Valuation models

CDO tranches and nth-to-default

Synthetic CDO and index tranches (CDX, iTraxx) and nth-to-default baskets, with defaults joined by a one-factor Gaussian, Student-t or NIG copula, a base correlation curve, or the large homogeneous pool limit; par spread, upfront, expected loss, the loss distribution, CS01 and correlation sensitivity.

Inputs

Results

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Methodology

Conventions

  • Each name's default time comes from its hazard curve; a one-factor copula joins them.
  • Losses are put on a grid and summed by recursion over names, given the common factor, then integrated over it.
  • Base correlation: each equity piece [0, K] takes the correlation quoted at K.

References

  • Li (2000), On Default Correlation
  • Andersen, Sidenius and Basu (2003), All Your Hedges in One Basket
  • Kalemanova, Schmid and Werner (2007), The NIG Distribution for Synthetic CDO Pricing
  • Vasicek (1987), Probability of Loss on Loan Portfolio

Formulas

Latent variable
Xᵢ = √ρ M + √(1 − ρ) Zᵢ
Tranche loss
EL = (E[min(L, D)] − E[min(L, A)]) / (D − A)
Premium
RPV01 = Σ τⱼ D(tⱼ) (1 − ½(ELⱼ₋₁ + ELⱼ))