What you don’t know cannot hurt you: On the detection of small jumps

With the availability of high frequency financial data, nonparametric estimation of volatility of an asset return process becomes feasible. One of the main issues is concerned with separating the volatility due to continuous evolution from the volatility due to jumps. This paper argues that the separation of continuous and jump evolution is soft, in addition, large biases can occur when estimating volatility in the presence of infinitely many small jumps. The paper advocates that for the purposes of application and analysis, one can identify small jumps with continuous evolution while treating large jumps as compound Poisson type. A main contiguity theorem is derived to relate compound poisson process to Lévy process with infinite activity in the asymptotic setting, and gives the cutoff below which the jumps are too small to be detectable. The theorem provides a straightforward tool to simulate a Lévy process with infinite activity, it also reveals why a “good” estimator can behave badly in finite samples. Some key words and phrases: Blumenthal-Getoor index, compound Poisson process, consistency, contiguity, discrete observation, infinite activity process, jumps, Lévy process, rate of convergence, volatility. What you don’t know cannot hurt you 1