MONOTONICITY IN THE VOLATILITY OF SINGLE-BARRIER OPTION PRICES

We generalize earlier results on barrier options for puts and calls and log-normal stock processes to general local volatility models and convex contracts. We show that Γ ≥ 0, that Δ has a unique sign and that the option price is increasing with the volatility for convex contracts in the following cases:• If the risk-free rate of return dominates the dividend rate, then it holds for up-and-out options if the contract function is zero at the barrier and for down-and-in options in general.• If the risk-free rate of return is dominated by the dividend rate, then it holds for down-and-out options if the contract function is zero at the barrier and for up-and-in options in general.We apply our results to show that a hedger who misspecifies the volatility using a time-and-level dependent volatility will super-replicate any claim satisfying the above conditions if the misspecified volatility dominates the true (possibly stochastic) volatility almost surely.