INCORPORATING RISK AND AMBIGUITY AVERSION INTO A HYBRID MODEL OF DEFAULT

It is well known that purely structural models of default cannot explain short‐term credit spreads, while purely intensity‐based models lead to completely unpredictable default events. Here we introduce a hybrid model of default, in which a firm enters a “distressed” state once its nontradable credit worthiness index hits a critical level. The distressed firm then defaults upon the next arrival of a Poisson process. To value defaultable bonds and credit default swaps (CDSs), we introduce the concept of robust indifference pricing. This paradigm incorporates both risk aversion and model uncertainty. In robust indifference pricing, the optimization problem is modified to include optimizing over a set of candidate measures, in addition to optimizing over trading strategies, subject to a measure dependent penalty. Using our model and valuation framework, we derive analytical solutions for bond yields and CDS spreads, and find that while ambiguity aversion plays a similar role to risk aversion, it also has distinct effects. In particular, ambiguity aversion allows for significant short‐term spreads.     

CREDIT SPREADS, OPTIMAL CAPITAL STRUCTURE, AND IMPLIED VOLATILITY WITH ENDOGENOUS DEFAULT AND JUMP RISK

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) Jumps 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; although in general credit spreads and implied volatility tend to move in the same direction under exogenous default models, this may not be true in presence of endogenous default and jumps. Pricing formulae of credit default swaps and equity default swaps are also given. In terms of mathematical contribution, we give a proof of a version of the "smooth fitting" principle under the jump model, justifying a conjecture first suggested by Leland and Toft under the Brownian model.     

Identifying the Factors that Affect Interest‐Rate Swap Spreads: Some Evidence from the United States and the United Kingdom

We assess the ability of the factors proposed in previous research to account for the stochastic evolution of the term structure of the U.S. and U.K. swap spreads. Using as factor proxies the level, volatility, and slope of the zero‐coupon government yield curve as well as the Treasury‐bill—London Interbank Offer Rate (LIBOR) spread and the corporate bond spread, we identify a procyclical behavior for the short‐maturity U.S. swap spreads and a countercyclical behavior for longer maturity U.S. swap spreads. Liquidity and corporate bond spreads are also significant, but their importance varies with maturity. The liquidity premium is more important for short‐maturity swap spreads, although the corporate bond spread affects long‐maturity swap spreads. For the United Kingdom, swap spreads are countercyclical across maturities. In addition, we find that shocks to the liquidity premium are more significant for long‐maturity swaps and that the links between corporate bond markets and swap markets are much stronger than in the United States. When we look at the links between U.S. and U.K. swap markets, we identify a significant influence of the U.S. factors on the U.K. swap spreads across maturities. © 2001 John Wiley & Sons, Inc. Jrl Fut Mark 21:737–768, 2001     

中国公司债券利差影响因素研究--基于评级信息的分析

本文以信用利差分解理论为基础,结合债券评级信息,基于市场实际数据实证研究了中国公司债券市场信用利差的决定因素。结果表明,预期违约损失在税后信用利差中只占很小的比例,无风险利率的期限结构、流动性风险因子、宏观经济指标等因子都对信用利差有较显著的解释力。     

Are credit spreads too low or too high? A hybrid barrier option approach for financial distress

Based on the works of Brockman, P. and Turtle, H. J. (2003) and Giesecke, K. (2004), we propose in this study a hybrid barrier option model to explain observed credit spreads. It is free of problems with the structural model, which underprescribed credit spreads for investment grade corporate bonds and overprescribed the high‐yield issues. Unlike the standard barrier option approach, our hybrid model does not imply, for high‐yield issues with firms under financial stress, a reduction of credit spreads while firm value actually falls. Our empirical analysis supports that when credit spreads are quoted abnormally higher or rising faster than expected, unexpected changes tend to persist. Otherwise a significant and prompt reversion to long‐term equilibrium takes place. This asymmetric pricing phenomenon is validated with a method introduced by Enders, W. and Granger, C. W. J. (1998) and Enders, W. and Siklos, P. L. (2001). The pricing asymmetry could not have been produced by a structural model employing only standard option. But it is consistent with a hybrid barrier option model. Our model characterizes the valuation of debt under financial stress and the asymmetric price pattern better than both the classical structural and the standard barrier option approaches. It can be extended to the study of individual CDS for its better liquidity than individual corporate bonds. This study provides helpful implications especially for the medium and high‐yield issues in pricing as well as portfolio diversification. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 29:1161–1189, 2009     

Identifying risks in emerging market sovereign and corporate bond spreads

This paper investigates the systematic risk factors driving emerging market (EM) credit risk by jointly modeling sovereign and corporate credit spreads at a global level. We use a multi-regional Bayesian panel VAR model, with time-varying betas and multivariate stochastic volatility. This model allows us to decompose credit spreads and build indicators of EM risks. A key result is that indices of EM sovereign and corporate credit spreads differ because of their specific reactions to global risks (risk aversion, liquidity and US corporate risk). For example, following Lehman's default, EM sovereign spreads ‘decoupled’ from the US corporate market, whereas EM corporates ‘recoupled.’     

Pricing credit spread options under a Markov chain model with stochastic default rate

This paper studies a Markov chain model that, unlike the existing models, has a stochastic default rate model so as to reflect real world phenomena. We extend the existing Markov chain models as follows: First, our model includes both the economy‐wide and the rating‐specific factors, which affect credit ratings. Second, our model allows both continuous and discrete movements in credit spreads, even when there exist no changes in credit ratings. Under these assumptions, we provide a valuation formula for a credit spread option, and examine its effects. This paper suggests a parsimonious model. As in J. Wei ( 2003 ), we find that rating‐specific factors are important. Also, discrete movements seem to play a larger role depending on the firm's credit rating. Finally, we show that a model, like the Kodera model, that uses only a common factor without allowing for discrete movements, may overprice credit spread put options. © 2004 Wiley Periodicals, Inc. Jrl Fut Mark 24:631–648, 2004     

What risks do corporate bond put features insure against?

Corporate bond prices are known to be influenced by default and term structure risk in addition to non‐default risks such as illiquidity. Putable corporate bonds allow investors to sell their holdings back to the issuer and may thus provide insurance against all of these risks. We first document empirically that embedded put option values are related to proxies for all three. In a second step, we develop a valuation model that simultaneously captures default and interest rate risk. We use this model to disentangle the reduction in yield spread enjoyed by putable bonds that can be attributed to each risk. Perhaps surprisingly, the most important reduction is due to mitigated default or spread risk, followed by term structure risk. The reduction in the non‐default component is present but rather small.     

Counter‐Credit‐Risk Yield Spreads: A Puzzle in China's Corporate Bond Market

In this paper, using China's risk‐free and corporate zero yields together with aggregate credit risk measures and various control variables from 2006 to 2013, we document a puzzle of counter‐credit‐risk corporate yield spreads. We interpret this puzzle as a symptom of the immaturity of China's credit bond market, which reveals a distorted pricing mechanism latent in the fundamental of this market. We also find interesting results about relationships between corporate yield spreads and interest rates and risk premia and the stock index, and these results are somewhat attributed to this puzzle.     

Equity volatility,bond yields,and yield spreads

This study examines the impact of implied and contemporaneous equity market volatility on Treasury yields, corporate bond yields, and yield spreads over Treasuries. The CBOE VIX is the measure of implied volatility, and the measure of contemporaneous volatility is constructed using intraday squared S&P 500 returns. We find that bond yields and spreads respond to changes in equity market volatility in a manner consistent with a flight‐to‐quality effect. Both short‐ and long‐term Treasury yields fall in response to increases in implied volatility, and the yield curve flattens modestly. Yields on short‐term investment grade bonds fall in response to contemporaneous volatility shocks, while long‐term spreads on low‐quality issues widen. This indicates that investors “look ahead” in anticipation of changes in equity market volatility but respond more strongly to changes in contemporaneous market activity. © 2011 Wiley Periodicals, Inc. Jrl Fut Mark     

Monotonicities in a Markov Chain Model for Valuing Corporate Bonds Subject to Credit Risk

In recent years, it has become common to use a Markov chain model to describe the dynamics of a firm's credit rating as an indicator of the likelihood of default. Such a model can be used not only for describing the dynamics but also for valuing risky discount bonds. The aim of this paper is to explain how the Markov chain model leads to the known empirical findings such that prior rating changes carry predictive power for the direction of future rating changes and a firm with low (high, respectively) credit rating is more likely to be upgraded (downgraded) conditional on survival as the time horizon lengthens. The model will also explain practically plausible statements such as that bond prices as well as credit risk spreads would be ordered according to their credit qualities. Stochastic monotonicities of absorbing Markov chains play a prominent role in these issues.     

The Co‐Movement of Credit Default Swap Spreads,Equity Returns and Volatility: Evidence from Asia‐Pacific Markets

We provide a comprehensive analysis of the co‐movement of credit default swap (CDS), equity, and volatility markets in four Asia‐Pacific countries at firm and index level during the period 2007–2010. First, we examine lead–lag relationships between CDS spread changes, equity returns, and changes in volatility using a vector autoregressive model. At the firm level equity returns lead changes in CDS spreads and realized volatility. However, at the index level the intertemporal linkages between the three markets are less clear‐cut. Second, we apply the measures proposed by Diebold and Yilmaz (2014) to an analysis of volatility spillovers among the CDS, equities, and volatility asset classes. The results suggest that realized volatility (at firm level) and implied volatility (at index level) are the main transmitters of cross‐market volatility spillovers. Third, we analyze the impact of various structural factors and confirm the importance of realized volatility of equity returns as a determinant of CDS spreads.     

A UNIFIED FRAMEWORK FOR PRICING CREDIT AND EQUITY DERIVATIVES

We propose a model which can be jointly calibrated to the corporate bond term structure and equity option volatility surface of the same company. Our purpose is to obtain explicit bond and equity option pricing formulas that can be calibrated to find a risk neutral model that matches a set of observed market prices. This risk neutral model can then be used to price more exotic, illiquid, or over‐the‐counter derivatives. We observe that our model matches the equity option implied volatility surface well since we properly account for the default risk in the implied volatility surface. We demonstrate the importance of accounting for the default risk and stochastic interest rate in equity option pricing by comparing our results to Fouque et al., which only accounts for stochastic volatility.     

ARBITRAGE‐FREE BILATERAL COUNTERPARTY RISK VALUATION UNDER COLLATERALIZATION AND APPLICATION TO CREDIT DEFAULT SWAPS

We develop an arbitrage‐free valuation framework for bilateral counterparty risk, where collateral is included with possible rehypothecation. We show that the adjustment is given by the sum of two option payoff terms, where each term depends on the netted exposure, i.e., the difference between the on‐default exposure and the predefault collateral account. We then specialize our analysis to credit default swaps (CDS) as underlying portfolios, and construct a numerical scheme to evaluate the adjustment under a doubly stochastic default framework. In particular, we show that for CDS contracts a perfect collateralization cannot be achieved, even under continuous collateralization, if the reference entity’s and counterparty’s default times are dependent. The impact of rehypothecation, collateral margining frequency, and default correlation‐induced contagion is illustrated with numerical examples.     

A two‐mean reverting‐factor model of the term structure of interest rates

This article presents a two‐factor model of the term structure of interest rates. It is assumed that default‐free discount bond prices are determined by the time to maturity and two factors, the long‐term interest rate, and the spread (i.e., the difference) between the short‐term (instantaneous) risk‐free rate of interest and the long‐term rate. Assuming that both factors follow a joint Ornstein‐Uhlenbeck process, a general bond pricing equation is derived. Closed‐form expressions for prices of bonds and interest rate derivatives are obtained. The analytical formula for derivatives is applied to price European options on discount bonds and more complex types of options. Finally, empirical evidence of the model's performance in comparison with an alternative two‐factor (Vasicek‐CIR) model is presented. The findings show that both models exhibit a similar behavior for the shortest maturities. However, importantly, the results demonstrate that modeling the volatility in the long‐term rate process can help to fit the observed data, and can improve the prediction of the future movements in medium‐ and long‐term interest rates. So it is not so clear which is the best model to be used. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23: 1075–1105, 2003     

Dividend policy and capital structure of a defaultable firm

Default risk significantly affects the corporate policies of a firm. We develop a model in which a limited liability entity subject to default at an exponential random time jointly sets its dividend policy and capital structure to maximize the expected lifetime utility from consumption of risk‐averse equity investors. We give a complete characterization of the solution to the singular stochastic control problem. The optimal policy involves paying dividends to keep the ratio of firm's equity value to investors' wealth below a critical threshold. Dividend payout acts as a precautionary channel to transfer wealth from the firm to investors for mitigation of losses in the event of default. Higher the default risk, more aggressively the firm leverages and pays dividends.     

基于灰关联度的企业信用风险评分方法及应用

基于企业具有“违约倾向”的假定,根据违约企业样本,构造了一个“典型违约企业”,并利用灰关联度确定了影响企业违约的各因素权重。最后给出了一种以“典型违约企业”为参照的企业信用风险评分方法。     

Hedging under counterparty credit uncertainty

This study investigates optimal production and hedging decisions for firms facing price risk that can be hedged with vulnerable contracts, i.e., exposed to nonhedgeable endogenous counterparty credit risk. When vulnerable forward contracts are the only hedging instruments available, the firm's optimal level of production is lower than without credit risk. Under plausible conditions on the stochastic dependence between the commodity price and the counterparty's assets, the firm does not sell its entire production on the vulnerable forward market. When options on forward contracts are also available, the optimal hedging strategy requires a long put position. This provides a new rationale for the hedging role of options in the over‐the‐counter markets exposed to counterparty credit risk. © 2008 Wiley Periodicals, Inc. Jrl Fut Mark 28: 248–263, 2008     

Valuing Seller‐Defaultable Options

This study analyzes seller‐defaultable options that allow option writers to have a free‐will right to default, along with some prespecified default mechanisms. We analytically and numerically examine the pricing, hedging, defaulting, and profitability of the seller‐defaultable options, considering three possible scenarios for seller default. Analyzing the essential implications of seller‐defaultable options, we show that the option price is positively correlated with the default fine, underlying asset price, and volatility. The seller‐defaultable option's Greeks appear more complicated than those of the plain vanilla options. The likelihood of sellers defaulting increases with the underlying asset price, interest rate, volatility, and maturity time. Subject to the default mechanism, the buyers’ trading involves a trade‐off between profits and costs. © 2012 Wiley Periodicals, Inc. Jrl Fut Mark 33:129–157, 2013     

PRICING EQUITY DERIVATIVES SUBJECT TO BANKRUPTCY

We solve in closed form a parsimonious extension of the Black–Scholes–Merton model with bankruptcy where the hazard rate of bankruptcy is a negative power of the stock price. Combining a scale change and a measure change, the model dynamics is reduced to a linear stochastic differential equation whose solution is a diffusion process that plays a central role in the pricing of Asian options. The solution is in the form of a spectral expansion associated with the diffusion infinitesimal generator. The latter is closely related to the Schrödinger operator with Morse potential. Pricing formulas for both corporate bonds and stock options are obtained in closed form. Term credit spreads on corporate bonds and implied volatility skews of stock options are closely linked in this model, with parameters of the hazard rate specification controlling both the shape of the term structure of credit spreads and the slope of the implied volatility skew. Our analytical formulas are easy to implement and should prove useful to researchers and practitioners in corporate debt and equity derivatives markets.