Anomalous Diffusions in Option Prices: Connecting Trade Duration and the Volatility Term Structure

33 Pages Posted: 12 Aug 2019 Last revised: 10 Apr 2020

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Lorenzo Torricelli

University of Bologna Department of Statistics

Date Written: August 7, 2019


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 Levy 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 Levy regimes, which in turn yields a declining implied volatility term structure and a slower decay of the skew.

Keywords: anomalous diffusions, volatility skew term structure, derivative pricing, CTRWs, inverse L\'evy subordinators, time changes, L\'evy processes, subdiffusions, Beta distribution, triangular arrays

JEL Classification: G20, H60

Suggested Citation

Jacquier, Antoine and Torricelli, Lorenzo, Anomalous Diffusions in Option Prices: Connecting Trade Duration and the Volatility Term Structure (August 7, 2019). Available at SSRN: or

Antoine Jacquier

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom


The Alan Turing Institute ( email )

British Library, 96 Euston Road
96 Euston Road
London, NW12DB
United Kingdom

Lorenzo Torricelli (Contact Author)

University of Bologna Department of Statistics ( email )

Via Belle Arti
Bologna, 40121

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