Discontinuous Continuity Approximations: Hedging the Next Move

Recognizing that in reality price processes display intervals of no price change we construct finite activity processes capable of synthesizing risk neutral distributions. The proposed construction combines three independent processes, X, N, and G where X and G are Lévy subordinators while N is a standard Poisson process. The finite activity process is then defined as X taken at the time given by N observed at time G, which we call here the XNG process. Ten examples of processes are introduced for different choices for X and G. Analytical results for the Lévy measure of the XNG processes are presented. In addition we evaluate the quality of the XNG processes as discontinuous continuity approximators. The use of a calibrated finite activity process gives access to the distribution of the next move. This distribution is employed to build next move hedging strategies that maximize a conservative financial valuation of the hedged position. With respect to continuity approximation and hedging sensitivities to the moments of the risk neutral distribution, two of the ten models may be distinguished. These are when G is a finite actvity process and X has either finite activity or is a gamma process.