On pricing and hedging basket credit derivatives with dependent structure

Whereas a widely-held current view is that the correlation parameters occurring in the risk-neutral procedure for the pricing of basket credit derivatives should, at least in principle, be set equal to their real-world values obtained from historical data, the conclusion of this article is that the correlation parameters are risk-neutral parameters with a range of allowable values. As a result, buyers and sellers of basket credit derivatives have a wide range of arbitrage-free prices to choose from, and it is the market, not risk-neutral pricing, that determines, both in principle and in practice, a definite price. keywords: basket credit derivatives, correlations, collateralized debt obligations, first-to-default swaps

Credit risk modeling is about modeling losses. These losses are typically coming unexpectedly and triggered by shocks. So any process modeling the stochastic nature of losses should reasonable include jumps. In this paper we review a few jump driven models for the valuation of Credit Default Swaps (CDSs) and show how under these dynamic models also pricing of (exotic) derivatives on single name CDSs is possible. More precisely, we set up fundamental firm-value models that allow for fast pricing of the ‘vanillas’ of the CDS derivative markets: payer and receiver swaptions. It turns out that the proposed model is able to produce realistic implied volatility smiles. Moreover, we detail how a CDS spread simulator can be set up under this framework and illustrate its use for the pricing of exotic derivatives with single name CDSs as underliers.

Default risk is one of the most important and fastest growing areas in derivatives. Single-name credit default swaps are now well established, and markets for contracts based on the default experience of a credit portfolio, like collateralized debt obligations (CDOs) and basket default swaps, are developing rapidly. Portfolio credit risk requires default correlation as a critical input. But defaults are rare, so default correlation must be estimated in another way, typically from correlations in stock returns. In a „structural” credit risk model, a credit event occurs when the net asset value of the firm falls below a critical threshold. Transitions among credit rating classes can be modeled with a set of thresholds corresponding to the different ratings. The widely used „normal copula” model treats the default probability for a single issuer, and default correlations among pairs of issuers, as arising from exposure to a set of common stochastic risk factors plus an idiosyncratic term. The resulting models are intuitive, but calibration to (implied) correlations embedded in market CDO prices can be difficult, especially as the number of factors increases. In this article, Overbeck and Schmidt offer an alternative structural approach that produces similar values to the normal copula model, with considerably less difficulty. Their trick is to alter the time scales of the Wiener processes driving the underlying asset values, which is equivalent to allowing time-varying volatilities, and permits a much easier calibration of the correlations.

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.

The previous chapters consider the risk that a counterparty to a derivative contract defaults on its contractual obligations. Additionally, in Section 5.6.2, we address the pricing of default-free derivatives on credit-risky bonds. In this chapter, we look at derivative instruments with credit risk as their underlying variable determining the payoff of the instrument. Such instruments are commonly called credit derivatives.

A Framework for Extracting the Probability of Default from Listed Stock Option Prices

Copula sensitivity analysis for portfolio credit derivatives

Hazardous Hedging: The (Unacknowledged) Risks of Hedging with Credit Default Swaps

On Weak Predictor-Corrector Schemes for Jump-Diffusion Processes in Finance -- Moving Least Squares for Arbitrage-Free Price and Volatility Surfaces -- Solving Impulse Control Problems with Control Delays -- FIX: The Fear Index ? Measuring Market Fear -- American Option Pricing using Simulation and Regression: Numerical Convergence Results -- The COS Method for Pricing Options under Uncertain Volatility -- Fast Fourier Transform Option Pricing: Efficient Approximation Methods under Multi-Factor Stochastic Volatility and Jumps -- Pricing Credit Derivatives in a Wiener-Hopf Framework -- The Evaluation of Gas Swing Contracts with Regime Switching -- A Linear and Nonlinear Review of the Arbitrage-Free Parity Theory for the CDS and Bond Markets.

Cross-Market Liquidity and Dealer Profitability: Evidence from the Bond and CDS Markets

Political uncertainty and a firm's credit risk: Evidence from the international CDS market

In this first chapter, we show that a CDO tranche payoff can be perfectly replicated with a self-financed strategy based on the underlying credit default swaps. This extends to any payoff which depends only upon default arrivals, such as basket default swaps. Clearly, the replication result is model dependent and relies on two critical assumptions. First, we preclude the possibility of simultaneous defaults. The other assumption is that credit default swap premiums are adapted to the filtration of default times which therefore can be seen as the relevant information set on economic grounds. Our framework corresponds to a pure contagion model, where the arrivals of defaults lead to jumps in the credit spreads of survived names, the magnitude of which depending upon the names in question, and the whole history of defaults up to the current time. These jumps can be related to the derivatives of the joint survival function of default times. The dynamics of replicating prices of CDO tranches follows the same way. In other words, we only deal with default risks and not with spread risks. Unsurprisingly, the possibility of perfect hedging is associated with a martingale representation theorem under the filtration of default times. Subsequently, we exhibit a new probability measure under which the short term credit spreads (up to some scaling factor due to positive recovery rates) are the intensities associated with the corresponding default times. For ease of presentation, we introduced first some instantaneous default swaps as a convenient basis of hedging instruments. Eventually, we can exhibit a replicating strategy of a CDO tranche payoff with respect to actually traded credit default swaps, for instance, with the same maturity as the CDOtranche. Letus note that no Markovian assumption is required for the existence of such a replicating strategy. However, the practical implementation of actual hedging strategies requires some extra assumptions. We assume that all pre-default intensities are equal and only depend upon the current number of defaults. We also assume that all recovery rates are constant across names and time. In that framework, it can be shown that the aggregate loss process is a homogeneous Markov chain, more precisely a pure death process. Thanks to these restrictions, the model involves as many unknown parameters as the number of underlying names. Such Markovian model is also known as a local intensity model, the simplest form of aggregate loss models. As in local volatility models in the equity derivatives world, there is a perfect match of unknown parameters from a complete set of CDO tranches quotes. Numerical implementation can be achieved through a binomial tree, well-known to finance people, or by means of Markov chain techniques. We provide some examples and show that the market quotes of CDOs are associated with pronounced contagion effects. We can therefore explain the dynamics of the amount of hedging CDS and relate them to deltas computed by market practitioners. The figures are hopefully roughly the same, the discrepancies being mainly explained by contagion effects leading to an increase of dependence between default times after some defaults.KeywordsCredit Default SwapCredit SpreadGaussian CopulaCredit DerivativeDefault TimeThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

this paper we present a tree model for defaultable bond prices which can be used for the pricing of credit derivatives. The model is based upon the two-factor Hull-White (1994) model for default-free interest rates, where one of the factors is taken to be the credit spread of the defaultable bond prices. As opposed to the tree model of Jarrow and Turnbull (1992), the dynamics of default-free interest rates and credit spreads in this model can have any desired degree of correlation, and the model can be fitted to any given term structures of default-free and defaultable bond prices, and to the term structures of the respective volatilities. Furthermore the model can accommodate several alternative models of default recovery, including the fractional recovery model of Duffie and Singleton (1994) and recovery in terms of equivalent default-free bonds (see e.g. Lando (1998)). Although based on a Gaussian setup, the approach can easily be extended to non-Gaussian processes that avoid negative interest-rates or credit spreads

An important but rather neglected input to the pricing of credit derivatives is the recovery rate. It is a priori not clear how an (expected) recovery rate should be chosen for standard credit derivatives pricing models. The problem is especially acute when quoted prices contradict previous recovery rate assumptions. Therefore, in this paper we consider a dynamic calibration of traditional static CDS models. We start from the usual recalibration and propose heuristics for a market-consistent calibration of the recovery rate. Thus the market-implied recovery rate that is now an output can be used as an input for stochastic recovery rate models.