June, 24, 2003. Jean-Paul Laurent. ISFA Actuarial School, University of Lyon. Paper « basket defaults swaps, CDO's and Factor Copulas » available on AFFI ...
Example: six names portfolio Changes in credit curves of individual names Amount of individual CDS to hedge the basket Semi-analytical more accurate than 105 Monte Carlo simulations. Much quicker: about 25 Monte Carlo simulations.
Risk Management of Basket Credit Derivatives !
Changes in credit curves of individual names !
Dependence upon the choice of copula for defaults
CDO Tranches «Everything should be made as simple as possible, not simpler» !
Explicit premium computations for tranches
!
Use of loss distributions over different time horizons
!
Computation of loss distributions from FFT
!
Involves integration par parts and Stieltjes integrals
Credit Loss Distributions !
Accumulated loss at t: !
Where
loss given default
!
Characteristic function
!
By conditioning
!
Distribution of L(t) is obtained by FFT
Credit Loss distributions !
! !
!
!
One hundred names, same nominal. Recovery rates: 40% Credit spreads uniformly distributed between 60 and 250 bp. Gaussian copula, correlation: 50% 105 Monte Carlo simulations
Valuation of CDO’s
!
Tranches with thresholds Mezzanine: pays whenever losses are between A and B Cumulated payments at time t: M(t)
!
Upfront premium:
! !
!
B(t) discount factor, T maturity of CDO
!
Stieltjes integration by parts
!
where
Valuation of CDO’s
! !
One factor Gaussian copula CDO tranches margins with respect to correlation parameter
Risk Management of CDO’s !
Hedging of CDO tranches with respect to credit curves of individual names
!
Amount of individual CDS to hedge the CDO tranche
!
Semi-analytic : some seconds
!
Monte Carlo more than one hour and still shaky
Conclusion !
Factor models of default times: ! !
simple computation of basket credit derivatives and CDO’s deal easily with a large range of names and dependence structures
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