Credit spreads, endogenous bankruptcy and liquidity risk

In this paper, we consider a bond valuation model with both credit risk and liquidity risk to show that credit spreads are not negligible for short maturities. We adopt the structural approach to model credit risk, where the default triggering barrier is determined endogenously by maximizing equity value. As for liquidity risk, we assume that bondholders may encounter liquidity shocks during the lifetime of corporate bonds, and have to sell the bond immediately at the price, which is assumed to be a fraction of the price in a perfectly liquid market. Under this framework, we derive explicit expressions for corporate bond, firm value and bankruptcy trigger. Finally, numerical illustrations are presented.     

违约风险的交换期权的定价模型与解法

含交易对手违约风险的交换期权采用混合模型定价,借助公司价值模型中的补偿率,同时采用以强度为基础的违约函数来确定违约的发生.假定违约强度遵从均值回复的重随机Poisson过程：且违约强度过程与标的资产,企业价值都相关.利用等价鞅测度变换方法导出含有违约风险的交换期权的价格闭解.     

一类杠杆公司的破产概率:回望期权定价方法

利用结构化方法构造了杠杆公司的金融资产组合,由于公司破产的不可逆性和不确定性,可以把公司破产理解为公司所发行的债券发生违约.通过求解回望期权所满足的抛物型随机偏微分方程,推导出了混合分数跳-扩散模型下杠杆公司的股票定价公式,给出了杠杆公司在财务出现危机时股东通过资本注入来弥补经营损失和清偿债务而没有导致公司破产的概率,...     

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.     

Valuing Credit Default Swap under a double exponential jump di?usion model简

This paper discusses the valuation of the Credit Default Swap based on a jump market, in which the asset price of a firm follows a double exponential jump diffusion process, the value of the debt is driven by a geometric Brownian motion, and the default barrier follows a continuous stochastic process. Using the Gaver-Stehfest algorithm and the non-arbitrage asset pricing theory, we give the default probability of the first passage time, and more, derive the price of the Credit Default Swap.     

增长期权对最优资本结构的影响研究：企业生命周期的视角

资产构成和生命周期的动态关系决定了企业最优资本结构的变化趋势。本文在利息税盾和破产成本的权衡理论框架下,运用实物期权方法,理论上揭示了增长期权动态执行导致最优资本结构随企业生命周期的变化规律,扩展分析了股东-债权人代理冲突对最优资本结构的影响,并提供了基于中国沪深A股上市公司的经验证据。结果表明：随着企业由年轻走向成熟,增长期权不断执行和在位资产逐渐累积会降低企业总体的破产风险,从而企业最优财务杠杆呈现上升趋势且增长速度逐渐放缓;股东和债权人在增长期权执行决策上的代理冲突导致的投资不足会降低最优财务杠杆,但代理成本随企业生命周期推进呈递减趋势。     

双指数跳扩散模型信用违约互换短期价格研究

假设参考实体没违约时信用违约互换保护买方连续支付互换价格,导出了信用违约互换价格的表达式;对标的资产价值服从双指数跳扩散模型,得到了条件违约风险率和信用违约互换的短期价格极限.这些结果比纯扩散模型假设更符合实际.     

股票流动性与资本结构—基于中国证券市场的分析

以Amihud(2002)的不流动性测度(ILLIQ)作为流动性的衡量指标,以账面杠杆比(DBA)和市值杠杆比(DMA)作为公司资本结构的衡量指标,探讨了流动性对资本结构的影响.实证结果表明:流动性与公司财务杠杆显著负相关,无论被解释变量是DBA还是DMA,ILLIQ都具有信息增量提供能力,但流动性对DMA的影响较DBA更大.这表明,由于股票流动性能降低股权融资成本,流动性高的公司会更偏好股权融资.     

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.     

Analysis of the expected discounted penalty function for a general jump-diffusion risk model and applications in finance

In this paper, we extend the Cramér-Lundberg risk model perturbed by diffusion to incorporate the jumps of surplus investment return. Under the assumption that the jump of surplus investment return follows a compound Poisson process with Laplace distributed jump sizes, we obtain the explicit closed-form expression of the resulting Gerber-Shiu expected discounted penalty (EDP) function through the Wiener-Hopf factorization technique instead of the integro-differential equation approach. Especially, when the claim distribution is of Phase-type, the expression of the EDP function is simplified even further as a compact matrix-type form. Finally, the financial applications include pricing barrier option and perpetual American put option and determining the optimal capital structure of a firm with endogenous default.     

Stochastic equity volatility related to the leverage effect II: valuation of European equity options and warrants

We propose a general framework to assess the value of the financial claims issued by the firm, European equity options and warrantsin terms of the stock price. In our framework, the firm's asset is assumed to follow a standard stationary lognormal process with constant volatility. However, it is not the case for equity volatility. The stochastic nature of equity volatility is endogenous, and comes from the impact of a change in the value of the firm's assets on the financial leverage. In a previous paper we studied the stochastic process for equity volatility, and proposed analytic approximations for different capital structures. In this companion paper we derive analytic approximations for the value of European equity options and warrants for a firm financed by equity, debt and warrants. We first present the basic model, which is an extension of the Black-Scholes model, to value corporate securities either as a function of the stock price, or as a function of the firm's total assets. Since stock prices are observable, then for practical purposes, traders prefer to use the stock as the underlying instrument, we concentrate on valuation models in terms of the stock price. Second, we derive an exact solution for the valuation in terms of the stock price of (i) a European call option on the stock of a levered firm, i.e. a European compound call option on the total assets of the firm, (ii) an equity warrant for an all-equity firm, and (iii) an equity warrant for a firm financed by equity and debt. Unfortunately, to compute these solutions we need to specify the function of the stock price in terms of the firm's assets value. In general we are unable to specify this expression, but we propose tight bounds for the value of these options which can be easily computed as a function of the stock price. Our results provide useful extensions of the Black-Scholes model.     

基于KMV模型的我国中小上市公司信用风险研究

经过提高股权价值波动率精度的KMV模型对我国中小上市公司有很强的识别信用风险状况的能力.中小上市公司违约的可能性大于我国大型企业,信用状况不容乐观,整体信用状况在近3年间表现波澜不惊,到2006年违约风险有增大趋势.通过设定两条信用预警线,来监控中小上市公司的信用危机.资产规模对信用风险有显著影响,2004年之后资产规模与违约风险显著负相关,总资产小于3亿元的小公司抗风险能力最差.股份分置改革引起了中小上市公司信用风险短时间的波动,是2006年中小上市公司违约风险变大的重要原因。     

中国新三板市场阶段性集合竞价制度的价格发现研究

市场微观结构理论表明交易机制对资产价格的形成过程具有重要影响。本文以中国新三板交易机制改革为背景,从理论上分析了阶段性集合竞价制度的市场出清过程。阶段性集合竞价制度的核心在于市场出清时间间隔的设定。本文构建了一个存在信息摩擦和知情交易者学习机制的集合竞价市场出清模型,讨论了市场出清时间间隔对价格发现效率、资产价值不确定性和流动性风险的影响。研究发现:(1)在完美信息条件下,如果对市场规模较大和价值波动率较高的资产设定较短的市场出清时间间隔,将会降低投资者的流动性风险,提升市场质量;(2)在不完美信息条件下,除市场规模和资产价值波动率之外,信息不对称程度和知情交易者比例也是影响最优市场出清频率的重要因素;(3)在不完美信息条件下,对价值波动率较低的资产缩短市场出清时间间隔才能降低流动性风险,这与完美信息条件下的结论相反。     

Predicting the event and time horizon of bankruptcy using financial ratios and the maturity schedule of long-term debt

We study the problem of simultaneous and coherent assessment the probability of a firm’s bankruptcy at various time horizons in future. In contrast with usual (one-period) formulations of the problem, such multi-period formulation better matches the nature of bankruptcy process (bankruptcy occurs in time) and allows an easier and more natural incorporation of bankruptcy (default) prognoses in valuation of risky debt and equity, optimization of corporate capital structure etc. The study uses a new mathematical apparatus—multi-alternative decision rules of statistical decision theory. We investigate a new type of predictive variables that can be extracted from the maturity schedule of a firm’s long-term debt. The study develops Bayesian-type forecasting rules that use both maturity schedule factors and traditional financial ratios. These rules noticeably enhance bankruptcy prediction (compared with the familiar one-period Z-score rules of Altman) for bankruptcy within the first 1, 2 or 3 years. Predictive factors derived from schedule information enhance bankruptcy prediction at distant time horizons.     

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.     

Price limits and capital requirements of futures clearinghouses

Futures clearinghouses need capital to provide liquidity in case of default by clearing members. Price limits truncate observed futures prices and prevent observation of clearinghouses’ true default risk exposure. We show how to estimate the true default risk exposure from observed futures prices and model capital requirements using an option pricing model, which accounts for non-normality of and truncation in observed futures returns. We apply the model to the clearinghouse associated with the Winnipeg Commodity Exchange, compare required capital levels with actual capital levels and show that ignoring non-normality of futures returns causes overall capital requirements to be significantly underestimated.     

Optimal portfolio strategies with a liability and random risk: The case of different lending and borrowing rates

This paper deals with two problems of optimal portfolio strategies in continuous time. The first one studies the optimal behavior of a firm who is forced to withdraw funds continuously at a fixed rate per unit time. The second one considers a firm that is faced with an uncontrollable stochastic cash flow, or random risk process. We assume the firm’s income can be obtained only from the investment in two assets: a risky asset (e.g., stock) and a riskless asset (e.g., bond). Therefore, the firm’s wealth follows a stochastic process. When the wealth is lower than certain legal level, the firm goes bankrupt. Thus how to invest in the fundamental problem of the firm in order to avoid bankruptcy. Under the case of different lending and borrowing rates, we obtain the optimal portfolio strategies for some reasonable objective functions that are the piecewise linear functions of the firm’s current wealth and present some interesting proofs for the conclusions. The optimal policies are easy to be operated for any relevant investor.     

一般均衡下基于产品市场和资本市场的单名CDS定价

次贷危机呼吁新的信用衍生品定价模型, 因此为存在产品市场和资本市场的经济结构建立一般均衡的单名CDS定价模型, 使用最优化求解一般均衡下的商品价格和CDS价格. 可以发现一般均衡的CDS定价具有资本市场和产品市场的因素, 这表示CDS的价格不再是由单纯的资本市场因素决定的, 而是由无风险利率、资本产出弹性、违约率、回收率同时决定的. 通过数量约束用模拟的方式研究多个均衡的动态变化, 发现违约风险的增加使得价格剧烈波动且市场交易萎缩. 在为以中国工商银行为参考资产的CDS定价过程中, 发现各种因素在不同的时期都可能成为定价的主要影响因素. 可以发现, 次贷危机的定价体系存在着信用调整问题和定价与实体经济脱节的问题. 可以认为, 一般均衡下基于产品市场和资本市场的单名CDS定价可以囊括多个市场的交叉影响, 为衍生品定价提供一个新的方向.     

股票价格遵循几何分式Brown运动的期权定价

讨论了股票价格过程遵循几何分式B row n运动的欧式期权定价.由于该过程存在套利机会使得传统的期权定价方法(如资本资产定价模型(CAPM),套利定价模型(APT),动态均衡定价理论(DEPT))不可能对该期权定价.利用保险精算定价法,在对市场无其它任何假设条件下,获得了欧式期权的定价公式.并讨论了在有效期内股票支付已知红利和红利率的推广公式.     

Modelling dependence structure with Archimedean copulas and applications to the iTraxx CDS index

In this paper we model the dependence structure between credit default swap (CDS) and jump risk using Archimedean copulas. The paper models and estimates the different relationships that can exist in different ranges of behaviour. It studies the bivariate distributions of CDS index spreads and the kurtosis of equity return distribution. To take into account nonlinear relationships and different structures of dependency, we employ three Archimedean copula functions: Gumbel, Clayton, and Frank. We adopt nonparametric estimation of copula parameters and we find an extreme co-movement of CDS and stock market conditions. In addition, tail dependence indicates the extreme co-movements and the potential for a simultaneous large loss in stock markets and a significant default risk. Ignoring the tail dependence would lead to underestimation of the default risk premium.