Discrete-Time Quadratic-Optimal Hedging for European Contingent Claims

Abstract

We revisit the problem of optimally hedging a European contingent claim (ECC) using a hedging portfolio consisting of a risky asset that can be traded at pre-specified discrete times. The objective function to be minimized is either the second-moment or the variance of the hedging error calculated in the market probability measure. The main outcome of our work is to show that unique solutions exist in a larger class of admissible strategies under integrability and non-degeneracy conditions on the hedging asset price process that are weaker than previously thought possible. Specifically, we do not require the hedging asset price process to be square-integrable, and do not use the bounded mean-variance trade off assumption. Our criterion for admissible strategies only requires the cumulative trading gain, and not the incremental trading gains, to be square integrable. We derive explicit expressions for the second-moment and the variance of the hedging error to arrive at the respective optimal hedging strategies. We use the expressions mentioned above to also give explicit solutions to two constrained mean-variance frontier problems, namely, minimizing the variance subject to a lower bound on the mean profit, and maximizing the mean profit subject to an upper bound on the variance. Further, we explain the connections between our solution and that of the previous formulations. Finally, we identify the associated variance-optimal martingale measure and provide an expression for the L2-approximation price of the hedged ECC in that measure.