## Abstract

We consider a re-scaled Riemann–Liouville (RL) process [Formula presented], and using Lévy's continuity theorem for random fields we show that Z^{H} tends weakly to an almost log-correlated Gaussian field Z as H→0. Away from zero, this field differs from a standard Bacry–Muzy field by an a.s.Hölder continuous Gaussian process, and we show that [Formula presented] tends to a Gaussian multiplicative chaos (GMC) random measure ξ_{γ} for γ∈(0,1) as H→0. We also show convergence in law for ξ_{γ}^{H} as H→0 for γ∈[0,2) using tightness arguments, and ξ_{γ} is non-atomic and locally multifractal away from zero. In the final section, we discuss applications to the Rough Bergomi model; specifically, using Jacod's stable convergence theorem, we prove the surprising result that (with an appropriate re-scaling) the martingale component X_{t} of the log stock price tends weakly to B_{ξγ([0,t])} as H→0, where B is a Brownian motion independent of everything else. This implies that the implied volatility smile for the full rough Bergomi model with ρ≤0 is symmetric in the H→0 limit, and without re-scaling the model tends weakly to the Black–Scholes model as H→0. We also derive a closed-form expression for the conditional third moment E((X_{t+h}−X_{t})^{3}|F_{t}) (for H>0) given a finite history, and E(X_{T}^{3}) tends to zero (or blows up) exponentially fast as H→0 depending on whether γ is less than or greater than a critical γ≈1.61711 which is the root of [Formula presented]. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which X_{t}/t tends weakly to a non-Gaussian random variable X_{1} with non-zero skewness as t→0.

Original language | English |
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Article number | 109265 |

Journal | Statistics and Probability Letters |

Volume | 181 |

Early online date | 28 Oct 2021 |

DOIs | |

Publication status | Published - Feb 2022 |

## Keywords

- Fractional Brownian motion
- Gaussian fields
- Gaussian multiplicative chaos
- Rough volatility