Shortfall-Risk Minimizing Hedging Based on MCMC Method

Many researchers have studied the multi-period shortfall-risk minimizing hedging problem under jump-diffusion settings, however, up to now, there has not been many literatures to directly present the analytic solution for such a hedging problem. In this paper, defining the terminal shortfall as hedging risk, we research the minimal shortfall-risk hedging problem of European option by means of MCMC method. Firstly, we look on the underlying asset´s positions held at each strategy rebalancing moment during option´s maturity as a random vector; then, we aptly construct this random vector´s conditional joint probability density function, from which we sample a Markov chain, according to the Bayes principle and Ergodic theorem, the drawn Markov chain will congregate around the optimal strategy with higher probability and converge to it, thus, it´s reasonable to substitute the optimal hedging position with the averaged value of this Markov chain; finally, experimental analysis results illustrate that our method is not only feasible but convenient to manipulate while being helpful and referential to hedging practice.