Multiscale Intensity Models for Single Name Credit Derivatives

We study the pricing of defaultable derivatives, such as bonds, bond options, and credit default swaps in the reduced form framework of intensity‐based models. We use regular and singular perturbation expansions on the intensity of default from which we derive approximations for the pricing functions of these derivatives. In particular, we assume an Ornstein‐Uhlenbeck process for the interest rate, and a two‐factor diffusion model for the intensity of default. The approximation allows for computational efficiency in calibrating the model. Finally, empirical evidence on the existence of multiple scales is presented by the calibration of the model on corporate yield curves.     

利用传染模型对含有对手风险的信用证券定价(英文)

本文利用传染模型研究了可违约债券和含有对手风险的信用违约互换的定价。我们在约化模型中引入具有违约相关性的传染模型,该模型假设违约过程的强度依赖于由随机微分方程驱动的随机利率过程和交易对手的违约过程.本文模型可视为Jarrow和Yu(2001)及Hao和Ye(2011)中模型的推广.进一步地,我们利用随机指数的性质导出了可违约债券和含有对手风险的信用违约互换的定价公式并进行了数值分析.     

On the simulation of portfolios of interest rate and credit risk sensitive securities

We discuss extensions of reduced-form and structural models for pricing credit risky securities to portfolio simulation and valuation. Stochasticity in interest rates and credit spreads is captured via reduced-form models and is incorporated with a default and migration model based on the structural credit risk modelling approach. Calculated prices are consistent with observed prices and the term structure of default-free and defaultable interest rates. Three applications are discussed: (i) study of the inter-temporal price sensitivity of credit bonds and the sensitivity of future portfolio valuation with respect to changes in interest rates, default probabilities, recovery rates and rating migration, (ii) study of the structure of credit risk by investigating the impact of disparate risk factors on portfolio risk, and (iii) tracking of corporate bond indices via simulation and optimisation models. In particular, we study the effect of uncertainty in credit spreads and interest rates on the overall risk of a credit portfolio, a topic that has been recently discussed by Kiesel et al. [The structure of credit risk: spread volatility and ratings transitions. Technical report, Bank of England, ISSN 1268-5562, 2001], but has been otherwise mostly neglected. We find that spread risk and interest rate risk are important factors that do not diversify away in a large portfolio context, especially when high-quality instruments are considered.     

Market implied volatilities for defaultable bonds

Typically, implied volatilities for defaultable instruments are not available in the financial market since quotations related to options on defaultable bonds or on credit default swaps are usually not quoted by brokers. However, an estimate of their volatilities is needed for pricing purposes. In this paper, we provide a methodology to infer market implied volatilities for defaultable bonds using equity implied volatilities and CDS spreads quoted by the market in relation to a specific issuer. The theoretical framework we propose is based on the Merton’s model under stochastic interest rates where the short rate is assumed to follow the Hull–White model. A numerical analysis is provided to illustrate the calibration process to be performed starting from financial market data. The market implied volatility calibrated according to the proposed methodology could be used to evaluate options where the underlying is a risky bond, i.e. callable bond or other types of credit-risk sensitive financial instruments.

     

Credit Derivatives Pricing Based on Lévy Field Driven Term Structure

We use Lévy random fields to model the term structure of forward default intensity, which allows to describe the contagion risks. We consider the pricing of credit derivatives, notably of defaultable bonds in our model. The main result is to prove the pricing kernel as the unique solution of a parabolic integro-differential equation by constructing a suitable contractible operator and then considering the limit case for an unbounded terminal condition. Finally, we illustrate the impact of contagious jump risks on the defaultable bond price by numerical examples.     

A tractable LIBOR model with default risk

We develop a model for the dynamic evolution of default-free and defaultable interest rates in a LIBOR framework. Utilizing the class of affine processes, this model produces positive LIBOR rates and spreads, while the dynamics are analytically tractable under defaultable forward measures. This leads to explicit formulas for CDS spreads, while semi-analytical formulas are derived for other credit derivatives. Finally, we give an application to counterparty risk.     

Impact of risk aversion and belief heterogeneity on trading of defaultable claims

This paper studies the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer’s optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty—single or multiple sellers—may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity. For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade.     

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.     

A reduced-form model with default intensities containing contagion and regime-switching Vasicek processes

The contagion credit risk model is used to describe the contagion effect among different financial institutions. Under such a model, the default intensities are driven not only by the common risk factors, but also by the defaults of other considered firms. In this paper, we consider a two-dimensional credit risk model with contagion and regime-switching. We assume that the default intensity of one firm will jump when the other firm defaults and that the intensity is controlled by a Vasicek model with the coefficients allowed to switch in different regimes before the default of other firm. By changing measure, we derive the marginal distributions and the joint distribution for default times. We obtain some closed form results for pricing the fair spreads of the first and the second to default credit default swaps (CDSs). Numerical results are presented to show the impacts of the model parameters on the fair spreads.     

Using Utility Functions to Model Risky Bonds

This paper prices defaultable bonds by incorporating inherent risks with the use of utility functions. By allowing risk preferences into the valuation of bonds, nonlinearity is introduced in their pricing. The utility‐function approach affords the advantage of yielding exact solutions to the risky bond pricing equation when familiar stochastic models are used for interest rates. This can be achieved even when the default probability parameter is itself a stochastic variable. Valuations are found for the power‐law and log utility functions under the interest‐rate dynamics of the extended Vasicek and CIR models.     

Modelling the evolution of credit spreads using the Cox process within the HJM framework: A CDS option pricing model

In this paper a simulation approach for defaultable yield curves is developed within the Heath et al. (1992) framework. The default event is modelled using the Cox process where the stochastic intensity represents the credit spread. The forward credit spread volatility function is affected by the entire credit spread term structure. The paper provides the defaultable bond and credit default swap option price in a probability setting equipped with a subfiltration structure. The Euler–Maruyama stochastic integral approximation and the Monte Carlo method are applied to develop a numerical scheme for pricing. Finally, the antithetic variable technique is used to reduce the variance of credit default swap option prices.     

An integrated pricing model for defaultable loans and bonds

In recent years, credit risk has played a key role in risk management issues. Practitioners, academics and regulators have been fully involved in the process of developing, studying and analysing credit risk models in order to find the elements which characterize a sound risk management system. In this paper we present an integrated model, based on a reduced pricing approach, for market and credit risk. Its main features are those of being mark to market and that the spread term structure by rating class is contingent on the seniority of debt within an arbitrage-free framework. We introduce issues such as, the integration of market and credit risk, the use of stochastic recovery rates and recovery by seniority. Moreover, we will characterize default risk by estimating migration risk through a “mortality rate”, actuarial-based, approach. The resultant probabilities will be the base for determining multi-period risk-neutral transition probability that allow pricing of risky debt in the trading and banking book.     

Liquidity and credit risk

The paper uses fuzzy measure theory to represent liquidity risk, i.e. the case in which the probability measure used to price contingent claims is not known precisely. This theory enables one to account for different values of long and short positions. Liquidity risk is introduced by representing the upper and lower bound of the price of the contingent claim computed as the upper and lower Choquet integral with respect to a subadditive function. The use of a specific class of fuzzy measures, known as g λ measures enables one to easily extend the available asset pricing models to the case of illiquid markets. As the technique is particularly useful in corporate claims evaluation, a fuzzified version of Merton's model of credit risk is presented. Sensitivity analysis shows that both the level and the range (the difference between upper and lower bounds) of credit spreads are positively related to the ‘quasi debt to firm value ratio’ and to the volatility of the firm value. This finding may be read as correlation between credit risk and liquidity risk, a result which is particularly useful in concrete risk-management applications. The model is calibrated on investment grade credit spreads, and it is shown that this approach is able to reconcile the observed credit spreads with risk premia consistent with observed default rate. Default probability ranges, rather than point estimates, seem to play a major role in the determination of credit spreads.     

基于双因子CIR强度式定价的信用债券投资组合优化

信用风险和利率风险是相互关联影响的。资产组合优化不能将这两种风险单独考虑或简单的相加,应该进行整体的风险控制,不然会造成投资风险的低估。本文的主要工作：一是在强度式定价模型的框架下,分别利用CIR随机利率模型刻画利率风险因素“无风险利率”和信用风险因素“违约强度”的随机动态变化,衡量在两类风险共同影响下信用债券的市场价值,从而构建CRRA型投资效用函数。以CRRA型投资效用函数最大化作为目标函数,同时控制利率和信用两类风险。弥补了现有研究中仅单独考虑信用风险或利率风险、无法对两种风险进行整体控制的弊端。二是将无风险利率作为影响违约强度的一个因子,利用“无风险利率因子”和“纯信用因子”的双因子CIR模型拟合违约强度,考虑了市场利率变化对于债券违约强度的影响,反映两种风险的相关性。使得投资组合模型中既同时考虑了信用风险和利率风险、又考虑了两种风险的交互影响。避免在优化资产组合时忽略两种风险间相关性、可能造成风险低估的问题。     

宏观因素对房地产行业信用风险的影响研究——基于BALQR模型的实证

以2008年1月至2014年9月相关数据为基础,利用贝叶斯adaptive Lasso分位数回归(BALQR)模型对影响房地产业信用风险的宏观经济因素进行了分析.结果表明,在任何分位点上,对我国房地产行业信用风险影响最大的均是GDP增长率,其次是CPI增长率和消费者信心指数增长率,其中前者为负向作用后两者为正向作用,而资本市场的景气状态则对房地产行业信用风险基本没有显著作用.不过,在不同分位点上,不同宏观因素在房地产行业的不同信用风险水平上的影响程度又存在差异性.     

Tail behaviour of credit loss distributions for general latent factor models

Using a limiting approach to portfolio credit risk, we obtain analytic expressions for the tail behavior of credit losses. To capture the co‐movements in defaults over time, we assume that defaults are triggered by a general, possibly non‐linear, factor model involving both systematic and idiosyncratic risk factors. The model encompasses default mechanisms in popular models of portfolio credit risk, such as CreditMetrics and CreditRisk⁺. We show how the tail characteristics of portfolio credit losses depend directly upon the factor model's functional form and the tail properties of the model's risk factors. In many cases the credit loss distribution has a polynomial (rather than exponential) tail. This feature is robust to changes in tail characteristics of the underlying risk factors. Finally, we show that the interaction between portfolio quality and credit loss tail behavior is strikingly different between the CreditMetrics and CreditRisk⁺ approach to modeling portfolio credit risk.     

违约强度由Lévy从属过程驱动的约化信用风险模型及信用违约互换的定价

本文引入一个约化信用风险模型,其中违约强度定义为从属过程,即非负增Lévy过程.用概率方法得到了违约时间分布的解析表达式.利用该解析表达式,给出了该信用风险模型下的信用违约互换(Credit Default Swaps)的闭形式的定价公式.     

The pricing of credit risky securities under stochastic interest rate model with default correlation

In this paper, we study the pricing of credit risky securities under a three-firms contagion model. The interacting default intensities not only depend on the defaults of other firms in the system, but also depend on the default-free interest rate which follows jump diffusion stochastic differential equation, which extends the previous three-firms models (see R.A. Jarrow and F.Yu (2001), S.Y.Leung and Y.K.Kwok (2005), A.Wang and Z.Ye (2011)). By using the method of change of measure and the technology (H. S.Park (2008), R.Hao and Z.Ye (2011)) of dealing with jump diffusion processes, we obtain the analytic pricing formulas of defaultable zero-coupon bonds. Moreover, by the “total hazard construction”, we give the analytic pricing formulas of credit default swap (CDS).     

On absolutely continuous compensators and nonlinear filtering equations in default risk models

We discuss the pricing of defaultable assets in an incomplete information model where the default time is given by a first hitting time of an unobservable process. We show that in a fairly general Markov setting, the indicator function of the default has an absolutely continuous compensator. Given this compensator we then discuss the optional projection of a class of semimartingales onto the filtration generated by the observation process and the default indicator process. Available formulas for the pricing of defaultable assets are analyzed in this setting and some alternative formulas are suggested.     

A pricing model for secondary market yield based floating rate notes subject to default risk

The purpose of this article is to price secondary market yield based floating rate notes (SMY-FRNs) subject to default risk. SMY-FRNs are derivatives on the default-free term structure of interest rates, on the term structures for default-risky credit classes, and on the structure of a determined pool of bonds. The main problem in SMY-FRN pricing (as compared to the pricing of standard interest rate or credit derivatives) is market incompleteness, which makes traditional no-arbitrage pricing by replication fail. In general, SMY-FRNs are subject to two types of default risk. First, the SMY-FRN issuer may go bankrupt (direct default risk). Second, the possibility of the bankruptcy of the issuers in the underlying pool has an influence on the SMY-FRN coupons (indirect default risk). This article is the first one which provides a no-arbitrage pricing model for SMY-FRNs with direct and indirect default risks. It is also the first article applying incomplete market pricing methodology to SMY-FRNs.