Time for a Time-Change: A new Approach to Multivariate Intensity Models of Credit Risk Philipp J. Sch¨ onbucher D-MATH, ETH Z¨ urich Princeton, May 2008
Outline Introduction 1 Pricing Single-Tranche CDOs Current Multivariate Intensity Models 2 Time-Changed Intensity Models 3 Possible Specifications of Time Changes 4 Implementation 5 P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 2
Introduction Pricing Overview Introduction 1 Current Multivariate Intensity Models 2 Time-Changed Intensity Models 3 Possible Specifications of Time Changes 4 Implementation 5 P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 3
Introduction Pricing Current Portfolio Credit Risk Models Gauss copula (current standard): Widespread dissatisfaction (i) Ad-hoc “fit” via base correlation / recovery curves (ii) Unrealistic term-structure properties. (iii) Instability of parameters (iv) No proper dynamics, no consistent hedging. (v) Various quick-fixes exist for (i) and (ii). Multivariate firm’s value models: (i) Too little flexibility: Bad fit to single-obligors already (ii) Numerically prohibitively intensive for portfolios (iii) Plausible story, fundamental link. Top-down models: (i) Excellent fit and dynamics for standard index portfolios (ii) Aggregated, hard to get hedges against individual obligors Multivariate intensity models: Probably best way forward (more later) P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 4
Introduction Pricing Requirements from a New Model Application: Bespoke Tranches: Extrapolation of structure from indices to other portfolios. Exotic credit derivatives: Forward-starting tranches, various options on tranches and index, Leveraged super-senior tranches. Hedging: Realistic CDS (individual) and CDO (portfolio) dynamics. Calibration: to single-obligor survival probability curves to CDO tranches on standard indices Numerical efficiency: fast calibration which (almost) requires conditional independence P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 5
Introduction Pricing Related Literature Multivariate intensity models: Duffie and Garleanu [2001], Gaspar and Schmidt [2005], Mortensen [2005] Time-changes in credit risk: Joshi and Stacey [2005]: special case and precursor of this paper Giesecke and Tomecek [2005]: time-changes in top-down approaches Time-changes in option pricing: Clark [1973], Madan et al. [1998], Geman et al. [2001], Cont and Tankov [2004], Carr et al. [2003] P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 6
Introduction Pricing Introduction 1 Pricing Single-Tranche CDOs Current Multivariate Intensity Models 2 Time-Changed Intensity Models 3 Possible Specifications of Time Changes 4 Implementation 5 P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 7
Introduction Pricing Obligors and Loss Process i = 1 , . . . , I obligors with default times τ i and default indicator processes D i ( t ) = 1 { τ i ≤ t } . The key quantity of the model is the default loss process I � L ( t ) := D i ( t ) . i =1 (losses given default are normalized to one) P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 8
Introduction Pricing A Typical Loss Process Cumulative loss process L C of a STCDO with lower and upper attachment points K 1 and K 2 L C ( t ) = ( L ( t ) − K 1 ) + − ( L ( t ) − K 2 ) + . P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 9
Introduction Pricing Market Quotes: STCDOs on iTraxx Europe Maturity 3Y 5Y 7Y 10Y Low High Bid Ask Bid Ask Bid Ask Bid Ask 0 3 6.0 7.5 29.50 30.25 47.1 48 58.25 59.25 3 6 18 28 96 100 193 200 505 520 6 9 6 13 33 36 52 57 100 106 9 12 13 15 29 34 48 55 12 22 7.50 8.75 12 15 22 25 22 100 2.25 4.00 5.25 7.25 8.25 10.75 Index 22 38 47 58 Quotes for loss protection on tranches of European iTraxx Series 4, on Sept. 26th, 2005. Lower and upper attachment points are in % of notional, base correlation (BC) is given in %. Prices for the 0-3 tranche are % of notional upfront plus 500bp running, all other prices are bp p.a.. Source (including BCs): JPMorgan. P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 10
Introduction Pricing Remarks Liquid markets exist for STCDOs on standard indices. Standard pricing model: 1-Factor Gauss copula. Widespread dissatisfaction with performance and properties of Gauss copula models. Inability to “fit”, problems when interpolating base correlations. Instability of parameters (GM/Ford May 2005) Unrealistic term-structure properties. No proper dynamics, no consistent hedging. Exotic credit derivatives: Forward-starting tranches, options on tranches and index, Leveraged super-senior tranches. P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 11
Introduction Pricing The Cumulative Loss The initial cumulative loss is zero: L (0) = 0 . At the j -th credit event τ ( j ) ( 1 ≤ j ≤ I ), the cumulative loss is increased by the loss at this default: I � dL ( t ) = E i (1 − R i ) dD i ( t ) . i =1 E i exposure, and R i recovery rate of obligor i . The cumulative loss of the tranche L C ( t ) is the amount by which the cumulative loss of the portfolio has exceeded the lower bound K 1 , capped at the upper bound K 2 : L C ( t ) = ( L ( t ) − K 1 ) + − ( L ( t ) − K 2 ) + . P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 12
Introduction Pricing Default and Fee Payment The default payment of the protection seller to the protection buyer at a default event is the increase in the cumulative loss of the tranche: L C ( τ i ) − L C ( τ i − ) The protection buyer pays a periodic protection fee of s of the remaining notional of the tranche. K 2 − K 1 − L C ( t ) � � s · dt. P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 13
Introduction Pricing Pricing Tranche Protection I L C cumulative loss of the tranche: Loss payment at time t = increment in L C at time t . The NPV of the loss payments of the tranche can be transformed using integration-by-parts: � T � T β ( t ) dL C ( t ) = β ( T ) L C ( T ) − L C ( t ) dβ ( t ) 0 0 � T = β ( T ) L C ( T ) + L C ( t ) β ( t ) r ( t ) dt. 0 � t β ( t ) = exp {− 0 r ( s ) ds } is the default-free discount factor. dβ ( t ) = − r ( t ) β ( t ) dt. This holds for each sample path (and not just on average). P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 14
Introduction Pricing Pricing Tranche Protection II Assume independence of defaults and default-free interest rates: � � T � E Q β ( t ) dL C ( t ) = B (0 , T ) E Q � L C ( T ) � 0 � T E Q � L C ( t ) � + f (0 , t ) B (0 , t ) dt, 0 f (0 , t ) = − ∂ ∂T ln B (0 , T ) are the default free forward rates. Note: We only need the distribution or the density f L ( x, t ) of the cumulative loss of the whole portfolio. The expected tranche loss for each tranche is then: � K 2 E Q [ L ( t ) ] = ( x − K 1 ) f L ( x, t ) dx + ( K 2 − K 1 ) F L ( K 2 , t ) . K 1 P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 15
Introduction Pricing The Fee Payment The NPV of the reduction of the fee payment in one given scenario is � T s · L C ( t ) β ( t ) dt. 0 Its value is � T � T E Q � L C ( t ) β ( t ) E Q � L C ( t ) � � s dt = s B (0 , t ) dt. 0 0 Again, only dependence on the value of L C ( t ) (or L ( t ) ) at all times t ≤ T . P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 16
Current Multivariate Intensity Models Overview Introduction 1 Current Multivariate Intensity Models 2 Time-Changed Intensity Models 3 Possible Specifications of Time Changes 4 Implementation 5 P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 17
Current Multivariate Intensity Models Typical Setup of Multivariate Intensity Models Each obligor i ≤ I has a default arrival process N i ( t ) with intensity λ i ( t ) . Conditional on the joint realisation of { λ 1 ( t ) , λ 2 ( t ) . . . , λ I ( t ) } t ≥ 0 , the N i ( t ) are indep. inhomog. Poisson processes with intensities λ i ( t ) . (Joint Cox process) Individual default intensities λ i ( t ) are modeled as a (weighted) sum of: a common stochastic factor λ G ( t ) and an independent idiosyncratic component λ id i ( t ) λ i ( t ) = w i λ G ( t ) + λ id i ( t ) . P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 18
Current Multivariate Intensity Models Additive Specification: General Remarks λ i ( t ) = w i λ G ( t ) + λ id i ( t ) . Interpretation as competing risks model. Intrinsic bounds on risks: w i λ G ( t ) is the lowest possible level that λ i can reach. In large (homogeneous) portfolios: Portfolio default rate is always larger than λ id i ( t ) . Initial Fit to Single-Name CDS High-quality obligors will need lower w i , will have (relatively) little systematic risk if downgraded Dynamics depend on quality of obligors. Specification: Strong co-movements of λ i are necessary to reach realistic default dependence (i.e. high volatility of λ G ). P. Sch¨ onbucher, D-MATH, ETH Z¨ urich Time-Changed Intensity 19
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