This paper empirically tests five structural models of corporate bond pricing: those of Merton (1974), Geske (1977), Leland and Toft (1996), Longsta# and Schwartz (1995), and Collin-Dufresne and Goldstein (2001). We implement the models using a sample of 182 bond prices from firms with simple capital structures during the period 1986-1997. The conventional wisdom is that structural models do not generate spreads as high as those seen in the bond market, and true to expectations we find that the predicted spreads in our implementation of the Merton model are too low. However, most of the other structural models predict spreads that are too high on average. Nevertheless, accuracy is a problem, as the newer models tend to severely overstate the credit risk of firms with high leverage or volatility and yet su#er from a spread underprediction problem with safer bonds. The Leland and Toft model is an exception in that it overpredicts spreads on most bonds, particularly those with high coupons. More accurate structural models must avoid features that increase the credit risk on the riskier bonds while scarcely a#ecting the spreads of the safest bonds.

This paper is a critical survey of models designed for pricing fixed income securities and their associated term structures of market yields. Our primary focus is on the interplay between the theoretical specification of dynamic term structure models and their empirical fit to historical changes in the shapes of yield curves. We begin by overviewing the dynamic term structure models that have been fit to treasury or swap yield curves and in which the risk factors follow diffusions, jump-diffusion, or have “switching regimes. ” Then the goodness-of-fits of these models are assessed relative to their abilities to: (i) match linear projections of changes in yields onto the slope of the yield curve; (ii) match the persistence of conditional volatilities, and the shapes of term structures of unconditional volatilities, of yields; and (iii) to reliably price caps, swaptions, and other fixed-income derivatives. For the case of defaultable securities we explore the relative fits to historical yield spreads. 1

This paper examines the default probabilities (DPs) that are generated by alternative “structural” models of risky corporate bonds. 1 We have three objectives: (i) To distinguish “exogenous default ” from “endogenous default ” models

ABSTRACT. The model introduced in this article is designed to provide a consistent representation for both the real-world and pricing measures for the credit process. We find that good agreement with historical and market data can be achieved across all credit ratings simultaneously. The model is characterized by an underlying stochastic process that represents credit quality and default events are associated to barrier crossings. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift and implied recovery rates are estimated to consistently match the average spread curves corresponding to all the various ratings. 1.

We propose a two-sided jump model for credit risk by extending the Leland-Toft endogenous default model based on the geometric Brownian motion. The model shows that jump risk and endogenous default can have significant impacts on credit spreads, optimal capital structure, and implied volatility of equity options: (1) The jump and endogenous default can produce a variety of non-zero credit spreads, including upward, humped, and downward shapes; interesting enough, the model can even produce, consistent with empirical findings, upward credit spreads for speculative grade bonds. (2) The jump risk leads to much lower optimal debt/equity ratio; in fact, with jump risk, highly risky firms tend to have very little debt. (3) The two-sided jumps lead to a variety of shapes for the implied volatility of equity options, even for long maturity options; and although in generel credit spreads and implied volatility tend to move in the same direction under exogenous default models, but this may not be true in presence of endogenous default and jumps. In terms of mathematical contribution, we give a proof of a version of the “smooth fitting ” principle for the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model. 1

We use information contained in yield spreads to recover investors ’ ex ante required rates of return on corporate securities, and then use these ex ante returns to study the pricing of risky assets. Differently from the standard approach, our asset pricing tests do not rely on the use of ex post average equity returns as proxies for expected equity returns. We find that: (i) the market beta plays a significant role in the cross-section of expected equity returns, and its role persists even after size and book-to-market factors are accounted for; (ii) the risk premia associated with size and book-to-market are positive, significant, and countercyclical; and (iii) there is little evidence on positive momentum profits. We also find that systematic risk, as captured by common equity factors, is the main driver of the cross-sectional variation in bond yield spreads. JEL Classification: G12, E44

A rm issues a convertible bond. At each subsequent time, the bondholder must decide whether to continue to hold the bond, thereby collecting coupons, or to convert it to stock. The rm may at any time call the bond. Because calls and conversions usually occur far from maturity, we model this situation with a perpetual convertible bond, i.e, a convertible coupon-paying bond without maturity. This model admits a relatively simple solution, under which the value of the perpetual convertible bond, as a function of the value of the underlying rm, is determined by a nonlinear ordinary dierential equation.

The primary purpose of this paper is to introduce a new methodology for measuring the implied market value of aggregate market debt and analyzing the resultant risk effects of stochastic market leverage on asset prices in the economy. To our knowledge this is the first paper to attempt to directly isolate and analyze the effects of the implied market value of aggregate market debt on equity index option prices. We present what we believe are the first implied measures of the market value of aggregate corporate debt. We derive the implied market value of aggregate debt from option theory using only two contemporaneous market prices for the index price level and index option price. We demonstrate that the inclusion of the implied market value of aggregate debt results in significant statistical and economic improvements in the pricing of S&P 500 index put options relative to more complex models which omit leverage. JEL Classification: G12

Some modified structural and reduced-form models of credit risk implicitly assume that the market has less information than managers who declare default on their outstanding debt. As a result the announcement of default or disclosure of information that indicates a firm is in distress comes as a “surprise ” to the market. In this paper, we study the extent to which private information is revealed about a firm when it announces information indicating distress. The presence of this private information can be inferred from the extent to which investors can earn abnormal returns on bonds or equities issued by firms announcing distress or default. We analyze how much of the information revealed through the declaration of a credit event is publicly available before a specific announcement of credit difficulties. Using default probabilities supplied by Moody’s KMV (MKMV), known as the Expected Default Frequency�or the EDF�credit measure, we model market expectations regarding the firm’s likelihood of default. We then measure the impact of information revealed through an adverse credit event conditional on this expectation. We find that conditioning on EDF credit measures, only 11 % of the distressed firms’ equities and 18 % of the distressed bonds (belonging to 25 % of the distressed firms) display a

We discuss the valuation of credit derivatives in extreme regimes such as when the time-tomaturity is short, or when payoff is contingent upon a large number of defaults, as with senior tranches of collateralized debt obligations. In these cases, risk aversion may play an important role, especially when there is little liquidity, and utility indifference valuation may apply. Specifically, we analyze how short-term yield spreads from defaultable bonds in a structural model may be raised due to investor risk aversion. 1