Pricing and Hedging Contingent Claims by Entropy Segmentation and Fenchel Duality

Abstract

We present a new approach to the problem of characterizing and choosing equivalent martingale pricing measures for a contingent claim, in a finite-state incomplete market. This is the entropy segmentation method achieved by means of convex programming, thanks to which we divide the claim no-arbitrage prices interval into two halves, the buyer’s and the seller’s prices at successive entropy levels. Classical buyer’s and seller’s prices arise when the entropy level approaches 0. Next, we apply Fenchel duality to these primal programs to characterize the hedging positions, unifying in the same expression the cases of super (resp. sub) replication (arising when the entropy approaches 0) and partial replication (when entropy tends to its maximal value). We finally apply linear programming to our hedging problem to find in a price slice of the dual feasible set an optimal partial replicating portfolio with minimal CVaR. We apply our methodology to a cliquet style guarantee, using Heston’s dynamic with parameters calibrated on EUROSTOXX50 index quoted prices of European calls. This way prices and hedging positions take into account the volatility risk.