The skewed multifractal random walk with applications to option smiles
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 ζq = p - 2p(p - 1)Λ2 for even q = 2p, as in the symmetric MRW, and as ζq = p + 1 - 2p2Λ2 - α (q = 2p + 1), where Λ and α 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.