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In this paper, we argue that, once the costs of maintaining the hedgingportfolio are properly taken into account, semi-static portfolios should moreproperly be thought of as separate classes of derivatives, with non-trivial,model-dependent payoff structures. We derive new integral representations forpayoffs of exotic European options in terms of payoffs of vanillas, differentfrom Carr-Madan representation, and suggest approximations of the idealizedstatic hedging/replicating portfolio using vanillas available in the market. Westudy the dependence of the hedging error on a model used for pricing and showthat the variance of the hedging errors of static hedging portfolios can besizably larger than the errors of variance-minimizing portfolios. We explainwhy the exact semi-static hedging of barrier options is impossible forprocesses with jumps, and derive general formulas for variance-minimizingsemi-static portfolio. We show that hedging using vanillas only leads to largererrors than hedging using vanillas and first touch digitals. In all cases,efficient calculations of the weights of the hedging portfolios are in the dualspace using new efficient numerical methods for calculation of the Wiener-Hopffactors and Laplace-Fourier inversion.

Static and semi-static hedging as contrarian or conformist bets