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CDO Valuation: Term Structure, Tranche Structure and Loss Distributions

CDO Valuation: Term Structure, Tranche Structure and Loss Distributions. Michael Walker Department of Physics University of Toronto walker@physics.utoronto.ca. Global Credit Derivatives Market US$ bn (from BBA Credit Derivatives Report 2006). Credit Derivatives Products.

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CDO Valuation: Term Structure, Tranche Structure and Loss Distributions

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  1. CDO Valuation:Term Structure, Tranche Structureand Loss Distributions Michael Walker Department of Physics University of Toronto walker@physics.utoronto.ca

  2. Global Credit Derivatives Market US$ bn (from BBA Credit Derivatives Report 2006)

  3. Credit Derivatives Products

  4. CDO’s – a simplistic view

  5. CDO contracts provide insurance against tranche losses • e.g. consider 3-6% tranche • Protection buyer buys insurance against all losses from 3 to 6% of total notional. • Protection buyer pays a regular quarterly premium to an investor • Investor pays any losses lying between 3% and 6% to the protection buyer

  6. 0-3% quoted as upfront; remaining in bps per year(data from Julien Houdain and Fortis Investments)

  7. Focus – The calibration problem • There can be 20 to 30 CDO contracts (differing in maturity and loss tranche) on the market that reference the same underlying portfolio. • The problem is to find a risk-neutral measure that can be calibrated to reproduce all available market prices. • This talk presents a simple solution to this calibration problem. • “base corr” can calibrate to only one maturity at a time (but to different tranches at that maturity). • It will be shown that accurate marking of tranches to market requires simultaneous calibration to all maturities. (Trading and RM)

  8. The Basic Pricing Equation • For a CDO contract on a given tranche and for a given maturity, a fair premium requires that: PV(Expected tranche losses) = PV(Expected premium payments) • Define f(k,t) = expected loss per unit tranche notional for tranche k at time t

  9. Expected loss for tranche k

  10. Tranche term structures

  11. Importance of accurate calibration • Market-standard copula and base correlations models don’t calibrate simultaneously to different maturities (i.e. to term structures). • Calibration across maturities is important because it fixes not only total losses, but the timing of the losses. • The timing of the losses has important effects on the mark-to-market values of CDO’s, and the values of forward-starting CDO’s, and options on CDO’s

  12. The loss distributionF(l,t)

  13. The ‘expected’ risk-neutral recovery rate for the basket as a function of time

  14. Marking CDO’s to market • V = [w(k,M) – wold(k,M)]Teff(k,M) • w(k,M) is the annualized premium paid for protection on tranche k of maturity M • Teff(k,M) is the risky duration of the premium payments • Timing of losses …

  15. Mark-to-market 10 yr maturity

  16. FCDO Term structures

  17. CDO options on 3-6% tranche

  18. Conclusions - Results • Perfect calibration to any set of market prices for CDO’s that is arbitrage-free • Mark-to-market prices for CDO tranches that are as reliable as possible • Pricing of bespoke CDO tranches on standard baskets has been carried out. • A recent extension incorporates dynamics and values FCDO’s and options on CDO’s

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