The skewed multifractal random walk with applications to option smiles
将多重分形随机游走推广到非对称情形，引入过去收益与未来波动率的相关性，并计算矩的标度指数，用于再现期权波动率微笑的各种形状。
Abstract We generalize the construction of the multifractal random walk (MRW) due to Bacry et al (Bacry E, Delour J and Muzy J-F 2001 Modelling financial time series using multifractal random walks Physica A 299 84) to take into account the asymmetric character of 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, which behave as power laws of the time lag with an exponent ζ q =p−2p(p−1)λ2 for even q=2p, as in the symmetric MRW, and as ζ q =p + 1−2p 2λ2−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, which we compare with approximate cumulant expansion formulae for the implied volatility. A large variety of smile surfaces can be reproduced.