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Monte Carlo simulation techniques that use function approximations have been successfully applied to approximately price multi-dimensional American options. However, for many pricing problems the time required to get accurate estimates can still be prohibitive, and this motivates the development of variance reduction techniques. In this paper, we describe a zero-variance importance sampling measure for American options. We then discuss how function approximation may be used to approximately learn this measure; we test this idea in simple examples. We also note that the zero-variance measure is fundamentally connected to a duality result for American options. While our methodology is geared towards developing an estimate of an accurate lower bound for the option price, we observe that importance sampling also reduces variance in estimating the upper bound that follows from the duality.

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Lockdown in cities across the globe has imposed severe travel restrictions to limit the spread of Coronavirus disease. The travel behavior and operations will not be the same as before due to requirements such as physical (social) distancing. This study analyzes the resulting shortage in supply of public transport (buses) that will likely widen the existing gap between demand and supply. In this work, system optimization models are developed to efficiently reallocate the bus fleet to routes for different levels of physical distancing gaps and travel demand. The proposed models are applied to a real-life network of 34 bus routes of Delhi, considering three types of scenarios: current, practical, and ideal. In the practical scenarios, the additional, idling bus fleets can be allocated to the routes efficiently while maintaining physical distancing. The results show that the Business-as-Usual (BAU) scenario involving the current allocation approach will make it impossible to use public buses even if the bare minimum physical distancing has to be maintained. Further, the models proposed in the study significantly improve the key performance indicators for all scenarios.