Research

Corporate defaults cluster. The clustering is induced by the correlation among firms. It is driven by firm sensitivity to common economic factors such as interest rates or inflation, but it can also come from the feedback of an individual firm event to the aggregate level. A single default often causes ripple effects across credit markets.

A credit investor such as a bank making loans to corporations or an asset manager buying corporate bonds is exposed to the risk of correlated default. Portfolio credit derivatives are instruments that allow investors to trade this risk, i.e. to buy or sell default protection on a pool of firms. They come in many forms, can be standardized such as index default swaps and tranches, or custom-made. According to a survey of the International Swaps and Derivatives Association, the notional amount outstanding of credit derivatives reached $26 trillion by mid-2006, up 52% from the end of last year. Much of the growth has been driven by increased trading of portfolio credit derivatives, a segment that now represents some 30% of the market.

The quantitative modeling of correlated default risk is a main theme in CreditLab's research program. We are particularly interested in the pricing, hedging and calibration of credit derivatives. We have developed a top-down approach to this problem, in which a credit derivative is a contingent claim on the aggregate loss in a portfolio of credit sensitive securities such as loans, bonds or credit swaps. The key to valuing and hedging these derivatives is to parsimoniously describe the stochastic point process that governs the portfolio loss. We are interested in point process models that capture empirical features such as feedback and lead to computationally tractable valuation and hedging relations.

We have proposed new parametric families of intensity based point processes. An affine point process is specified by an intensity that is driven by affine jump diffusion risk factors. The portfolio loss itself is a risk factor so past defaults and their recoveries influence future loss dynamics. This specification incorporates feedback from events and a dependence structure among default and recovery rates. It also leads to semi-analytical pricing, hedging and calibration of credit derivatives based on the characteristic function of the point process. At present we study the calibration of an affine point process model to single name and tranche spread data, and we compare the model's hedging performance with that of conventional copula models.

Affine point processes as well as many other intensity based point processes can be obtained by time-changing a simple Poisson process, i.e. by running the Poisson process on a stochastic clock. An application of this insight leads to point processes that have interesting attributes, including event feedback. An example that facilitates closed form pricing and hedging of some credit derivatives is the time-changed birth process, which is generated by time-changing a birth process. The distribution of a time-changed birth process can be calculated explicitly in terms of the Laplace transform of the time change. The calculation of the Laplace transform is analogous to the calculation of the price of a zero coupon bond, which is well understood for a wide range of parametric interest rate models. If we adopt any of these models for the time change, then we obtain an analytical expression for the Laplace transform that facilitates an analytical treatment of the credit derivatives pricing problem.

While time change methods are very powerful, they require a special structure of the point process. Surprisingly general results can be obtained if we shift our attention from the distribution of the process to its Fourier transform, or characteristic function. Using a complex-valued measure change, we can derive a formula for a Fourier transform of any counting process that describes the arrival of unpredictable events. This Fourier transform is expressed in terms of a Laplace transform of the counting process compensator, or cumulative intensity. The Laplace transform is a familiar expression in the defaultable term structure literature: it represents the zero-recovery price of a defaultable security. Calculating this price is standard for a wide range of parametric intensity models. Affine and quadratic specifications yield closed form expressions. If we adopt any of these models, then our formula provides a closed form expression for the Fourier transform of the point process. The Fourier transform leads directly to pricing relations for a number of single- and multi-name derivatives. It also generates explicit expressions for risk measures of corporate debt portfolios. Furthermore, this transform approach supports a stochastic interest rate specification that can incorporate a dependence structure among default and interest rates. This dependence structure becomes important in the presence of flights to quality, i.e. when investors shed default risk after a significant credit event and seek less risky investments.

Monte Carlo simulation is the method of choice if an instrument has a complex payoff profile. An example is a cash collateralized debt obligation. Here we generate sample paths of the loss point process and average over the corresponding discounted payoffs to obtain an estimate of the derivative price. We showed that if the loss process is specified by an intensity, then an exact simulation algorithm for the loss process exists and leads to an unbiased estimate of the derivative price. It turns out that exact simulation algorithms for point processes can be constructed by intensity projection. At present we develop an importance sampling scheme for the exact algorithm. We also examine an alternative approximate algorithm that greatly reduces the time needed to generate a path.

The point process models and methods we develop also have applications in the risk management of corporate debt portfolios. We propose maximum-likelihood estimators and goodness-of-fit tests for affine and other point processes, and implement them on a sequence of historical default events that goes back to 1970. The estimates lead to a forecast distribution of future defaults and losses for portfolios of corporate debt, and risk measures such as value at risk and expected shortfall. The loss distribution is at the center of the risk management process. The estimates also form the basis for extracting from market prices of credit indexes and tranches the risk premia investors demand for bearing exposure to correlated corporate default risk. An understanding of the risk premium is the key to several important applications, including the construction of trading strategies that seek to exploit excessive premia.