Pricing and integration of credit default swap index tranches

• Author: Andrew Carverhill,Dan Luo
• Published date: 01 April 2020
• Date: 01 April 2020
• DOI: http://doi.org/10.1002/fut.22082

J Futures Markets. 2020;40:503–526. wileyonlinelibrary.com/journal/fut © 2019 Wiley Periodicals, Inc.

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503

Received: 25 February 2019

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Accepted: 21 November 2019

DOI: 10.1002/fut.22082

RESEARCH ARTICLE

Pricing and integration of credit default swap index

tranches

Andrew Carverhill

1

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Dan Luo

2,3

1

Department of Economics and Finance,

City University of Hong Kong, Hong

Kong, China

2

School of Finance, Shanghai University

of Finance and Economics, Shanghai,

China

3

Shanghai Key Laboratory of Financial

Information Technology, Shanghai, China

Correspondence

Dan Luo, School of Finance, Shanghai

University of Finance and Economics, 777

Guoding Road, 200433 Shanghai, China.

Email: luo.dan@mail.shufe.edu.cn

Funding information

Natural Science Foundation of China,

Grant/Award Number: 71972123/G0205

Abstract

This paper first designs an efficient procedure to value Credit Default Swap

Index tranches using an intensity‐based model. The tranche spreads are

effectively explained by a three‐factor version of this model, both before and

during the financial crisis of 2008. We then construct tradable tranche portfolios

to track the intensity factors and compare the pricing of the tranches with

equities and their derivatives. Our results show that the senior tranche spreads

do not offer returns in excess of the common risk compensations in the equity

and derivatives markets, while the junior tranche is not spanned by these

standard factors.

KEYWORDS

collateralized debt obligation tranches, credit default swap index, default intensity,

market integration, option smirk

JEL CLASSIFICATION

G11; G12; G13

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INTRODUCTION

The Financial Crisis of 2008 has been widely attributed to the explosion in the use of credit derivatives. Can such

instruments be accurately priced? Have the actual market prices or associated spread quotations been reasonable?

These questions have been intensively studied, notably in the contributions of Collin‐Dufresne, Goldstein, and Yang

(CDGY, 2012) and Coval, Jurek, and Stafford (CJS, 2009a). CJS make the point that the senior collateralized debt

obligation (CDO) tranches and the S&P 500 index (SPX) options are exposed to the same market‐wide catastrophic

default risk. However, when they price the credit default swap index (CDX) tranches using the pricing kernel inferred

from the index options, they find that spreads on the senior CDX tranches are too low to compensate the systematic risk

they are bearing.

1

CDGY argue that CJS’s copula type CDO pricing model is not flexible enough to capture firms’joint

default dynamics. They provide a fully dynamic and integrated model for both the market and each individual firm.

Fitting this model to the CDX and index options data, they find no evidence of market segmentation either before or

during the financial crisis period.

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Given the complexity of these structural credit products and the debate on the

pricing they receive in the market, further analysis using different approaches may bring new insights into the

valuation of those products from a perspective of their integration with other sectors of the financial market, especially

the equity and equity derivatives which are subject to the same firm default risk.

1

See CJS table VI. Except for the junior 0–3 tranche, the average model implied spreads are 2–5 times as large as actual spreads for all other tranches.

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Li and Zhao (2012) also study the modeling and efficiency of CDX tranche prices. These authors directly modify the model of CJS, so that it can price the price the CDX tranches and index options

accurately and simultaneously.

In this paper, we first implement an efficient tranche pricing procedure, based on the reduced‐form top‐down credit

risk model of Longstaff and Rajan (LR, 2008). This model can encompass a number of factors representing default

intensities, and as do LR, we implement it for three factors. We also design an efficient procedure to evaluate the model,

exploiting its affine structure, as in Duffie, Pan, and Singleton (DPS, 2000). We fit the model to the well‐established

spread quotations of the tranches associated with the standard CDX credit index, using the Monte Carlo Expectation

Maximization algorithm of Duffie, Eckner, Horel, and Saita (DEHS, 2009). This algorithm is a hybrid. It efficiently

combines a maximum likelihood estimation of the model parameters and a Bayesian filtration of the latent default

intensity factors. However, LR and CDGY calibrate their models by minimizing the pricing errors hence the dynamic

evolution of the risk factors in their models are left out in the estimation. We then provide a method to form tradable

tranche portfolios to represent the intensity factors, and relate them to factor‐mimicking portfolios established in the

equity derivatives and equities markets. We differ from CJS in finding that the senior tranche spreads are consistently

priced across the markets. We also differ from CDGY since the junior tranche offers returns in excess of the standard

risk factors in the equity and equity derivatives markets.

The LR Model that we implement directly models losses to the underlying portfolio of a CDO. Total loss on the

portfolio is assumed to be driven by a single or multiple risk factor(s). A three‐factor version of this model fits the actual

CDX tranche spreads to high precision, with pricing errors typically around several basis points. By contrast, the

bottom‐up model of Duffie and Gärneanu (2001), begins by modeling the dynamics of each underlying firm, and then

aggregates over the portfolio to characterize the joint default behavior. Default intensity of a firm, in this model, consists

of an idiosyncratic component, as well as broader sectoral, regional and global components, and default arrivals are

assumed to be independent, conditioned on the intensity processes. Feldhütter (2008) fits this model to the CDX

tranche spreads and concludes that the model exhibits too little variation in the senior tranches, although it matches the

average spreads well.

After fitting the LR Model, we then apply it to investigating the CDX market itself. LR note that the three default

intensity factors can be interpreted as representing single firm default, default of a number of firms together, and

market‐wide catastrophic default, respectively, and the CDO spreads give the prices of insuring against such types of

default. Have these prices been reasonable, in the crisis period, and is there evidence that the CDX market has been

segmented away from the rest of the capital market? In particular, have the writers of these instruments set the prices in

their own favor?

We address this issue raised in CJS and CDGY, however, following a less structural approach, similar to that of Fama

and French (1993, 1996). Our approach is to construct portfolios representing our three CDO factors, and to see, using

OLS regression, whether the returns of these can be explained/hedged using established market factors. Successful

hedging corresponds to a significant regression coefficient. However, if the residual return is significant, revealed by the

t‐statistic on the unit constant in the regression, then we conclude that the market factors have not accounted for all the

priced sources of risk in the CDO spreads.

We first test our CDO factor portfolios against three factors extracted using principal components analysis (PCA),

from the SPX option returns. We refer to these as the market, the volatility, and the smirk factors, and they represent an

overwhelming fraction of the option price dynamics. The market factor here is essentially just the market index itself,

and the volatility factor is essentially the same as that in Bakshi and Kapadia (2003) and Coval and Shumway (2001).

The smirk factor broadly represents the return from being short OTM puts and long OTM calls, and it can be

interpreted in terms of the price of insurance against market‐wide losses. This factor was analyzed for the S&P 500

futures options by Carverhill, Cheuk, and Dyrting (CCD, 2009). CJS are using the options smirk to infer the price of

market‐catastrophe risk in their CDO pricing procedure.

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We then test our CDO factor portfolios against the Fama–French equity portfolios, representing the market, size

effect (small minus big [SMB]), and book‐to‐market [BTM] effect (high minus low [HML]).

The results are as follows. Our CDO factor portfolios experience high returns at the onset of the Financial Crisis in

September 2007, and the junior tranche also has a high return in May 2005, coinciding with the GM crisis. Testing the

factors against the option factors, we see that they are all negatively exposed to the market factor, as expected. Also,

consistent with CJS, the senior tranche is exposed to the smirk factor. Having hedged all the options factors, there is a

residual Sharpe ratio for the first (junior tranche) factor, but not the other factors. Our results suggest that the senior

3

The options smirk is the stylized fact that out of the money (OTM) puts have higher implied volatility than OTM calls. Intuitively, this reflects demand for insurance against falls in the equity index,

that OTM puts can provide.

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CARVERHILL AND LUO