Hedging Default Risks of CDOs in Markovian Contagion Models

Transcription

1 Hedging Default Risks of CDOs in Markovian Contagion Models Second Princeton Credit Risk Conference 24 May 28 Jean-Paul LAURENT ISFA Actuarial School, University of Lyon, Presentation related to the paper Hedging default risks of CDOs in Markovian contagion models (28) Available on Joint work with Areski Cousin (Univ. Lyon) and Jean-David Fermanian (BNP Paribas)

2 Preliminary or or obituary? On human grounds, shrinkage rather than enlargement of the job market On scientific grounds, collapse of the market standards for risk managing CDOs Thanks to the crisis, our knowledge of the flaws of the various competing models has dramatically improved We know that we don t know and why No new paradigm has yet emerged (if ever) Paradoxically, academic research is making good progress but at its own pace Model to be presented is low tech, unrealistic, nothing new But deserves to be known (this is pure speculation).

3 Overview CDO Business context Decline of the one factor Gaussian copula model for risk management purposes Recent correlation crisis Unsatisfactory credit deltas for CDO tranches Risks at hand in CDO tranches Tree approach to hedging defaults From theoretical ideas To practical implementation of hedging strategies Robustness of the approach?

4 CDO Business context CDS hedge ratios are computed by bumping the marginal credit curves In F Gaussian copula framework Focus on credit spread risk individual name effects Bottom-up approach Smooth effects Pre-crisis Poor theoretical properties Does not lead to a replication of CDO tranche payoffs Not a hedge against defaults Unclear issues with respect to the management of correlation risks

5 CDO Business context We are still within a financial turmoil Lots of restructuring and risk management of trading books Collapse of highly leveraged products (CPDO) February and March crisis on itraxx and CDX markets Surge in credit spreads Extremely high correlations Trading of [6-%] tranches Emergence of recovery rate risk uestions about the pricing of bespoke tranches Use of quantitative models? The decline of the one factor Gaussian copula model

6 CDO Business context

7 CDO Business context Recovery rates Market agreement of a fixed recovery rate of 4% is inadequate Currently a major issue in the CDO market Use of state dependent stochastic recovery rates will dramatically change the credit deltas

8 CDO Business context Decline of the one factor Gaussian copula model Credit deltas in high correlation states Close to comonotonic default dates (current market situation) Deltas are equal to zero or one depending on the level of spreads Individual effects are too pronounced Unrealistic gammas Morgan & Mortensen

9 CDO Business context The decline of the one factor Gaussian copula model + base correlation This is rather a practical than a theoretical issue Negative tranche deltas frequently occur Which is rather unlikely for out of the money call spreads Though this could actually arise in an arbitrage-free model Schloegl, Mortensen and Morgan (28) Especially with steep base correlations curves In the base correlation approach, the deltas of base tranches are computed under different correlations And with thin tranchelets Often due to numerical and interpolation issues

10 CDO Business context No clear agreement about the computation of credit deltas in the F Gaussian copula model Sticky correlation, sticky delta? Computation wrt to credit default swap index, individual CDS? Weird effects when pricing and risk managing bespoke tranches Price dispersion due to projection techniques Negative deltas effects magnified Sensitivity to names out of the considered basket

11 Risks at at hand in in CDO tranches Default risk Default bond price jumps to recovery value at default time. Drives the CDO cash-flows Credit spread risk Changes in defaultable bond prices prior to default Due to shifts in credit quality or in risk premiums Changes in the marked to market of tranches Interactions between credit spread and default risks Increase of credit spreads increases the probability of future defaults Arrival of defaults may lead to jump in credit spreads Contagion effects (Jarrow & Yu) Enron failure was informative Not consistent with the conditional independence assumption

12 Risks at at hand in in CDO tranches Parallel shifts in credit spreads As can be seen from the current crisis On March, 28, the 5Y CDX IG index spread quoted at 94 bp pa starting from 3 bp pa on February 27 See grey figure this is also associated with a surge in equity tranche premiums

13 Risks at at hand in in CDO tranches Changes in the dependence structure between default times In the Gaussian copula world, change in the correlation parameters in the copula The present value of the default leg of an equity tranche decreases when correlation increases Dependence parameters and credit spreads may be highly correlated

14 Risks at at hand in in CDO tranches The ultimate step : complete markets As many risks as hedging instruments News products are only designed to save transactions costs and are used for risk management purposes Assumes a high liquidity of the market Perfect replication of payoffs by dynamically trading a small number of «underlying assets» Black-Scholes type framework Possibly some model risk This is further investigated in the presentation Dynamic trading of CDS to replicate CDO tranche payoffs

15 Tree approach to to hedging defaults What are we trying to achieve? Show that under some (stringent) assumptions the market for CDO tranches is complete CDO tranches can be perfectly replicated by dynamically trading CDS Exhibit the building of the unique risk-neutral measure Display the analogue of the local volatility model of Dupire or Derman & Kani for credit portfolio derivatives One to one correspondence between CDO tranche quotes and model dynamics (continuous time Markov chain for losses) Show the practical implementation of the model with market data Deltas correspond to sticky implied tree

16 Tree approach to to hedging defaults Main theoretical features of the complete market model No simultaneous defaults Unlike multivariate Poisson models Credit spreads are driven by defaults Contagion model Jumps in credit spreads at default times Credit spreads are deterministic between two defaults Bottom-up approach Aggregate loss intensity is derived from individual loss intensities Correlation dynamics is also driven by defaults Defaults lead to an increase in dependence

17 Tree approach to to hedging defaults Without additional assumptions the model is intractable Homogeneous portfolio Only need of the CDS index No individual name effect Top-down approach Only need of the aggregate loss dynamics Markovian dynamics Pricing and hedging CDO tranches within a binomial tree Easy computation of dynamic hedging strategies Perfect calibration the loss dynamics from CDO tranche quotes Thanks to forward Kolmogorov equations Practical building of dynamic credit deltas Meaningful comparisons with practitioner s approaches

18 Tree approach to to hedging defaults We will start with two names only Firstly in a static framework Look for a First to Default Swap Discuss historical and risk-neutral probabilities Further extending the model to a dynamic framework Computation of prices and hedging strategies along the tree Pricing and hedging of tranchelets Multiname case: homogeneous Markovian model Computation of risk-neutral tree for the loss Computation of dynamic deltas Technical details can be found in the paper: hedging default risks of CDOs in Markovian contagion models

19 Some notations : τ, τ 2 default times of counterparties and 2, H t available information at time t, P historical probability, α, α P P 2 Tree approach to to hedging defaults : (historical) default intensities: [ [ P P τ t, t+ H = α, i =,2 i t i Assumption of «local» independence between default events Probability of and 2 defaulting altogether: [ [ [ [ ( ) 2 P τ + τ2 + t = α α2 P P t, t, t, t H in Local independence: simultaneous joint defaults can be neglected

20 Tree approach to to hedging defaults Building up a tree: Four possible states: (D,D), (D,ND), (ND,D), (ND,ND) Under no simultaneous defaults assumption p (D,D) = Only three possible states: (D,ND), (ND,D), (ND,ND) Identifying (historical) tree probabilities: α P ( DND, ) α P 2 ( α α2 ) + P P P p( ) = p( ) = p( ) + p( ) = p( ) = α DD, DND, DD, DND, D,. P p(, ) = p DD ( NDD, ) = p( DD, ) + p( NDD, ) = p(., D) = α2 p(, ) = p ND ND ( D,. ) p(., D)

21 Tree approach to to hedging defaults Stylized cash flows of short term digital CDS on counterparty : α CDS premium α α P ( D, ND) α P 2 ( α α2 ) + P P α α Stylized cash flows of short term digital CDS on counterparty 2: α P α 2 ( DND, ) α P 2 ( α α2 ) + P P α 2 α 2

22 Tree approach to to hedging defaults Cash flows of short term digital first to default swap with premium α F : α α F P ( D, ND) α P 2 ( α α2 ) + P P α F α F Cash flows of holding CDS + CDS 2: P ( ) α α + α2 ( D, ND) α P 2 ( α α2 ) + P P ( α ) α2 + ( α ) α2 + Perfect hedge of first to default swap by holding CDS + CDS 2 Delta with respect to CDS =, delta with respect to CDS 2 =

23 Tree approach to to hedging defaults Absence of arbitrage opportunities imply: α = α + α F 2 Arbitrage free first to default swap premium Does not depend on historical probabilities α, α P P 2 Three possible states: (D,ND), (ND,D), (ND,ND) Three tradable assets: CDS, CDS2, risk-free asset ( α α2 ) + α P + r α P 2 P P + r + r For simplicity, let us assume ( D, ND) r =

24 Three state contingent claims Example: claim contingent on state Can be replicated by holding CDS + α risk-free asset α α P 2 ( α α2 ) + Replication price = Tree approach to to hedging defaults α P α P P α α ( DND, ) α α + ( D, ND)? α P 2 ( α α2 ) + ( α α2 ) + α P α P 2 α P 2 ( α α2 ) + P P α P α P P P ( D, ND) P P α α α ( DND, ) ( D, ND)

25 Tree approach to to hedging defaults Similarly, the replication prices of the ( ND, D ) and claims α 2 α P α P 2 ( α α2 ) + P P Replication price of: ( D, ND)? α P α P 2 ( α α2 ) + ( α ) α2 + a b P P c ( DND, ) α P α P 2 ( α α2 ) + P P ( D, ND) Replication price = ( ( ) ) α a + α b + α + α c 2 2

26 Tree approach to to hedging defaults Replication price obtained by computing the expected payoff Along a risk-neutral tree ( ( ) ) α a + α b + α + α c 2 2 α α 2 ( α α2 ) + ( D, ND) Risk-neutral probabilities Used for computing replication prices Uniquely determined from short term CDS premiums No need of historical default probabilities a b c

27 Tree approach to to hedging defaults Computation of deltas Delta with respect to CDS : Delta with respect to CDS 2: δ δ 2 Delta with respect to risk-free asset: p p also equal to up-front premium payoff CDS payoff CDS 2 ( ) ( a = p + δ ) α + δ2 α2 ( ) ( b = p + δ ) α + δ2 α2 ( ) ( c = p + δ ) α + δ2 α2 payoff CDS payoff CDS 2 As for the replication price, deltas only depend upon CDS premiums

28 Dynamic case: Tree approach to to hedging defaults α α 2 ( α α2 ) + ( DND, ) λ 2 λ ( π π2 ) + π π 2 λ 2 CDS 2 premium after default of name κ CDS premium after default of name 2 π CDS premium if no name defaults at period π 2 CDS 2 premium if no name defaults at period Change in CDS premiums due to contagion effects Usually, π < α < κ and π < α < λ κ κ ( DD, ) ( D, ND) ( DD, ) ( DND, )

29 Tree approach to to hedging defaults Computation of prices and hedging strategies by backward induction use of the dynamic risk-neutral tree Start from period 2, compute price at period for the three possible nodes + hedge ratios in short term CDS,2 at period Compute price and hedge ratio in short term CDS,2 at time Example: term structure of credit spreads computation of CDS premium, maturity = 2 p will denote the periodic premium Cash-flow along the nodes of the tree

30 Tree approach to to hedging defaults Computations CDS on name, maturity = 2 λ α α 2 ( α α2 ) + p p p ( DND, ) 2 λ ( π π2 ) + 2 κ κ π π 2 p p p p p ( DD, ) ( D, ND) ( DD, ) ( DND, ) Premium of CDS on name, maturity = 2, time =, p solves for: ( ) ( p) α p ( p) κ p( κ ) α2 + ( p+ ( p) π pπ2 p( π π2 ))( α α2 ) = + +

31 Tree approach to to hedging defaults Stylized example: default leg of a senior tranche Zero-recovery, maturity 2 Aggregate loss at time 2 can be equal to,,2 Equity type tranche contingent on no defaults Mezzanine type tranche : one default Senior type tranche : two defaults α κ2 + α2 κ up-front premium default leg α α 2 ( α α2 ) + ( DND, ) λ 2 λ ( π π2 ) + 2 κ κ π π 2 ( DD, ) ( D, ND) ( DD, ) ( DND, ) senior tranche payoff

32 α Stylized example: default leg of a mezzanine tranche Time pattern of default payments λ 2 + α 2 ( ( ) )( ) α α 2 + α + α2 π + π2 up-front premium default leg + Tree approach to to hedging defaults ( α α2 ) ( DND, ) λ ( π π ) + π π 2 2 Possibility of taking into account discounting effects The timing of premium payments Computation of dynamic deltas with respect to short or actual CDS on names,2 2 κ κ ( DD, ) ( D, ND) ( DD, ) ( DND, ) mezzanine tranche payoff

33 Tree approach to to hedging defaults In theory, one could also derive dynamic hedging strategies for standardized CDO tranches Numerical issues: large dimensional, non recombining trees Homogeneous Markovian assumption is very convenient CDS premiums at a given time t only depend upon the current number of defaults CDS premium at time (no defaults) CDS premium at time (one default) CDS premium at time (no defaults) Nt () ( ) α = α2 = αi t =, N() = λ2 = κ = αi ( t =, N( t) = ) ( ) π = π2 = αi t =, N( t) =

34 Tree in the homogeneous case Tree approach to to hedging defaults α i (,) α i (,) ( ) 2α, ( DND, ) α, (, ) α i (, ) 2α i (,) If we have N () =, one default at t= The probability to have N (2) =, one default at t=2 Is α i (,) and does not depend on the defaulted name at t= Nt () is a Markov process Dynamics of the number of defaults can be expressed through a binomial tree α i α i α i (,) i (,) α i ( ), ( DD, ) ( )( D, ND) ( DD, ) ( DND, )

35 From name per name to number of defaults tree, ( DD, ) N () = N () = N () = 2α, ( ) 2, α i Tree approach to to hedging defaults ( ) α i i ( ) ( ) ( ) 2α, i α i (,) α i (,) ( ) 2α, N (2) = ( DND, ) α, α i ( ) (, ) α i ( )( D, ND) ( DND, ) (, ) 2α i (,) N (2) = 2 α i (,) number α, N (2) = of defaults 2, tree α i α i i (,) α i ( ), ( DD, )

36 Easy extension to n names Predefault name intensity at time t for Nt () defaults: α i tnt, ( ) Number of defaults intensity : sum of surviving name intensities: λ ( tnt, ( )) = ( n Nt ( )) α i ( tnt, ( )) ( n 2) α ( 2,2) N () = nα i (,) nα i (, ) nα i ( ) (,) N () = N () = nα, Tree approach to to hedging defaults on marginal distributions of ( ) ( n ) α, ( ) ( n ) αi, i N (2) = 2 N (2) = N (2) = ( ) ( ) ( ) ( ) ( ) α,, α,, α,, α 2,, α 2,, i i i i i Nt () nα i by forward induction. ( ) ( ) nα 2, i ( 2,) ( ) ( ) ( n ) α 2, ( n ) αi 2, i ( ) ( n ) αi 2,2 i N (3) = 3 N (3) = 2 N (3) = N (3) = can be easily calibrated

37 Calibration of the tree example Number of names: 25 Default-free rate: 4% 5Y credit spreads: 2 bps Recovery rate: 4% Tree approach to to hedging defaults Loss intensities with respect to the number of defaults For simplicity, assumption of time homogeneous intensities Increase in intensities: contagion effects Compare flat and steep base correlation structures

38 Tree approach to to hedging defaults Dynamics of the credit default swap index in the tree The first default leads to a jump from 9 bps to 3 bps The second default is associated with a jump from 3 bps to 95 bps Explosive behavior associated with upward base correlation curve

39 Tree approach to to hedging defaults What about the credit deltas? In a homogeneous framework, deltas with respect to CDS are all the same Perfect dynamic replication of a CDO tranche with a credit default swap index and the default-free asset Credit delta with respect to the credit default swap index = change in PV of the tranche / change in PV of the CDS index

40 Tree approach to to hedging defaults Dynamics of credit deltas: Deltas are between and Gradually decrease with the number of defaults Concave payoff, negative gammas When the number of defaults is > 6, the tranche is exhausted Credit deltas increase with time Consistent with a decrease in time value

41 Tree approach to to hedging defaults Market and tree deltas at inception Market deltas computed under the Gaussian copula model Base correlation is unchanged when shifting spreads Sticky strike rule Standard way of computing CDS index hedges in trading desks [-3%] [3-6%] [6-9%] [9-2%] [2-22%] market deltas model deltas NA Smaller equity tranche deltas for in the tree model How can we explain this?

42 Tree approach to to hedging defaults Smaller equity tranche deltas in the tree model (cont.) Default is associated with an increase in dependence Contagion effects Increasing correlation leads to a decrease in the PV of the equity tranche Sticky implied tree deltas Recent market shifts go in favour of the contagion model

43 Tree approach to to hedging defaults The current crisis is associated with joint upward shifts in credit spreads Systemic risk And an increase in base correlations Sticky implied tree deltas are well suited in regimes of fear (Derman)

44 Tree approach to to hedging defaults What do we learn from this hedging approach? Thanks to stringent assumptions: credit spreads driven by defaults homogeneity Markov property It is possible to compute a dynamic hedging strategy Based on the CDS index That fully replicates the CDO tranche payoffs Model matches market quotes of liquid tranches Very simple implementation Credit deltas are easy to understand Improve the computation of default hedges Since it takes into account credit contagion Provide some meaningful results in the current credit crisis