Price Comparison Results and Super-replication: An Application to Passport Options

In this paper, we provide a new proof of the result that option prices are increasing in volatility when the underlying is a diffusion process. This has been shown to hold for convex payoff, path-independent options by El Karoui et al [7], Hobson [12] amongst others. The advantage of the new proof is that it can be extended to establish monotonicity results for path-dependent payoffs where the payoff depends on the maximum (or minimum) of the asset price process. The techniques used to prove each of these results are mean comparison theorems of Hajek [9] and coupling of stochastic processes. Using these results, and the connection between passport and lookback options, we prove that the price of a passport option is increasing in volatility for general diffusion models for the asset price. It is shown that the seller of a passport option can super-replicate if the volatility is overestimated, regardless of the strategy followed by the holder.