n- 6 I CDS Spreads m = 5 groups I I 5 such that I = {,..., 6}, I 2 = {,..., 9}, I 3 = {,..., 25}, I 4 = {,..., 6}, I 5 = {,..., 25} Piecewise-constant intensities λ {},..., λ {25}, λ I,..., λ I5 with grid points corresponding to CDS pillars Homogeneous and constant recovery rates: 40% Constant short-term interest rate: 3%

14 Calibration on CDX index

15 Calibration on CDX index Calibration results: Tranche [0,3] [3,7] [7,0] [0,5] [5,30] Model spread in bps Market spread in bps Abs. Err. in bps % Rel. Err Names in the set I 5 \ I 4 are excluded from the calibration constraints (they can only default within the Armageddon shock I 5)

16 Calibration on CDX index Implied 5-year loss distribution: Loss distribution at t=5 with constant recovery of 40% log of P[N =k] for t=5 0. t P[Nt=k] for t= Number of defaults

17 Min-variance hedging strategies 0.4 Hedging with the 3 riskiest names CDS nominal exposure [0 3%] tranche [3 7%] tranche [7 0%] tranche [0 5%] tranche [5 30%] tranche year CDS spread

18 Min-variance hedging strategies 0.4 Hedging with the 4 riskiest names CDS nominal exposure [0 3%] tranche [3 7%] tranche [7 0%] tranche [0 5%] tranche [5 30%] tranche year CDS spread

19 Min-variance hedging strategies 0.25 Hedging with the 5 riskiest names 0.2 CDS nominal exposure [0 3%] tranche [3 7%] tranche [7 0%] tranche [0 5%] tranche [5 30%] tranche year CDS spread

20 Min-variance hedging strategies 0.2 Hedging with the 6 riskiest names (group ) 0.5 CDS nominal exposure [0 3%] tranche [3 7%] tranche [7 0%] tranche [0 5%] tranche [5 30%] tranche year CDS spread

21 Conclusion Thank you for your attention!

22 References Bielecki, T.R., Vidozzi, A. and Vidozzi, L.: A Markov Copulae Approach to Pricing and Hedging of Credit Index Derivatives and Ratings Triggered Step Up Bonds, J. of Credit Risk, Brigo, D., Pallavicini, A., Torrential, R.: Calibration of CDO Tranches with the Dynamical Generalized-Poisson Loss Model. Working Paper, Elouerkhaoui, Y.: Pricing and Hedging in a Dynamic Credit Model. International Journal of Theoretical and Applied Finance, Vol. 0, Issue 4, , Lindskog, F. and McNeil, A. J.: Common Poisson Shock Models: Applications to Insurance and Credit Risk Modelling. ASTIN Bulletin, 33(2), , 2003