7th Edition Out! January 30, 2007 Data CDO Models Dynamic Models as long as Portfolio Losses: Prior Research Our Objective

7th Edition Out! January 30, 2007 Data CDO Models Dynamic Models as long as Portfolio Losses: Prior Research Our Objective www.phwiki.com

7th Edition Out! January 30, 2007 Data CDO Models Dynamic Models as long as Portfolio Losses: Prior Research Our Objective

Sauceda, Mike, Executive Producer has reference to this Academic Journal, PHwiki organized this Journal Copyright © John Hull 2008 Dynamic Models of Portfolio Credit Risk: A Simplified Approach John Hull Princeton Credit Conference May 2008 7th Edition Out! Copyright © John Hull 2008 Copyright © John Hull 2008 January 30, 2007 Data

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Copyright © John Hull 2008 CDO Models St in addition to ard market model is one-factor Gaussian copula model of time to default Alternatives that have been proposed: t-, double-t, Clayton, Archimedian, Marshall Olkin, implied copula All are static models. They provide a probability distribution as long as the loss over the life of the model, but do not describe how the loss evolves Dynamic models are needed to value options in addition to structured deals such as leveraged super seniors Copyright © John Hull 2008 Dynamic Models as long as Portfolio Losses: Prior Research Structural: Albanese et al; Baxter (2006); Hull et al (2005) Reduced Form: Duffie in addition to Gârleanu (2001), Chapovsky et al (2006), Graziano in addition to Rogers (2005), Hurd in addition to Kuznetsov (2005), in addition to Joshi in addition to Stacey (2006) Top Down: Sidenius et al (2004), Bennani (2005), Schonbucher (2005), Errais, Giesecke, in addition to Goldberg (2006), Longstaff in addition to Rajan (2006), Putyatin et al (2005), in addition to Walker (2007) Copyright © John Hull 2008 Our Objective Build a simple dynamic model of the evolution of losses that is easy to implement in addition to easy to calibrate to market data The model is developed as a reduced as long as m model, but can also be presented as a top down model

Copyright © John Hull 2008 Specific vs General Models Specific models track the evolution of default risk on a single name or portfolio that remains fixed (e.g. describes how credit spread as long as a particular company evolves) General models track the evolution of default risk on a single name or portfolio that is updated so that it always has certain properties (e.g. describes how the average spread as long as an A-rated company evolves) We focus on specific models Copyright © John Hull 2008 CDO Valuation Key to valuing a CDO lies in the calculation of expected tranche principal on payment dates Expected payment on a payment date equals spread times expected tranche principal on that date Expected payoff between payment dates equals reduction in expected tranche principal between the dates Expected accrual payments can be calculated from expected payoffs Expected tranche principal at time t can be calculated from the cumulative default probabilities up to time t in addition to recovery rates of companies in the portfolio The Model Instead of modeling the hazard rate, h(t) we model This is –ln[S(t)] where S(t) is the survival probability calculated from the path followed by the hazard rate between times 0 in addition to t Filtration: We assume that at time t we know the path followed by S between time zero in addition to time t in addition to the number of defaults up to time t Copyright © John Hull 2008

Copyright © John Hull 2008 The Model (Homogeneous Case) where dq represents a jump that has intensity l in addition to jump size H m in addition to l are functions only of time in addition to H is a function of the number of jumps so far. m > 0, H > 0. X The Hazard Rate Process The hazard rate process is where dI is an impulse that has intensity l Copyright © John Hull 2008 Copyright © John Hull 2008 Analytic Results in addition to Binomial Trees Once m(t), l(t), in addition to the the size of the jth jump, Hj, have been specified the model can be used to value analytically CDOs Forward CDOs Options on CDOs For other instruments a binomial tree can be used

Copyright © John Hull 2008 Binomial Tree as long as Model Copyright © John Hull 2008 Simplest Version of Model Jump size is constant in addition to m(t), is zero Jump intensity, l(t) is chosen to match the term structure of CDS spreads There is then a one-to-one correspondence between tranche quotes in addition to jump size Implied jump sizes are similar to implied correlations Copyright © John Hull 2008 Comparison of Implied Jump Sizes with Implied Tranche Correlations

Copyright © John Hull 2008 Possible Extensions to Fit All Market Data Multiple processes each with its own jump size in addition to intensity Intensity in addition to jump size changing in different intervals: 0 to 5 yrs, 5 to 7 yrs, in addition to 7 to 10 yrs Model can (in principle) fit all quotes simultaneously We have chosen to focus on extensions where there are relatively few parameters Copyright © John Hull 2008 Extension Involving Three Parameters The size of the Jth jump is HJ = H0 ebJ The three parameters are The initial jump size The growth in the jump size The jump intensity The model reflects empirical research showing that correlation is higher in adverse market conditions Copyright © John Hull 2008 Variation of best fit jump parameters, H0 in addition to b, across time. (10-day moving averages)

Copyright © John Hull 2008 Variation of jump intensity (10-day moving averages) Copyright © John Hull 2008 Evolution of Loss Distribution on January 30, 2007 as long as 3-parameter model. Copyright © John Hull 2008 Valuation of Forward Contracts on CDOs that End in Five Years Using 3-Parameter Model on January 30, 2007

Copyright © John Hull 2008 Valuation of At-The-Money Options (in basis points) on CDOs that End in Five Years Using 3-Parameter Model on January 30, 2007 Copyright © John Hull 2008 Implied Volatilities from Black’s Model as Option Maturity is Changed Copyright © John Hull 2008 Implied Volatilities as long as 2 yr Option from Black’s Model as Strike Price is Changed

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Copyright © John Hull 2008 Leverage Super Senior with Loss Trigger Total exposure of seller of protection is limited to a fraction x of the tranche notional When losses reach some level the buyer of protection can cancel the deal in addition to seller has to pay the value of the tranche to the buyer. Define n as the number of losses triggering cancellation Copyright © John Hull 2008 Breakeven LSS spread on Jan 30, 2007 as a function of the maximum percentage loss by protection seller, x%, in addition to the number of defaults triggering close out, n Copyright © John Hull 2008 Extensions Model can be extended so that Different companies have different CDS spreads The recovery rate is negatively correlated with the default rate iTraxx in addition to CDX jumps are modeled jointly

Bespokes Calibrate homogeneous model to iTraxx in addition to CDX IG If all names are North American, use a non-homogeneous model where underlying companies have the the CDX IG jumps in addition to their own drifts. If there are a mixture of European in addition to North American names use a non-homogeneous model where the iTraxx in addition to CDX IG jumps are modeled jointly Copyright © John Hull 2008 Copyright © John Hull 2008 Conclusions It is possible to develop a simple dynamic model as long as losses on a portfolio by modeling the cumulative default probability as long as a representative company The only way of fitting the market appears to be by assuming that there is a small probability of a series of progressively bigger jumps in the cumulative probability. As credit market deteriorates default correlation becomes higher

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