Interacting default intensity with a hidden Markov process

In this paper we consider a reduced-form intensity-based credit risk model with a hidden Markov state process. A filtering method is proposed for extracting the underlying state given the observation processes. The method can be applied to a wide range of problems. Based on this model, we derive the joint distribution of multiple default times without imposing stringent assumptions on the form of default intensities. Closed-form formulas for the distribution of default times are obtained which are then applied to solve a number of practical problems such as hedging and pricing credit derivatives. The method and numerical algorithms presented can be applicable to various forms of default intensities.     

Credit derivatives with multiple debt issues

We evaluate the most actively traded types of credit derivatives within a unified pricing framework that allows for multiple debt issues. Since firms default on all of their obligations, total debt is instrumental in the likelihood of default and therefore in credit derivatives valuation. We use a single factor interest rate model where the exponential default frontier is based on total debt and is made coherent with observed bond prices. Analytical formulae are derived for credit default swaps, total return swaps (both fixed-for-fixed and fixed-for-floating), and credit risk options (CROs). Price behaviors and hedging properties of all these credit derivatives are investigated. Simulations document that credit derivatives prices may be significantly affected by terms of debt other than those of the reference obligation. The analysis of CROs indicates their superior ability to fine-tune the hedging of magnitude and arrival risks of default.     

Hedging bonds subject to credit risk

This paper provides a simple and practical approach to hedging bonds that are subject to credit risk. Three new hedge ratios are derived and tested and the roles of basis risk and diversification is investigated. Empirical tests reveal that basis risk is an important factor in hedging corporate bonds. These tests identify a need for new interest rate derivatives where the underlying asset is subject to credit risk.     

Derivative Pricing with Liquidity Risk: Theory and Evidence from the Credit Default Swap Market

We derive an equilibrium asset pricing model incorporating liquidity risk, derivatives, and short‐selling due to hedging of nontraded risk. We show that illiquid assets can have lower expected returns if the short‐sellers have more wealth, lower risk aversion, or shorter horizon. The pricing of liquidity risk is different for derivatives than for positive‐net‐supply assets, and depends on investors' net nontraded risk exposure. We estimate this model for the credit default swap market. We find strong evidence for an expected liquidity premium earned by the credit protection seller. The effect of liquidity risk is significant but economically small.     

A Simple Approach to Valuing Risky Fixed and Floating Rate Debt

We develop a simple approach to valuing risky corporate debt that incorporates both default and interest rate risk. We use this approach to derive simple closed-form valuation expressions for fixed and floating rate debt. The model provides a number of interesting new insights about pricing and hedging corporate debt securities. For example, we find that the correlation between default risk and the interest rate has a significant effect on the properties of the credit spread. Using Moody's corporate bond yield data, we find that credit spreads are negatively related to interest rates and that durations of risky bonds depend on the correlation with interest rates. This empirical evidence is consistent with the implications of the valuation model.     

Credit risk modeling with affine processes

This article combines an orientation to credit risk modeling with an introduction to affine Markov processes, which are particularly useful for financial modeling. We emphasize corporate credit risk and the pricing of credit derivatives. Applications of affine processes that are mentioned include survival analysis, dynamic term-structure models, and option pricing with stochastic volatility and jumps. The default-risk applications include default correlation, particularly in first-to-default settings. The reader is assumed to have some background in financial modeling and stochastic calculus.     

Pricing and managing risks of European-style options in a Markovian regime-switching binomial model

We discuss the pricing and risk management problems of standard European-style options in a Markovian regime-switching binomial model. Due to the presence of an additional source of uncertainty described by a Markov chain, the market is incomplete, so the no-arbitrage condition is not sufficient to fix a unique pricing kernel, hence, a unique option price. Using the minimal entropy martingale measure, we determine a pricing kernel. We examine numerically the performance of a simple hedging strategy by investigating the terminal distribution of hedging errors and the associated risk measures such as Value at Risk and Expected Shortfall. The impacts of the frequency of re-balancing the hedging portfolio and the transition probabilities of the modulating Markov chain on the quality of hedging are also discussed.     

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.     

Risk and hedging: Do credit derivatives increase bank risk?

The objective of this paper is to investigate whether financial innovation of credit derivatives makes banks more exposed to credit risk. Although credit derivatives are important for hedging and securitizing credit risk – and thereby likely to enhance the sharing of such risk – some commentators have raised concerns that they may destabilize the banking sector. This paper investigates this issue in a simple model driven by costs of financial distress. The analysis identifies two effects of credit derivatives innovation – they enhance risk sharing as suggested by the hedging argument – but they also make further acquisition of risk more attractive. The latter effect, if dominant, can therefore destabilize the banking sector. The critical factor is, perhaps surprisingly, the competitive nature of the existing underlying credit markets. As these markets become more elastic the threat of destabilization increases. The paper discusses issues related to bank regulation within the context of the model.     

Up and down credit risk

This paper discusses the main modeling approaches that have been developed for handling portfolio credit derivatives, with a focus on the question of hedging. In particular, the so-called top, top down and bottom up approaches are considered. We give some mathematical insights regarding the fact that information, namely the choice of a relevant model filtration, is the major modeling issue. In this regard, we examine the notion of thinning that was recently advocated for the purpose of hedging a multi-name derivative by single-name derivatives. We then illustrate by means of numerical simulations (semi-static hedging experiments) why and when the portfolio loss process may not be a ‘sufficient statistic’ for the purpose of valuation and hedging of portfolio credit risk.     

Pricing Derivatives on Financial Securities Subject to Credit Risk

This article provides a new methodology for pricing and hedging derivative securities involving credit risk. Two types of credit risks are considered. The first is where the asset underlying the derivative security may default. The second is where the writer of the derivative security may default. We apply the foreign currency analogy of Jarrow and Turnbull (1991) to decompose the dollar payoff from a risky security into a certain payoff and a “spot exchange rate.” Arbitrage-free valuation techniques are then employed. This methodology can be applied to corporate debt and over the counter derivatives, such as swaps and caps.     

Option pricing and hedging in different cyclical structures: a two-dimensional Markov-modulated model

The critical role of interest rate risk and associated regime-switching risk in pricing and hedging options is examined using a closed-form valuation model. Equity call options are valued under the proposed 2-dimensional Markov-modulated model in which asset prices and interest rates exhibit Markov regime-switching features. In addition, the relationship between cyclical structures and option prices are analyzed using a time-varying transition probability matrix. The proposed model can enhance the forecast transition probabilities in an out-sample period. The cycle-stylized effect of an economy exhibits different impacts on option prices and hedging strategies in a short- and a long-cycle economy. Our closed-form formula based on more realistic specifications with respect to business-cyclical structures in various financial markets is more appropriate for pricing and hedging options.     

Pricing and hedging of credit derivatives via?the?innovations approach to nonlinear filtering

In this paper, we propose a new, information-based approach for modelling the dynamic evolution of a portfolio of credit risky securities. In our setup, market prices of traded credit derivatives are given by the solution of a nonlinear filtering problem. The innovations approach to nonlinear filtering is used to solve this problem and to derive the dynamics of market prices. Moreover, the practical application of the model is discussed: we analyse calibration, the pricing of exotic credit derivatives and the computation of risk-minimizing hedging strategies. The paper closes with a few numerical case studies.     

Pricing Weather Derivatives Using the Indifference Pricing Approach

Abstract

This paper adopts an incomplete market pricing model–the indifference pricing approach–to analyze valuation of weather derivatives and the viability of the weather derivatives market in a hedging context. It incorporates price risk, weather/quantity risk, and other risks in the financial market. In a mean-variance framework, the relationship between the actuarial price and the indifference price of weather derivatives is analyzed, and conditions are obtained concerning when the actuarial price does not provide an appropriate valuation for weather derivatives. Conditions for the viability of the weather derivatives market are examined. This paper also analyzes the effects of partial hedging, natural hedges, basis risk, quantity risk, and price risk on investors’ indifference prices by examining the distributional impacts of the stochastic variables involved.     

信用违约风险传染模型的比较研究

本文试图对几种有代表性的模型进行比较,来分析由于建模方式的不同,而导致的对信用期权定价和对冲的结果的不同.如果将违约风险传染考虑进去,类似德隆帝国崩溃的事件,或许就能避免.     

The analytic pricing of asymmetric defaultable swaps

Swaps where both parties are exposed to credit risk still lack convincing pricing mechanisms. This article presents a reduced-form model where the event of default is related to structural characteristics of each party. The cash flows submitted to credit risk are identified before the swap is priced. Analytical pricing formulas for interest rate and currency swaps are computed using a Gaussian model for risky bonds. Currency swaps exhibit additional correlation risk. The benefits from netting depend on the balance between exposures and market conditions in valuation. We show that sources of credit risk asymmetries are also likely to impact on credit spreads.     

Handling Weather Related Risks Through the Financial Markets: Considerations of Credit Risk, Basis Risk, and Hedging

The profits of many businesses are strongly affected by weather related events, and insurance against weather related risks (acts of God) has been a traditional domain for transfer of (certain) of these risks. Recent innovations in the capital market have now provided financial instruments to transfer and hedge some of these risks. Unlike insurance solutions, however, using these financial derivative instruments creates a situation in which the return to the purchaser of the instrument is no longer perfectly correlated with the loss experienced. Such a mismatch creates new risks which must be examined and evaluated as part of ascertaining cost effective risk management plans. Two newly engendered risks, basis risk (the risk created by the fact that the return from the financial derivative is a function of weather at a prespecified geographical location which may not be identical to the location of the firm) and credit risk (the risk that the counterparty to the derivative contract may not perform), are analyzed in this article. Using custom tailored derivatives from the over the counter market can decrease basis risk, but increases credit risk. Using standardized exchange traded derivatives decreases credit risk but increases basis risk. Here also the effectiveness of using hedging methods involving forwards and futures having linear payoffs (linear hedging) and methods using derivatives having nonlinear payoffs such as those involving options (nonlinear hedging) for the purpose of hedging basis risk are examined jointly with credit risk.     

基于套利理论与ICIR模型的债券市场发行定价偏离研究

基于套利定价理论与利率期限结构理论,运用Tobit多元线性回归模型,得出债券发行定价的主要影响因素为债券无风险利率、债券期限溢价、债项信用评级、债券主体信用评级和债券赎回风险溢价,在此基础上再通过改进的CIR定价模型(ICIR)对2006～2010年各债券定价偏离现象进行研究的结果表明,在1%的显著性水平上,ICIR模型测算的债券理论价格通过了二级市场的定价检验,ICIR模型对债券发行定价偏离进行检验具有较强的合理性;同时,从发行年份来看,近五年来,债券定价偏离总体呈逐年下降趋势,债券发行定价与ICIR定价与二级市场定价逐步接轨,市场化程度越来越高。     

Extreme Events and the Copula Pricing of Commercial Mortgage-Backed Securities

Commercial mortgage-backed securities (CMBS), as a portfolio-based financial product, have gained great popularity in financial markets. This paper extends Childs, Ott and Riddiough’s (J Financ Quant Anal, 31(4), 581–603, 1996) model by proposing a copula-based methodology for pricing CMBS bonds. Default on underlying commercial mortgages within a pool is a crucial risk associated with CMBS transactions. Two important issues associated with such default—extreme events and default dependencies among the mortgages—have been identified to play crucial roles in determining credit risk in the pooled commercial mortgage portfolios. This article pays particular attention to these two issues in pricing CMBS bonds. Our results show the usefulness and potential of copula-based models in pricing CMBS bonds, and the ability of such models to correctly price CMBS tranches of different seniorities. It is also important to sufficiently consider complex default dependence structure and the likelihood of extreme events occurring in pricing various CMBS bonds.     

Pricing credit derivatives under incomplete information: a nonlinear-filtering approach

This paper considers a general reduced-form pricing model for credit derivatives where default intensities are driven by some factor process X. The process X is not directly observable for investors in secondary markets; rather, their information set consists of the default history and of noisy price observations for traded credit products. In this context the pricing of credit derivatives leads to a challenging nonlinear-filtering problem. We provide recursive updating rules for the filter, derive a finite-dimensional filter for the case where X follows a finite-state Markov chain, and propose a novel particle-filtering algorithm. A numerical case study illustrates the properties of the proposed algorithms.