Basket Default Swaps Research Articles

Credit Derivatives : Main Concepts and Pricing

The investors have the possibility to structure and to optimize the risk adjusted performance of their liabilities by diversifying them among several markets and instruments. Credit derivatives are financial instruments which allow a restructuring of the risk/return profiles of credits and permit investors to access new markets. They facilitate the trading of credit risk, its transfer and hedging. They can reduce transaction costs and provide managers a way to "short sell" loans or bonds. They offer to investors the possibility to leverage bond or loan positions. These products complete also the market since they provide a mean to synthesize assets which are not available in the market. This paper surveys the main types of credit derivatives and their valuation. Forward contracts on bonds, Total Return Swaps, Credit Default Swaps, Basket Default Swaps, Credit Default Exchange Swaps, Credit Linked Notes, Basket Default Notes, Levered Portfolio Notes, and Credit Spread Derivatives are subsequently studied in terms of definition, overview, structure, benefits, and pricing.

Copula sensitivity analysis for portfolio credit derivatives

Modeling dependence among input random variables is often critical for performance evaluation of stochastic systems, and copulas provide one approach to model such dependence. In the financial industry, the dependence among default times of different assets drives the pricing of portfolio credit derivatives, including basket default swaps and collateralized debt obligations. Copula sensitivities provide information about how changes in the dependence level affect the output. We study sensitivity analysis of elliptical copulas and Archimedean copulas using infinitesimal perturbation analysis, and an unbiased estimators is derived using conditional Monte Carlo (CMC) to address discontinuities that arise in portfolio credit derivatives. In simulation experiments, the new estimators have smaller variance than other applicable methods.

Elliptical and Archimedean copula models: an application to the price estimation of portfolio credit derivatives

This paper explores the impact of elliptical and Archimedean copula models on the valuation of basket default swaps. We employ Monte Carlo simulation, in connection with the copula models, to estimate the default times and to calculate the swap payment legs and the cumulative swap premium. The numerical experiments reveal some sensitivity analysis on the impact of swap parameters on the fair prices of the 𝑛th-to-default swaps. Finally, using the results presented, an appropriate choice of copula model can be made based on the computation time of the valuation process, and such a choice hugely affects the quantitative risk analysis of the portfolio.

Sensitivity Analysis of Portfolio Credit Derivatives by Conditional Monte Carlo Simulation

We study sensitivity analysis of portfolio credit derivatives, including basket default swaps and collateralized debt obligations. An unbiased estimator is derived using conditional Monte Carlo for sensitivities with respect to systemic parameters (parameters that influence some or all the entities). Copula-based methods are used to model the joint distribution of the default times. Simulation experiments demonstrate the advantages of the proposed derivative estimator over other methods.

On an automatic and optimal importance sampling approach with applications in finance

Calculating high-dimensional integrals efficiently is essential and challenging in many scientific disciplines, such as pricing financial derivatives. This paper proposes an exponentially tilted importance sampling based on the criterion of minimizing the variance of the importance sampling estimators, and its contribution is threefold: (1) A theoretical foundation to guarantee the existence, uniqueness, and characterization of the optimal tilting parameter is built. (2) The optimal tilting parameter can be searched via an automatic Newton’s method. (3) Simplified yet competitive tilting formulas are further proposed to reduce heavy computational cost and numerical instability in high-dimensional cases. Numerical examples in pricing path-dependent derivatives and basket default swaps are provided.

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.

Joint survival probability via truncated invariant copula

Given an intensity-based credit risk model, this paper studies dependence structure between default intensities. To model this structure, we use a multivariate shot noise intensity process, where jumps occur simultaneously and their sizes are correlated. Through very lengthy algebra, we obtain explicitly the joint survival probability of the integrated intensities by using the truncated invariant Farlie–Gumbel–Morgenstern copula with exponential marginal distributions. We also apply our theoretical result to pricing basket default swap spreads. This result can provide a useful guide for credit risk management.

Pricing of Basket Default Swaps Based on Factor Copulas and NIG

Abstract Due to the European debt crisis, the credit default swap (CDS) has been brought back to the spotlight of the financial market. At the meantime, the basket default swaps (BDS) emerges as the hottest issue amid the growing researches about CDS. It is extremely significant to define the correlations between the underlying assets and the default time. The copula approach can accurately specify the joint distribution. In this paper, the factors affecting the company's valuation are classified into systematic factors and non-systematic factors. The fact-based statistics are utilized to analyze the distribution of systematic factors. Then we obtain the samples’ default information from the bond market and the correlation of the samples from stock market. We use the factor copula model to simulate the whole sample's distribution and the correlation between the underlying assets, and then give the price for a BDS.

Change Analysis for the Dependence Structure and Dynamic Pricing of Basket Default Swaps

AbstractIn this paper we use a type of dynamic copula method to characterise the dependence structure between financial assets and price basket default swaps (BDSs). We first employ a goodness‐of‐fit test and a binary segmentation procedure to analyse the change of dependence structure between the obligations underlying a BDS, then present a numerical example to demonstrate the change analysis and BDS pricing process. We find that in different time periods, the best copula fitting the data is not the same; therefore the tranche spreads of the BDS are also different. We also compare our results with those obtained from static copulas and dynamic Gaussian copulas. The results show that the static copula and dynamic Gaussian copula methods underestimate the spreads for riskier tranches and overestimate those for less risky tranches.

Numerical methods to quantify the model risk of basket default swaps

The valuation of basket default swaps depends crucially on the joint default probability of the underlying assets in the basket. It is known that this probability can be modeled by means of a copula function which links the marginal default probabilities to a joint probability. The valuation bears risk due to the uncertainty of the copula, the relation of the assets to each other and the marginal distributions which we call together the model risk. To value basket default swaps and to compute model risk parameters we present an efficient numerical approach based on importance sampling and applicable to different classes of copula models. Our numerical findings show that the choice of the underlying copula model influences strongly the risk profile of the basket and should be tailored advisedly.

On a reduced form credit risk model with common shock and regime switching

Reduced form credit risk models are important ones in credit risk theory. In such a model, certain correlated relations are constructed to represent the default dependence structure among the default intensity processes. In this paper, we introduced a reduced form credit risk model in which the default dependence structures among default intensity processes are described by the so-called common shocks with regime-switching. We derive some closed-form expressions for the joint distribution of the default times and for the pricing formulas of the basket default swaps. We also give numerical results to show the applicable aspects of the proposed model.

Pricing basket default swaps in a tractable shot noise model

We value CDS spreads and k th-to-default swap spreads in a tractable shot noise model. The default dependence is modelled by letting the individual jumps of the default intensity be driven by a common latent factor. The arrival of the jumps is driven by a Poisson process. By using conditional independence and properties of the shot noise processes we derive tractable closed form expressions for the default distribution and the ordered survival distributions. These quantities are then used to price k th-to-default swap spreads. We calibrate a homogeneous version of the model to the term structure on market data from the iTraxx Europe index series sampled during the period 2008-01-14 to 2010-02-11. We perform 435 calibrations in this turbulent period and almost all calibrations yield very good fits. Finally we study k th-to-default spreads in the calibrated model.

Analytical pricing of basket default swaps in a dynamic Hull-White framework

In this paper, some analytical results related to the Hull & White dynamic model of credit portfolio of N obligors in the case of constant jump size are provided. For instance, this specific assumption combined with the moment generating function of the Poisson process lead to analytical calibration for the model with respect to the underlying CDSs. Further, extremely simple analytical expressions are obtained for first-to-default swaps; the more general case of quantities related to nth-to-default swaps also have a closed form and remain tractable for small n. Similarly, pairwise correlation between default indicators also proves to be simple. Although the purpose of this note is not to compare models, we compare the shape of pairwise default correlations of the Hull & White, the Gaussian copula and the Mai & Scherer model with compound Poisson process as Levy subordinator. It is shown that only the models including jumps can lead to non-vanishing default correlation for short-term maturities. Further, these models can generate higher default correlation levels compared to the Gaussian one. When calibrated on default probability of first default time, Jump-based models also lead to much higher default probability for the last obligor to default. Finally, we tackle the problem of simultaneous jumps, which prevent the above class of models to be usable when recoveries are name-specific. To that end, we propose a tractable compromise to deal with baskets being non-homogeneous recovery-wise under the Hull & White model by splitting isolated and non-isolated default events.

Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.

The kth default time distribution and basket default swap pricing

We propose an alternative method for finding the kth default time distribution in a homogeneous portfolio with dependency. Analysing order statistics of default times with a one-factor Gaussian copula model, we explicitly derive the probability distribution. Moreover, we compute the prices of basket default swaps such as the kth to default swaps and m out of n default swaps within our framework. To test the efficiency and accuracy of our method we compare the theoretical prediction with existing methods.

Analytical Pricing of Basket Default Swaps in a Dynamic Hull & White Framework

In this paper, some analytical results related to the Hull & White dynamic model of credit portfolio of N obligors in the case of constant jump size are provided. For instance, this specific assumption combined with the moment generating function of the Poisson process lead to analytical calibration for the model with respect to the underlying CDSs. Further, extremely simple analytical expressions are obtained for first-to-default swaps; the more general case of quantities related to 'n'th-to-default swaps also have a closed form and remain tractable for small n. Similarly, pairwise correlation between default indicators also proves to be simple. Although the purpose of this note is not to compare models, we compare the shape of pairwise default correlations of the Hull & White, the Gaussian copula and the Mai & Scherer model with compound Poisson process as Levy subordinator. It is shown that only the models including jumps can lead to non-vanishing default correlation for short-term maturities. Further, these models can generate higher default correlation levels compared to the Gaussian one. When calibrated on default probability of first default time, Jump-based models also lead to much higher default probability for the last obligor to default. Finally, we tackle the problem of simultaneous jumps, which prevent the above class of models to be usable when recoveries are name-specific. To that end, we propose a tractable compromise to deal with baskets being non-homogeneous recovery-wise under the Hull & White model by splitting isolated and non-isolated default events.

FAST VALUATION OF FORWARD-STARTING BASKET DEFAULT SWAPS

A basket default swap (BDS) is a credit derivative with contingent payments that are triggered by a combination of default events of the reference entities. A forward-starting basket default swap (FBDS) is a BDS starting at a specified future time. Existing analytic or semi-analytic methods for pricing FBDS are time consuming due to the large number of possible default combinations before the BDS starts. This paper develops a fast approximation method for FBDS based on the conditional independence framework. The method converts the pricing of a FBDS to an equivalent BDS pricing problem and combines Monte Carlo simulation with an analytic approach to achieve an effective method. This hybrid method is a novel technique which can be viewed either as a means to accelerate the convergence of Monte Carlo simulation or as a way to estimate parameters in an analytic method that are difficult to compute directly. Numerical results demonstrate the accuracy and efficiency of the proposed hybrid method.

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.

Efficient Algorithms for Basket Default Swap Pricing with Multivariate Archimedean Copulas

We introduce a new importance sampling method for pricing basket default swaps based on exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structure than the existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of our proposed algorithms, we demonstrate several numerical examples compared with the crude Monte Carlo simulation, and show that our proposed estimators produce remarkably small variance with accurately expected values in pricing basket default swaps.

The kth Default Time Distribution and Basket Default Swap Pricing

We propose an alternative method of finding the kth default time distribution in a homogeneous portfolio with dependency. Analyzing order statistics of default times with one factor Gaussian copula model, we explicitly derive probability distribution. Moreover we compute the prices of basket default swaps such as kth to default swaps and m out of n default swaps within our framework. To test efficiency and accuracy of our method we compare the theoretical prediction with some of existing methods.