Stochastic Evolution Equations in Portfolio Credit Modelling

Abstract

We consider a structural credit model for a large portfolio of credit risky assets. By considering the large portfolio limit we introduce a stochastic partial differential equation which describes the evolution of the density of asset values. The loss function of the portfolio is then a function of the evolution of this density at the default boundary. We develop numerical methods for pricing and calibration of the model to credit indices and consider its performance pre and post credit crunch. We also use it to price dynamic credit products such as forward starting CDO tranches.

Numerical valuation of basket credit derivatives in structural jump-diffusion models

“…We now derive a formula for c CDO , following Bush et al (2011). It is assumed that default risk is hedgeable, eg, with defaultable bonds, which translates into the existence of a pricing measure Q under which the discounted values of traded credit risky assets are martingales.…”

“…In practice, a default is announced on a daily basis, as Fang et al (2010) argue. In order to make models computationally more tractable, some studies, eg, Hull et al (2010) and Bush et al (2011), assume that default is detected only on spread payment dates. We follow this line and thus assume that defaults are monitored quarterly.…”

“…In contrast, Kiesel & Scherer (2007) simulate the asset value process for each company to estimate joint default probabilities, which is not subject to model simplifications, but computationally expensive, since the basket size is typically large, eg, 125 for the iTraxx index. We make a virtue of this by using a large basket approximation in the spirit of Bush et al (2011), extending the model there to jump-diffusions. Using this approach, the computational effort is essentially independent of the number of companies.…”

“…Bush et al (2011) propose a new approach to the pricing of large portfolios of credit risky assets. The authors consider a structural model where firm values are driven by exchangeable diffusion processes where the dependence between firms is due to exposure to a common market factor.…”

“…Motivated by ideas in Bush et al (2011), where the joint density of asset values is evolved in a large basket approximation, we develop an algorithm for the efficient estimation of CDO index and tranche spreads consistent with underlying CDSs, through a finite difference simulation of the resulting SPDE. We verify the validity of this approximation numerically by comparison to results obtained by direct Monte Carlo simulation of the basket constituents.…”