Shortfall Risk Minimisation vs. Symmetric (Quadratic) Hedging

The seller of a contingent claim H can always find a self-financing investment strategy that (super)hedges the claim H. When the seller wants to endow an initial capital x less than the one required to get perfect (super)hedging, the shortfall risk minimisation problem arises in a natural way. The aim is to find the strategy that minimises E{`([H(ST )−V x,φ T ])} (shortfall risk), where V x,φ t is the value at time t of the portfolio obtained according to the strategy φ and ` is a loss function (increasing on R+ and such that `(0) = 0) describing the attitude of investor towards taking risks. Another approach studied in the literature is that of quadratic hedging. Here, the investor wants to determine the initial capital x and the strategy φ to minimize E{(H(ST )− V x,φ T )}. This paper investigates a comparison between these two approaches. At first, the second problem is generalised to the case when an expression of the form E{`(H(ST )−V x,φ T )} is considered, with ` a suitable loss function: the resulting problem is called symmetric hedging problem. Then, we consider the case when the investor only wants to minimize the criterion w.r.t. the strategy φ, starting with a given initial capital x. This way, every triple (H, x, φ) defines two optimisation problems (shortfall risk minimization and symmetric hedging) which are compared.