Explicit robust hedging strategies for convex or concave payoffs under a
continuous semimartingale model with uncertainty and small transaction costs
are constructed. In an asymptotic sense, the upper and lower bounds of the
cumulative volatility enable us to super-hedge convex and concave payoffs
respectively. The idea is a combination of Mykland's conservative delta hedging
and Leland's enlarging volatility. We use a specific sequence of stopping times
as rebalancing dates, which can be superior to equidistant one even when there
is no model uncertainty.

We study specific nonlinear transformations of the Black-Scholes implied
volatility to show remarkable properties of the volatility surface. Model-free
bounds on the implied volatility skew are given. Pricing formulas for the
European options which are written in terms of the implied volatility are
given. In particular, we prove elegant formulas for the fair strikes of the
variance swap and the gamma swap.