Building on recent work incorporating recovery risk into structural models by Cohen & Costanzino (2015), we consider the Black-Cox model with an added recovery risk driver. The recovery risk driver arises naturally in the context of imperfect information implicit in the structural framework. This leads to a two-factor structural model we call the Stochastic Recovery Black-Cox model, whereby the asset risk driver A_t defines the default trigger and the recovery risk driver R_t defines the amount recovered in the event of default. We then price zero-coupon bonds and credit default swaps under the Stochastic Recovery Black-Cox model. Finally, we compare our results with the classic Black-Cox model, give explicit expressions for the recovery risk premium in the Stochastic Recovery Black-Cox model, and detail how the introduction of separate but correlated risk drivers leads to a decoupling of the default and recovery risk premiums in the credit spread. We conclude this work by computing the effect of adding coupons that are paid continuously until default, and price perpetual (consol bonds) in our two-factor firm value model, extending calculations in the seminal paper by Leland (1994).

In this work we incorporate recovery risk into Merton's original credit risk model by introducing a separate risk driver for the recovery process and rationalize this new model within a "partial information" perspective. We show that while adding the recovery risk driver has no impact on probabilities of default (PD), it does have an impact on loss given default (LGD), and on quantities that depend on LGD such as credit prices and spreads. In fact, the addition of recovery risk allows for a mechanism to increase credit spreads, and therefore may account for some of the bond mispricings typical when using Merton's model. Finally, using the new model we price both bonds and CDS, and explicitly compute the price of recovery risk.

In this work, we introduce a general framework for incorporating stochastic recovery into structural models. The framework extends the approach to recovery modeling developed in Cohen and Costanzino (2015, 2017) and provides for a systematic way to include different recovery processes into a structural credit model. The key observation is a connection between the partial information gap between firm manager and the market that is captured via a distortion of the probability of default. This last feature is computed by what is essentially a Girsanov transformation and reflects untangling of the recovery process from the default probability. Our framework can be thought of as an extension of Ishizaka and Takaoka (2003) and, in the same spirit of their work, we provide several examples of the framework including bounded recovery and a jump-to-zero model. One of the nice features of our framework is that, given prices from any one-factor structural model, we provide a systematic way to compute corresponding prices with stochastic recovery. The framework also provides a way to analyze correlation between Probability of Default (PD) and Loss Given Default (LGD), and term structure of recovery rates.

In this work we propose and analyze a model which addresses the pulsing behavior of sellers in an online auction (store). This pulsing behavior is observed when sellers switch between advertising and processing states. We assert that a seller switches her state in order to maximize her profit, and further that this switch can be identified through the seller’s reputation. We show that for each seller there is an optimal reputation, i.e., the reputation at which the seller should switch her state in order to maximize her total profit. We design a stochastic behavioral model for an online seller, which incorporates the dynamics of resource allocation and reputation. The design of the model is optimized by using a stochastic advertising model from [1] and used effectively in the Stochastic Optimal Control of Advertising [2]. This model of reputation is combined with the effect of online reputation on sales price empirically verified in [3]. We derive the Hamilton-Jacobi-Bellman (HJB) differential equation, whose solution relates optimal wealth level to a seller’s reputation. We formulate both a full model, as well as a reduced model with fewer parameters, both of which have the same qualitative description of the optimal seller behavior. Coincidentally, the reduced model has a closed form analytical solution that we construct.

In this note, we revisit the derivation of Merton’s original credit model and remove the assumption that recovery is a constant fraction of assets. We do so by introducing a separate risk driver for the recovery process. The recovery process is correlated to the asset process so as to have the correct PD-LGD dynamics empirically observed in the literature. We show that while adding the recovery risk driver has no impact on probabilities of default, it does have an impact on LGD, and on quantities that depend on LGD such as bond prices and spreads. In fact, the addition of recovery risk allows for a mechanism to increase credit spreads, and therefore may account for some of the bond mispricings typical when using Merton’s model.