Composite Basket Model-Part II
Last week we introduced the Composite Basket Model (or CBM) and discussed its implementation. This week we present some of the results obtained with the CBM and discuss how to calibrate and price with this model.
The Composite Basket Model
The CBM, described in detail in part 1, acknowledges the fact that the Gaussian copula or similar methods are widely used to price basket products while it also attempts to rationalise how market participants adjust their prices to accommodate their own views of risk. It does this by combining the Gaussian copula with an exponential multivariate copula (see  for a description of exponential copulas). This approach offers computational tractability and, as we will see next, captures much of the structure exhibited by market prices.
Calibration & Results
We chose to use a fixed idiosyncratic intensity (floored as required to ensure that the probabilities are not negative) and two separate systemic intensities for the US and EU markets. We then calibrated these parameters to fit two different sets of quotes: DJ.CDX.NA.2 (in this note referred to as iTraxx North America) and iTraxx Europe 5 year tranches, as of the 14th of July 2004 (10 data points). The calibration was configured to favour results that fall inside or closer to the bid/offer range but to select weakly inside that range. The parameters are listed in table 1 and the two data sets in tables 2 and 3.
The equity tranches are quoted in terms of an upfront fee and a fixed running coupon of 5% in all cases. These were converted into an effective coupon (with no upfront fee) using the standard Gaussian copula model with a fitted single correlation parameter.
If the model result lies outside the bid/offer range, the errors shown are relative errors to the closest quote (negative if below bid, positive if above offer). If inside this range, no error is reported. These error values are shown only to assist in comparing results. During the calibration a suitable error measure was used.
The idiosyncratic intensity shown is a maximum value. For each asset the actual intensity used will be capped in such a way that the survival probabilities of each asset are correctly calibrated to the single name market.
We then applied the same set of parameters to the corresponding 10 year quotes and to a second set of US quotes corresponding to a different date. These are shown in tables 4 through 7.
We find a reasonable agreement between the model results and the published quotes across different markets and different dates. These results are much better than is possible using the Gaussian copula. Naturally the results obtained are preliminary as there is limited data with which to test the method. Using historical data, we observe only small variations in underlying spreads for the period in which reliable data can be obtained. Further work will be required to establish the model stability through time. However, the demonstrated ability to match prices for the European and US markets (which have fairly different average spreads) is significant and suggests that the model would display some robustness with respect to movements in the underlying spreads. Additionally, a similar conclusion can be drawn from the model's ability to match prices across the five and ten year maturities.
The comparisons yield one other interesting result. If we split the systemic risk into Europe and the US then the natural interpretation for this value is that it captures the default risk inherent in that particular economy. The argument then implies that the fitted intensities should be comparable to market quotes on sovereign debt. Quotes on government debt are approximately 0.5-2bp for the US and 3-7bp for Europe. On the other hand, assuming a recovery of 30%, the systemic intensities correspond to spreads of 3bp and 6bp respectively for the US and Europe. These are in reasonable agreement with the quoted values and raise the possibility of reducing the number of free parameters and calibrating the systemic terms independently of the baskets.
It is also of interest to note that these results seem to be applicable to the first-to-default market. Restricting ourselves to the four FTD standardised baskets (launched recently) that are generally seen as uncorrelated both competitively and geographically then:
Further work is required to establish whether this first-to-default result is meaningful. However, the fact that parameters fitted to tranche instruments seem also to apply in the instances shown above is by itself a very interesting observation.
In this paper we focussed on value calculations but first order risks are equally critical aspects of derivative trading. Although this is not shown in the results above, we have compared the credit spread deltas produced by the Composite Basket Model against those produced by the Gaussian copula calibrated to first-loss tranches (i.e. using base correlations as discussed in ). We found the results to be very similar for the most traded mezzanine tranches. Equity tranches exhibit slightly less leverage and senior tranches more leverage in the CBM than is obtained with the base correlation approach.
In this and last week's article we introduced a new proposal for dealing with the Gaussian copula skew and the pricing of bespoke baskets in the single tranche market. The Composite Basket Model acknowledges the fact that the Gaussian copula or similar methods are widely used by market participants to price these products while it also attempts to rationalise how market participants adjust their prices to accommodate their own views of risk.
The results presented indicate that this approach is capable of capturing much of the structure implied by the published quotes: once the parameters are calibrated, the model fits the 5yr and 10yr iTraxx Europe, the 5yr and 10yr iTraxx North America and the standardised FTD baskets. These results, obtained for different terms and valuation dates, indicate that a trivial generalisation to bespoke baskets is possible with this method at least as long as our baskets are as diversified as the indices themselves are.
|Table 1: Composite basket model parameters|
|U.S. systemic intensity||5.6bps|
|E.U. systemic intensity||9.0bps|
|Table 2: Traxx Europe five year as of July 14 (average spread 46bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 30/32.5%.|
|12 – 22||0.21%||0.26%||0.21%||-2%|
|9 – 12||0.50%||0.57%||0.45%||-11%|
|6 – 9||0.82%||0.90%||0.96%||6%|
|3 – 6||1.95%||2.10%||2.14%||2%|
|0 – 3||13.69%||14.57%||13.95%||—|
|Table 3: iTraxx North America five year as of July 14 (average spread 62bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 40/42%.|
|15 – 30||0.13%||0.17%||0.17%||—|
|10 – 15||0.44%||0.54%||0.66%||22%|
|7 – 10||1.31%||1.38%||1.31%||—|
|3 – 7||3.43%||3.65%||3.44%||—|
|0 – 3||17.82%||18.70%||18.61%||—|
|Table 4: iTraxx Europe 10 year as of July 14 (average spread 65bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 49.7/54.7%|
|12 – 22||0.70%||0.90%||0.68%||-3%|
|9 – 12||1.31%||1.61%||1.63%||1%|
|6 – 9||2.08%||2.48%||2.45%||—|
|0 – 3||15.98%||17.79%||16.64%||—|
|Table 5: iTraxx North America 10 year as of July 14 (average spread 83bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 56.5/60%.|
|15 - 30||N/A||N/A||0.57%||N/A|
|0 – 3||19.32%||20.90%||20.81%||—|
|Table 6: iTraxx North America five year as of May 26 (average spread 66bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 44.5/45.5%.|
|15 – 30||0.12%||0.20%||0.20%||—|
|10 – 15||0.53%||0.68%||0.79%||16%|
|7 – 10||1.46%||1.57%||1.44%||-1%|
|3 – 7||3.70%||4.05%||3.83%||—|
|0 – 3||19.61%||20.08%||19.38%||-1%|
|Table 7: iTraxx North America 10 year as of May 26 (average spread 88bps). Equity tranche (0-3) spreads correspond to the effective spread equivalent to an upfront payment of 57/67%.|
|15 - 30||N/A||N/A||0.66%||N/A|
|0 – 3||19.53%||25.08%||21.52%||—|
|Table 8: First-to-default baskets as of July 14.|
This week's Learning Curve was written by Pedro Tavares, head of quantitative analytics for credit derivatives, Thu-Uyen Nguyen, director responsible for new product development, Alexander Chapovsky, quantitative analytics and Igor Vaysburd, quantitative analytics at Merrill Lynch.