Copula based simulation procedures for pricing basket Credit Derivatives

This paper deals with the impact of structure of dependency and the choice of procedures for rare-event simulation on the pricing of multi-name credit derivatives such as nth to default swap and Collateralized 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 basket default swaps and 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 is widely used to measure the default risk in basket credit derivatives. Default risk is often considered as a rare-event and then, many studies have shown that many distributions have fatter tails than those captured by the normal distribution. Subsequently, the choice of copula and the choice of procedures for rare-event simulation govern the pricing of basket credit derivatives. An alternative to the Gaussian copula is Clayton copula and t-student copula under importance sampling procedures for simulation which captures the dependence structure between the underlying variables at extreme values and certain values of the input random variables in a simulation have more impact on the parameter being estimated than others .Collateralized Debt Obligations, Basket Default Swaps, Monte Carlo method, One factor Gaussian copula, Clayton copula, t-student copula, importance sampling

THE APPLICATION OF COPULAS IN PRICING DEPENDENT CREDIT DERIVATIVES INSTRUMENTS

The aim of this paper is to use copulas functions to capture the different structures of dependency when we deal with portfolios of dependent credit risks and a basket of credit derivatives. We first present the wellknown result for the pricing of default risk, when there is only one defaultable firm. After that, we expose the structure of dependency with copulas in pricing dependent credit derivatives. Many studies suggest the inadequacy of multinormal distribution and then the failure of methods based on linear correlation for measuring the structure of dependency. Finally, we use Monte Carlo simulations for pricing Collateralized debt obligation (CDO) with Gaussian an Student copulas.default risk, credit derivatives, CDO, copulas functions, Monte Carlo simulations.

Price Calibration of basket default swap: Evidence from Japanese market

The aim of this paper is the price calibration of basket default swap from Japanese market data. The value of this instruments depend on the number of factors including credit rating of the obligors in the basket, recovery rates, intensity of default, basket size and the correlation of obligors in the basket. A fundamental part of the pricing framework is the estimation of the instantaneous default probabilities for each obligor. Because default probabilities depend on the credit quality of the considered obligor, well-calibrated credit curves are a main ingredient for constructing default times. The calibration of credit curves take into account internal information on credit migrations and default history. We refer to Japan Credit Rating Agency to obtain rating transition matrix and cumulative default rates. Default risk is often considered as a rare-event and then, many studies have shown that many distributions have fatter tails than those captured by the normal distribution. Subsequently, the choice of copula and the choice of procedures for rare-event simulation govern the pricing of basket credit derivatives. Joshi and Kainth (2004) introduced an Importance Sampling technique for rare-event that forces a predetermined number of defaults to occur on each path. We consider using Gaussian copula and t-student copula and study their impact on basket credit derivative prices. We will present an application of the Canonical Maximum Likelihood Method (CML) for calibrating t-student copula to Japanese market data.Basket Default Swaps, Credit Curve, Monte Carlo method, Gaussian copula, t-student copula, Japanese market data, CML, Importance Sampling

Hedging residual value risk using derivatives

Abstract: In the leasing industry the lessor faces a risk, at the end of the contract, in not recovering sufficient capital value from resale of the asset. We propose a model to hedge residual value risk using the Gaussian copula methodology. After discussing residual value risk and credit risk modelization, a new derivative product is introduced and analyzed; the Collateralized Residual Values (CRV). The model is applied to an European auto lease portfolio of operating lease contracts pertaining to a major company. Our results indicate that the financial product is easy to customize, and to implement through the contract characteristics and the level of correlation.Residual value risk, credit risk, credit derivatives, factor modeling, copula

Main Flaws of The Collateralized Debt Obligation‘s: Valuation Before And During The 2008/2009 Global Turmoil

As a result of the 2008 financial crisis, the world credit markets stalled significantly and raised the doubts of market participants and policymakers about the proper and fair valuation of financial derivatives and structured products such as collateralized debt obligations (CDOs). The aim of the paper is to contribute to the understanding of CDOs and shed light on CDO valuation based on data before and during the current financial upheaval. We present the One Factor Model based on a Gaussian Copula and test five hypothesizes. Based on the results we discovered four main deficiencies of the CDO market. For our modelling we used data of the CDX NA IG 5Y V3 index from 20 September 2007 until 27 February 2009 and its quotes we appropriately transform into CDO quotes. Based on the results we discovered four main deficiencies of the CDO market: i) an insufficient analysis of underlying assets by both investors and rating agencies; ii) the valuation model was usually based only on expected cash-flows when neglecting other factors such mark-to-market losses or correlation risk; iii) mispriced correlation; and finally iv) the mark-to-market valuation obligation for financial institutions should be reviewed. Based on the mentioned recommendations we conclude that the CDO market has a chance to be regenerated. However, the future CDO market would then be more conscious, driven by smarter motives rather than by poor understanding of risks involved in CDOs.collateralized debt obligations, Gaussian Copula, valuation, securitization

Credit risk tools: an overview

This document presents several Credit Risk tools which have been developed for the Credit Derivatives Risk Management. The models used in this context are suitable for the pricing, sensitivity/scenario analysis and the derivation of risk measures for plain vanilla credit default swaps (CDS), standardized and bespoke collateralized debt obligations (CDO) and, in general, for any credit risk exposed A/L portfolio.\\ In this brief work we compute the market implied probability of default (PD) from market spreads and the theoretical CDS spreads from historical default frequencies. The loss given default (LGD) probability distribution has been constructed for a large pool portfolio of credit obligations exploiting a single-factor gaussian copula with a direct convolution algorithm computed at several default correlation parameters. Theoretical CDO tranche prices have been calculated. We finally design stochastic cash-flow stream model simulations to test fair pricing, compute credit value at risk (CV@R) and to evaluate the one year total future potential exposure (FPE) and derive the value at risk (V@R) for a CDO equity tranche exposure.interest rate swap, spot rate term structure, credit default swap, probability of default, copula function, direct convolution, loss given default, collateralized debt obligation, exposure at default, stochastic cash-flow stream model, value at risk, credit value at risk, future potential exposure, Monte Carlo simulation.

Pricing contingent claims on credit and carbon single and multiple underlying assets

This thesis proposes alternative ways to price contingent claims written on portfolios of credit instruments as well as on carbon underlying assets. On the first topic of this research we tackle the pricing of Collateralized Debt Obligations (CDOs) by introducing two different approaches through the application of respectively Johnson SB distributions and entropy optimization principles, in contrast to market standard pricing approaches based on variations of the Gaussian copula model. The relevance of this topic is in line with the events that unfolded during the “credit crunch” of mid-2007 to early 2009, when CDOs made headlines as being responsible for more than $542 billion in losses through writedowns by financial institutions. On the second topic we propose a pricing methodology for Emission Reduction Purchase Agreement (ERPA) contracts. These are instruments based on carbon as an asset class and created by the emergence of an international carbon market that followed the adoption of the Kyoto Protocol (KP) to the United Nations Framework Convention on Climate Change (UNFCCC) in December 1997. ERPAs are of vital importance to the function of KP’s market mechanisms and the carbon markets at large as they formalize transactions of emissions reduction offsets between sellers and buyers, more specifically transactions involving Certified Emission Reductions (CERs). We propose a pricing methodology based on stochastic modeling of CER volume delivery risk and carbon prices as the two main drivers underlying ERPAs, and apply it to a case study on a run-of-river hydro power CDM project activity in China

CDO and HAC

Modelling portfolio credit risk is one of the crucial challenges faced by financial services industry in the last few years. We propose the valuation model of collateralized debt obligations (CDO) based on copula functions with up to three parameters, with default intensities estimated from market data and with a random loss given default that is correlated with default times. The methods presented are used to reproduce the spreads of the iTraxx Europe tranches. We apply hierarchical Archimedean copulae (HAC) whose construction allows for the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependence among the assets from the same group is specified with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. The copula with two and three parameters models the relation between the loss given default and the default times. Our approach describes the market prices better than the standard pricing procedure based on the Gaussian distribution.CDO, CDS, multivariate distributions, Copulae, correlation smile, loss given default

CDO Pricing with Copulae

Modeling the portfolio credit risk is one of the crucial issues of the last years in the financial problems. We propose the valuation model of Collateralized Debt Obligations based on a one- and two-parameter copula and default intensities estimated from market data. The presented method is used to reproduce the spreads of the iTraxx Europe tranches. The two-parameter model incorporates the fact that the risky assets of the CDO pool are chosen from six different industry sectors. The dependency among the assets from the same group is described with the higher value of the copula parameter, otherwise the lower value of the parameter is ascribed. Our approach outperforms the standard market pricing procedure based on the Gaussian distribution.CDO, CDS, multifactor models, multivariate distributions, Copulae, correlation smile

Dynamic hedging of portfolio credit derivatives

We compare the performance of various hedging strategies for index collateralized debt obligation (CDO) tranches across a variety of models and hedging methods during the recent credit crisis. Our empirical analysis shows evidence for market incompleteness: a large proportion of risk in the CDO tranches appears to be unhedgeable. We also show that, unlike what is commonly assumed, dynamic models do not necessarily perform better than static models, nor do high-dimensional bottom-up models perform better than simpler top-down models. When it comes to hedging, top-down and regression-based hedging with the index provide significantly better results during the credit crisis than bottom-up hedging with single-name credit default swap (CDS) contracts. Our empirical study also reveals that while significantly large moves—“jumps”—do occur in CDS, index, and tranche spreads, these jumps do not necessarily occur on the default dates of index constituents, an observation which shows the insufficiency of some recently proposed portfolio credit risk models.hedging, credit default swaps, portfolio credit derivatives, index default swaps, collateralized debt obligations, portfolio credit risk models, default contagion, spread risk, sensitivity-based hedging, variance minimization