Modeling of Measurement Error in Financial Returns Data

In this article, we consider the modeling of measurement error for fund returns data. In particular, given access to a time series of discretely observed log-returns and the associated maximum over the observation period, we develop a stochastic model that models the true log-returns and maximum via a L & eacute;vy process and the data as a measurement error thereof. The main technical difficulty of trying to infer this model, for instance Bayesian parameter estimation, is that the joint transition density of the return and maximum is seldom known, nor can it be simulated exactly. Based upon the novel stick-breaking representation of [1 c1], we provide an approximation of the model. We develop a Markov chain Monte Carlo (MCMC) algorithm to sample from the Bayesian posterior of the approximated posterior and then extend this to a multilevel MCMC method, which can reduce the computational cost to approximate posterior expectations, relative to ordinary MCMC. We prove that the computational complexity of our multilevel MCMC scheme is optimal for L & eacute;vy models with increments that can be sampled in constant time. We implement our methodology on several applications, including for real data.