# Static and semi-static hedging as contrarian or conformist bets

## TL;DR

This paper explores the complexities of semi-static hedging, emphasizing model-dependent payoffs, deriving new integral representations, and analyzing hedging errors, especially for exotic options and jump processes.

## Contribution

It introduces new integral formulas for exotic option payoffs, analyzes hedging errors, and develops efficient numerical methods for semi-static hedging with vanillas and digital options.

## Key findings

- Hedging errors can be significantly larger for static portfolios than for variance-minimizing portfolios.
- Exact semi-static hedging of barrier options is impossible with jump processes.
- Using vanillas and first touch digitals improves hedging accuracy.

## Abstract

In this paper, we argue that, once the costs of maintaining the hedging portfolio are properly taken into account, semi-static portfolios should more properly be thought of as separate classes of derivatives, with non-trivial, model-dependent payoff structures. We derive new integral representations for payoffs of exotic European options in terms of payoffs of vanillas, different from Carr-Madan representation, and suggest approximations of the idealized static hedging/replicating portfolio using vanillas available in the market. We study the dependence of the hedging error on a model used for pricing and show that the variance of the hedging errors of static hedging portfolios can be sizably larger than the errors of variance-minimizing portfolios. We explain why the exact semi-static hedging of barrier options is impossible for processes with jumps, and derive general formulas for variance-minimizing semi-static portfolio. We show that hedging using vanillas only leads to larger errors than hedging using vanillas and first touch digitals. In all cases, efficient calculations of the weights of the hedging portfolios are in the dual space using new efficient numerical methods for calculation of the Wiener-Hopf factors and Laplace-Fourier inversion.