Multi-name Credit Derivatives Research Articles

FACTOR COPULA MODEL FOR PORTFOLIO CREDIT RISK

A critical aspect in the valuation and risk management of multi-name credit derivatives is the modeling of the dependence among sources of credit risk. The dependence modeling poses difficulties in the pricing of a multi-name credit derivatives, in the estimation of the value-at-risk of a portfolio, or in the pricing of some other basket credit derivative as the description not only on the default arrival in an individual reference entity but on the default dependence among entities in the portfolio should be considered. Although the elliptical models have been widely used due to their mathematical tractability, the dependence modeling using the multi-dimensional Lévy process has shown growing interest among researchers despite its complexity. In this paper, we introduce one factor copula model for portfolio credit risk based on Normal Tempered Stable (NTS) distribution and calibrate the model through 5-year synthetic Collateralized Debt Obligation (CDO) tranche spreads under a large homogeneous portfolio approximation. The calibration results show that the one factor copula model based on NTS distribution is more flexible and provides a dependence structure fitting market CDO tranche spreads. As one of the major applications of the dependence modeling in credit risk, this model shares the advantage of the Gaussian one factor model, and all extensions and implementation methods used for it can be utilized.

Increment Variance Reduction Techniques with an Application to Multi-name Credit Derivatives

Increment variance reduction techniques are add-ons to Monte Carlo (MC) simulations. They make MC simulations converging faster by repeating the number of simulations with an incremental rate derived from mathematical functions. Besides speeding up MC simulations, the major advantage of increment techniques is their ability to handle large numbers of simulations avoiding memory saturation and overflow which occur when a plain MC simulation is involved in the pricing of multi-name credit derivatives. A trend among authors pricing financial securities with MC simulation has been to choose Quasi-Monte Carlo (QMC) methods using deterministic sequences instead of MC methods involving pseudorandom generators such as congruential generator and Mersenne twister. The Increment family models circumvent the constraint of identifying the optimal QMC sequence to price a given security by using a common generator such as Matlab-LCG-Xor RNG and determining the optimal mathematical function of incrementation of MC simulations that will make the pricing of the security adequate. Market participants in need of selecting a reliable numerical method for pricing complex financial securities such as multi-name credit derivatives will find our paper appealing.

Extending the intensity model with joint defaults to incorporate the lasting effects from common credit events

AbstractThis paper studies how the lasting effects of common credit events influence default probability distribution and the prices of multiname credit derivatives. Based on a joint defaults model where common credit events are used to generate simultaneous defaults, we extend the model to allow for their impacts to last for a longer while. The default intensity of each entity is heightened significantly while the impact still has an influence, until some time later when this effect fades away. Incorporating these lasting effects helps to generate higher default correlation, which is more consistent with today's highly correlated financial markets. The proposed model can be either formulated as a Markov chain or implemented by Monte Carlo simulation in order to calculate the default probability distributions and multiname derivatives prices. Our numerical results demonstrate the strong influences from the lasting effects and provide a justification of their incorporation.

Dependent defaults and losses with factor copula models

Abstract We present a class of flexible and tractable static factor models for the term structure of joint default probabilities, the factor copula models. These high-dimensional models remain parsimonious with paircopula constructions, and nest many standard models as special cases. The loss distribution of a portfolio of contingent claims can be exactly and efficiently computed when individual losses are discretely supported on a finite grid. Numerical examples study the key features affecting the loss distribution and multi-name credit derivatives prices. An empirical exercise illustrates the flexibility of our approach by fitting credit index tranche prices.

Contemporary Risk Management

The purpose of this section on risk management is to analyze the contemporary research in this field. Fifteen selected research studies were compared and contrasted to provide a better understanding on how to improve hedging performance and reduce risk to meet corporate financial goals. An investigation and evaluation of recent trends with risk management to understand best practices in developing hedging and risk mitigation techniques was conducted. The first research study explored was about complex economic systems from Smith. The second pertained to tail risk in Asian markets by So and Tsu. The third investigated index volatility with the S&P 500 using different GARCH models by Srinivasan. The fourth study involved extreme returns probed by Viebig and Poddig. The next investigation was with concerns in operation risk by Neil, Hager and Anderson. The sixth research topic proposed by Reimann and Tupak explored evolutionary financial stylized facts. Seventh, potential future exposure relating to credit risk was inspected by Ng, Peterson, and Rodriguez. The eighth article considered the defaultable term structure in the context of credit events by Bruti-Liberati et al. The ninth study illustrated defaultable swaps modeling of multi-name credit derivatives in a default basket by Jabbour, Kramin, and Young. The tenth research study explored commodity price risk management in terms of Value-at-Risk and Liquidity-adjusted Value-at-Risk by Al Janabi. Chang, McAleer, and Tansuchat was cited in the eleventh research study involving correlations in the crude oil markets. For the twelfth category static hedging in the hydropower markets was explored by Fleten, Brathen, and Nissen-Meyer. For the thirteenth study investigated, which was cite from Masood, Aktan and Chandhary, involved investment decisions in risk management by Saudi risk managers. The fourteenth study discussed hedging credit and interest rate derivatives by Errais. Lastly, portfolio optimization and dynamic hedging using receding horizontal control methods was engaged by Meindl. In the conclusion of this Depth component a discussion on the synthesis of relevant research related to risk management modeling and techniques was conducted. Comments on the financial crisis of 2008 and how to monitor or possibly mitigate such a crisis will be mentioned. Within the conclusion section each research category mentioned above was summarized within the framework of risk management which promotes social change by reducing portfolio risk and asset depreciation. In addition to the summary of these research studies, some questions were explored to provide possible investigational paths on hedging performance and risk mitigation. For example, can risk management techniques and models be useful to the average investor or trader? Should a risk manager implement hedging strategies in a portfolio? For complicated firms is there a graphical way to monitor risk? It is within this framework that current research in corporate finance related to risk management was explored.

Pricing Credit Risk Derivatives with R

This paper shows the implementation of some pricing techniques for multiname credit derivatives. Among these financial instruments we consider Basket Default Swaps (BDS) and Collateralised Debt Obligations (CDO). The pricing methodologies put forward are based on Monte Carlo simulations. These techniques are written in the R programming language. This software framework provides an ample set of functions for credit risk modelling. Two main issues are put forward in this paper: the first one is the employment of Quasi Random Number (QRN) for reducing the variance of price estimate in the Monte Carlo experiment; the second one is the code parallelization of some pricing techniques using R end-user techniques.

Modeling Term Structure of Default Correlation

Students of option theory have spent decades trying to find the most plausible way to explain the implied volatility smile. Multi-name credit derivatives feature a similar problem in the form of the correlation skew. Prices for tranche securities with different exposures to the credit risk in an underlying pool of assets are greatly affected by the expected default correlation. But implying that correlation from the market prices of the tranches yields a different value from each one, in a pattern that generally increases with seniority. As with the volatility smile, much modeling effort has gone into trying to smooth out the correlation skew, with only limited success. In this article, the author proposes a new idea—that the correlation skew may be mainly a term structure effect. Because the earliest defaults in a pool will affect the equity tranche first, while the senior tranches can only be hit later, after many defaults have already happened, there is a clear connection between maturity and a tranche’s default risk exposure. This article develops a model of the correlation term structure and shows that it can explain tranche prices surprisingly well, without requiring a very different correlation for each tranche.

Analysis of Systematic Risks in Multi-Name Credit and Equity Markets

In this paper, we present an intensity-based common factor model that is used to analyze the valuation of common systematic risks in multi-name credit and equity markets. In particular, we use a hybrid intensity model to price single-name credit instruments such as credit default swaps (CDSs), multi-name credit derivatives such as collateralized debt obligations (CDOs), and equity index options such as calls and puts on the S&P 500. Once we have the expressions for the model prices of these instruments, we then calibrate the model parameters to fit the market data. We study two problems, the “forward” and “backward” problems: in the former, we start from equity index options and then compute the CDO tranche spreads, while in the latter, we fit the parameters to the CDO tranche spreads and then compute the equity index option prices and implied volatilities. We find that, based on our hybrid model, the systematic risks in the two markets were similar from 2004 to 2007, while the credit market incorporated far greater systematic risk than the equity market during the financial crisis from 2008 to 2010.

A Multivariate Default Model with Spread and Event Risk

We present a new portfolio default model based on a conditionally independent and identically distributed (CIID) structure of the default times. It combines an intensity-based ansatz in the spirit of Duffie and Gârleanu (2001). Risk and valuation of collateralized debt obligations. Financial Analysts Journal, 57(1), 41–59. with the Lévy subordinator concept introduced in Mai and Scherer (2009). A tractable multivariate default model based on a stochastic time-change. International Journal of Theoretical and Applied Finance, 12(2), 227–249. We aim at exploiting the computational advantages of the CIID framework for evaluating multiname credit derivatives, while incorporating two central drivers for credit products. More precisely, we allow for both a dynamic evolution of the portfolio credit default swap (CDS) spread (unlike static copula models) and cataclysmic events allowing for simultaneous defaults (unlike intensity-based portfolio loss processes). While the former feature is considered to be crucial for consistently hedging credit products, the second property is supposed to take into account default clusters and the market’s fear of extreme events. For applications, the model is approximated by a related top-down representation of the portfolio loss process. It is shown how to coherently calculate hedging deltas for collateralized debt obligations (CDOs) w.r.t. portfolio CDS and how to consistently calibrate the model to the two products. Both tasks solely require the computation of one-dimensional (Laplace inversion) integrals and can be carried out within fractions of a second. Illustrating the stability and functionality of the pricing approach, the new model and the models it is related to are calibrated to a daily time-series of iTraxx Europe index CDS and CDOs. We find the fitting results of the presented model to be very promising and conclude that it may be used for the dynamic pricing and hedging of credit derivatives.

PRICING CREDIT DERIVATIVES IN A MARKOV-MODULATED REDUCED-FORM MODEL

Numerous incidents in the financial world have exposed the need for the design and analysis of models for correlated default timings. Some models have been studied in this regard which can capture the feedback in case of a major credit event. We extend the research in the same direction by proposing a new family of models having the feedback phenomena and capturing the effects of regime switching economy on the market. The regime switching economy is modeled by a continuous time Markov chain. The Markov chain may also be interpreted to represent the credit rating of the firm whose bond we seek to price. We model the default intensity in a pool of firms using the Markov chain and a risk factor process. We price some single-name and multi-name credit derivatives in terms of certain transforms of the default and loss processes. These transforms can be calculated explicitly in case the default intensity is modeled as a linear function of a conditionally affine jump diffusion process. In such a case, under suitable technical conditions, the price of credit derivatives are obtained as solutions to a system of ODEs with weak coupling, subject to appropriate terminal conditions. Solving the system of ODEs numerically, we analyze the credit derivative spreads and compare their behavior with the nonswitching counterparts. We show that our model can easily incorporate the effects of business cycle. 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 floating interest rate, high default intensity rate, and high volatility. We also model the effects of firm restructuring on the credit spread, in case of a default.

Credit Portfolios, Credibility Theory, and Dynamic Empirical Bayes

We begin with a review of (a) the pricing theory of multiname credit derivatives to hedge the credit risk of a portfolio of corporate bonds and (b) current approaches to modeling correlated default intensities. We then consider pricing of insurance contracts using credibility theory in actuarial science. After a brief discussion of the similarities and differences of both pricing theories, we propose a new unified approach, which uses recent advances in dynamic empirical Bayes modeling, to evolutionary credibility in insurance rate-making and default modeling of credit portfolios.

RISK PREMIA AND OPTIMAL LIQUIDATION OF CREDIT DERIVATIVES

This paper studies the optimal timing to liquidate credit derivatives in a general intensity-based credit risk model under stochastic interest rate. We incorporate the potential price discrepancy between the market and investors, which is characterized by risk-neutral valuation under different default risk premia specifications. We quantify the value of optimally timing to sell through the concept of delayed liquidation premium, and analyze the associated probabilistic representation and variational inequality. We illustrate the optimal liquidation policy for both single-name and multi-name credit derivatives. Our model is extended to study the sequential buying and selling problem with and without short-sale constraint.

A Top-Down Approach to Multiname Credit

A multiname credit derivative is a security that is tied to an underlying portfolio of corporate bonds and has payoffs that depend on the loss due to default in the portfolio. The value of a multiname derivative depends on the distribution of portfolio loss at multiple horizons. Intensity-based models of the loss point process that are specified without reference to the portfolio constituents determine this distribution in terms of few economically meaningful parameters and lead to computationally tractable derivatives valuation problems. However, these models are silent about the portfolio constituent risks. They cannot be used to address applications that are based on the relationship between portfolio and component risks, for example, constituent risk hedging. This paper develops a method that extends these models to the constituents. We use random thinning to decompose the portfolio intensity into a sum of constituent intensities. We show that a thinning process, which allocates the portfolio intensity to constituents, uniquely exists, and is a probabilistic model for the next-to-default. We derive a formula for the constituent default probability in terms of the thinning process and the portfolio intensity, and develop a semi-analytical transform approach to evaluate it. The formula leads to a calibration scheme for the thinning processes and an estimation scheme for constituent hedge sensitivities. An empirical analysis for September 2008 shows that the constituent hedges generated by our method outperform the hedges prescribed by the Gaussian copula model, which is widely used in practice.

A Copula Approach to Default Correlation and the Pricing of Basket Default Swap

SYNOPTIC ABSTRACTIn recent years, rapidly growing credit derivative market produces many complex credit derivatives, such as multi-name credit derivatives whose trade is very active and the pricing of which is more complicated than single-name credit derivatives. In this paper, we first construct the hazard rate function considering the practice of discrete credit spread, and derive the joint default distribution by applying copulas to characterize the default correlation; then we use three copula functions, Gaussian, Gumbel and Clayton, to simulate the correlated time to default of obligors, and price a basket default swap and make sensitive analysis under different Kendall tau.

Copula based simulation procedures for pricing collateralised debt obligations

The aim of this paper is to study the impact of structure of dependency on the pricing of multi-name credit derivatives such as collateralised debt obligations (CDO). The correlation between names defaulting has an effect on the value of the basket credit derivatives. We present a copula based simulation procedure for pricing CDO under different structure of dependency and assessing the influence of different price drivers (correlation, hazard rates and recovery rates) on modelling portfolio losses. Gaussian copulas and Monte Carlo simulation are widely used to measure the default risk in basket credit derivatives. Many studies have shown that many distributions have fatter tails than those captured by the normal distribution. We use distributions with fat tails such as the t-student distribution. The paper has several practical implications that are of value for financial hedgers and engineers, financial regulators, central banks, and financial risk managers.

Multi-Scale Time-Changed Birth Processes for Pricing Multi-Name Credit Derivatives

We develop two parsimonious models for pricing multi-name credit derivatives. We derive closed form expression for the loss distribution, which then can be used in determining the prices of tranche and index swaps and more exotic derivatives on these contracts. Our starting point is the model of Ding et al., 2008, which takes the loss process as a time-changed birth process. We introduce stochastic parameter variations into the intensity of the loss process and use the multi-time scale approach of Fouque et al., 2003 and obtain explicit perturbation approximations to the loss distribution. We demonstrate the competence of our approach by calibrating it to the CDX index data.

Utility valuation of multi-name credit derivatives and application to CDOs

We study the impact of risk-aversion on the valuation of credit derivatives. Using the technology of utility-indifference pricing in intensity-based models of default risk, we analyse resulting yield spreads in multi-name credit derivatives, particularly CDOs. We study first the idealized problem with constant intensities where solutions are essentially explicit. We also give the large portfolio asymptotics for this problem. We then analyse the case where the firms have stochastic default intensities driven by a common factor, which can be viewed as another extreme from the independent case. This involves the numerical solution of a system of reaction-diffusion PDEs. We observe that the nonlinearity of the utility-indifference valuation mechanism enhances the effective correlation between the times of the credit events of the various firms leading to non-trivial senior tranche spreads, as often seen from market data.

Multiscale Intensity Models and Name Grouping for Valuation of Multi-Name Credit Derivatives

The pricing of collateralized debt obligations (CDOs) and other basket credit derivatives is contingent upon (i) a realistic modelling of the firms' default times and the correlation between them, and (ii) efficient computational methods for computing the portfolio loss distribution from the individual firms' default time distributions. Factor models, a widely used class of pricing models, are computationally tractable despite the large dimension of the pricing problem, thus satisfying issue (ii), but to have any hope of calibrating CDO data, numerically intense versions of these models are required. We revisit the intensity-based modelling setup for basket credit derivatives and, with the aforementioned issues in mind, we propose improvements (a) via incorporating fast mean-reverting stochastic volatility in the default intensity processes, and (b) by considering homogeneous groups within the original set of firms. This can be thought of as a hybrid of top-down and bottom-up approaches. We present a calibration example, including data in the midst of the 2008 financial credit crisis, and discuss the relative performance of the framework.

An approximation for credit portfolio losses

Mixture models play an important role in the modeling of portfolio losses. In these models the risk of default of individual obligors (indexed by i ∈ {1, . . . , m}) depends on an underlying set of common economic factors, denoted Ψ. Given these factors, the losses due to default li of individual obligors are assumed to be stochastically independent. Dependence between different obligors stems only from dependence of the individual default probabilities on the set of factors. These models are used for both risk management of credit portfolios and valuation of multi-name credit derivatives. The current article investigates both issues. The numerical evaluation of the portfolio loss distribution is usually based on the two-stage structure of mixture models. For instance, in order to sample from the loss distribution by standard Monte Carlo, one generates first a realization of the systematic factor variable Ψ. In a second step one generates a sequence of independent variates li, 1 ≤ i ≤ m, according to the conditional distribution of (li)1≤i≤m given Ψ. Standard Monte Carlo can be quite slow, and so various numerical techniques for estimating the distribution of the total loss of a portfolio in mixture models have been developed. In this paper we focus on the the second stage, i.e. the conditional loss distribution given the underlying factors, and propose an alternative way for evaluating the conditional distribution of the total loss L = ∑m i=1 li. Our approximation is based on a central limit theorem; error bounds can be derived from the Berry-Esseen-inequality. We compare the numerical performance of our method relative to the standard Vasicek-approximation and the true loss distribution obtained by standard Monte Carlo methods. It turns out that our suggested approximation technique often provides more accurate results than the Vasicek-approximation while being computationally less expensive

Modeling correlated defaults: first passage model under stochastic volatility

Default dependency structure is crucial in pricing multi-name credit derivatives as well as in credit risk management. In this paper, we extend the first passage model for one name with stochastic volatility (Fouque-Sircar-Solna, Applied Mathematical Finance 2006) to the multi-name case. Correlation of defaults is generated by correlation between the Brownian motions driving the individual names as well as through common stochastic volatility factors. A numerical example for the loss distribution of a portfolio of defaultable bonds is examined after stochastic volatility is incorporated.