In this paper we present our ongoing effort to use importance sampling to develop unbiased, bounded estimators of densities, distribution functions and expectations of functions of a random vector, when the characteristic function of the (multi-dimensional) random vector is available in analytic or semi-analytic form. This is especially of interest in options pricing as stochastic processes such as affine jump processes and Levy processes are ubiquitous in financial modeling and typically have characteristic functions (of their value at a given time) that are easily evaluated while their density or distribution functions have no readily computable closed form. Typically, for pricing options via Monte Carlo, a discretized version of the underlying SDE is simulated using Euler or a related method and the resultant estimator has a discretization bias. A noteworthy feature of our Monte Carlo approach is that, when applicable, it provides unbiased estimators.
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