Numerical Computation of Worst-Case Distributions in Monte Carlo Portfolio Risk Management

Stress testing is a central tool in financial risk management and is widely used to assess the resilience of portfolios under adverse conditions. Traditional stress tests typically rely on predefined scenarios in which selected risk factors are shocked by predetermined amounts. Such approaches may fail to identify the most harmful but still plausible situations for a given portfolio. This paper considers systematic stress testing in which the search for adverse outcomes is extended from individual scenarios to the space of probability distributions. Following the framework of Breuer and Csiszár, the worst-case distribution is defined within a plausibility region around a reference distribution using the Kullback–Leibler divergence. Although this framework provides an analytical characterization of worst-case distributions, practical applications often require numerical evaluation of portfolio losses within simulation-based models. We propose a numerical method that allows the worst-case distribution to be computed directly within Monte Carlo or quasi-Monte Carlo portfolio models. The approach approximates the reference distribution by a discrete set of scenarios and reduces the optimization problem to a one-dimensional root finding problem that can be solved efficiently using robust numerical algorithms. The proposed method is illustrated in a Monte Carlo risk analysis framework and applied at three different stages of the risk measurement process: the risk-neutral density used for valuation of financial instruments, the distribution of risk factors used in scenario generation, and the resulting profit and loss distribution of the portfolio. The numerical results show that worst-case distributions within a given plausibility constraint can lead to substantially larger losses than standard stress scenarios while remaining statistically plausible relative to the reference model.