Sharpe-ratio pricing and hedging of contingent claims in incomplete markets by convex programming

We analyze the problem of pricing and hedging contingent claims in a financial market described by a multi-period, discrete-time, finite-state scenario tree using an arbitrage-adjusted Sharpe-ratio criterion. We show that the writer’s and buyer’s pricing problems are formulated as conic convex optimization problems which allow to pass to dual problems over martingale measures and yield tighter pricing intervals compared to the interval induced by the usual no-arbitrage price bounds. An extension allowing proportional transaction costs is also given. Numerical experiments using S & P 500 options are given to demonstrate the practical applicability of the pricing scheme.