### Executive Summary

In this paper we describe a calibration routine created to price complex bespoke tranche CDOs using liquidly-traded tranche CDOs. We demonstrate the use of a mapping technique called tranche loss proportion that is used to identify the equivalent risk of the standard tranche’s annex.

### Instrument Description

[latexpage]

Single name CDS offer a buyer protection against a credit event arising for an an underlying credit. Many single name CDS contracts are standardized and cleared, often on ICE. An index CDS offers protection against a credit event of a portfolio of credits, also known as the annex of constituents. There are many index CDS that are standardized, and cleared on ICE. For example, CDX are index CDS available on investment-grade and high-yield portfolios of North American corporate debt. Each index CDS has a label such as IG7 or IG32. There is liquid trade in European index CDS such as iTraxx.

A tranche CDO offers protection against a credit event for a specified slice of an index CDS when the portfolio loss is between bounded strikes, described as the attachment and detachment points. When there are fewer defaults than the attachment point, the tranche CDO does not pay out. Beyond the detachment point, the tranche CDO ceases to payout. CDX indices have standard tranches such as 0-3%, 3-7%, etc.

A bespoke tranche CDO can offer an investor or speculator a custom attachment and detachment point, a custom portfolio of debt instruments, or custom maturity date. For example, the attachment and detachment points could be 5.2% to 6.2%. The portfolio of debt instruments could include some North American instruments, and some European instruments. A bespoke tranche could also define a non-standard term to maturity.

## Pricing Standard Contracts

Single name CDS pricing is related to the probability of default, and the recovery rate, in addition to discount factors.

The probability of default is calculated for a projected term based on credit spreads. Higher credit spreads indicate a higher probability of default. There is an accepted method available to convert a credit spread for a term into a probability of default for the term.

The expected recovery rate in the event of default is usually a flat rate such as 40%, but need not be restricted to that value.

Pricing of index CDS and tranche CDOs also requires default correlations between the constituents in the annex. The correlation is used to build the loss distribution for the portfolio to the maturity date of the deal. Pricing models simplify the correlation between the credits to be constant for all constituents. Pricing models can vary the correlations throughout the term of the deal.

Pricing of a tranche CDO, since it involves an exposure only to the losses that occur on the portfolio between the attachment and detachment points can be modeled as a synthetic long position and short position in two base tranches. For example, the pricing of the 3% to 7% tranche can be replicated as a long position in the 7% index equity tranche (0% to 7%) and a short position in the 3% index equity tranche with the same terms and conditions including underlying portfolio and maturity.

Finally, for tranche CDO pricing, there are pricing models that support stochastic recovery rates. Findur supports one such pricing model. Stochastic recovery is implemented as an additional ‘volatility’ surface, simply based on the convenience of using the vol software infrastructure, not because it is truly a volatility pricing model input. The stochastic recovery pricing model introduces an additional recovery factor term, which is applied by the model to the recovery rates during pricing. The factor is usually 1.0, although there have been historical circumstances when it has been necessary to lower the factor in order to calibrate the CDO correlations.

Pricing bespoke tranche CDOs is more complex than any of the other credit derivatives types described here.

## Calibration of Standard Tranche CDO Correlations

The implied correlations required to price tranche CDOs can be solved by calibration using a bootstrapping technique. The technique will resolve a surface in space defined by the CDO strike and contract maturity. For example, a CDX IG index has liquid tranches with strikes at 3%, 7%, 10%, 15% and 30% across maturities of 5 years, 7 years and 10 years.

It is possible to obtain the implied correlation for a 0 to 3% CDX maturing in 5 years by pricing iteratively until the solved theoretical price matches the observed market price, using credit spreads obtained from single-name CDS. After performing the same routine for the 0 to 7% CDX with the same term, the 3% to 7% tranche can be calibrated to obtain the implied correlation on the tranche.

After solving for the 5 year maturities, the base correlations can be solved for the 7 year and then the 10 year maturities.

## Pricing Bespoke Tranche CDOs

Bespoke tranche CDOs are less liquid than standard tranche CDOs, and their prices are not observed in the market.

Pricing of bespoke tranche CDOs relies on the pricing of liquid tranche CDOs. Once the pricing inputs and market price of the liquid tranche CDOs are available, the correlations of the issues in the portfolio of the bespoke tranche CDO may be solved.

The challenge to the pricing model is to adjust the inputs to the bespoke tranche CDO’s pricing model to appropriately account for the differences between the bespoke tranche and the standard contracts. There are three dimensions on which the bespoke tranche may differ: contract maturity date, credit portfolio, tranche attachment and detachment points.

### Bespoke Contract Maturity Date

Pricing of bespoke CDO tranches with non-standard maturity dates is the least complex customization of the index for which it is necessary to account. For example, suppose we need to price a 6 year IG CDX. We just need to interpolate correlations between the standard 5 year IG CDX and the 7 year IG CDX.

### Bespoke Portfolio from One Reference Annex

The standard CDX and iTraxx indices contain 125 constituents. First, let’s assume we are pricing a bespoke portfolio of constituents that are a subset of the 125 constituents in an IG reference annex. In this case, we usually try to price the bespoke CDO by identifying the terms of an standard CDX that has equivalent ‘riskiness.’ Riskiness is usually considered to be a different detachment point such that the bespoke equity tranche [0, Kbespoke] is equivalent to the standard index equity tranche [0, Kindex].

There are several methods available to map the bespoke tranche CDO’s riskiness to the standard index: no mapping, moneyness, probability, equity spread, and tranche loss proportion. In this paper, we will document the details of the tranche loss proportion method, which we have found produced the most reliable results.

### Tranche Loss Proportion: One Reference Annex

The tranche loss proportion (TLP) method defines the riskiness of the bespoke tranche CDO being equivalent to the riskiness of a standard tranche CDO when the following holds: the ratios of the expected loss on the tranche to the expected loss on the portfolio are equal for the bespoke and reference index.

\[

\frac{\mathbb{E}(L^{bespoke}[0,K^{bespoke}])}{\mathbb{E}(L^{bespoke}[0,100]})} = \frac{\mathbb{E}(L^{index}[0,K^{index}])}{\mathbb{E}(L^{index}[0,100]})}

\]

In this section we outline the method used to obtain the correlations required to price a bespoke CDO using the TLP technique, derived from more liquidly-traded tranche CDOs.

- The first part of this section demonstrates the approach to calculate the $TLP$ denominator: expected portfolio loss ($EPL$) of the bespoke and reference index CDOs.
- The second part demonstrates the approach to calculate the expected tranche loss ($ETL$) used as the $TLP$ numerator.

#### Calculating Expected Portfolio Loss

Findur can price the $EPL$ by overriding the CDO’s attachment and detachment points, leaving all else equal. In memory, the calibration routine sets the attachment point to 0 and the detachment point to 100, and the correlations to 0. The results can be replicated easily in the GUI

The $EPL^{bespoke}$ is the MTM result for the default leg.

A similar process is followed for the reference index deal. Make sure that the notional and maturity dates match the bespoke deal’s notional and maturity date. Then set the attachment point to 0, the detachment point to 100, and the correlation surface values to 0.

The $EPL^{index}$ is the MTM result for the default leg.

#### Calculating Expected Tranche Loss

The objective of the solver is to find the reference index deal’s strike that has riskiness equivalent to the bespoke tranche’s strike. That does not necessarily occur at the same strike because the underlying constituents will have different credit spreads.

We need an initial guess to use for the risk-equivalent strike of the reference index. 5% is a reasonable initial guess, so calculate the MTM result on the insurance leg of the reference index swap that attaches at 0 and detaches at 5%. This produces $ETL^{(index, 5)}$.

Calculate the $TLP^{(index, 5)}$ as the ratio of the $ETL^{(index, 5)}$ to the $EPL^{index}$.

Using that initial guess, we lookup the correlation of the reference index’s equity tranche for the same maturity. We use that correlation to price the bespoke tranche. In Findur, this can be done most easily by setting the value of all elements of the correlation surface to the reference index correlation. This removes the possibility of any effects of interpolation or extrapolation across the correlation surface.

We next calculate $ETL^{(bespoke, attach)}$ on an equity tranche of the bespoke portfolio using an attachment point of zero, and a detachment point equal to the bespoke contract’s *attachment* point. The MTM result for the default leg is $ETL^{(bespoke, attach)}$. For example, if the bespoke tranche contract confirmation attaches at 5.2% and detaches at 6.2%, we first obtain the MTM result on the bespoke portfolio for 0% to 5.2%. Use $ETL^{(bespoke, attach)}$ to calculate $TLP^{(bespoke, attach)}$.

Calculate the $TLP$ error as

\[

error = TLP^{(bespoke, attach)} – TLP^{(index, 5)}

\]

Repeat the process using a second guess for the equivalent strike of the reference index, for example 7%. Use the insurance leg MTM to calculate $TLP^{(index, 7)}$ as the ratio of the $ETL^{(index, 7)}$ to the $EPL^{index}$. Use the correlation for the reference index equity tranche that detaches at 7% to reprice the bespoke tranche and obtain $TLP^{(bespoke, attach)}$.

Calculate the $TLP$ error as

\[

error = TLP^{(bespoke, attach)} – TLP^{(index, 7)}

\]

Use an interpolation method, such as Newton-Raphson, to identify a third guess based on the previous reference index detachment point guesses (5% and 7%) and the resulting errors. Repeat the $TLP$ error calculations until the error is smaller than a threshold value, $epsilon$.

When the residual error is close enough to zero, the routine has solved for the reference index equivalent strike that equates the ratio of the reference index tranche loss to the reference index expected portfolio loss with the same calculation for the bespoke tranche’s *attachment* point.

The routine should be repeated to solve for the reference index equivalent strike that matches the bespoke tranche’s *detachment* point. In one example when this solution was run in production, the bespoke tranche that attaches at 5.2% and detaches at 6.2%, the reference index equivalent riskiness, when using the TLP methodology, attaches at 6.195% and detaches at 7.224%.

When the routine is complete, the solver has identified the correlations of the equivalent strikes of the reference index. The two correlations {equivalent attachment point, equivalent detachment point} should be saved to the bespoke deal’s correlation surface against the bespoke deal’s attachment and detachment lookup keys.

### Tranche Loss Proportion: Two Reference Annexes

In the previous section, we assumed the bespoke portfolio included constituents that were a subset of the 125 credits in one standard reference annex. In this section, we extend the example to a common situation where the bespoke index includes a mix of names from different regions, similar to a synthesis of North American CDX constituents and European iTraxx constituents.

It is customary to consider the correlation of the bespoke tranche to be related to the correlations for a standard CDX IG index, and a standard iTraxx index. We start by assuming a weight criterion between the reference indices. For example, we assume that the bespoke portfolio is approximately represented by a synthetic portfolio of 57% CDX and 43% iTraxx. This may be implemented in Findur by creating a composite ‘volatility’ definition to store the correlations. The composite definition has the standard CDX and iTraxx correlations as parents, using a formula of $0.57*CDX + 0.43*iTraxx$.

Just as before, we use a similar approach to price the bespoke CDO by identifying the terms of standard CDX and iTraxx deals that have equivalent ‘riskiness’ where riskiness is defined based on matching the expected TLP of the reference index and bespoke CDO.

The pricing routine uses a similar iterative method that was described in the previous section. A difference arises because it is necessary to identify the equivalent strikes on two reference indices, doubling the number of iterations.

Perform all of the steps described in the previous section to identify the equivalent attachment and detachment points for the CDX reference index. When it is necessary to calculate $TLP^{(bespoke, attachment)}$, be sure to apply the correlation used to price the CDX index deal to the iTraxx correlation surface. This is necessary because the bespoke deal, in memory, uses a composite correlation surface that has parents that include both the CDX and iTraxx correlations.

When the routine is complete, the solver has identified the correlations of the equivalent strikes of the CDX and iTraxx indices. These four correlations should be saved to the bespoke deal’s composite correlation surface’s parents.

\begin{equation}

\begin{matrix}

(\emph{CDX parent, equivalent attachment point}) \\

(\emph{CDX parent, equivalent detachment point})

\end{matrix}

\end{equation}

\begin{equation}

\begin{matrix}

(\emph{iTraxx parent, equivalent attachment point}) \\

(\emph{iTraxx parent, equivalent detachment point})

\end{matrix}

\end{equation}

## Results

A client that implemented this calibration routine in Findur had acceptance criteria defined based on spread delta residual to a legacy system pricing and to an external consultant’s results. At the time, CDOs were trading deep out of the money to pre-financial crisis levels (the client sold protection under the CDOs).

Implementation of the TLP calibration routine described here resulted in pricing tight enough to satisfy these acceptance criteria and complete the project deliverable.

## Additional Reading

Additional details about mapping algorithms for bespoke tranche CDOs are available in these papers.

- Baheti, P and Morgan S. (2007), ‘Base Correlation Mapping,’
*Lehman Brothers Quantitative Credit Research Quarterly*, March 2007. - Li, C (2008), ‘Pricing of Bespoke CDOs‘, MSc Thesis Stochastics and Financial Mathematics
- Prampolini, A and Dinnis, M (2009), ‘CDO Mapping with Stochastic Recovery,’ working paper, HSH Nordbank, available on www.defaultrisk.com