Approximate hedging for nonlinear transaction costs on the volume of traded assets

This paper is dedicated to the replication of a convex contingent claim h(S1) in a financial market with frictions, due to deterministic order books or regulatory constraints. The corresponding transaction costs rewrite as a non linear function G of the volume of traded assets, with G′(0) > 0. For a stock with Black-Scholes mid-price dynamics, we exhibit an asymptotically convergent replicating portfolio, defined on a regular time grid with n trading dates. Up to a well chosen regularization h of the payoff function, we first introduce the frictionless replicating portfolio of h(S 1 ), where S n is a fictive stock with enlarged local volatility dynamics. In the market with frictions, a proper modification of this portfolio strategy provides a terminal wealth, which converges in probability to the claim of interest h(S1), as n goes to infinity. In terms of order book shapes, the exhibited replicating strategy only depends on the size 2G′(0) of the bid-ask spread. The main innovation of the paper is the introduction of a ’Leland type’ strategy for non-vanishing (non-linear) transaction costs on the volume of traded shares, instead of the commonly considered traded amount of money. This induces lots of technicalities, that we pass through using an innovative approach based on the Malliavin calculus representation of the Greeks.