Imputation, Estimation and Missing Data in Finance
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Suppose <em>X</em> is a diffusion process, possibly multivariate, and suppose that there are various segments of the components of <em>X</em> that are missing. This happens, for example, if <em>X</em> is the price of various assets and these prices are only observed at specific discrete trading times. Imputation (or conditional simulation) of the missing pieces of the sample paths of <em>X</em> is discussed in several settings. When <em>X</em> is a Brownian motion the conditioned process is a tied down Brownian motion or a Brownian bridge process. In the special case of Gaussian stochastic processes the problem is simplified since the conditional finite dimensional distributions of the process are multivariate Normal. For more general diffusion processes, including those with jump components, an acceptance-rejection simulation algorithm is introduced which enables one to sample from the exact conditional distribution without appealing to approximate time step methods such as the popular Euler or Milstein schemes. The method is referred to as <em>pathwise imputation</em>. Its practical implementation relies only on the basic elements of simulation while its theoretical justification depends on the pathwise properties of stochastic processes and in particular Girsanov's theorem. The method allows for the complete characterization of the bridge paths of complicated diffusions using only Brownian bridge interpolation. The imputation methods discussed are applied to estimation, variance reduction and exotic option pricing.