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W e show the existence of Nash equilibria in a Bertrand oligopoly price competition game using a possibly asymmetric attraction demand model with convex costs under mild assumptions. We show that the equilibrium is unique and globally stable. To our knowledge, this is the first paper to show the existence of a unique equilibrium with both nonlinear demand and nonlinear costs. In addition, we guarantee the linear convergence rate of tatônnement. We illustrate the applicability of this approach with several examples arising from operational considerations that are often ignored in the economics literature.

This paper develops rare-event simulation methods for the estimation of portfolio credit risk-the risk of losses to a portfolio resulting from defaults of assets in the portfolio. Portfolio credit risk is measured through probabilities of large losses, which are typically due to defaults of many obligors (sources of credit risk) to which a portfolio is exposed. An essential element of a portfolio view of credit risk is a model of dependence between these sources of credit risk: large losses occur rarely and are most likely to result from systematic risk factors that affect multiple obligors. As a consequence, estimating portfolio credit risk poses a challenge both because of the rare-event property of large losses and the dependence between defaults. To address this problem, we develop an importance sampling technique within the widely used Gaussian copula model of dependence. We focus on difficulties arising in multifactor models-that is, models in which multiple factors may be common to multiple obligors, resulting in complex dependence between defaults. Our importance sampling procedure shifts the mean of the common factor to increase the frequency of large losses. In multifactor models, different combinations of factor outcomes and defaults can produce large losses, so our method combines multiple importance sampling distributions, each associated with a shift in the mean of common factors. We characterize "optimal" mean shifts. Finding these points is both a combinatorial problem and a convex optimization problem, so we address computational aspects of this step as well. We establish asymptotic optimality results for our method, showing that-unlike standard simulation-it remains efficient as the event of interest becomes rarer.

The measurement of portfolio credit risk focuses on rare but significant large-loss events. This paper investigates rare event asymptotics for the loss distribution in the widely used Gaussian copula model of portfolio credit risk. We establish logarithmic limits for the tail of the loss distribution in two limiting regimes. The first limit examines the tail of the loss distribution at increasingly high loss thresholds; the second limiting regime is based on letting the individual loss probabilities decrease toward zero. Both limits are also based on letting the size of the portfolio increase. Our analysis reveals a qualitative distinction between the two cases: in the rare-default regime, the tail of the loss distribution decreases exponentially, but in the large-threshold regime the decay is consistent with a power law. This indicates that the dependence between defaults imposed by the Gaussian copula is qualitatively different for portfolios of high-quality and lower-quality credits.