  Anomalous diffusions arise as scaling limits of continuous-time random walks
(CTRWs) whose innovation times are distributed according to a power law. The
impact of a non-exponential waiting time does not vanish with time and leads to
different distribution spread rates compared to standard models. In financial
modelling this has been used to accommodate for random trade duration in the
tick-by-tick price process. We show here that anomalous diffusions are able to
reproduce the market behaviour of the implied volatility more consistently than
usual L\'evy or stochastic volatility models. We focus on two distinct classes
of underlying asset models, one with independent price innovations and waiting
times, and one allowing dependence between these two components. These two
models capture the well-known paradigm according to which shorter trade
duration is associated with higher return impact of individual trades. We fully
describe these processes in a semimartingale setting leading no-arbitrage
pricing formulae, and study their statistical properties. We observe that
skewness and kurtosis of the asset returns do not tend to zero as time goes by.
We also characterize the large-maturity asymptotics of Call option prices, and
find that the convergence rate is slower than in standard L\'evy regimes, which
in turn yields a declining implied volatility term structure and a slower decay
of the skew.
