Exact and Efficient Simulation of Correlated Defaults

Correlated default risk plays a significant role in financial markets. Dynamic intensity-based models, in which a firm default is governed by a stochastic intensity process, are widely used to model correlated default risk. The computations in these models can be performed by Monte Carlo simulation. The standard simulation method, which requires the discretization of the intensity process, leads to biased simulation estimators. The magnitude of the bias is often hard to quantify. This paper develops an exact simulation method for intensity-based models that leads to unbiased estimators of credit portfolio loss distributions, risk measures, and derivatives prices. In a first step, we construct a Markov chain that matches the marginal distribution of the point process describing the binary default state of each firm. This construction reduces the original estimation problem to one involving a simpler Markov chain expectation. In a second step, we estimate the Markov chain expectation using a simple acceptance/rejection scheme that facilitates exact sampling. To address rare event situations, the acceptance/rejection scheme is embedded in an overarching selection/mutation scheme, in which a selection mechanism adaptively forces the chain into the regime of interest. The resulting simulation engine can treat, without customization, many intensity-based models in the literature. Numerical experiments demonstrate the effectiveness of our method. ∗Corresponding author. Department of Management Science & Engineering, Stanford University, Stanford, CA 94305-4026, USA, Phone (650) 723 9265, Fax (650) 723 1614, email: [email protected], web: www.stanford.edu/∼giesecke. †The Perot Group. ‡Department of Management Science & Engineering, Stanford University. §Mizuho-DL Financial Technology, Tokyo. ¶We are grateful to Mizuho-DL Financial Technology for a grant that supported this work, and to Apaar Sadhwani for research assistance. This paper was presented at seminars at the Chinese University of Hong Kong, Columbia University, Mizuho-DL Financial Technology, the 2009 ICBI Global Derivatives and Risk Management Conference, and the 15th INFORMS Applied Probability Society Conference. We thank seminar participants for comments. We also thank Jose Blanchet, Rama Cont, Jean-Pierre Fouque, Josselin Garnier, Paul Glasserman, Martin Haugh, Baeho Kim and Steve Kou for comments.