  We apply rough-path theory to study the discrete-time gamma-hedging strategy.
We show that if a trader knows that the market price of a set of European
options will be given by a diffusive pricing model, then the discrete-time
gamma-hedging strategy will enable them to replicate other European options so
long as the underlying pricing signal is sufficiently regular. This is a sure
result and does not require that the underlying pricing signal has a quadratic
variation corresponding to a probabilisitic pricing model. We show how to
generalise this result to exotic derivatives when the gamma is defined to be
the Gubinelli derivative of the delta by deriving rough-path versions of the
Clark--Ocone formula which hold surely.
  We illustrate our theory by proving that if the stock price process is
sufficiently regular, as is the implied volatility process of a European
derivative with maturity $T$ and smooth payoff $f(S_T)$ satisfying $f^{\prime
\prime}>0$, one can replicate with certainty any European derivative with
smooth payoff and maturity $T$.
