Managing Synthetic CDO Tranches using Base Correlations
Managing Synthetic CDO Tranches using Base Correlations Managing Synthetic CDO Tranches using Base Correlations
Consider linear interpolation iTraxx 18-Sep-2007 1 0.9 0.8 0.7 1200 1000 800 Fair spreads of tranchlets Base Correlation 0.6 0.5 0.4 Fair spread (bp) 600 400 Arbitrage Negative spreads 0.3 0.2 200 0.1 0 0 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Strike 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Strike Increase X, fair spread decreases Low gradient, fair spread increases 34
Instead, interpolate expected tranche loss Parcel and Wood (2007): Easier to formulate no-arbitrage conditions with expected tranche loss ETL( X, ρX ) = ∫ Pr( L > s) ds 0 ∂ETL( X, ρX ) Pr( L > x) = > 0 ∂X 2 ∂ ETL( X, ρX ) fL( x) = − 2 ∂X X Eliminates negative spreads Eliminates fair spreads increasing Turn extrapolation of base correlations into an interpolation exercise Expected tranche loss of zero strike is zero Expected tranche loss of 100% detachment point is equal to the expected portfolio loss 35
- Page 2 and 3: Managing synthetic CDO tranches usi
- Page 4 and 5: Synthetic CDO mechanics losses Pro
- Page 6 and 7: Protection buyer can hedge by selli
- Page 8 and 9: Agenda Synthetic CDO mechanics Ba
- Page 10 and 11: Gaussian Copula Model (con’t) S
- Page 12 and 13: Base correlation framework Each tr
- Page 14 and 15: Stress tests Base correlations c
- Page 16 and 17: Subprime crisis has spread to … L
- Page 18 and 19: Stress Test Example for CDX.NA.IG S
- Page 20 and 21: So which stress test makes sense I
- Page 22 and 23: Agenda Synthetic CDO mechanics Ba
- Page 24 and 25: Mapping base correlations between b
- Page 26 and 27: Base correlation mapping methods Fi
- Page 28 and 29: Method 4: Expected Tranche Loss Pro
- Page 30 and 31: Comparison of mappings for iTraxx S
- Page 32 and 33: Heterogeneous portfolios Can we fin
- Page 36: Summary Base correlation are usef
Instead, interpolate expected tranche loss<br />
Parcel and Wood (2007):<br />
<br />
Easier to formulate no-arbitrage conditions with expected tranche loss<br />
ETL(<br />
X,<br />
ρX<br />
) = ∫ Pr( L > s)<br />
ds<br />
0<br />
∂ETL(<br />
X,<br />
ρX<br />
)<br />
Pr( L > x)<br />
=<br />
> 0<br />
∂X<br />
2<br />
∂ ETL(<br />
X,<br />
ρX<br />
)<br />
fL(<br />
x)<br />
= −<br />
2<br />
∂X<br />
X<br />
Eliminates negative spreads<br />
Eliminates fair spreads<br />
increasing<br />
<br />
Turn extrapolation of base correlations into an interpolation exercise<br />
Expected tranche loss of zero strike is zero<br />
Expected tranche loss of 100% detachment point is equal to the expected portfolio<br />
loss<br />
35
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