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

## TL;DR

This paper introduces anomalous diffusion models based on CTRWs to better capture market implied volatility and trade duration effects, providing a more accurate framework than traditional models.

## Contribution

It develops new asset price models incorporating anomalous diffusions with dependent and independent components, describing their properties and implications for option pricing.

## Key findings

- Anomalous diffusions better fit implied volatility patterns.
- Skewness and kurtosis persist over time in these models.
- Large-maturity option prices decay slower than in Le9vy models.

## Abstract

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.