We propose a versatile Monte-Carlo method for pricing and hedging options
when the market is incomplete, for an arbitrary risk criterion (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 to significantly reduce extreme
risks, and generically leads to low Gamma hedging. Similarly, the inclusion of
transaction costs reduces the Gamma of the optimal strategy.Comment: 23 pages, 7 figures, 8 table