Expectations are important measures of random performances. They are widely used in practice. For instance, in financial industry, probabilities and the so-called conditional Value-at-Risk (CVaR) are expectations that are used to characterize financial risk. Sensitivities of the expectations provide information on how changes in input parameters will affect the output performance. They play important roles in modeling, analysis and management of random performances....[ Read more ]

Expectations are important measures of random performances. They are widely used in practice. For instance, in financial industry, probabilities and the so-called conditional Value-at-Risk (CVaR) are expectations that are used to characterize financial risk. Sensitivities of the expectations provide information on how changes in input parameters will affect the output performance. They play important roles in modeling, analysis and management of random performances.

Calculating the sensitivities of expectations could be difficult, especially when the expectations are taken for discontinuous functions, and hence one may have to resort to Monte Carlo simulations. In this thesis we develop a Monte Carlo approach to estimating sensitivities of expectations of discontinuous functions. We prove that these sensitivities can be written as conditional expectations, and devise pathwise sensitivity estimators based on the conditional-expectation forms. We show that the proposed estimators are consistent and follow central limit theorems. Applications of this approach include sensitivity analysis of probabilities and CVaRs, and estimating price sensitivities (or called Greeks) of financial options. We discuss in details each of these applications, and illustrate the performance of the proposed approach using numerical examples. Empirical results show that the proposed approach works well for practical problems.