Ebook Maximum Likelihood Estimation of Structural Credit Spread Models - Deterministic and Stochastic Interest Rates
In Merton (1974), a pricing model for corporate liabilities was developed using an option valuation approach. In his setting, the unobserved asset value of the firm is governed by a geometric Brownian motion. Subsequently, many variants of this model have been proposed in the literature. Merton’s and its extended models are typically referred to as structural credit spread (or risky bond) models. Examples abound; Longstaff and Schwartz (1995), Madan and Unal (1998) and Collin-Dufresne and Goldstein (2001). This paper develops a maximum likelihood estimation method for this class of models.
As pointed out in Jarrow and Turnbull (2000) among others, there are several limitations associated with the implementation of structural credit spread models. First, the asset value is an unobserved quantity. This in turn creates problems with the estimation of the various required parameters such as the drift and volatility of the asset value process and the correlation among different asset value processes. Other limitations are related to default threshold and stochastic interest rates because the parameter values specific to these features are also unobserved.
In the academic literature, two approaches have been proposed for dealing with the estimation problem when the underlying asset value is unobserved. The first approach, which we will refer to as the implicit estimation method, uses some observed quantities and the corresponding restrictions derived from the theoretical model to extract point estimates for the model parameter(s) and the unobserved asset value. Take the univariate case of Merton’s (1974) model as an example.
The implicit estimation method relies on two equations: one relating the equity value to the asset value and the other relating the equity volatility to the asset volatility. The two-equation system can then be solved for the two unknown variables: the asset value and volatility. The implicit estimation method has been adopted by Ronn and Verma (1984) to implement the deposit insurance pricing model of Merton (1978) and by Jones, Mason and Rosenfeld (1984) to conduct an empirical study of Merton’s (1974) risky bond pricing model. A three-equation extension of the implicit estimation method was used in Duan, Moreau and Sealey (1995) to implement their deposit insurance model with stochastic interest rate where the third equation relates the equity duration to the asset duration.
The second estimation approach was proposed by Duan (1994, 2000). In these papers, a likelihood function based on the observed equity values is derived by employing the transformed data principle in conjunction with the equity pricing equation. With the likelihood function in place, maximum likelihood estimation and statistical inference become straightforward. The maximum likelihood method was applied to Merton’s (1978) deposit insurance pricing model in Duan (1994), Duan and Yu (1994), and Laeven (2002). Later, Duan and Simonato (2002) extended the method to deposit insurance pricing under stochastic interest rate. For credit risk, the estimation method has been applied to a strategic corporate bond pricing model by Ericsson and Reneby (2001).
Theoretically, the maximum likelihood estimation method has several advantages relative to the implicit estimation method. First, the maximum likelihood method provides an estimate of the drift of the unobserved asset value process under the physical probability measure. This can in turn be used to obtain an estimate of the default probability of the firm. Such an estimate is not available within the context of the implicit estimation method since the theoretical equity pricing equation typically does not contain the drift of the asset value process under the physical probability measure.
The second advantage is associated with the asymptotic properties of the maximum likelihood estimator such as consistency and asymptotic normality, which in turn allows for statistical inference to assess the quality of parameter estimates and/or perform testing on the hypotheses of interest. In contrast, consistency is unattainable with the implicit estimation method because it erroneously forces a stochastic variable to be a constant (see Duan (1994)). Consequently, no reliable statistical inference using the implicit estimation method can be expected.
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