Price comparison results and super-replication: An application to passport options (Q2722281)

In reality option sellers do not know the ``true'' dynamics of the underlying asset and any model used is an approximation. One way of dealing with this is to overestimate volatility. The results of this paper show that for a wide class of path-dependent options (including the passport option), overestimating volatility leads to super replication. It is provided a short alternative proof of the result that convex, path-independent option prices are increasing in volatility when the underlying is a diffusion process. This is a well-known result but the author's alternative proof is more general in that it can be extended to cover exotic options. This monotonicity result has been extended to path-dependent options whose payoff is a function of the maximum (or minimum) of the diffusion process. Using the link between passport and lookback options, this result is applied directly to the passport option. The extended monotonicity result is used to obtain a comparison for passport option prices. For general diffusion models, the price of a passport option is increasing in the volatility of the asset price. Following from this, the author shows that regardless of the strategy followed by the option holder, the sellers will always super-replicate when they overestimate volatility. This again holds for general diffusion models and is proved using a coupling argument.