Advances in Risk Management Simulation

Evren Baysal

doi:https://doi.org/10.21985/N2CM9J

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Advances in Risk Management Simulation

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Risk measurement involves estimating some functional of a loss distribution. This calls for nested simulation, in which risk factors are sampled at an outer level of simulation, while the inner level of simulation provides estimates of loss given each realization of the risk factors. Assessing the statistical uncertainty of estimates of risk measures at the outer simulation level is crucial in designing computationally efficient two-level simulation procedures for risk management applications. Confidence intervals for the risk measure of interest provide information about this statistical uncertainty and we provide asymptotically valid confidence intervals and confidence regions involving value at risk, conditional tail expectation, and expected shortfall (conditional value at risk), based on three different methodologies. One is an extension of previous work based on robust statistics, the second is a straightforward application of bootstrapping, and we derive the third using empirical likelihood. We then evaluate the small-sample coverage of the confidence intervals and regions in simulation experiments using financial examples. We find that the coverage probabilities are approximately nominal for large sample sizes, but are noticeably low when sample sizes are too small (roughly, less than 500 here). The new empirical likelihood method provides the highest coverage at moderate sample sizes in these experiments. Nested simulations can also be used in evaluating trading and hedging strategies. Suppose that one wishes to evaluate the distribution of profit and loss (P&L) resulting from a dynamic trading strategy. A straightforward method, then, is to simulate thousands of paths (i.e., time series) of relevant financial variables and to track the resulting P&L at every time at which the trading strategy rebalances its portfolio. In many cases, this requires numerical computation of portfolio weights at every rebalancing time on every path by a nested simulation performed conditional on market conditions at that time on that path. Such a two-level simulation could involve many millions of simulations to compute portfolio weights, and thus be too computationally expensive to attain high precision. We show that response surface methodology enables a more efficient simulation procedure: in particular, it is possible to do far fewer simulations by using kriging to model portfolio weights as a function of underlying financial variables.

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