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