Minimizing Expected Loss of Hedging in Incomplete and Constrained Markets

Abstract

We study the problem of minimizing the expected discounted loss $$ E\left[e^{-\int_0^Tr(u)du}( C- X^{x,\pi}(T))^+\right] $$ when hedging a liability C at time t=T, using an admissible portfolio strategy $\pi(\cdot)$ and starting with initial wealth x. The existence of an optimal solution is established in the context of continuous-time Ito process incomplete market models, by studying an appropriate dual problem. It is shown that the optimal strategy is of the form of a knock-out option with payoff C, where the "domain of the knock-out" depends on the value of the optimal dual variable. We also discuss a dynamic measure for the risk associated with the liability C, defined as the supremum over different scenarios of the minimal expected loss of hedging C.

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