"Monte Carlo Methods in Finance, Statistics and Biostatistics"

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This research concerns the applications of Monte Carlo methods to problems in Finance and Statistics. We frequently wish to estimate the expected value of a function of a stochastic process, whose distribution Q is intractible or difficult to simulate from. We investigate the use of "randomized importance sampling". We simulate from an alternate simpler distribution P and then weight the observations using (random) importance sampling (IS) weights W, which compensate for the fact that the wrong distribution was used to conduct the simulation. The estimator is a weighted average with random weights W. W estimates the relative likelihood of the observations under the two measures and can be chosen to take only integer values or rescaled, observations with weight 0 discarded. There are emany potential applications since for stochastic processes, likelihoods ("Radon Nikodym derivatives") are difficult to compute exactly, but unbiased estimators of them are tractable. For example if the process is a diffusion or jump-diffusion process, evaluating the likelihood requires evaluating an integral of the process, i.e. simulating it at every time point in an interval. However it is easy to obtain an unbiased estimator which samples the process only at finitely many points. Simulations using such estimators will be used to assess risk or evalute option prices. We can also obtain our sampling weights using only the characteristic function of a distribution. We apply this methodology to the analysis of stochastic volatilitiy models, which provide a critical improvement over the ill-fitting Black-Scholes model in finance. We will consider other problems in which the characteristic function of the log-spot price is known, such as affine stochastic volatility models with and without stochastic interest rates, time-changed Levy processes, and other complex models. Using simulation-based randomized importance sample weights permits adoption of broader classes of models, or accommodating incomplete observed data, since it allows imputation to be carried out under the simple measure P. This work will also facilitate the analysis of more complex missing data problems in continuous-time longitudinal models in statistics and biostatistics.