Infinite-dimensional linear programming and model-independent hedging of contingent claims

Abstract

We consider model-independent pathwise hedging of contingent claims in discrete-time markets, in the framework of infinite-dimensional linear programmes (LP). The dual problem can be formulated as optimization over the set of martingale measures subject to market constraints. Absence of model-independent arbitrage plays a crucial role in ensuring that both the primal and the dual problems are well posed and there is no duality gap. In fact we show that different notions of model-independent arbitrage are required to prove duality results in various settings. We then specialize this duality theory to the situation where European Call options are traded on the market. In particular we consider hedging portfolios that consist of static positions in traded options and a dynamic trading strategy. The dual variables are then constrained to martingale measures consistent with prices of traded options. When only finitely many Call options are traded, the notion of weak arbitrage introduced in Davis and Hobson (2007) is sufficient to ensure absence of duality gap between the primal and the dual problems. In this case the set of feasible dual variables is not closed, and extrapolation of Call option prices (equivalently of the implied volatility smile) is required. We finally provide numerical examples to support our theoretical claims. By discretizing the infinite-dimensional LPs, we compute arbitrage-free price bounds for Forward-Start options. We further perform a sensitivity analysis of the aforementioned extrapolation and find that in the case of Forward- Start options it does not significantly influence arbitrage bounds obtained by numerically solving discretized problems.