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Article Excerpt
1. Introduction
Several financial securities depend on more than just one category of risk. Prominent among these are corporate bonds (which depend on interest-rate risk and on credit risk of the issuing firm) and convertible bonds (which depend, in addition, on equity risk). In this paper, we develop and implement a model for the pricing of securities whose values may depend on one or more of three sources of risk: equity risk, credit risk, and interest-rate risk.
Our framework is based on generalizing the reduced-form approach to credit risk (Duffie and Singleton 1999, Madan and Unal 2000) to include a process for equity. The typical reduced-form model involves two components, one describing the evolution of (riskless) interest rates, and the other, an intensity process that captures the likelihood of firm default; equity is not modeled explicitly. But any default process for a company's debt must obviously also apply to that company's equity. That is, when debt is in default, equity must also go into some post-default value. Motivated by this, we knit an equity process into a reduced-form model in an arbitrage-free manner; equity in the integrated model now follows a jump-to-default process, i.e., it gets absorbed at zero when a default happens. (1) The resulting framework captures simultaneously the three sources of risk mentioned above, and can be calibrated to market data to extract default probabilities or price hybrid securities.
Although our model is anchored in the reduced-form approach, the specifics draw on insights gained from the structural approach to credit risk (cf. Merton 1974, Black and Cox 1976, and others). Our starting point, the idea that default is associated with an absorbing value for equity, is itself borrowed from structural models. The process we posit for the evolution of equity prices prior to default--a constant elasticity of variance (CEV) process--is also motivated by structural models. An important characteristic of the Merton (1974) model is its generation of the so-called leverage effect, a negative relationship between equity prices and equity volatility. The leverage effect has also been documented empirically (e.g., Christie 1982). The CEV specification for equity prices generates a leverage effect in our reduced-form setting. Finally, we take the default intensity in our model to vary inversely with equity prices (and, therefore, directly with equity volatility). This specification is also motivated by the existence of a similar relation in the Merton (1974) model between default likelihood, equity prices, and equity volatility.
Our final framework, then, involves the following components. We have a CEV model describing the evolution of equity prices prior to default, an intensity process for default, and a riskless interest-rate model (for which purpose we use the Heath-Jarrow-Morton 1990 (HJM) model, although any other interest-rate model could be used). The result is a single parsimonious model accounting for correlations that combines the three major sources of risk.
We implement the model in a discrete-time setting, using the Nelson and Ramaswamy (1990) approach to discretize the CEV model. Rather than specifying an exogenous process for the default probability, we make it a dynamic function of both equity and interest-rate information. This enables us to derive default probabilities as endogenous functions of the information on the lattice, jointly calibrated to equity prices and default spreads. As a consequence, default information in the model is extracted from both equity- and debt-market information rather than from just debt-market information (as in reduced-form credit-risk models) or from just equity-market information (as in structural credit-risk models). This allows valuation, in a single consistent framework, of hybrid debt-equity securities such as convertible bonds that are vulnerable to default, as well as of derivatives on interest rates, equity, and credit. Our model can also serve as a basis for valuing credit portfolios where correlated default is an important source of risk. Finally, the model enables the extraction of credit-risk premia.
Our framework has several antecedents and points of reference in the literature. We have already mentioned the connection to both reduced-form and structural models. Jump-to-default equity models, in which equity gets absorbed at zero following a default, have also been examined in Davis and Lischka (1999), Carayannopoulos and Kalimipalli (2003), Campi et al. (2005), Carr and Linetsky (2006), and Le (2006). (2) The first two papers use the Black-Scholes (1973) model for the equity-price process prior to default which is a special case of the CEV model we use, and which does not admit the leverage effect; the other three, like ours, use the CEV process. (3) The specification of the default intensity process in Davis and Lischka (1999) is somewhat more restrictive than ours; their default intensity is perfectly correlated with the equity process, whereas we allow it to depend on both equity returns and interest rates and other information as well. Carayannopoulos and Kalimipalli (2003) use a default intensity specification similar to ours but their model does not allow for stochastic interest rates.
Campi et al. (2005) assume a constant intensity process for default; they do not allow for stochastic interest rates either. Le (2006) and Carr and Linetsky (2006) endogenize the default probability in a manner similar to our paper. Le calibrates the model to option prices to recover default probabilities in the model. Then he applies these default probabilities to credit spreads to identify implied recovery rates. Carr and Linetsky, working in a continuous-time setting but without interest-rate risk, are able to provide explicit closed-form solutions for survival probabilities, credit default swaps (CDSs) spreads, and European option prices.
Also related to our paper are the reduced-form models in Schonbucher (1998, 2002) and Das and Sundaram (2000), which study "defaultable HJM" models. These are HJM models with a default process tacked on. Our model generalizes these to include equity processes as well. In particular, the Das and Sundaram (2000) model results as a special case of our framework if the equity process is switched off. Our framework may also be viewed as a generalization of Amin and Bodurtha (1995) (see also Brenner et al. 1987). The Amin-Bodurtha model combines interest-rate risk and equity risk (in the form of a Black-Scholes model) but does not incorporate credit risk. Because there is no default, equity in their model is necessarily infinitely lived and never gets "absorbed" in a postdefault value. Other frameworks are nested within our model too. For example, if the equity and hazard-rate processes are switched off, we obtain the HJM model, whereas if the interest-rate and hazard-rate processes are switched off, we obtain a discrete-time CEV tree, as described by Nelson and Ramaswamy (1990).
Our lattice design allows recombination, making the implementation of the model simple and efficient; indeed, the model is fully implementable on a spreadsheet. Unlike many earlier models, we are able to (a) price derivatives on equity and interest rates with default risk; (b) extract probabilities of default (PD) endogenously in the model; (c) provide for the risk-neutral simulation of correlated default risk in a manner consistent with no arbitrage and consistent with equity correlations (which we believe, has not been undertaken in any model so far); and (d) extract credit risk premia.
The rest of this paper proceeds as follows. In [section]2 we develop the pricing lattice in the state variables of the model in a manner that allows for additional structure to accommodate default risk. Section 3 deals with implementation issues, including a discussion of how default swaps may be used to calibrate the model for subsequent use. We show that the model may generate a wide range of spread curve shapes. Empirical calibration to markets is undertaken to evidence the ease of implementation. This section also explores the impact of default risk on embedded options within classic bond structures. Section 4 applies the model to the extraction of credit risk premia and uses data from Dow Jones CDX index firms to examine the principal components of these premia. Finally, an analysis of the model application to correlated default products is provided. Section 5 concludes by summarizing the economic and technical benefits of the model.
2. The Model
As we have noted, the motivation for our model is simple. If the default process for a company's debt is described by a hazard rate [lambda] (as in the standard reduced-form model...
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