The skewed multifractal random walk with applications to option smiles

TL;DR

This paper extends the multifractal random walk model to incorporate asymmetry and correlation in financial returns, enabling better modeling of option implied volatility smiles and capturing complex market phenomena.

Contribution

The authors introduce a skewed multifractal random walk model that preserves scale invariance while capturing asymmetry and correlation effects in financial data.

Findings

The model reproduces the HARCH effect and causal cascade phenomena.

It generates a wide variety of volatility smile shapes.

The extended MRW aligns well with empirical option market data.

Abstract

We generalize the construction of the multifractal random walk (MRW) due to Bacry, Delour and Muzy to take into account the asymmetric character of the financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, that behave as power-laws of the time lag with an exponent zeta_q=p-2p(p-1) lambda^2 for even q=2p, as in the symmetric MRW, and as zeta_q=p(1-2p lambda^2)+1-alpha (q=2p+1), where lambda and alpha are parameters. We show that this extended model reproduces the `HARCH' effect or `causal cascade' reported by some authors. We illustrate the usefulness of this skewed MRW by computing the resulting shape of the volatility smiles generated by such a process, that we compare to approximate cumulant expansions formulas for the implied volatility. A large variety of smile surfaces can be reproduced.