Credit risk pricing in a consumption‐based equilibrium framework with incomplete accounting information

We present a consumption‐based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where the default time is modeled as the first hitting time of a default boundary and bond investors have imperfect/partial information about the firm value. The imperfect information is generated by the underlying observed state variables and a noisy observation process of the firm value. In addition, the consumption, the volatility, and the firm value process are modeled to follow affine diffusion processes. Using the EZ equilibrium solution as the pricing kernel, we provide an equivalent pricing measure to compute the prices of financial derivatives as discounted values of the future payoffs given the incomplete information. The price of a zero‐coupon bond is represented in terms of the solutions of a stochastic partial differential equation (SPDE) and a deterministic PDE; the self‐contained proofs are provided for both this representation and the well‐posedness of the involved SPDE. Furthermore, this SPDE is numerically solved, which yields some insights into the relationship between the structure of the yield spreads and the model parameters.

[1]  Junchi Ma.  Credit Risk Pricing based on Epstein-Zin Preference , 2019 .

[2]  G. Karniadakis,et al.  Numerical Methods for Stochastic Partial Differential Equations with White Noise , 2018 .

[3]  Jinniao Qiu,et al.  Maximum Principle for Quasi-linear Reflected Backward SPDEs , 2016, 1604.02425.

[4]  Shanjian Tang,et al.  Strong solution of backward stochastic partial differential equations in C2 domains , 2011, Probability Theory and Related Fields.

[5]  Shanjian Tang,et al.  Strong solution of backward stochastic partial differential equations in C2 domains , 2010, 1006.2185.

[6]  Shanjian Tang,et al.  Lp Theory for Super-Parabolic Backward Stochastic Partial Differential Equations in the Whole Space , 2010, 1006.1171.

[7]  L. Evans,et al.  Partial Differential Equations , 1941 .

[8]  Ravi Bansal,et al.  An Empirical Evaluation of the Long-Run Risks Model for Asset Prices , 2009 .

[9]  P. Kloeden,et al.  The Numerical Approximation of Stochastic Partial Differential Equations , 2009 .

[10]  Xin Guo,et al.  Credit Risk Models with Incomplete Information , 2009, Math. Oper. Res..

[11]  Harjoat S. Bhamra,et al.  The Levered Equity Risk Premium and Credit Spreads: A Unified Framework , 2009 .

[12]  Ivan Shaliastovich,et al.  AN EQUILIBRIUM GUIDE TO DESIGNING AFFINE PRICING MODELS , 2008 .

[13]  Long Chen,et al.  On the Relation between the Credit Spread Puzzle and the Equity Premium Puzzle , 2008 .

[14]  M. Röckner,et al.  A Concise Course on Stochastic Partial Differential Equations , 2007 .

[15]  Philip Protter,et al.  Information reduction via level crossings in a credit risk model , 2007, Finance Stochastics.

[16]  M. Yor,et al.  Equivalent and absolutely continuous measure changes for jump-diffusion processes , 2005, math/0508450.

[17]  D. Dufresne.  The integrated square-root process , 2001 .

[18]  D. Duffie,et al.  Term Structures of Credit Spreads with Incomplete Accounting Information , 2001, World Scientific Reference on Contingent Claims Analysis in Corporate Finance.

[19]  Ravi Bansal,et al.  Risks for the Long Run: A Potential Resolution of Asset Pricing Puzzles , 2000 .

[20]  Robert J. Elliott,et al.  On Models of Default Risk , 2000 .

[21]  D. Duffie,et al.  Modeling term structures of defaultable bonds , 1999 .

[22]  David Lando,et al.  On cox processes and credit risky securities , 1998 .

[23]  E. Dynkin.  Semilinear Parabolic Equations, Diffusions, and Superdiffusions , 1998 .

[24]  R. Jarrow,et al.  Pricing Derivatives on Financial Securities Subject to Credit Risk , 1995 .

[25]  E. Fama,et al.  Common risk factors in the returns on stocks and bonds , 1993 .

[26]  Larry G. Epstein,et al.  Stochastic differential utility , 1992 .

[27]  Xun Yu Zhou,et al.  A duality analysis on stochastic partial differential equations , 1992 .

[28]  E. Fama,et al.  BUSINESS CONDITIONS AND EXPECTED RETURNS ON STOCKS AND BONDS , 1989 .

[29]  Larry G. Epstein,et al.  Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework , 1989 .

[30]  Evan L. Porteus,et al.  Temporal Resolution of Uncertainty and Dynamic Choice Theory , 1978 .

[31]  M. Zakai.  On the optimal filtering of diffusion processes , 1969 .

[32]  H. Kunita.  Some extensions of Ito's formula , 1981 .