CDO Valuation Term Structure Tranche Structure and Loss

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

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