Dynamic hedging of synthetic CDO tranches

Transcription

1 ISFA, Université Lyon 1 Young Researchers Workshop on Finance 2011 TMU Finance Group Tokyo, March 2011

2 Introduction In this presentation, we address the hedging issue of CDO tranches in a market model where pricing is connected to the cost of the hedge In credit risk market, models that connect pricing to the cost of the hedge have been studied quite lately Discrepancies with the interest rate or the equity derivative market Model to be presented is not new, require some stringent assumptions, but the hedging can be fully described in a dynamical way

3 Introduction Presentation related to the papers : Hedging default risks of CDOs in Markovian contagion models (2008), Quantitative Finance, with Jean-Paul Laurent and Jean-David Fermanian Delta-hedging correlation risk? (2010), submitted, with Stéphane Crépey and Yu Hang Kan

4 Contents 1 Theoretical framework 2 3

5 Default times n credit references τ 1,...,τ n : default times defined on a probability space (Ω, G, P) N i t = 1 {τi t}, i = 1,..., n : default indicator processes H i = (H i t) t 0, H i t = σ(n i s, s t), i = 1,..., n : natural filtration of N i H = H 1 H n : global filtration of default times

6 Default times No simultaneous defaults : P(τ i = τ j ) = 0, i j Default times admit H-adapted default intensities For any i = 1,..., n, there exists a non-negative H-adapted process α i,p such that is a (P, H)-martingale. M i,p t α i,p t = 0 on the set {t > τ i} := N i t t 0 αs i,p ds M i,p, i = 1,..., n will be referred to as the fundamental martingales

7 Market Assumption Instantaneous digital CDS are traded on the names i = 1,..., n Instantaneous digital CDS on name i at time t is a stylized bilateral agreement Offer credit protection on name i over the short period [t, t + dt] Buyer of protection receives 1 monetary unit at default of name i In exchange for a fee equal to α i tdt 0 t 1 α i tdt : default of i between t and t + dt α i tdt : survival of name i t + dt Cash-flow at time t + dt (buy protection position) : dn i t α i tdt α i t = 0 on the set {t > τ i } (Contrat is worthless)

8 Market Assumption Credit spreads are driven by defaults : α 1,..., α n are H-adapted processes Payoff of a self-financed strategy V 0 e rt + n i=1 r : default-free interest rate V 0 : initial investment T δ i, i = 1,..., n, H-predictable process 0 δse i r(t s) ( dns i αsds i ). }{{} CDS cash-flow

9 Hedging and martingale representation theorem Theorem (Predictable representation theorem) Let A H T be a P-integrable random variable. Then, there exists H-predictable processes θ i, i = 1,..., n such that and E P ( T 0 A = E P [A] + = E P [A] + ) θi s αs i,p ds <. n T i=1 0 T n i=1 0 θs i ( dn i s αs i,p ds ) θ i sdm i,p s

10 Hedging and martingale representation theorem Theorem (Predictable representation theorem) Let A H T be a Q-integrable random variable. Then, there exists H-predictable processes ˆθ i, i = 1,..., n such that and E Q ( T 0 A = E Q [A] + = E Q [A] + ) θi s αs i,p ds <. n T i=1 0 T n i=1 0 ˆθ s i ( dn i s αsds i ) }{{} CDS cash-flow ˆθ i sdm i s

11 Hedging and martingale representation theorem Building a change of probability measure Describe what happens to default intensities when the original probability is changed to an equivalent one From the PRT, any Radon-Nikodym density ζ (strictly positive (P, H)-martingale with expectation equal to 1) can be written as dζ t = ζ t n i=1 π i tdm i,p t, ζ 0 = 1 where π i, i = 1,..., n are H-predictable processes

12 Hedging and martingale representation theorem Conversely, the (unique) solution of the latter SDE is a local martingale (Doléans-Dade exponential) ( n ) t n ζ t = exp πsα i s i,p ds (1 + πτ i i ) N i t i=1 0 i=1 The process ζ is non-negative if π i > 1, for i = 1,..., n The process ζ is a true martingale if E P [ζ t ] = 1 for any t or if π i is bounded, for i = 1,..., n

13 Hedging and martingale representation theorem Theorem (Change of probability measure) Define the probability measure Q as where ζ t = exp ( dq Ht = ζ t dp Ht. n i=1 t Then, for any i = 1,..., n, the process M i t := M i,p t t 0 0 ) n πsα i s i,p ds (1 + πτ i i ) N i t πsα i s i,p ds = Nt i i=1 t 0 (1 + π i s)α i,p s is a Q-martingale. In particular, the (Q, H)-intensity of τ i is (1 + π i t)α i,p t. ds

14 Hedging and martingale representation theorem From the absence of arbitrage opportunity { α i t > 0 } { } P a.s. = α i,p t > 0 For any i = 1,..., n, the process ˆπ i defined by : ( ) ˆπ t i αt i = α i,p 1 (1 Nt ) i t is an H-predictable process such that ˆπ i > 1 The process ζ defined with π 1 = ˆπ 1,..., π n = ˆπ n is an admissible Radon-Nikodym density Under Q, credit spreads α 1,..., α n are exactly the intensities of default times

15 Hedging and martingale representation theorem The predictable representation theorem also holds under Q In particular, if A is an H T measurable payoff, then there exists H-predictable processes ˆθ i, i = 1,..., n such that A = E Q [A H t ] + n T i=1 t ˆθ i s ( dn i s αsds i ). }{{} CDS cash-flow Starting from t the claim A can be replicated using the self-financed strategy with [ ] the initial investment V t = E Q e r(t t) A H t in the savings account the holding of δs i = ˆθ se i r(t s) for t s T and i = 1,..., n in the instantaneous CDS As there is no charge to enter a CDS, the replication price of A at time t is V t = E Q [ e r(t t) A H t ]

16 Hedging and martingale representation theorem A depends on the default indicators of the names up to time T includes the cash-flows of CDO tranches or basket credit default swaps, given deterministic recovery rates The latter theoretical framework can be extended to the case where actually traded CDS are considered as hedging instruments See Cousin and Jeanblanc (2010) for an example with a portfolio composed of 2 names or in a general n-dimensional setting when default times are assumed to be ordered

17 Hedging and martingale representation theorem Risk-neutral measure can be explicitly constructed We exhibit a continuous change of probability measure Completeness of the credit market stems from a martingale representation theorem Perfect replication of claims which depend only upon the default history using CDS written on underlying names and default-free asset Provide the replication price at time t But does not provide any operational way of constructing hedging strategies Markovian assumption is required to effectively compute hedging strategies

18 Markovian contagion model Pre-default intensities only depend on the current status of defaults αt i = α i ( t, Nt 1,..., Nt n ) 1t<τi, i = 1,..., n Ex : Herbertsson - Rootzén (2006) α i ( t, Nt 1,..., Nt n ) = ai + j i b i,jn j t Ex : Lopatin (2008) α i (t, N t) = a i(t) + b i(t)f(t, N t) with N t = Connection with continuous-time Markov chains ( ) N 1 t,..., Nt n Markov chain with possibly 2 n states Default times follow a multivariate phase-type distribution n i=1 N i t

19 Pre-default intensities only depend on the current number of defaults All names have the same pre-default intensities α αt i = α (t, N t ) 1 t<τi, i = 1,..., n where N t = n i=1 This model is also referred to as the local intensity model N i t

20 No simultaneous default, the intensity of N t is equal to λ(t, N t ) = (n N t ) α(t, N t ) N t is a continuous-time Markov chain (pure birth process) with generator matrix : λ(t, 0) λ(t, 0) λ(t, 1) λ(t, 1) 0 Λ(t) = λ(t, n 1) λ(t, n 1) Model involves as many parameters as the number of names

21 Replication price of a European type payoff ] V (t, k) = E Q [e r(t t) Φ(N T ) N t = k V (t, k), k = 0,..., n 1 solve the backward Kolmogorov differential equations : δv (t, k) δt = rv (t, k) λ(t, k) (V (t, k + 1) V (t, k)) Approach also puts in practice by Schönbucher (2006), Herbersson (2007), Arnsdorf and Halperin (2007), Lopatin and Misirpashaev (2007), Cont and Minca (2008), Cont and Kan (2008), Cont, Deguest and Kan (2009)

22 Computation of credit deltas V (t, N t), price of a CDO tranche (European type payoff) V I (t, N t), price of the CDS index (European type payoff) [ ] V (t, N t) = E Q e r(t t) Φ(N T ) N t V I (t, N t) = E Q [ e r(t t) Φ I (N T ) N t ] Using standard Itô s calculus ( ) dv (t, N t) = V (t, N t) δ I (t, N t)v I (t, N t) rdt + δ I (t, N t)dv I (t, N t) where δ I (t, N t) = V (t, Nt + 1) V (t, Nt) V I (t, N t + 1) V I (t, N. t) Perfect replication with the index and the risk-free asset

23 CDO tranches on standard Index Performance analysis of alternative hedging strategies developed for the correlation market CDO tranches on standard Index such as CDX North America Investment Grade index 100% CDX North America Main 125 CDS, Investment Grade Second Super Senior First Super Senior Senior Senior Mezzanine Junior Mezzanine Equity 30% 15% 10% 7% 3% 0% Spreads, level of subordination

24 Data set Series 5 of the 5-year CDX NA IG from 20 September 2005 to 20 March 2006 Series 9 of the 5-year CDX NA IG from 20 September 2007 to 20 March 2008 Series 10 of the 5-year CDX NA IG from 21 March 2008 to 20 September CDX5 CDX9 CDX10 Index spread CDX5 CDX9 CDX10 Base correlation at 3% strike bps Observation day Observation day

25 Model Specifications Gauss : Base correlation approach based on the standard one-factor Gaussian copula pricing device Para : Local intensity model parametric specification of local itensities (Herbertsson (2008)) λ(t, k) = λ(k) = (n k) EM : Local intensity model local itensities λ(t, k) obtained by minimizing a relative entropy distance with respect to a prior distribution [ ( )] inf Q Λ EQ 0 dq dq ln dq 0 dq 0 (Cont and Minca (2008)) k i=0 b i

26 Calibration results Root mean squared calibration errors (in percentage) : CDX5 CDX9 CDX10 Tranche Gauss Para EM Gauss Para EM Gauss Para EM Index %-3% %-7% %-10% %-15% %-30% Comparison of typical shapes of local intensities λ(t, k), Para (left), EM (right)

27 Hedging ratios Comparison of three alternative hedging methods Gauss delta : Index spread sensitivity computed in a one-factor Gaussian copula model calibrated at time t Gauss t = V(t, St + ε, ρt) V(t, St, ρt) V I (t, S t + ε) V I (t, S t) where V and V I are the Gaussian copula pricing function associated with (resp.) the tranche and the CDS index. S t is the Index spread at time t and ρ t is the time-t base correlation. Local intensity delta : δ I (t, N t) = V (t, Nt + 1) V (t, Nt) V I (t, N t + 1) V I (t, N. t) with both Parametric (Param) and Entropy Minimisation (EM) calibration methods

28 Hedging ratios Time series of equity tranche deltas, CDX.NA.IG series 5, 9 and Gauss Para EM Tranche [0%,3%] deltas CDX Tranche [0%,3%] deltas CDX9 Gauss Para EM Oct05 Dec05 Feb Tranche [0%,3%] deltas CDX10 0 Apr08 Jun08 Aug08 0 Oct07 Dec07 Feb08 Gauss Para EM

29 Hedging performance Back-testing hedging experiments on series 5, 9 and 10 (1-day rebalancing) Relative hedging error = Residual volatility = Average P&L increment of the hedged position, Average P&L increment of the unhedged position P&L increment volatility of the hedged position P&L increment volatility of the unhedged position. Relative hedging errors (in percentage) CDX5 CDX9 CDX10 Tranche Li Para EM Li Para EM Li Para EM 0%-3% %-7% %-10% %-15% %-30%

30 Hedging performance Residual volatilities (in percentage) CDX5 CDX9 CDX10 Tranche Gauss Para EM Gauss Para EM Gauss Para EM 0%-3% %-7% %-10% %-15% %-30% Conclusion : Hedging based on local intensity model with Entropy Minimisation calibration gives poor performance Before the crisis (CDX5), Gauss delta outperforms local intensity deltas During the crisis (CDX9 & CDX10), no clear evidence to discriminate between Gauss delta and Para local intensity delta

31 Conclusion Thank you for your attention!

32 References Arnsdorf, M. and Halperin, I. : BSLP : Markovian bivariate spread-loss model for portfolio credit derivatives, working paper, JP Morgan, 2007 Cont, R., Deguest, R. and Kan, Y. H. : Recovering Default Intensity from CDO Spreads : Inversion Formula and Model Calibration, SIAM Journal on Financial Mathematics, 2010 Cont, R. and Kan, Y.H. : Dynamic Hedging of Portfolio Credit Derivatives, Financial Engineering Report , Columbia University, Cont, R. and Minca, A. : Recovering portfolio default intensities implied by CDO quotes, Columbia University, Cousin, A. and Jeanblanc, M. : 2010 Hedging portfolio loss derivatives with CDSs, working paper, 2010

33 References Cousin, A., Jeanblanc, M. and Laurent, J.-P. : Hedging CDO tranches in a Markovian environment, book chapter, Paris Princeton-Lectures in Mathematical Finance, 2010 Cousin, A. and Laurent, J.-P. : Dynamic hedging of synthetic CDO tranches : Bridging the gap between theory and practice, book chapter, 2010 Frey, R., Backhaus, J. : Dynamic hedging of synthetic CDO-tranches with spread- and contagion risk, Journal of Economic Dynamics and Control. Frey, R., Backhaus, J. : Pricing and hedging of portfolio credit derivatives with interacting default intensities. International Journal of Theoretical and Applied Finance, 11 (6), , Frey, R., Backhaus, J. : Pricing and hedging of portfolio credit derivatives with interacting default intensities. International Journal of Theoretical and Applied Finance, 11 (6), , 2008.

34 References Herbertsson, A. and Rootzén, H. : Pricing k-th to default swaps under default contagion, the matrix-analytic approach, Journal of Computational Finance, 2006 Herbertsson, A. : Pricing synthetic CDO tranches in a model with default contagion using the matrix-analytic approach, Journal of Credit Risk, 2008 Lopatin, A. V. : A simple dynamic model for pricing and hedging heterogeneous CDOs, working paper, Numerix, 2008 Lopatin, A. V. and Misirpashaev, T. : Two-dimensional Markovian model for dynamics of aggregate credit loss, working paper, Numerix, 2008 Schönbucher, P.J. : Portfolio losses and the term-structure of loss transition rates : a new methodology for the pricing of portfolio credit derivatives, working paper, ETH Zürich, 2006