Defaultable Bonds Research Articles

A Default Risk Model under Macroeconomic Conditions

This article presents a structural model of default risk under macroeconomic conditions. The macroeconomic conditions are assumed to be a finite state of a Markov chain. The innovation of the author’s model is to characterize the firm default, the default-free pure discount bond price, the defaultable bond price, and the credit spread associated with macroeconomic conditions. By using the Wiener–Hopf factorization, the author shows that there is a risk-neutral measure under regime switching. Under macroeconomic conditions, he derives both the default-free bond price and the price of the defaultable bond governed by fundamental systems of partial differential equations. The defaultable yield to maturity, credit spread, and duration are related to the finite state of the Markov chain.

DEFAULTABLE BOND PRICING USING REGIME SWITCHING INTENSITY MODEL

In this paper, we are interested in finding explicit numerical formulas to evaluate defaultable bonds prices of firms. For this purpose, we use a default intensity whose values depend on the credit rating of these firms. Each credit rating corresponds to a state of the default intensity. Then, this regime switches as soon as one of the credit rating of a firm also changes. Moreover, this regime switching default intensity model allows us to capture well some market features or economics behaviors. Thus, we obtain two explicit different formulas to evaluate the conditional Laplace transform of a regime switching Cox Ingersoll Ross model. One using the property of semi-affine of the model and the other one using analytic approximation. We conclude by giving some numerical illustrations of these formulas and real data estimation results.

General Properties of Solutions to Inhomogeneous Black-Scholes Equations with Discontinuous Maturity Payoffs and Application

We provide representations of solutions to terminal value problems of inhomogeneous Black-Scholes equations and studied such general properties as min-max estimates, gradient estimates, monotonicity and convexity of the solutions with respect to the stock price variable, which are important for financial security pricing. In particular, we focus on finding representation of the gradient (with respect to the stock price variable) of solutions to the terminal value problems with discontinuous terminal payoffs or inhomogeneous terms. Such terminal value problems are often encountered in pricing problems of compound-like options such as Bermudan options or defaultable bonds with discrete default barrier, default intensity and endogenous default recovery. Our results are applied in pricing defaultable discrete coupon bonds.

Regime Switching and Bond Pricing

This article proposes an overview of the usefulness of the regime switching approach for building various kinds of bond pricing models and of the roles played by the regimes in these models. Both default-free and defaultable bonds are considered. The regimes can be used to capture stochastic drifts and/or volatilities, to represent discrete target rates, to incorporate business cycles or crises, to introduce contagion, to reproduce zero lower bound spells, or to evaluate the impact of standard or nonstandard monetary policies. From a technical point of view, we stress the key role of Markov chains, Compound Autoregressive (Car) processes, Regime Switching Car processes and multihorizon Laplace transforms.

Regime Switching and Bond Pricing

This article proposes an overview of the usefulness of the regime switching approach for building various kinds of bond pricing models and of the roles played by the regimes in these models. Both default-free and defaultable bonds are considered. The regimes can be used to capture stochastic drifts and/or volatilities, to represent discrete target rates, to incorporate business cycles or crises, to introduce contagion, to reproduce zero lower bound spells, or to evaluate the impact of standard or nonstandard monetary policies. From a technical point of view, we stress the key role of Markov chains, Compound Autoregressive (Car) processes, Regime Switching Car processes and multihorizon Laplace transforms.

MICRO CREDIT RISK METRICS: A COMPREHENSIVE REVIEW

SUMMARYDefault modelling is a general term used for several interrelated fields of risk management. Bond defaults, credit (loan) defaults, firm defaults and country defaults are examples of this kind. The scope and reason for existence of this study is to focus mainly on firm default. The purpose of this review is to shed light on the development and evaluation of the models proposed for predicting bankruptcy in terms of conceptualization, country distribution, sector specification, time dimension, variables used and findings reported. The current review includes firm default studies published in business fields such as accounting, economics, finance and management science. This review is distinct in that it seeks (i) to give a comprehensive examination of the models, (ii) to compare and contrast the features of the models and (iii) to show with a solid argument where future research should be focused. Copyright © 2013 John Wiley & Sons, Ltd.

Rare Disasters, Credit, and Option Market Puzzles

We embed systematic default, procyclical recovery rates, and external habit persistence into a model with a slight possibility of a macroeconomic disaster of reasonable magnitude. We derive analytical solutions for defaultable bond prices and show that a single set of structural parameters calibrated to the real economy can simultaneously explain several key empirical regularities in equity, credit, and options markets. Our model captures the empirical level and volatility of credit spreads, generates a flexible credit risk term structure, and provides a good fit to a century of observed spreads. The model also matches high-yield and collaterized debt obligation tranche spreads, equity market moments, and index option skewness. Finally, our model implies a time-varying relationship between bond and option prices that depends on the state of the economy and that explains the conflicting empirical evidence found in the literature. This paper was accepted by Jerome Detemple, finance.

Practical development of a CCP risk management system for institutional investor settlement in the Korean stock market

The authors have redesigned the logic for setting the net debit cap and provided the architecture for a credit and market value at risk (VaR) measuring system for risk mitigation in institutional investor settlement in the Korean stock market, where the Korea Securities Depository (KSD) as a central counterparty (CCP) assumes the payment obligation of participants and settles by DVP2 on settlement date T + 2. In total, 6,345 data points of net settlement funds from 16th January to 27th August, 2012, were used for 89 members to redesign the logic of the net debit cap and 11,541 data points of net settlement funds from 16th January to 27th August, 2012, were used for the 125 account groups (a set of securities settlement accounts designated for participants’ convenience, such as management of margin surplus value and payment and delivery of settlement fund) to calculate risk exposure. For setting the net debit cap, all data were divided into four different groups by quartile and four different net debit caps on quartile 3 were set, with 75 per cent points for each group. According to the simulation result, the total amount of the guaranteed net settlement fund could be reduced by 52.22 per cent while maintaining operational efficiency of members. In addition, 62 artificial default rates for credit VaR were generated by using the bond default rates of the Korea Financial Investment Association (KOFIA) and the global and Asia-Pacific corporate default rates of Standard & Poor’s (S&P). For market VaR, the International Monetary Fund (IMF) crisis, 9/11, sub-prime crisis and Lehman’s bankruptcy were chosen to build the ‘stress-test’ environment. Four risk exposures — ‘high risk’, ‘risk’, ‘semi risk’ and ‘low risk’ — of the net settlement fund were devised for measuring the degree of risk exposure of the net settlement fund. For further study, measurement of a net debit cap to be calculated with an average of three highest values may not be sufficient, so, alternative methods to measure the optimal number of settlement funds to represent the whole settlement fund may be needed.

The effect of unfunded pension liabilities on corporate bond ratings, default risk, and recovery rate

Unfunded pension liabilities lower ratings of non-senior secured bonds but do not affect ratings of senior secured bonds due to their higher seniority. Pension funding improvement (deterioration) is associated with bond rating upgrade (downgrade). Moreover, large unfunded liabilities increase bond default risk and reduce the recovery rate of bondholders after controlling for credit ratings, suggesting that bond ratings do not fully capture pension underfunding risk. Overall, our results highlight the important effects of unfunded pension obligations on bond ratings, default risk, and creditors’ payoff, and suggest that investors should look beyond bond ratings in making investment decisions.

On the conditional default probability in a regulated market with jump risk

In this paper, we introduce tractable dynamic models for financial variables (such as interest rates, foreign exchange rates, commodity prices, etc.) with capturing both jump risk and boundedness of the price fluctuation in a regulated market. For the jump risk, we use a compound Poisson process with double exponential jump amplitude to describe the jumps. As for the boundedness of the fluctuation, we introduce two regulators to regulate (or control) the price dynamics. Based on such a dynamics, we then study the default events under the structural framework for credit risk. The explicit expression for the Laplace transform of the default time is derived. For exploring the effect of the jump risk and the regulation, we perform some numerical simulation. As a practical application, an analytic valuation of defaultable zero-coupon bonds is presented. It turns out that our model can produce a variety of strictly positive credit spread term structures, including downward, humped and upward styles.

CREDIT DERIVATIVES PRICING WITH STOCHASTIC VOLATILITY MODELS

This paper proposes a model for pricing credit derivatives in a defaultable HJM framework. The model features hump-shaped, level dependent, and unspanned stochastic volatility, and accommodates a correlation structure between the stochastic volatility, the default-free interest rates, and the credit spreads. The model is finite-dimensional, and leads (a) to exponentially affine default-free and defaultable bond prices, and (b) to an approximation for pricing credit default swaps and swaptions in terms of defaultable bond prices with varying maturities. A numerical study demonstrates that the model captures stylized various features of credit default swaps and swaptions.

Method of Paired Contours and Pricing Barrier Options and CDs of Long Maturities

For prices of options with barrier and lookback features, defaultable bonds and CDS, and probability distribution functions in Levy models, joint probability distributions of the process and its supremum or/and infimum, one can derive explicit analytical formulas in terms of the Laplace inversion, Fourier inversion and the Wiener-Hopf factorization. Unless the characteristic exponent is rational, the main examples being BM, DEJD and HEJD models, accurate numerical realizations of these formulas are difficult or very time consuming, for options of very long and very short maturities especially. In the paper, a systematic approach to contour deformations in pricing formulas is developed, which greatly increases the accuracy and speed of calculations; the efficiency of the method is demonstrated with numerical examples. For options and CDS of moderate and long maturities, much faster asymptotic formulas of comparable level of accuracy are developed.

Analytical Pricing of Defaultable Bond with Stochastic Default Intensity

We provide analytical pricing formula of corporate defaultable bond with both expected and unexpected default in the case with stochastic default intensity. In the case with constant short rate and exogenous default recovery using PDE method, we gave some pricing formula of the defaultable bond under the conditions that 1)expected default recovery is the same with unexpected default recovery; 2) default intensity follows one of 3 special cases of Willmott model; 3) default intensity is uncorrelated with firm value. Then we derived a pricing formula of a credit default swap. And in the case of stochastic short rate and exogenous default recovery using PDE method, we gave some pricing formula of the defaultable bond under the conditions that 1) expected default recovery is the same with unexpected default recovery; 2) the short rate follows Vasicek model; 3) default intensity follows one of 3 special cases of Willmott model; 4) default intensity is uncorrelated with firm value; 5) default intensity is uncorrelated with short rate. Then we derived a pricing formula of a credit default swap. We give some credit spread analysis, too.

Asymptotic Expansion for Term Structures of Defaultable Bonds with Non-Gaussian Dependent Innovations

In the context of credit risk, the term structure models that have been studied in the literature are typically models driven by Brownian motion or standard jump diffusions. These models provide coherent modeling that is straightforward to implement. To make these models more flexible, we develop a discrete-time approximation of a continuous-time Vasicek term structure analysis with non-Gaussian and dependent innovations. Higher-order asymptotic theory enables us to evaluate the term structures of defaultable bonds. Numerical examples show that the effects of non-Gaussianity and the dependency of both risk-free rate and default process strongly influence the evaluation of defaultable bonds. As an application, we estimate the parameters of our proposed models for the Japanese corporate credit default swap market.

A Family of Density Expansions for Lévy-Type Processes with Default

We consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Levy-type martingale subject to default. This class of models allows for local volatility, local default intensity, and a locally dependent Levy measure. Generalizing and extending the novel adjoint expansion technique of Pagliarani, Pascucci, and Riga (2013), we derive a family of asymptotic expansions for the transition density of the underlying as well as for European-style option prices and defaultable bond prices. For the density expansion, we also provide error bounds for the truncated asymptotic series. Additionally, for pure diffusion processes, we derive an asymptotic expansion for the implied volatility induced by European calls/puts. Our method is numerically efficient; approximate transition densities and European option prices are computed via Fourier transforms; approximate bond prices are computed as finite series. Additionally, as in Pagliarani et al. (2013), for models with Gaussian-type jumps, approximate option prices can be computed in closed form. Numerical examples confirming the effectiveness and versatility of our method are provided, as is sample Mathematica code.

Fuzzy semi-Markov migration process in credit risk

We explore the rating system used by credit agencies with a focus on problems that justify the use of fuzzy set theory. We prove that a fuzzy market is viable if and only if an equivalent martingale measure exists, from which we construct the forward probability measure and under which the discounted price of a default-free bond is a martingale. We model the evolution of credit migration of a defaultable bond as an inhomogeneous semi-Markov process with fuzzy states. We study the effects of changing the real probability measure to a forward probability measure. In addition, we investigate the asymptotic behaviour of the survival probability in each fuzzy state given in the absence of default. Finally, we discuss parameter estimation and calibration of the inhomogeneous Markov chain with fuzzy states.

Optimal Investment and Consumption with Default Risk: HARA Utility

In this paper, we consider a portfolio optimization problem in a defaultable market. The representative investor dynamically allocates his or her wealth among the following securities: a perpetual defaultable bond, a money market account and a default-free risky asset. The optimal investment and consumption policies that maximize the in nite horizon expected discounted HARA utility of the consumption are explicitly derived. Moreover, numerical illustrations are also presented.

Asymptotic behaviour of the survival probabilities in an inhomogeneous semi-Markov model for the migration process in credit risk

We start with the stochastic foundation of the general discrete-time Market of defaultable bonds. We prove that the above market is viable, if and only if there exists an equivalent martingale measure, from which we construct the forward probability measure and under which the discounted default free bond price is a martingale. Assuming that the migration process of defaultable bonds evolves as an inhomogeneous semi-Markov process, we study the asymptotic behaviour of survival probabilities. We provide a method of estimating real-world transition probability sequences for the semi-Markov process, and statistics for testing their homogeneity over time.

Higher Order Binaries with Time Dependent Coefficients and Two Factors - Model for Defaultable Bond with Discrete Default Information

In this article, we consider a 2 factors-model for pricing defaultable bond with discrete default intensity and barrier where the 2 factors are stochastic risk free short rate process and firm value process. We assume that the default event occurs in an expected manner when the firm value reaches a given default barrier at predetermined discrete announcing dates or in an unexpected manner at the first jump time of a Poisson process with given default intensity given by a step function of time variable. Then our pricing model is given by a solving problem of several linear PDEs with variable coefficients and terminal value of binary type in every subinterval between the two adjacent announcing dates. Our main approach is to use higher order binaries. We first provide the pricing formulae of higher order binaries with time dependent coefficients and consider their integrals on the last expiry date variable. Then using the pricing formulae of higher binary options and their integrals, we give the pricing formulae of defaultable bonds in both cases of exogenous and endogenous default recoveries and credit spread analysis.

Asymptotic Analysis for One-Name Credit Derivatives

We propose approximate solutions to price defaultable zero-coupon bonds as well as the corresponding credit default swaps and bond options. We consider the intensity-based approach of a two-correlated-factor Hull-White model with stochastic volatility of interest rate process. Perturbations from the stochastic volatility are computed by using an asymptotic analysis. We also study the sensitive properties of the defaultable bond prices and the yield curves.