We develop BayesianMarkov chain Monte Carlo methods for inferences of continuous-time models with stochastic volatility and infinite-activity Lévy jumps using discretely sampled data. Simulation studies show that (I) our methods provide accurate joint identification of di.usion, stochastic volatility, and Lévy jumps, and (ii) a.ne jump-di.usion models fail to adequately approximate the behavior of infinite-activity jumps. In particular, the affine jump-diffusion models fail to capture the “infinitely many”small Lévy jumps which are too big for Brownian motion to model and too small for compound Poisson process to capture. Empirical studies show that infinite-activity Lévy jumps are essential for modeling the S&P 500 index returns.
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