Anomalous diffusions in option prices: Connecting trade duration and the volatility term structure

Anomalous diffusions arise as scaling limits of continuous-time random walks whose innovation times are distributed according to a power law. The impact of a nonexponential waiting time does not vanish with time and leads to different distribution spread rates compared to standard models. In financial modeling this has been used to accommodate random trade duration in the tick-by-tick price process. We show here that anomalous diffusions are able to reproduce the market behavior of the implied volatility more consistently than the usual L'evy or stochastic volatility models. Two distinct classes of underlying asset models are analyzed: one with independent price innovations and waiting times, and one allowing dependence between these two components. These 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 to no-arbitrage pricing formulas, study their statistical properties, and in particular observe that skewness and kurtosis of asset returns do not tend to zero as time goes by. We finally characterize the large-maturity asymptotics of call option prices, and find that the convergence rate to the spot price is slower than in standard L'evy regimes, which in turn yields a declining implied volatility term structure and a slower time decay of the skew.