Semi-static hedging of path-dependent barrier options by non-vanilla European ones has caught quite some attention among both academics and practitioners. In fact, the common practice is questionable since it would only work out in case the stochastic log-returns are conditionally symmetric which is unrealistic. We show in the context of continuous stochastic volatility models how to obtain the hedging strategy via a change to an 'asymmetric risk measure' under which all characteristics can be obtained easily. These considerations lead to a general as well as a dual general form of self-duality. In practice, the hedging strategy would involve instruments written on the cumulative variance.

In the Lévy world, the notion of quasi self-duality is linked to an exponentially decaying factor when comparing the right- and left tails of the Lévy measure density. Lifting symmetric processes to their stochastic exponentials, a certain Möbius transform is involved. Since for non-vanishing order parameter of quasi self-duality two martingale properties have to be satisfied simultaneously, there is a non-trivial relation between the order and shift parameter representing carrying costs in financial applications. This leads to an equation containing an integral term which has to be inverted in applications. We discuss several important properties of this equation and, for some well-known models, we derive a family of closed-form inversion formulae leading to parametrisations of sets of possible combinations in the corresponding parameter spaces of Lévy driven models. As for continuous stochastic volatility models, we provide a structure result for quasi self-dual processes. Moreover, we give a characterisation of continuous Ocone martingales via a strong version of self-duality of their stochastic exponentials. Finally, we discuss an example of a non-Ocone martingale which is process, but not conditionally symmetric, and whose stochastic exponential is a strict local martingale.

The talk is based on joint works with Michael Schmutz, University of Bern, and on ongoing work with Zhanyu Chen, London School of Economics.