CDO Valuation: Term Structure, Tranche Structure and Loss Distributions
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Transcript of 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
Global Credit Derivatives Market US$ bn (from BBA Credit Derivatives Report 2006)
Credit Derivatives Products
CDO’s – a simplistic view
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
0-3% quoted as upfront; remaining in bps per year(data from Julien Houdain and Fortis Investments)
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)
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
Expected loss for tranche k
Tranche term structures
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
The loss distribution F(l,t)
The ‘expected’ risk-neutral recovery rate for the basket as a function of time
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 …
Mark-to-market 10 yr maturity
FCDO Term structures
CDO options on 3-6% tranche
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