Option pricing and hedging with minimum local expected shortfall

Abstract

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 us to significantly reduce extreme risks, and generically leads to low Gamma hedging. We 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.

Notes

E-mail: [email protected], [email protected] and [email protected]

 If the volatility was stochastic, we should assume that the option price also depends on the current level of volatility σ _k and rather write mcC_k (x_k ,σ _k ).

 A different possibility, that we use in section 4 below, is to write ϕ _k and mcC_k as simple functions (with correct asymptotic behaviours), parameterized by a few numbers that are determined by the optimization.

 A better approximation that takes into account the (already known) time derivative of mcC_k (x_k ) is to write

.

 The aim of the present paper is not to discuss in detail the optimal choice of p and of the form of the basis functions, but rather to demonstrate the overall feasability of the method.