On nonlinear ill-posed inverse problems with applications to pricing of defaultable bonds and option pricing

This paper considers the estimation of an unknown function h that can be characterized as a solution to a nonlinear operator equation mapping between two infinite dimensional Hilbert spaces. The nonlinear operator is unknown but can be consistently estimated, and its inverse is discontinuous, rendering the problem ill-posed. We establish the consistency for the class of estimators that are regularized using general lower semicompact penalty functions. We derive the optimal convergence rates of the estimators under the Hilbert scale norms. We apply our results to two important problems in economics and finance: (1) estimating the parameters of the pricing kernel of defaultable bonds; (2) recovering the volatility surface implied by option prices allowing for measurement error in the option prices and numerical error in the computation of the operator.

PRICING OF MULTIPLE DEFAULTABLE BOND

In this paper a generalized defaultable bond pricing formula is derived by assuming that there exists a defaultable forward rate term structure and that firms in the economy interact when default occurs.Generally,The risk-neutral default intensity λ Q is not equal to the empirical or actual default intensity λ.This paper proves that multiple default intensities are invariant under equivalent martingale transformation,given a well-diversified portfolio corresponding to the defaultable bond.Thus one can directly apply default intensities and fractional losses empirically estimated to the evaluation of defaultable bonds or contingent claims.

A Study on The Timeliness of Credit Rating on Bond Defaults--Evidence from Chinese Bond Market

With the increase of China’s bond issuance and slowdown of the economic growth,the potential credit risks such as bond default in the bond market are gradually emerging.The frequent occurrence of bond defaults and the problem of false credit ratings make bond investors and market participants more cautious about the credit ratings issued by rating agencies.Based on the default bonds from 2016 to 2019,this paper analyzes the adjustment of rating of defaulted bonds by rating agencies before default.It also compares the impact of both the regulatory events and the entrance of international agencies on timeless of credit ratings on default bonds.At the same time,the divergence of rating timeliness between different rating agencies is compared.The research shows that after the unified supervision of regulators and the punishment of Dagong Global Credit Rating Co.Ltd in 2018,the timeliness of rating agencies'downgrading of defaulted bonds has increased significantly;Compared with other rating agencies,the timeliness of rating agencies owned by international rating agencies are better.

A Heston local-stochastic volatility model for optimal investment-reinsurance strategy with a defaultable bond in an ambiguous environment

This study considers an optimal investment and reinsurance problem involving a defaultable security for an insurer in an ambiguous environment.In other words,the insurer is ambiguous about the insurance claim that is exponentially distributed with an uncertain rate parameter.The insurer can purchase proportional reinsurance and invest its wealth in three assets:a risk-free asset,a risky asset,the price process of which satisfies the Heston local-stochastic volatility model,and a defaultable corporate bond.For the optimal investment–reinsurance objective with a smooth ambiguity utility proposed by Klibanoff,P.,Marinacci,M.,and Mukerji,S.[A smooth model of decision making under ambiguity,Econometrica,2005,73(6):1849-1892],the equilibrium strategy is introduced and the extended Hamilton–Jacobi–Bellman equation is established through a stochastic control approach.However,the analytical solution of the strategy under the Heston local-stochastic volatility model cannot be obtained because of the complicated nonlinearity of the partial differential equation.In this study,we employ a perturbation method to derive an asymptotic solution for the post-and pre-default cases.In addition,we present a sensitivity analysis to explain the impact of model parameters on the equilibrium investment–reinsurance strategy.

Predicting Credit Bond Default with Deep Learning: Evidence from China

China’s credit bond market has rapidly expanded in recent years.However,since 2014,the number of credit bond defaults has been increasing rapidly,posing enormous potential risks to the stability of the financial market.This study proposed a deep learning approach to predict credit bond defaults in the Chinese market.A convolutional neural network(CNN)was selected as the classification model and to reduce the extreme imbalance between defaulted and non-defaulted bonds,and a generative adversarial network(GAN)was used as the oversampling model.Based on 31 financial and 20 non-financial indicators,we collected Wind data on all credit bonds issued and matured or defaulted from 2014 to 2021.The experimental results showed that our GAN+CNN approach had superior predictive performance with an area under the curve(AUC)of 0.9157 and precision of 0.8871 compared to previous research and other commonly used classification models-including the logistic regression,support vector machine,and fully connected neural network models-and oversampling techniques-including the synthetic minority oversampling technique(SMOTE)and Borderline SMOTE model.For one-year predictions,indicators of solvency,capital structure,and fundamental properties of bonds are proved to be the most important indicators.

Downside risk and defaultable bond returns

This paper analyzes the influence of downside risk on defaultable bond returns.By introducing a defaultable bond-trading model,we show that the decline in market risk tolerance and information accuracy leads to trading loss under downside conditions.Our empirical analysis indicates that downside risk can explain a large proportion of the variation in yield spreads and contains almost all valid information on liquidity risk.As the credit level decreases,the explanatory power of downside risk increases significantly.We also investigate the predictive power of downside risk in cross-sectional defaultable bond excess returns using a portfolio-level analysis and Fama-Mac Beth regressions.We find that downside risk is a strong and robust predictor for future bond returns.In addition,due to the higher proportion of abnormal transactions in the Chinese bond market,downside risk proxy semi-variance can better explain yield spreads and predict portfolio excess returns than the proxy value at risk.

Smooth-pasting property on reflected Lévy processes and its applications in credit risk modeling

We study the smooth-pasting property for a class of conditional expectations with reflected Levy process as underlying state process. A relationship between local times and regulators for the doubly reflected Levy process is established. As applications, we derive the analytic pricing formula for a zero-coupon defaultable bond when the default intensity （resp. the stochastic loss rate） is modeled as one-sided （resp. double-sided） reflected Levy processes. Finally, some numerical illustrations are provided.

Forecasting Chinese Corporate Bond Defaults: Comparative Study of Market- vs. Accounting-Based Models

This paper provides the first empirical study on bond defaults in the Chinese market.It overcomes the deficiencies of existing methods,which suffer from lack of actual default data for back testing.With newly available bond default data,we analyze the roles of market variables against accounting variables under various models.While we find that Merton's market-based structural model and KMV's Distance to Default exhibit languid discriminating power compared with hazard models that have carefully constructed predictors,other market variables carry significant information about bond defaults and could help improve on models with only the accounting variables.This implies that the collective intelligence of the market could somehow mitigate the distortion caused by misreported accounting information.Further,model performance can be significantly improved by adding predicting variables that link an individual financial measure to the broader market performance,such as the relative margin—a business environment proxy introduced in this study.We not only shed light on the default behavior of the Chinese bond market,but also provide a promising approach to improve the variable selection process.