The Riemann–Liouville field and its GMC as H→0, and skew flattening for the rough Bergomi model

We consider a re-scaled Riemann–Liouville (RL) process ZtH=∫0t(t−s)H−12dWs, and using Lévy’s continuity theorem for random fields we show that ZH 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 ξγH(dt)=eγZtH−12γ2V ar(ZtH)dt tends to a Gaussian multiplicative chaos (GMC) random measure ξγ for γ∈(0,1) as H→0. We also show convergence in law for ξγH as H→0 for γ∈[0,2) using tightness arguments, and ξγ 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 Xt of the log stock price tends weakly to Bξγ([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((Xt+h−Xt)3|Ft) (for H>0) given a finite history, and E(XT3) 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 14+12logγ−316γ2. We also briefly discuss the pros and cons of a H=0 model with non-zero skew for which Xt/t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t→0.