Option pricing and hedging with minimum local expected shortfall

We propose a versatile Monte-Carlo method for pricing and hedging options when the market is incomplete, for an arbitrary risk critcrion (chosen here to be the expected shortfall), for a large class of stochastic processes, and in the presence of transaction costs. We illustrate the method on plain vanilla options when the price returns follow a Student -t distribution. We show that in the presence of fat-tails, our strategy allows us to significantly reduce extreme risks, and generically loads to low Gamma hedging. He also find that using an asymmetric risk function generates option skews, even when the underlying dynamics is unskewed. Finally, we show the proper accounting of transaction costs leads to an optimal strategy with reduced Gamma, which is found to outperform Leland's hedge.