On the shortfall risk control -- a refinement of the quantile hedging method

The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minimizing setting is ruled out. The existence and concrete forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests are proven. The well known quantile hedging method as well as the classical Neyman-Pearson lemma are generalized. Optimal hedging strategies with shortfall constraints in the Black-Scholes and exponential Poisson model are explicitly determined.