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

In this paper, we study the problem of hedging a basket credit derivatives, in particular, we are interested in basket default swaps. For the pricing of credit derivatives, we consider a factor Copula approach. Single-name credit default swaps will be chosen as the hedging instruments. The hedging mechanism is tested using simulated data with a given measure. Numerical results reveal the efficiency of our proposed hedging method.

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.

This chapter presents the Heath–Jarrow–Morton (HJM) (Heath et al, Econometrica 60(1):77–105, 1992) model for pricing interest rate derivatives. Given frictionless and competitive markets, and assuming a complete market, this is the most general arbitrage-free pricing model possible with a stochastic term structure of interest rates. This model, with appropriate modifications, can also be used to price derivatives whose values depend on a term structure of underlying assets, examples include exotic equity derivatives where the underlyings are call and put options, commodity options where the underlyings are futures prices, and credit derivatives where the underlyings are risky zero-coupon bond prices.

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 derivatives are among the most important new financial instruments, but also among the most complicated. Each individual issuer is continuously exposed to default risk, and default intensity looking forward is not constant. It typically has a term structure, as revealed in the CDS market. A portfolio of risky bonds, as in a CDO, aggregates the individual risks, and now the correlations among them also become important. CDO tranches then redistribute and split up this aggregate exposure among a set of new securities. Evaluating the resulting tranche exposures requires a model for the individual default risks and their correlations, but even in the industry-standard Gaussian copula model, the problem is computationally intractable without heroic simplifying assumptions. The plainest vanilla model assumes correlations are equal for all pairs of credits. Then, analogous to the way an implied volatility can be extracted from an option9s market price, the implied correlation can be extracted from a CDO tranche price. But, as with implied volatility, the resulting tranche correlations differ widely for different tranches, leading to the use of „base correlation,” a different implied correlation concept. Base correlation is still inconsistent with the model it is derived from, but it is not quite as badly behaved as tranche correlations. In this article, Hull and White offer an alternative approach that considerably reduces the inconsistencies in calibrating a copula to a set of CDO tranche prices. The secret is to make default intensities and recovery rates stochastic, rather than requiring a single value. By imposing the restrictions that the single-name CDS and the CDO tranches must all be priced by the model just as they are in the market, and that the probabilities for the set of possible individual default intensities must sum to 1 and exhibit maximum smoothness, Hull and White are able to imply tranche correlations that are much better behaved than the standard approach. The last part of the article extends their procedure in a number of directions, to nonstandard attachment points, bespoke portfolios, and CDO-squared securities.

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

This paper introduces the “Smile to Credit” model, which allows pricing and hedging equity and credit derivatives in a single framework. In this model, default intensity is a function of the ratio of market cap to debt. Once fitted to a volatility surface and a credit curve, the model provides fairly reliable estimates of the equity-credit delta.

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.

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.

This study constructs a credit derivative pricing model using economic fundamentals to evaluate CDX indices and quantify the relationship between credit conditions and the economic environment. Instead of selecting specific economic variables, numerous economic and financial variables have been condensed into a few explanatory factors to summarize the noisy economic system. The impacts on default intensity processes are then examined based on no‐arbitrage pricing constraints. The approximated results show that economic factors indicated credit problems even before the recent subprime mortgage crisis, and economic fundamentals strongly influenced credit conditions. Testing of out‐of‐sample data shows that credit evolution can be identified by dynamic explanatory factors. Consequently, the factor‐based pricing model can either facilitate the evaluation of default probabilities or manage default risks more effectively by quantifying the relationship between economic environment and credit conditions. © 2010 Wiley Periodicals, Inc. Jrl Fut Mark

Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

We analyze a general but parsimonious price competition model for an oligopoly in which each firm offers any number of products. The demand volumes are general piecewise affine functions of the full price vector, generated as the “regular” extension of a base set of affine functions. The model specifies a product assortment, along with their prices and demand volumes, in contrast to most commonly used demand models, such as the multinomial logit model or any of its variants. We show that a special equilibrium in this model has global robust stability. This means that, from any starting point, the market converges to this equilibrium when firms use a particular response mapping to dynamically adjust their own prices in response to their competitors’ prices. The mapping requires each firm to only know the demand function and cost structure for its own products (but not for other firms’ products).

Modelling Simultaneous Defaults: A Top-Down Approach

Multiscale Intensity Models for Single Name Credit Derivatives