Minimizing shortfall risk using duality approach: An application to partial hedging in incomplete markets

Option pricing and hedging in a complete market are well-studied with nice results using martingale theories. However, they remain as open questions in incomplete markets. In particular, when the underlying processes involve jumps, there could be infinitely many martingale measures which give an interval of no-arbitrage prices instead of a unique one. Consequently, there is often no martingale representation theorem to produce a perfect hedge. The question of picking a particular price and executing a hedging strategy according to some reasonable criteria becomes a non-trivial issue and an interesting question.; In this paper, we study the duality approach in minimizing the shortfall risk proposed by Follmer and Leukert (2000). First we extend the duality results in Kramkov and Schachermayer (1999) to utility functions which are state dependent and not necessarily strictly concave, as our model requires, and in the generality of a semimartingale setting. Then we specialize the duality results to the problem of minimizing shortfall. We next focus on the mixed diffusion case where we explicitly characterize the primal and dual sets in terms of the characteristics. We provide upper bounds for the value function using duality results. Each upper bound produced in this way corresponds to a dual element. For lower bounds, we pick a particular strategy which we call the 'bold strategy' and compute the corresponding value function. In the cases of bonds and call options and constant parameters, closed form solutions for the upper and lower bounds are computed and numerical examples given. This research provides for the first time a method of checking the quality of a hedging strategy according to the principle of minimizing shortfall in an incomplete market model.