Hedging credit index tranches Investigating versions of the standard model Christopher C. Finger chris.fi[email protected]
Risk Management Subtle company introduction
www.riskmetrics.com Risk Management 22 Motivation
A standard model for credit index tranches exists. It is commonly acknowledged that the common model is flawed. Most of the focus is on the static flaw: the failure to calibrate to all tranches on a single day with a single model parameter. But these are liquid derivatives. Models are not used for absolute pricing, but for relative value and hedging. We will focus on the dynamic flaws of the model.
www.riskmetrics.com Risk Management 32 Outline
1 Standard credit derivative products
2 Standard models, conventions and abuses
3 Data and calibration
4 Testing hedging strategies
5 Conclusions
www.riskmetrics.com Risk Management 42 Standard products
Single-name credit default swaps Contract written on a set of reference obligations issued by one firm Protection seller compensates for losses (par less recovery) in the event of a default. Protection buyer pays a periodic premium (spread) on the notional amount being protected. Quoting is on fair spread, that is, spread that makes a contract have zero upfront value at inception.
www.riskmetrics.com Risk Management 52 Standard products
Credit default swap indices (CDX, iTraxx) Contract is essentially a portfolio of (125, for our purposes) equally weighted CDS on a standard basket of firms. Protection seller compensates for losses (par less recovery) in the event of a default. Protection buyer pays a periodic premium (spread) on the remaining notional amount being protected. New contracts (series) are introduced every six months. Standardization of premium, basket, maturity has created significant liquidity. Quoting is on fair spread, with somewhat of a twist. Pricing depends only on the prices of the portfolio names . . . almost.
www.riskmetrics.com Risk Management 62 Standard products
Tranches on CDX Protection seller compensates for losses on the index in excess of one level (the attachment point) and up to a second level (the detachment point). For example, on the 3-7% tranche of the CDX, protection seller pays losses over 3% (attachment) and up to 7% (detachment). Protection buyer pays an upfront amount (for most junior tranches) plus a periodic premium on the remaining amount being protected. Standardization of attachment/detachment, indices, maturity. Not strictly a derivative on the index, in that payoff does not reference the index price Pricing depends on the distribution of losses on the index, not just the expectation.
Also, options on CDX, but we will not consider these.
www.riskmetrics.com Risk Management 72 CDX history
200
160
120
Index spread (bp) 80
40
0 Mar05 Sep05 Mar06 Sep06 Mar07 Sep07 Mar08
www.riskmetrics.com Risk Management 82 CDX tranche history
0−3% (LHS) 3−7% (RHS) 7−10% (RHS)
500 60
50 400
40 300
30 Tranche fair spread (bp) Tranche upfront price (%) 200
20
100 10
0 0 Mar05 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 92 What do we want from a model?
Fit market prices, but why? Extrapolate, i.e. price more complex, but similar, structures Non-standard attachment points Customized baskets Hedge risk due to underlying Dealers provide liquidity in tranches, but want to control exposure to underlyings. Speculators want to make relative bets on tranches without a view on underlyings. Risk management Aggregate credit exposures across many product types. Recognize risk that is truly idiosyncratic. For anything other than extrapolation, we care about how prices evolve in time, so we should look at the dynamics.
www.riskmetrics.com Risk Management 102 Standard pricing models—basic stuff
Price for a tranche is the difference of Expectation of discounted future premium payments, and Expectation of discounted future losses. Boils down to the distribution of the loss process on the index portfolio, in particular things like
E min{(d − a), max{0, Lt − a}}.
Suffices to specify the joint distribution of times to default Ti for all names in the basket. CDS (or CDX) quotes imply the marginal distributions for time to default for individual names Fi (t) = P{Ti < t}.
www.riskmetrics.com Risk Management 112 Standard pricing models—specific stuff
Dependence structure is a Gaussian copula:
Let Zi be correlated standard Gaussian random variables. −1 Default times are given by Ti = Fi (Φ(Zi )). Correlation structure is pairwise constant . . . √ p Zi = ρ Z + 1 − ρ εi .
For a single period, just count the number of Zi that fall −1 below the default threshold αi = Φ (pi ).
www.riskmetrics.com Risk Management 122 Criticisms of the standard model
Does not fit all market tranche prices on a given day. No dynamics, so no natural hedging strategy. Link to any observable correlation is tenuous. At best, model is “inspired” by Merton framework, so correlation is on equities.
www.riskmetrics.com Risk Management 132 Model flavors
Most numerical techniques rely on integrating over Z, given which all defaults are conditionally independent. Granular model—use full information on underlying spreads, and model full discrete loss distribution. Homogeneity assumption—assume all names in the portfolio have the same spread; use index level. Fine grained limit—continuous distribution, easy integrals Large pool model—combine homogeneity and fine grained assumptions. Also the question of whether to use full spread term structures or a single point
www.riskmetrics.com Risk Management 142 Correlation conventions
Start to introduce model abuses, ala the B-S volatility smile.
Compound correlations Price each individual tranche with a distinct correlation. Not all tranches are monotonic in correlation. Trouble calibrating mezzanine tranches, especially in 2005
Base correlations Decompose each tranche into “base” (i.e. 0-x%) tranches. Bootstrap to calibrate all tranches. Base tranches monotonic, but calibration not guaranteed.
www.riskmetrics.com Risk Management 152 Correlation conventions
Constant moneyness (ATM) correlations Some movements in implied correlation are due to changing “moneyness” as the index changes. Examine correlations associated with a detachment point equal to implied index expected loss. If base correlations are “sticky strike”, then ATM correlations are closer to “sticky delta”.
www.riskmetrics.com Risk Management 162 CDX correlation structure
100 Mar05 Mar06 90 Mar07 Mar08
80
70
60
50
40 Base correlation (%)
30
20
10
0 0 5 10 15 20 25 30 Detachment point (%)
www.riskmetrics.com Risk Management 172 CDX base correlations, granular model
100
80
60
40 Base correlation (%)
20
0 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 182 CDX base correlations, large pool model
100
80
60
40 Base correlation (%)
20
0 Mar05 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 192 CDX base correlations, large pool model, with DJX option-implied correlation
100
80
60
40 Base correlation (%)
20
0 Mar05 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 202 CDX Series 4-7, GR model
70
60
50
40
30 Base correlation (%)
20
10
0 4 6 8 10 12 14 16 18 Detachment plus index EL (%)
www.riskmetrics.com Risk Management 212 CDX Series 4-7, LP model
70
60
50
40
30 Base correlation (%)
20
10
0 4 6 8 10 12 14 16 18 Detachment plus index EL (%)
www.riskmetrics.com Risk Management 222 CDX Series 8-9, GR model
70
60
50
40
30 Base correlation (%)
20
10
0 4 6 8 10 12 14 16 18 Detachment plus index EL (%)
www.riskmetrics.com Risk Management 232 CDX Series 8-9, LP model
70
60
50
40
30 Base correlation (%)
20
10
0 4 6 8 10 12 14 16 18 Detachment plus index EL (%)
www.riskmetrics.com Risk Management 242 Time series properties
Examine the correlation between various implied correlations and the index. We would like this to be low. Why? For risk, we have identified idiosyncratic risk correctly. For hedging, we have captured what we are able to from the underlying.
Start with statistics on daily changes.
www.riskmetrics.com Risk Management 252 CDX 0-3%, large pool model, base correlations
60 Flat curve Full curve
50
40
30 Correlation to index (%) 20
10
0 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 262 CDX 0-3%, base correlations
60 Large pool Granular 50
40
30
20
10 Correlation to index (%)
0
−10
−20 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 272 CDX 0-3%, large pool model
60 Base Corr ATM Corr
40
20
0 Correlation to index (%) −20
−40
−60 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 282 CDX 0-3%
60 LP, base Gran, ATM 50
40
30
20
10 Correlation to index (%) 0
−10
−20
−30 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 292 CDX 3-7%
60 LP, base Gran, ATM 50
40
30
20
10 Correlation to index (%)
0
−10
−20 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 302 Time series properties
Now look at statistics on weekly changes.
www.riskmetrics.com Risk Management 312 CDX 0-3%, large pool model, base correlations
80 Flat curve Full curve
60
40
20
0 Correlation to index (%)
−20
−40
−60 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 322 CDX 0-3%, base correlations
80 Large pool Granular
60
40
20
0 Correlation to index (%)
−20
−40
−60 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 332 CDX 0-3%, large pool model
80 Base Corr ATM Corr 60
40
20
0
−20 Correlation to index (%)
−40
−60
−80 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 342 CDX 0-3%
80 LP, base Gran, ATM 60
40
20
0
−20 Correlation to index (%)
−40
−60
−80 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 352 CDX 3-7%
80 LP, base Gran, ATM
60
40
20
0 Correlation to index (%)
−20
−40
−60 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 362 Time series conclusions
Looks like granular model produces “more idiosyncratic” implied correlations. The ATM approach looks appears to improve things, but seems to overcorrect in the most volatile periods. High correlations overall suggest index moves may have some predictive power for implied correlations. Differences are much more pronounced at a one-day horizon than a one-week horizon.
www.riskmetrics.com Risk Management 372 Hedging experiment
On day zero, we have all prices (spreads and tranches) plus history. At a future date, we are told how the underlying spreads (index or single-name CDS) have moved, and are asked to guess what the new tranche prices should be. Compare the predicted tranche price moves to the actual ones, over time and across different modeling approaches. Another approach would be to look at delta hedge performance, but this mixes in the error from linearization.
www.riskmetrics.com Risk Management 382 Hedging experiment candidates
Models Regression—fit linear relationship between tranche price changes and index changes, using prior six months of data. Large pool model—calibrate based on current index levels. Granular model—calibrate based on current single-name CDS levels.
Correlation approaches Base—use most recent calibrated correlation. ATM—move along the most recent correlation structure according to change in the index-implied expected loss. Regression—fit linear relationship between base correlation changes and index changes, using prior six months of data.
www.riskmetrics.com Risk Management 392 CDX 0-3%, Regression
80 8
70
60 4
50
40 0 Tranche price (%) 30 Prediction error (%)
20 −4
10
0 −8 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 402 CDX 0-3%, LP model, base correlations
80 8
70
60 4
50
40 0 Tranche price (%) 30 Prediction error (%)
20 −4
10
0 −8 Sep05 Mar06 Sep06 Mar07 Sep07
www.riskmetrics.com Risk Management 412 CDX 0-3%, Regression
8 Ser 4 Ser 5 Ser 6 6 Ser 7 Ser 8 Ser 9 4
2
0
Predicted change (%) −2
−4
−6
−8 −8 −6 −4 −2 0 2 4 6 8 Tranche price change (%)
www.riskmetrics.com Risk Management 422 CDX 0-3%, LP model, base correlations
8 Ser 4 Ser 5 Ser 6 6 Ser 7 Ser 8 Ser 9 4
2
0
Predicted change (%) −2
−4
−6
−8 −8 −6 −4 −2 0 2 4 6 8 Tranche price change (%)
www.riskmetrics.com Risk Management 432 Statistics on hedging approaches
Examine standard deviation of tranche forecast error. Daily horizon
Bars are for correlation approaches: Base (blue), ATM (green), Regression (red). Curve is for simple regression model.
www.riskmetrics.com Risk Management 442 CDX 0-3%, LP model
0.03
0.025
0.02
0.015 STD of prediction error 0.01
0.005
0 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 452 CDX 0-3%, GR model
0.03
0.025
0.02
0.015 STD of prediction error 0.01
0.005
0 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 462 CDX 3-7%, LP model
−3 x 10 3.5
3
2.5
2
1.5 STD of prediction error
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 472 CDX 3-7%, GR model
−3 x 10 3.5
3
2.5
2
1.5 STD of prediction error
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 482 CDX 7-10%, LP model
−3 x 10 2.5
2
1.5
1 STD of prediction error
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 492 CDX 7-10%, GR model
−3 x 10 2.5
2
1.5
1 STD of prediction error
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 502 Statistics on hedging approaches
Examine correlation of tranche forecast error with actual tranche move. Daily horizon
Bars are for correlation approaches: Base (blue), ATM (green), Regression (red). Curve is for simple regression model.
www.riskmetrics.com Risk Management 512 CDX 0-3%, LP model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 522 CDX 0-3%, GR model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 532 CDX 3-7%, LP model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 542 CDX 3-7%, GR model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 552 CDX 7-10%, LP model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 562 CDX 7-10%, GR model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 572 Statistics on hedging approaches
Examine standard deviation of tranche forecast error. Weekly horizon
Bars are for correlation approaches: Base (blue), ATM (green), Regression (red). Curve is for simple regression model.
www.riskmetrics.com Risk Management 582 CDX 0-3%, LP model
0.045
0.04
0.035
0.03
0.025
0.02 STD of prediction error 0.015
0.01
0.005
0 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 592 CDX 0-3%, GR model
0.045
0.04
0.035
0.03
0.025
0.02 STD of prediction error 0.015
0.01
0.005
0 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 602 CDX 3-7%, LP model
−3 x 10 5
4.5
4
3.5
3
2.5
2 STD of prediction error
1.5
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 612 CDX 3-7%, GR model
−3 x 10 5
4.5
4
3.5
3
2.5
2 STD of prediction error
1.5
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 622 CDX 7-10%, LP model
−3 x 10 3.5
3
2.5
2
1.5 STD of prediction error
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 632 CDX 7-10%, GR model
−3 x 10 3.5
3
2.5
2
1.5 STD of prediction error
1
0.5
0 4 5 6 7 8 9 All Series www.riskmetrics.com Risk Management 642 Statistics on hedging approaches
Examine correlation of tranche forecast error with actual tranche move. Weekly horizon
Bars are for correlation approaches: Base (blue), ATM (green), Regression (red). Curve is for simple regression model.
www.riskmetrics.com Risk Management 652 CDX 0-3%, LP model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 662 CDX 0-3%, GR model
0
−0.1
−0.2
−0.3
−0.4
−0.5 Corr(Act,Err) −0.6
−0.7
−0.8
−0.9
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 672 CDX 3-7%, LP model
0.1
0
−0.1
−0.2
−0.3
−0.4 Corr(Act,Err) −0.5
−0.6
−0.7
−0.8
−0.9 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 682 CDX 3-7%, GR model
0.1
0
−0.1
−0.2
−0.3
−0.4 Corr(Act,Err) −0.5
−0.6
−0.7
−0.8
−0.9 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 692 CDX 7-10%, LP model
0.5
0 Corr(Act,Err)
−0.5
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 702 CDX 7-10%, GR model
0.5
0 Corr(Act,Err)
−0.5
−1 4 5 6 7 8 9 All Series
www.riskmetrics.com Risk Management 712 Conclusions
Overall, hedging errors are surprisingly large. For the equity tranche, a simple regression works quite well, with its underhedging problems reduced at a slightly longer horizon. There is a benefit to using the granular model, but is it worth the cost? For more senior tranches, the ATM approach appears to capture some of the link to credit spreads, but does not markedly reduce the hedging error.
Any candidate for a new standard model should be able to do better in this experiment.
www.riskmetrics.com Risk Management 722 Further investigations
Can we use these results to learn more about how our model might be misspecified? Richer correlation structure? Different copula? Better dynamics? How much worse (or better) are compound correlations? Can the regression be improved by including the HiVol index? Can the LP model be improved by adding a simple correction for heterogeneity? Does any of this change at or close to defaults?
www.riskmetrics.com Risk Management 732 Further reading
Couderc, F. (2007) Measuring Risk on Credit Indices: On the Use of the Basis. RiskMetrics Journal. Finger, C. (2004) Issues in the Pricing of Synthetic CDOs. RiskMetrics Journal. Finger, C. (2005) Eating our Own Cooking. RiskMetrics Research Monthly. June.
All are available at www.riskmetrics.com
www.riskmetrics.com Risk Management 742