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Portfolio Losses and the Term Structure of Loss Transition Rates- A New Methodology for the Pricing of Portfolio Credit Derivates

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    Working Paper Series    _______________________________________________________________________________________________________________________ National Centre of Competence in Research Financial Valuation and Risk Management Working Paper No. 264 Portfolio Losses and the Term Structure of Loss Transition Rates: A New Methodology for the Pricing of Portfolio Credit Derivates Philipp J. Schönbucher First version: February 2005 Current version: September 2005 This research has been carried out within the NCCR FINRISK project on “Credit Risk”  ___________________________________________________________________________________________________________  PORTFOLIO LOSSES AND THE TERM STRUCTURE OF LOSSTRANSITION RATES: A NEW METHODOLOGY FOR THE PRICING OFPORTFOLIO CREDIT DERIVATIVES PHILIPP J. SCH¨ONBUCHERDepartment of Mathematics, ETH Z¨urichFebruary 2005this version: September 2005 Abstract.  In this paper, we present a model for the joint stochastic evolution of the cu-mulative loss process of a credit portfolio and of its probability distribution. At any giventime, the loss distribution of the portfolio is represented using  forward transition rates , i.e.the transition rates of a hypothetical time-inhomogeneous Markov chain which reproduces thedesired transition probability distribution. This approach allows a straightforward calibrationof the model (e.g. to a full initial term- and strike structure of synthetic CDOs includingthe correlation smile) and it is shown that (except for regularity restrictions) every arbitrage-free loss distribution admits such a representation with forward transition rates. To capturethe  stochastic   evolution of the loss distribution, the transition rates are then equipped withstochastic dynamics of their own, and martingale / drift restrictions on these dynamics arederived which ensure absence of arbitrage in the model. Furthermore, we analyze the dy-namics of spreads and STCDO-prices that are implied by the model and show that the inputparameters can be viewed as  spread move   parameters and  correlation move   parameters. Wealso show how every dynamic model for correlated individual defaults can be cast into thisframework. 1.  Introduction The markets for portfolio credit derivatives have become more standardized. Two importantreference indices of CDS portfolios have been created (iTraxx for Europe and CDX for theUSA) and there are now liquid markets for index-CDS and for standardized synthetic single-tranche CDOs (STCDOs) on these index portfolios. Simultaneously, more exotic portfolio credit JEL Classification.  G 13. Key words and phrases.  Default Correlation, Stochastic Correlation, CDO Pricing, HJM-Models. Author’s Address:  ETH Zurich, D-MATH, R¨amistr. 101, CH-8092 Zurich, Switzerland.p@schonbucher.de, www.schonbucher.deThe author would like to thank the participants at the 2nd Inaugural Fixed-Income Conference in Prague, and atthe JPMorgan Workshop on Credit Risk Modelling, New York, for their feedback. All errors are my own. Furthercomments and suggestions are welcome.Financial support by the National Centre of Competence in Research “Financial Valuation and Risk Management”(NCCR FINRISK), Project 5:  Credit Risk   is gratefully acknowledged. The NCCR FINRISK is a research programsupported by the Swiss National Science Foundation. 1  2 PHILIPP J. SCH¨ONBUCHER Attachment 3Y 5Y 7Y 10YLow High Bid O ff  er BC Bid O ff  er BC Bid O ff  er BC Bid O ff  er BC0 3 6.0 7.5 14.12 29.50 30.25 12.05 47.1 48 9.51 58.25 59.25 9.303 6 18 28 24.40 96 100 23.13 193 200 20.96 505 520 13.346 9 6 13 31.25 33 36 31.43 52 57 30.84 100 106 23.909 12 13 15 38.91 29 34 38.87 48 55 32.6112 22 7.50 8.75 57.49 12 15 59.97 22 25 55.8522 100 2.25 4.00 5.25 7.25 8.25 10.75Index 22 38 47 58 Table 1.  Market quotes for tranched loss protection of di ff  erent maturities onEuropean iTraxx Series 4, on Sept. 26th, 2005. Lower and upper attachmentpoints are in % of notional, base correlation (BC) is given in %. Prices for the0-3 tranche are % of notional upfront plus 500bp running, all other prices arebp p.a.. Source (including BC calculations): JPMorgan’s Bloomberg page.derivatives have arisen, in particular so-called  bespoke STCDOs  , CDOs of CDOs (or CDO 2 s),forward starting STCDOs, STCDOs with embedded options, and outright options on STCDOsor on indices.The fact that prices on STCDOs on standardized portfolios are quoted very frequently and withrelatively narrow bid-ask spreads has uncovered several shortcomings of the existing pricing mod-els for CDOs. In particular, it turned out that the standard one-factor Gauss copula model (seeLi (2000)) was unable to simultaneously price all traded STCDOs with the same correlationparameter: Di ff  erent STCDOs require di ff  erent correlation parameters giving rise to the “corre-lation smile”. A typical examples are the quotes and “base correlation” parameters in table 1.To account for this e ff  ect, several modifications of the one-factor Gauss copula model have beenproposed, see e.g. Andersen and Sidenius (2004), Hull and White (2003).One reason for the introduction of CDS indices and the corresponding index tranches was thecreation of   hedge instruments   for the management of the risk of the more exotic portfolio creditderivatives. But when it comes to the dynamic hedging of transactions, the Gauss copula and itsextensions have a problem because these models are essentially static. As shown by Sch¨onbucherand Schubert (2001), copula models can be equipped with consistent intensity dynamics but theresulting dynamics are not necessarily realistic (in particular if a Gaussian copula or one of itsmodifications is used), and the necessary analytics can become very involved very quickly. Fur-thermore, it turned out that Gauss copula models and their modifications had highly unrealistic  forward   prices for STCDO tranches which is problematic if these models are to be used to priceoptions or forward contracts on CDOs. In this respect, alternative approaches may be moresuccessful, e.g. the frailty approach (Sch¨onbucher (2003)), fully multivariate intensity models(Du ffi e et al. (2000)), or the intensity-Gamma model by Joshi and Stacey (2005).As opposed to the approaches described above which start from a specification of the individualobligors’ default processes, the model presented in this paper takes a top-down approach whichfocuses on the  cumulative   loss process of the  whole   portfolio. To motivate this approach, considerthe situation in equity markets: If we have to price an option on the S&P 500  index  , it is natural  TERM STRUCTURE OF LOSS TRANSITION RATES 3 to model the S&P 500 index directly instead of modelling the 500 individual share price processesof its constituents. Although some information on the individual components of the index is lostin the transition to the aggregate top-down index model, an aggregate index model is oftenbetter able to capture other features such as a volatility smile or the empirically observed indexdynamics.The situation in credit index markets is very similar, with liquid markets for the indices (iTraxxand CDX) and a set of traded derivatives on the index loss distributions. If there are liquidmarkets for STCDOs of all strikes 1 , we can even infer an  implied distribution   for the cumulativeloss process of the portfolio at all time-horizons at which we have a full strike structure of STCDOs. Thus, in order to provide a framework which allows to capture the price dynamicsof STCDOs we directly model the stochastic evolution of the distribution of the cumulative lossprocess of the underlying reference portfolio, and we defer the link back to the individual obligors’defaults to a later stage.The idea of a top-down approach for credit portfolio losses is not new. First, any model whichassumes homogeneity of obligors like the popular “large-pool” approximations (e.g. Vasicek(1987), Lucas et al. (2001) or Gordy (2003)) follows the same philosophy, and also more recentpapers, e.g. Giesecke and Goldberg (2005), Frey and Backhaus (2004) are inspired by similarideas. The distinguishing feature of the model presented here is that we model the  full forward distribution   of the loss process, i.e. we provide a framework which we can model the forwardlooking loss probabilities  simultaneously   for  all   possible time-horizons and loss levels. All othertop-down models that we are aware of do not attempt to model forward loss distributions butmodel the “spot” loss process. Unfortunately, a “spot”-modelling approach quickly leads intolarge complications when one tries to calibrate the model to a correlation smile.The  forward  -modelling approach of this paper is inspired by the famous Heath et al. (1992)(HJM) approach in interest-rate models where a full forward-looking term structure of interest-rates is modelled. While Heath et al. (1992) model a single term structure of zero couponbond prices, here we shall model the evolution of a loss process, and we have to find a suitableparametric representation for a full term- and strike structure of prices of STCDOs on this lossprocess. Essentially, we first have to invent the “forward rates” for portfolio losses, before wecan make them stochastic.We propose to to represent the loss distribution using  forward transition rates   of a hypotheticaltime-inhomogeneous Markov chain which is constructed in such a way that its distribution co-incides with the given loss distribution. It is shown that (up to very weak regularity conditions)the set of loss distributions that can be reached based upon such forward transition-rates coin-cides with the set of   all   arbitrage-free loss distributions, i.e. we can truly say that  without loss of generality   we represent the loss distribution with a set of forward transition rates. Further-more, this representation with  forward   transition rates allows a straightforward calibration of themodel to a full initial term- and strike structure of synthetic CDOs on the underlying portfolio(including the correlation smile), and (under the additional assumption of one-step transitions)this representation is even unique. This representation is introduced and analyzed in section 2. 1 In this paper we talk of “strikes” for STCDOs to emphasize their similarity to options. The technical termin the market is “attachment point” and “exhaustion point”.
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