Default Intensity Processes Research Articles

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.

Counterparty Risk for Credit Default Swaps: Markov Chain Interacting Intensities Model With Stochastic Intensity

We analyze the counterparty risk for credit default swaps using the Markov chain model of portfolio credit risk of multiple obligors with interacting default intensity processes. The default correlation between the protection seller and underlying entity is modeled by an increment in default intensity upon the occurrence of an external shock event. The arrival of the shock event is a Cox process whose stochastic intensity is assumed to follow an affine diffusion process with jumps. We examine how the correlated default risks between the protection seller and the underlying entity may affect the credit default premium in a credit default swap.

Counterparty Risk for Credit Default Swaps: Markov Chain Interacting Intensities Model with Stochastic Intensity

We analyze the counterparty risk for credit default swaps using the Markov chain model of portfolio credit risk of multiple obligors with interacting default intensity processes. The default correlation between the protection seller and underlying entity is modeled by an increment in default intensity upon the occurrence of an external shock event. The arrival of the shock event is a Cox process whose stochastic intensity is assumed to follow an affine diffusion process with jumps. We examine how the correlated default risks between the protection seller and the underlying entity may affect the credit default premium in a credit default swap.

Credit Risk Model with Lagged Information

In this paper, I propose a structural credit risk model with lagged information on a firm. Under the simple assumption that information on a firm's asset value is observed with a time lag, I show the model also has the properties of a reduced form model with a default intensity process. In contrast to some previous studies in which both periodic and noisy accounting report assumptions are necessary to derive the same result, I argue that lagged information is sufficient to generate a default intensity process. This difference in the assumptions has fundamentally different implications for the time-series of credit spreads; under both the noisy accounting and lag information assumptions, the model-implied credit spreads exhibit periodic patterns whereas under my model assumptions they do not.

From Default Correlation to Implied Hazard Rate Densities

In this paper I show that the standard deviation of a CDO implied hazard rate density increases with increased correlation between firms' default intensity processes. This is done by generating CDO tranches spreads from a model where firms' defaults are modelled by jump-diffusion hazard rate processes that are correlated through systematic jumps, correlation in the processes' random shocks, or both. The increase in the standard deviation of a CDO implied hazard rate density holds irrespective of how the jump-diffusion hazard rate processes' are correlated. In addition, model simulations indicate that systematic jumps that are rare and with large mean size are required to yield senior CDO tranche model spreads that are consistent with market quotes. This observations supports similar conclusions reported in the existing literature. Further to this, I propose a model where the default of one firm triggers the default of other firms. When the number of firms hit by this default is approximately 1/3 of a CDO's reference portfolio, I am able to produce model spreads that are within range of the extreme spreads observed during the 2007-2008 credit crisis. In addition, using a vector autoregression on daily observations, I provided evidence that lags of the standard deviation of the implied hazard rate density and the spread of the 3-month Eurodollar and 3-month T-Bill rates (TED spread) are factors for one-period ahead prediction of each other.

Recovering Portfolio Default Intensities Implied by CDO Quotes

We propose a stable non-parametric algorithm for the calibration of pricing models for portfolio credit derivatives: given a set of observations of market spreads for CDO tranches, we construct a risk-neutral default intensity process for the portfolio underlying the CDO which matches these observations, by looking for the risk neutral loss process 'closest' to a prior loss process, verifying the calibration constraints. We formalize the problem in terms of minimization of relative entropy with respect to the prior under calibration constraints and use convex duality methods to solve the problem: the dual problem is shown to be an intensity control problem, characterized in terms of a Hamilton-Jacobi system of differential equations, for which we present an analytical solution. Given a set of observed CDO tranche spreads, our method allows to construct an implied intensity process consistent with the observed spreads. We illustrate our method on ITRAXX index data: our results reveal strong evidence for the dependence of loss transitions rates on the past number of defaults, thus offering quantitative evidence for contagion effects in the risk-neutral loss process.

An Integrated Model for Hybrid Securities

We develop a model for pricing securities whose value may depend simultaneously on equity, interest-rate, and default risks. The framework may also be used to extract probabilities of default (PD) functions from market data. Our approach is entirely based on observables such as equity prices and interest rates, rather than on unobservable processes such as firm value. The model stitches together in an arbitrage-free setting a constant elasticity of variance (CEV) equity model (to represent the behavior of equity prices prior to default), a default intensity process, and a Heath-Jarrow-Morton (HJM) model for the evolution of riskless interest rates. The model captures several stylized features such as a negative relation between equity prices and equity volatility, a negative relation between default intensity and equity prices, and a positive relationship between default intensity and equity volatility. We embed the model on a discrete-time, recombining lattice, making implementation feasible with polynomial complexity. We demonstrate the simplicity of calibrating the model to market data and of using it to extract default information. The framework is extensible to handling correlated default risk and may be used to value distressed convertible bonds, debt-equity swaps, and credit portfolio products such as collateralized debt obligations (CDOs). Applied to the CDX INDU (credit default index–industrials) Index, we find the S&P 500 index explains credit premia.

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

The pricing of collateralized debt obligations and other basket credit derivatives is contingent upon (i) a realistic modeling 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 modeling 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, and discuss the relative performance of the framework.

Information reduction via level crossings in a credit risk model

This paper provides an alternative credit risk model based on information reduction where the market only observes the firm’s asset value when it crosses certain levels, interpreted as changes significant enough for the firm’s management to make a public announcement. For a class of diffusion processes we are able to provide explicit expressions for the firm’s default intensity process and its zero-coupon bond prices.

The Market Price of Risk in Interest Rate Swaps: The Roles of Default and Liquidity Risks*

We study how the market prices the default and liquidity risks incorporated into interest rate swap spreads. We jointly model the Treasury, repo, and swap term structures using a five‐factor affine framework and estimate the model by maximum likelihood. The credit spread is driven by a persistent liquidity process and a rapidly mean‐reverting default intensity process. The credit premium for all but the shortest maturities is primarily compensation for liquidity risk. The term structure of liquidity premia increases steeply, while that of default premia is almost flat. Both liquidity and default premia vary significantly over time.

Credit Events and the Valuation of Credit Derivatives of Basket Type

Thispaper provides a simple model for valuing a credit derivativewhose payoff depends on the identity (or identities) of the first(or first two) to occur of a given list of credit events, suchas defaults. The joint survival probability of occurrence timesof credit events is formulated in terms of stochastic intensityprocesses under the assumption of conditional independence. Basedon the joint survival probability, we can easily obtain the pricingformulas of such credit derivatives under the risk-neutral valuationframework. When the default intensity processes follow the extendedVasicek model, closed-form solutions of the pricing formulasare given.

Valuation of a Credit Swap of the Basket Type

This article provides a simple model to value a credit swap ofthe basket type. Unlike the previous literature, we considerthe joint survival probability of occurrence times of creditevents in terms of stochastic intensity processes under the assumptionof conditional independence. Based on the joint survival probability,such a credit swap can be valued under the risk-neutral valuationframework. Assuming that the default intensity processes followthe extended Vasicek model with a correlation structure, an analyticexpression of the valuation formula is derived. Some numericalexample is given to demonstrate the usefulness of our model.