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

Abstract We consider a re-scaled Riemann–Liouville (RL) process Z t H = ∫ 0 t ( t − s ) H − 1 2 d W s , and using Levy’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.  Holder continuous Gaussian process, and we show that ξ γ H ( d t ) = e γ Z t H − 1 2 γ 2 V ar ( Z t H ) d t 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 ξ γ ( [ 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 1 4 + 1 2 log γ − 3 16 γ 2 . 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 .