The Rough Bergomi model as H → 0 - skew

flattening/blow up and non-Gaussian rough volatility

Martin Forde∗ Masaaki Fukasawa† Stefan Gerhold‡ Benjamin Smith§

March 28, 2020

Abstract

Using Jacod’s stable convergence theorem, we prove the surprising result that the martingale component Xtof the log stock price for the popular Rough Bergomi stochastic volatility model (when suitably re-scaled) tendsweakly to Bξγ([0,t]) as H → 0, where ξγ is the Gaussian multiplicative chaos measure of the Riemann-Liouvilleprocess which drives the model in the H → 0 limit, and 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 theH → 0 limit, and without re-scaling the model tends weakly to the Black-Scholes model as H → 0. We alsoderive a closed-form expression for the conditional third moment E((Xt+h −Xt)3|Ft) (for H > 0) given a finitehistory, and E(X3

T ) tends to zero (or blows up) exponentially fast as H → 0 depending on whether γ is lessthan or greater than a critical γ ≈ 1.61711. We describe how to use this equation to calibrate a time-dependentcorrelation function ρ(t) to the observed skew term structure (ρ(t) satisfies an Abel integral equation for whichwe establish existence and uniqueness) and briefly discuss the pros and cons of a H = 0 model with non-zeroskew for which Xt/

√t tends weakly to a non-Gaussian random variable X1 with non-zero skewness as t → 0.

Finally, we also define a generalized family of non-Gaussian rough volatility models, with an associated non-Gaussian multiplicative chaos in the H → 0 limit, using a similar construction to [BM03] with an independentlyscattered infinitely divisible random measure, and we show that the resulting GMC in the H → 0 limit is locallymultifractal for integer moments with a non-quadratic multifractal exponent.1

1 Introduction

[FGS20] consider a re-scaled Riemann-Liouville process ZHt =∫ t

0(t − s)H−

12 dWs in the H → 0 limit. Using’s

Levy’s continuity theorem for tempered distributions, they show that ZH tends weakly to an almost log-correlatedGaussian field Z as H → 0, which is a random tempered distribution, i.e. a random element of the dual of theSchwartz space S and can also be viewed as a Gaussian in the fractional Sobolev Hilbert space H− 1

2−ε, and one canalso show convergence of (Z − ZH)/H to some Γ ∈ H− 1

2−ε, the so-called first order correction field. As a corollary

ξHγ (dt) = eγZHt − 1

2γ2Var(ZHt )dt is shown to tend to a Gaussian multiplicative chaos (GMC) random measure ξγ for

γ ∈ (0, 1) as H → 0. Unlike standard constructions of GMC, ZHt is not a backwards martingale in H so one cannotappeal to the martingale convergence theorem. We later address the more difficult “L1-regime” where γ ∈ [1,

√2)

using Strassen’s theorem and we show that our limiting GMC object using the RL process falls within the generalsetup of Shamov[Sha16] using randomized shifts on the Cameron-Martin space of the underlying Brownian motion.Our more involved analysis for γ ∈ (0,

√2) using Strassen’s theorem and [Sha16] also covers the simpler case

γ ∈ (0, 1) but we have included the simpler L2 analysis for the latter for pedagogical reasons. We also construct acandidate GMC for the super-critical phase γ >

√2, using an independent stable subordinator time-changed by our

Riemann-Liouville GMC (similar to section 3 in [BJRV14]) to construct an atomic GMC with the correct (locally)multifractal exponent for γ-values greater than

√2, which is closely related to the non-standard branch of gravity

in conformal field theory.

These results have a natural application to the popular Rough Bergomi stochastic volatility model, since ξHγ is thequadratic variation of the log stock price for this model and values of H as low as .03 have been reported in empiricalstudies of this model (see e.g. Fukasawa et al.[FTW19]). In this article, using the aforementioned Riemann-LiouvilleGMC and Jacod’s stable convergence theorem, we prove the surprising result that the martingale component Xt

of the log stock price for the Rough Bergomi model tends weakly to Bξγ([0,t]) as H → 0 where B is a Brownianmotion independent of everything else, which means the smile for the rBergomi model with ρ ≤ 0 is symmetric in

∗Dept. Mathematics, King’s College London, Strand, London, WC2R 2LS (Martin.Forde@kcl.ac.uk)†Graduate School of Engineering Science, Osaka University 1-3 Machikaneyama, Toyonaka, Osaka, Japan

(Fukasawa@sigmath.es.osaka-u.ac.jp)‡TU Wien, Financial and Actuarial Mathematics, Wiedner Hauptstraße 8/105-1, A-1040 Vienna, Austria

(sgerhold@fam.tuwien.ac.at)§Dept. Mathematics, King’s College London, Strand, London, WC2R 2LS (Benjamin.Smith@kcl.ac.uk)1We would like to thank Juhan Aru, Nathanael Berestycki, Paul Hager, Janne Junnila, Remi Rhodes, Alex Shamov and Vincent

Vargas for clarifying many points in their articles.

1

the H → 0 limit for γ ∈ (0, 1). More generally for any γ > 0, we derive a closed-form expression for the conditionalthird moment E((Xt+h − Xt)

3|Ft) (for H > 0, and we find that E(X3t ) decays exponentially fast or blows up

exponentially fast depending on whether γ is less than or greater than a critical γ ≈ 1.61711. The expressionfor E(X3

t ) can then be used to calibrate a time-dependent correlation function ρ(t) to the observed skew termstructure, which satisfies an Abel integral equation (for which we establish existence and uniqueness) and we canalso define 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. Finally, we define a new family of non-Gaussian rough volatility models,with an associated non-Gaussian multiplicative chaos in the H → 0 limit, using a similar approach to [BM03] viaindependently scattered infinitely divisible random measures, and we show that the resulting GMC in the H → 0limit is locally multifractal for integer moments with a non-quadratic multifractal exponent.

2 The Rough Bergomi model in the H → 0 limit

2.1 The skew flattening phenomenon

We consider the standard Rough Bergomi model for a stock price process XHt :

dXHt = − 1

2

√V Ht +

√V Ht dWt ,

V Ht = eγZHt − 1

2γ2Var(ZHt )

ZHt =∫ t

0(t− s)H− 1

2 (ρdWs + ρdW⊥t )

(1)

where γ ∈ (0, 1), |ρ| ≤ 1 and W , W⊥ are independent Brownian motions, and (without loss of generality) we setXH

0 = 0. We let

XHt =

∫ t

0

√V Ht dWt (2)

denote the martingale part of XH .

Theorem 2.1 For γ ∈ (0, 1), XH tends to B⊥ξγ([0,(.)]) stably (and hence weakly) in law on any finite interval [0, T ],

where B⊥ is a Brownian motion independent of everything else.

Corollary 2.2 From the weak convergence established in the theorem we see that

limH→0

E(eikXHt ) = lim

H→0E(e−

12 (ik+k2)

∫ t0

√V Hs dWs))

= E(e−12 (ik+k2)ξγ([0,t]))

= E(eik(− 12 ξγ([0,t])+Bξγ ([0,t])))

which (by a well known result in Renault&Touzi[RT96]) implies that implied volatility smile for the true RoughBergomi model in (1) is symmetric in the log-moneyness k = log K

S0.

Remark 2.1 We call this the skew flattening phenomenon, so in particular XHt (for a single fixed t) tends weakly

to some weakly symmetric distribution µ.

Proof. From Theorem 2.2 in [FGS20], we know that 〈XH〉t tends to a random variable ξγ([0, t]) in L2 (and hence

in probability), and 〈XH ,W 〉t = ρ∫ t

0

√V Hu du . But

E((V Ht )12 ) = E(e

12 (γZHt − 1

2γ2 1

2H t2H))

= E(e12γZ

Ht − 1

2 ·14γ

2· 12H+ 1

2 ·14γ

2· 12H−

12γ

2 14H t

2H) = e−1

16H γ2t2H → 0

as H → 0, so (by Markov’s inequality) P(√V Ht > δ) ≤ 1

δE(√V Ht )→ 0, so

√V Ht tends to zero in probability, and

hence

Gt := 〈XH ,W 〉tp→ 0 . (3)

Moreover, for any bounded martingale N orthogonal to W

〈XH , N〉t = 0 . (4)

Thus setting Zt = Wt and applying Theorem IX.7.3 in Jacod&Shiryaev[JS03] (see also Proposition II.7.5 andDefinition II.7.8 in [JS03]), we can construct an extension (Ω, F , (Ft), P) of our original filtered probability space(Ω,F ,Ft,P) and a continuous Z-biased F-progressive conditional PII martingale X on this extension (see Definition

2

7.4 in chapter II in [JS03] for definition), such that XH converges stably (and hence weakly) to X (see Definition5.28 in chapter XIII in [JS03] for definition of stable convergence) for which

〈X〉t = ξγ([0, t])

〈X,M〉t = 0

for all continuous (bounded) martingales M with respect to the original filtration Ft. From Proposition 7.5 andDefinition 7.8 in Chapter 2 in [JS03], this means that

Xt = X ′t +

∫ t

0

usdWs

where X ′ is an Ft-local martingale and u is a predictable process on the original space (Ω,F ,P). One such M isMt = Wt∧τb∧τ−b , where τb = inft : Wt = b, so we have a pair of continuous local martingales (M,X) with

〈X,M〉t = 〈X,W 〉t =

∫ t

0

usds = 0

for t ≤ τb ∧ τ−b, so in fact ut ≡ 0. Then applying F.Knight’s Theorem 3.4.13 in [KS91] with M (1) = X andM (2) = W , if Tt = infs ≥ 0 : 〈X〉s > t, then XTt is a Brownian motion independent of W . Hence X has thesame law as B⊥ξγ([0,t]) for any Brownian motion B⊥ independent of W .

2.2 H → 0 behaviour for the non re-scaled rough Bergomi model

If we replace the definition of ZH with the usual RL process ZHt =√

2H∫ t

0(t− s)H− 1

2 ds (as is usually done), thenfrom Remark 2.4 in [FGS20], we know that ξHγ (A) tends Leb(A) in L2 for any Borel set A ⊆ [0, 1], so adapting

Theorem 2.1 for this case, we see that XH tends weakly to a standard Brownian motion, which means the roughBergomi model tends weakly to the Black-Scholes model in the H → 0 limit.

2.3 A closed-form expression for the skewness of XHt

In this subsection we compute an explicit expression for the skewness of XHt (conditioned on its history), which (as

a by-product) gives a more “hands-on” proof as to why the skew tends to zero as H → 0, and also allows us to seehow fast the skew decays and the H -value in (0, 1

2 ) where the absolute value of the skew is maximized.

We first note that (trivially) XH has the same law as XH defined bydXH

t =√V Ht (ρdBt + ρdWt) ,

V Ht = eγZHt − 1

2γ2Var(ZHt )

ZHt =∫ t

0(t− s)H− 1

2 dBt

(5)

where B is independent of W , and this is the version of the model we use in this subsection. We henceforthuse Et((.)) as shorthand for the conditional expectation E((.)|FBt ), and we now replace the constant ρ with atime-dependent ρ(t), and replace our original V Ht process with

V Ht = ξ0(t) eγZHt − 1

2γ2Var(ZHt )

to incorporate a non-flat initial variance term structure.

Proposition 2.3

Et0((XHT − XH

t0 )3) = 3γ

∫ T

t0

∫ t

0

ρ(s) ξ12t0(s)ξt0(t) e

12γ

2Covt0 (ZHs ZHt )− 1

8γ2Vart0 (ZHs )(t− s)H− 1

2 dsdt (6)

where ξt0(t) = ξ0(t)eγ∫ t00 (t−u)H−

12 dBu− γ2

4H [t2H−(t−t0)2H ]. This simplifies to

E((XHT )3) = 3ργV

32

0

∫ T

0

∫ t

0

e12γ

2(RH(s,t)− s2H8H )(t− s)H− 12 dsdt < ∞ (7)

if t0 = 0, ρ is constant and ξ0(t) = V0 for all t (i.e. flat initial variance term structure).

Proof. See Appendix B.

3

Remark 2.2 Using that RH(s, t)→ RfBM(s, t) as s, t→ 0 (for H > 0 fixed), where RfBM(s, t) = 12H

12 (t2H + s2H −

|t − s|2H) is the covariance function of 1√2HWH where WH is a standard (one or two-sided) fractional Brownian

motion, we find that the exponent in (7) behaves like 116H (s2H + 2t2H − 2(t− s))2H) for s < t as s, t→ 0, and thus

can effectively be ignored, so (for ρ constant)

E((XHT )3) ∼ 3ργV

32

0

∫ T

0

∫ t

0

e12γ

2(RH(s,t)− s2H8H )(t− s)H− 12 dsdt =

3ργV32

0

(H + 12 )(H + 3

2 )TH+ 3

2

as T → 0.

Remark 2.3 We have tested (7) against Monte Carlo estimates for E((XHT )3) and the results are in very close

agreement. Note that XH is driftless so (5) is only a toy model at the moment, but we easily adapt Proposition

2.3 and the two remarks above to incorporate the additional − 12 〈X

H〉t drift term required to make St = eXHt a

martingale. However, the relative contribution from this drift will disappear in the small-time limit, so we omitthe tedious details, since rough stochastic volatility models are generally used (and considered more realistic) oversmall time horizons.

2.4 Convergence of the skew to zero

Corollary 2.4 For γ ∈ (0, 1) and 0 ≤ t ≤ T ≤ 1, Et0((XHT − XH

t0 )3)→ 0 a.s. as H → 0.

Proof. See Appendix A.

2.5 Speed of convergence of the skew to zero

The following corollary quantifies the speed at which the skew goes to zero.

Proposition 2.5 Let ρ(.) be continuous and bounded away from zero with constant sign for t sufficiently small.Then

− limH→0

H log[sgn(ρ)E((XHT )3)] = r(γ) =

116γ

2 0 ≤ γ ≤ 1 ,14 + 1

2 log γ − 316 γ ≥ 1

(8)

Proof. The proof involves many intermediate lemmas, see [Ger20] for details.

Remark 2.4 Note that r(γ) ≤ r(γ) ≤ 116γ

2, where r(γ) = 116γ

21γ2≤8 + 12 (1 + log γ2

8 )1γ2≥8 and r(γ) is negative forγ larger than the root at ≈ 1.61711, which makes the integral explode as H → 0 for such values of γ.

2.6 Calibrating a time-dependent correlation function to the skew term structure

From (7) we see that

∂

∂TE((XH

T )3) = V32

0

∫ T

0

ρ(s)e12γ

2(RH(s,T )− s2H8H )(T − s)H− 12 ds . (9)

If g(T ) = ddT E((XH)3

T ) is known e.g. from the observed prices of call options in the market, this becomes a Volterranon-convolution integral equation of the first kind (also known as an Abel equation) for the unknown ρ(.). Thenfrom e.g. the first Theorem in [Atk74], we know that if g(T ) = T β g(T ) with α + β > 0 (where α = 1

2 − H) and

g ∈ C1[0, T ], then (9) admits a unique solution of the form ρ(t) = tα+β−1f(t) for some f ∈ C1[0, T ]. For ourapplication, we have to then choose β = H + 1

2 to ensure that ρ(t) = O(1), and then choose V0 to ensure that|ρ(t)| ≤ 1. If the calibrated |ρ(t)| exceeds 1 for some t > 0 for all choices of V0 this means either the model is wrongor we need a different H-value. We could also solve for (9) numerically using an Adams scheme.

2.7 A H = 0 model - pros and cons

Returning to Section 4.1, we can circumvent the problem of vanishing skew, by considering a toy model of the form

Xt = σ(ρWt + ρB⊥ξγ([0,t])) (10)

where ρ =√

1− ρ2, W and ξγ([0, t]) are defined as in Section 2.1 with γ ∈ (0, 1), and B⊥ is a Brownian motionindependent of W . Then from the tower property we see that

E(eikXt) = E(E(eik(αWt+βBξγ ([0,t]))|W )) = E(eikαWt− 12k

2β2ξγ([0,t])))

4

0.1 0.2 0.3 0.4 0.5

0.02

0.04

0.06

0.08

0.1 0.2 0.3 0.4 0.5

0.5

1.0

1.5

2.0

2.5

3.0

Figure 1: Here we plotted the (re-normalized) skew 1ρE((XH

T )3) using (7) (which is independent of ρ) as a function

of H for T = .1 (left) an T = 1 (right), for γ = .2 (blue), γ = .4 (grey), γ = .6 (blue dashed), and γ = .8 (greydots). The numerical results indicate that there is a unique H ∈ (0, 1

2 ) where the absolute value of the skewness ismaximized.

and (from Corollary 2.3 in [FGS20]) we know that ξγ([0, t]) ∼ tξγ([0, 1]) (i.e. self-similarity), so

E(eik√tXt) = E(eikαWt/

√t− 1

2k2β2ξγ([0,t])/t) = E(eikαW1− 1

2k2β2ξγ([0,1]))

so X is self-similar with Xt/√t ∼ X1 for all t > 0, and X1 (and hence Xt) has non-zero skewness for α 6= 0; more

specifically

E((Xt√t)3) = 4σ3ρ(1− ρ2)γ (11)

and E(X21 ) = σ2, and we can derive a similar (slightly more involved) expression for E(X4

1 ). We note that theskewness (11) is minimized (resp. maximized) at ρ∗± = ± 1√

3≈ ±0.577 (this does not imply that the density of Xt

is symmetric when ρ = ±1 even though the skewness is zero in this case). The ρ component achieves the goal of aH = 0 model with non-zero skewness, and following a similar argument to Lemma 5 in [MT16] one can establishthe following small-time behaviour for European put options in the Edgeworth Central limit theorem regime:

1√tE((ex

√t − eXt)+) ∼ ex

√t E((x− Xt√

t)+) ∼ E((x− Xt√

t)+) ∼ E((x− X1)+)

and limt→0 σt(x√t, t) = CB(x, .)−1(C(x)) for x > 0, where σt(x, t) denotes the implied volatility of a European

call option with strike ex√t maturity t and S0 = 1 (CB(x, σ) is the Bachelier model call price formula). Hence we

see the full smile effect in the small-time FX options Edgeworth regime unlike the H > 0 case discussed in e.g.[Fuk17], [EFGR19], [FSV19], where the leading order term is just Black-Scholes, followed by a next order skewterm, followed by an even higher order convexity term.

We can go from a toy model to a real model adding back the usual − 12 〈X〉t drift term for the log stock price

X so St = eXt is a martingale, and in this case we lose self-similarity for X but Xt/√t still tends weakly to a

non-Gaussian random variable, and in particular limt→0 E((Xt√t)3) = 4σ3ρρ2γ. 2. This model overcomes two of the

main drawbacks of the original Bacry et al.[BDM01] multifractal random walk (i.e. a Brownian motion evaluatedat an independent MRW MT

γ ([0, (.))), namely zero skewness and unrealistic small-time behaviour.

However, the property in (11) does not appear to be time-consistent, since if we define ηht := E((Xt+h−Xt√h

)3|Ft)for t > 0, then E((ηht )2) = O(h−γ

2

) (and not O(1) as we would want), so we do not pursue this model further atthe present time.

Remark 2.5 It is interesting to compare (11) against the small-time behaviour for a local vol model with σ(S) =σ−1S<S0 + σ+1S≥S0 , for which Pigato[Pig19] shows that the short-time implied volatility skew behaves like

√π√2

σ+ − σ−σ+ − σ−

1√t

as t → 0, which attains the model-free upper bound in [Fuk10] in the limit as σ− → 0. Nevertheless, the extremeskew seen in [Pig19] will not hold as soon as S moves away from S0, so this property is also not time-consistent(see also Remarks 4.3 and 4.4 in [Fuk20].

2We can also replace the ρWt component of X with a second rBergomi component with a non-zero H-value, and derive similarresults

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-5 -4 -3 -2 -1 0 1 2 3 4 5

0

5

10

15

20

25

Figure 2: Here we have plotted the density of X1 for the H = 0 model with skew model in (10) , for ρ = ρ∗− = − 1√3

(left) and ρ = −1 (right) for γ = .5, using the Karhunen-Loeve expansion described in [FGS20].

3 Non-Gaussian rough models, non-Gaussian multifractal exponentsand the H → 0 limit

We now let P be an independently scattered infinitely divisible random measure (see [BM03] for details) with

E(eiqP (A)) = eϕ(q)µ(A)

for q ∈ R, where µ(du, dw) = 1w2 dwdu denotes the Haar measure, and ϕ is the characteristic exponent of an infinitely

divisible distribution. Let AHt := 0 ≤ u ≤ t, w ≥ gH(u, t) for some family of functions which satisfy the followingcondition:

Condition 1 gH(., t) ≥ 0 with gH(u, t) increasing in t and H.

We define the process

ωHt = P (AHt )

with filtration FH = σ(P (A×B) : B ⊆ [H,∞], A,B ∈ B(R)) (compare to a similar filtration on page 17 in [RV10]).Then

E(eiq1ωHs +iq2ω

Ht ) = E(eiq1P (AHs )+iq2P (AHt ))

= E(eiq1P (AHs ∩AHt )+iq1P (AHs ∩(AHt )c)+iq2P (AHt ∩A

Hs )+iq2P (AHt ∩(AHs )c))

= eϕ(q1)µ(AHs ∩(AHt )c)+ϕ(q1+q2)µ(AHs ∩AHt )+ϕ(q2)P (AHt ∩A

cs)) .

Similarly

E(eiq1ωHt +iq2ω

H2t ) = E(eiq1P (AHt )+iq2P (A

H2t ))

= E(eiq1P (AHt ∩AH2t )+iq1P (AHt ∩(A

H2t )c)+iq2P (A

H2t ∩A

Ht )+iq2P (A

H2t ∩(AHt )c))

= eϕ(q1)µ(AHt ∩(AH2t )c)+ϕ(q1+q2)µ(AHt ∩A

Ht )+ϕ(q2)P (AHt ∩(AHt )c)) .

Differentiating once in q1 and once in q2 and then setting q1 = q2 = 0 (and assuming ϕ′(0) = 0 so E(P (A)) = 0 forall A) we find that

E(ωHs ωHt ) = γ2µ(AHs ∩AHt )

E(ωHt ωH2t ) = γ2µ(AHt ∩A

H2t )

where γ2 = ϕ′′(0), and we can re-write the covariance of ωH as

E(ωHs ωHt ) = γ2

∫ s

0

∫ ∞gH(u,t)

1

w2dwdu = γ2

∫ s

0

1

gH(u, t)du

for 0 ≤ s ≤ t. We now want to construct the g function so E(ωHs ωHt ) matches the covariance KH(s, t) of some

given family of Gaussian process ZH with ZH0 = 0 (e.g. the family of Riemann-Liouville processes indexed by H).Differentiating with respect to s, we see this is accomplished if

1

gH(s, t)= Rs(s, t) .

6

-

ϵ

Figure 3: Here we have plotted gH(s, t) for different t values for the RL process/field with H = .25 (left) and H = 0(middle). The middle and right plots show the corresponding Bi,jn sets using our method (middle) and the [BM03]construction with truncated triangles (right). Note we have dropped the n subscript when labelling the sets Bi,jn ,and n = 4.

We further assume that KH(s, t) ∼ log 1|t−s| + h(s, t) as H → 0 from above, for some function h bounded away

from zero, so as to ensure interesting GMC-type behaviour for the measure eωHt

E(eωHt )dt as H → 0.

We can do this explicitly for the re-scaled Riemann-Liouville process ZHt = γ∫ t

0(t− s)H− 1

2 dWs to obtain

gH(s, t) =1

γ

2s12−Ht

32−H

Γ( 12 +H)(t(1 + 2H) 2F1(1, 1

2 −H,32 + H, st ) + (1− 2H)s2F1(2, 3

2 −H,52 +H, st ))

where 2F1(a, b, c, z) is the regularized hypergeometric function3 (see first two plots in Figure 3), and in AppendixC we verify that Condition 1.1 in (1) is satisfied. For H = 0 we get

g0(s, t) =

√s (t− s)√

t.

Now let ψ(z) = ϕ(−iz) as in [BM03] and

ξHϕ (dt) =eω

Ht

E(eωHt )dt .

For H2 < H1, ωH2t − ωH1

t = P (AH2t \ A

H1t ) and ωHt = P (AHt ) are independent for any H ≥ H1, so ωH2

t is anFH -martingale since we imposed that ϕ′(0) so E(P (A)) = 0 for all A, and from this one can easily verify thatξHϕ (A) is an FH -martingale for any Borel set A.

Remark 3.1 Note that although

E(ωHs ωHt ) = E(ZHs Z

Ht )

this does not imply that E(ωHs ωH2t ) = E(ZHs Z

H2t ) for H 6= H2.

3.1 Computing the E(ξϕ[0, T ]q) for q ∈ N and local multifractlity with non-Gaussianmultifractal exponent

Let ωHt = ωHt − E(ωHt ), and define In(...) as follows

E(e∑ni=1 ω

Hti ) = eI

Hn (t1,...,tn)

for 0 = t0 < t1 < ... < tn. Then from e.g. the middle plot in Figure 3 we see that

IHn (t1, ..., tn) =

n∑i=1

n∑j=i

ψ(j)µ(Bi,jn ) −n∑i=1

ψ(1)KH(ui, ui)

where the final sum here comes from the E(ωHt ) term in the definition of ωHt . Let

Bi,jn = (u,w) : ti−1 ≤ u ≤ ti, gH(u, tj) ≤ w ≤ gH(u, tj+1) .3we are using Mathematica’s definition here

7

for i = 1..n, j = i..n, where tn+1 is understood to be +∞. Then

µ(Bi,jn ) =

∫ ti

ti−1

∫ gH(u,tj+1)

gH(u,tj)

1

w2dwdu = µ(Ai,j \ Ai−1,j \ (Ai,j+1 \ Ai−1,j+1))

=

∫ ti

ti−1

(1

gH(u, tj)− 1

gH(u, tj+1))du

= KH(ti, tj)−KH(ti−1, tj)− (KH(ti, tj+1)−KH(ti−1, tj+1))

for i = 1..n and j = i..n − 1, and for j = n, µ(Bi,jn ) =∫ titi−1

1gH(u,tj)

du = KH(ti, tj) − KH(ti−1, tj), where

Ai,j = AHti ∩AHtj . Thus we see that

IHn (t1, ..., tn) =

n∑i=1

n∑j=i

ψ(j)(KH(ti, tj)−KH(ti−1, tj)−KH(ti, tj+1) +KH(ti−1, tj+1)) −n∑i=1

ψ(i)KH(ui, ui)

and note that all diagonal terms cancel.

By visual inspection of the right plot in Figure 3, we see we get the exact same formula for µ(Bi,jn ) for the[BM03] construction as well if we now define Bi,jn as

Bi,jn = Ai,j \ Ai−1,j \ (Ai,j+1 \ Ai−1,j+1)

with KH(., .) replaced by KTε (., .), as defined in section 2.4 in [FGS20]. Note KH(ti, tj) = 0 if ti or tj = 0, but this

is not the case for KTε .

For the special cases n = 1, 2, 3, 4 we have

IH(t1) = 0

IH2 (t1, t2) = (ψ(2)− 2ψ(1))KH(t1, t2)

IH3 (t1, t2, t3) = (ψ(2)− 2ψ(1))(KH(t1, t2) +KH(t2, t3)) + (ψ(3)− 2ψ(2) + ψ(1))KH(t1, t3)

IH4 (t1, t2, t3, t4) = (ψ(2)− 2ψ(1))(KH(t1, t2) +KH(t2, t3) +KH(t3, t4))

+ (ψ(3)− 2ψ(2) + ψ(1))(KH(t1, t3) +KH(t2, t4))

+ (ψ(4)− 2ψ(3) + ψ(2))KH(t1, t4)

and (noting that ψ(0) = 0), the general formula is given by

IHn (t1, ..., tn) =

n−1∑i=1

(ψ(i+ 1)− 2ψ(i) + ψ(i− 1))

n−i∑j=1

KH(tj , tj+i) (12)

and note that ψ(i+1)−2ψ(i)+ψ(i−1) ≥ 0 since ψ is convex (see e.g. Lemma 2.2.31 in [DZ98]). For the log-normalcase ψ(p) = 1

2γ2p2, the general formula simplifies to the well known Selberg formula

IHn (t1, t2, .., tn) = γ2n∑i=1

i−1∑j=1

KH(ti, tj)

(see also section 2.4 in [FGS20]). Returning to the general case now but specializing to the case when KH(s, t) =RH(s, t), we see that

E(ξHϕ ([0, T ]n)) = n!

∫ T

0

∫ t1

0

...

∫ tn−1

0

eIn(t1,...,tn)dt1...dtn . (13)

Theorem 3.1 For any integer q ∈ [1, q∗] (where q∗ is the largest q-value for which ψ(q) > 1 as in [BM03]) andany interval I ⊆ [0, 1], ξHϕ (I) tends to some non-negative random variable ξϕ(I) as H → 0 a.s. and in Lq, and

E[ξHϕ (I)q]→ E[ξϕ(I)q].

Proof. From the upper bound part of the sandwich equation Eq 26 in [FGS20], we have the following inequalityfor 0 < s < t < 1:

RH(s, t) ≤ Kθl∗(H)(s, t)

where θ = 4 · sup(I) and KTl (s, t) is the covariance of the model in [BM03], and l∗(H) ↓ 0 as H ↓ 0. Combining

this with (12) and (13) with RH(s, t) = KH(s, t) and using that all coefficients of KH in (12) are non-negative, wesee that for integer q ∈ [1, q∗)

E[ξHϕ (I)q] ≤ E[Mθl∗(H)(I)q] (14)

8

where MTl is the [BM03] regularized GMC measure as defined as in Eq 17 in [FGS20], and note we are not using

Kahane’s inequality here since for integer q ∈ [1, q∗) we have a Selberg-type formula as in (13) for both cases beingcompared, and Kahane’s inequality is only known to hold for the Gaussian case. Moreover, from Lemma 3 in[BM03] we know that

supl>0

E[Mθl (I)q] < ∞

for q ∈ [1, q∗), so we have the uniform bound supH>0 E[ξHϕ (I)q] < ∞. We know from the end of the previous

subsection that ξHϕ (I) is an FH -martingale. Then (by Doob’s martingale convergence theorem for continuous

martingales) ξHϕ (I) tends to some random variable which we call ξϕ(I) as H → 0 a.s. and in Lq for integerq ∈ [1, q∗). Moreover, from the reverse triangle inequality, the aforementioned Lq-convergence implies that

E(ξHϕ (I)q) → E(ξϕ(I)q)

as H → 0, for integer q ∈ [1, q∗).

Corollary 3.2 We can then mimick the remaining arguments in Theorem 2.2 in [FGS20] to show that there existsa random measure ξϕ such that ξϕ(A) = ξϕ,A a.s. for any Borel set A.

(13) combined with the upper and lower bounds on RH(, , ) in Eq 26 in [FGS20] implies that ξϕ is locallymultifractal for integer q ∈ [1, q∗) with the same multifractal exponent as the [BM03] MRM, namely ζ(q) = q−ψ(q),i.e.

− limδ→0

logE(ξϕ([t, t+ δ])q)

log δ= ζ(q)

for t > 0.Note that the multifractality for the [BM03] MRM is exact, whereas we only have local multifractality for ξϕ, but

ξϕ has the advantage that it is built from the family of Gaussian processes ωHt which have zero boundary conditionwhich is much more natural in a financial modelling context; the [BM03] MRM only makes sense financially if wecondition on the prior history and use the prediction formula obtained in [FS20] which is rather cumbersome toapply in practice.

3.2 A simple multifractal model with zero boundary condition

If we instead let AHt := At = 0∨ t−T ≤ u ≤ t, t−u ≤ w ≤ T, then we find that E(ωsωt) = log 1|t−s|+log(s∨ t) for

0 ≤ s < t ≤ T and E(ωsωt) = log+ T|t−s| for t ≥ T , where we have dropped the dependence on H here since clearly

At does not depend on H here. This simple model has the advantage that ω0 = 0 but mimicks the behaviour ofthe usual Bacry-Muzy log-correlated Gaussian field for t ≥ T .

3.3 Non-Gaussian generalized rough Bergomi models

We can now consider a non-Gaussian extension of the rough Bergomi modeldXH

t =√V Ht dWt ,

V Ht = eωHt +γZHt

E(eωHt +γZHt )

ZHt =∫ t

0(t− s)H− 1

2 (ρdWs + ρdW⊥t )

with XH0 = 0, |ρ| ≤ 1 and W , W⊥ are independent Brownian motions, and ωH independent of both Brownian

motions, and the KH associated with ωH not necessarily equal to the covariance RH of the RL process (but stillsatisfying the zero boundary condition KH(0, t) = 0) for all t. Then by trivial modifications of the argument above,we can define a GMC as the a.s. limit ξϕ of the measure V Ht dt as H → 0.

We can then also consider the following toy model

Xt = σ(ρWt + ρBξHϕ ([0,t]))

where B is an Brownian independent of P and W . Then this model will have skew in H → 0 limit, but (unlike themodel in (10)) will not be self-similar in general since ξHϕ ([0, t]) is not self-similar for general ϕ.

9

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[BDM01b] Bacry, E., J.Delour, and J.Muzy, “Modelling Financial time series using multifractal random walks”,Physica A, 299, 84-92, 2001.

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[BDW17] Bierme, H., O.Durieu, and Y.Wang, “Generalized Random Fields and Levy’s continuity Theorem on thespace of Tempered Distributions”, preprint, 2017.

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[EFGR19] El Euch, O., M.Fukasawa, J.Gatheral and M.Rosenbaum, “Short-term at-the-money asymptotics understochastic volatility models”, SIAM J. Finan. Math., 10 (2019), 491-511.

[FZ17] Forde, M. and H.Zhang, “Asymptotics for rough stochastic volatility models”, SIAM J. Finan. Math., 8,114-145, 2017.

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[FGS20] Forde, M., Gerhold, B.Smith, “The Riemann-Liouville process as H → 0 - sub and super critical GMCand simulation via a new Karhunen-Loeve theorem”, 2020, preprint.

[FSV19] Forde, M., B.Smith and L.Viitasaari, “Rough volatility and CGMY jumps with a finite history and theRough Heston model - small-time asymptotics in the k

√t regime, preprint, 2019.

[Fuk17] Fukasawa, M., “Short-time at-the-money skew and rough fractional volatility”, Quant. Finance, 17, No. 2,189-198, 2017.

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[FTW19] Fukasawa, M., T.Takabatake, and R.Westphal, “Is Volatility Rough”, preprint, 2019.

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[Fuk20] Fukasawa, M., “Volatility has to be rough”, preprint, 2020.

[Ger20] Gerhold, S., “Asymptotic analysis of a double integral occurring in the rough Bergomi model”, 2020, toappear in Mathematical Communications.

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11

A Proof of Corollary 2.4

For T ≤ 1, using that RH(s, t) ↑ R(s, t) and (t− s)H− 12 ↑ (t− s)− 1

2 we see that

|Et0((XHT − XH

t0 )3)| ≤ 3|ρ|γ∫ T

0

∫ t

0

ξ12t0(s)ξt0(t)e

12γ

2(Rt0 (s,t)− s2H8H )(t− s)− 12 dsdt

≤ 3|ρ|γ∫ T

0

∫ t

0

ξ12t0(s)ξt0(t)e

12γ

2(R(s,t)− s2H8H )− 12 log(t−s)dsdt

≤ 3ξ12t0(s)ξt0(t) |ρ|γ

∫ T

0

∫ t

0

e12 (1+γ2) log 1

t−s + 12γ

2gdsdt

≤ const. × E(M√ 12 (1+γ2)

([0, T ])2) < ∞

for γ ∈ (0, 1) where Mγ(dt) is the usual [BM03] GMC, and R0(s, t) = Et0(ZsZs) =∫ st0

(s − u)−12 (t − u)−

12 duds,

g = 2 log(2√

2), ξt = sup0≤s≤t ξs. The result then follows from the dominated convergence theorem.

B Proof of Proposition 2.3

We first recall that for any continuous martingale M , using Ito’s lemma and integrating by parts we know thatE(M3

t ) = 3E(∫ t

0Msd〈M〉s) = 3E(Mt〈M〉t). Thus we see that

Et0((XHT −XH

t0 )3)

= 3Et0((XHT −XH

t0 )(〈XHT 〉 − 〈XH

t0 〉))

= 3Et0(

∫ T

t0

ρ(s)√V Hs dBs ·

∫ T

t0

V Ht dt)

= 3Et0(

∫ T

t0

ρ(s)ξ12t0(s) e

12γ

∫ st0

(s−u)H−12 dBu− 1

2 ·12γ

2∫ st0

(s−u)2H−1dudBs ·

∫ T

t0

ξt0(t) eγ∫ tt0

(t−u)H−12 dBu− 1

2γ2∫ tt0

(t−u)2H−1dudt) .

So we (formally) need to compute

δI = Et0(e12γ

∫ st0

(s−u)H−12 dBu− 1

2 ·12γ

2∫ st0

(s−u)2H−11s<tdudBs · eγ∫ tt0

(t−u)H−12 dBu− 1

2γ2∫ tt0

(t−u)2H−1du)

= Et0(eγ∫ tt0

(t−u)H−12 dBu + 1

2γ∫ st0

(s−u)H−12 dBu− (...)

dBs)

where (...) refers to the non-random terms. To this end, let X = γ∫ tt0

(t− u)H−12 dBu + 1

2γ∫ st0

(s− u)H−12 dBu and

Y = dBs. Then E(XY ) = γ(t− s)H− 12 ds 1s<t (since formally E( 1

2γ∫ st0

(s− u)H−12 dBu · dBs) = 0, see end of proof

for discussion on how to make this argument rigorous) and

E(Y eX) = e12E(X2)E(XY ) = e

12VH(s,t)γ(t− s)H− 1

2 ds 1s<t

⇒ δI = e− 1

2γ2∫ tt0

(t−u)2H−1du− 12 ·

12γ

2∫ tt0

(s−u)2H−11s<tdu e12VH(s,t)γ(t− s)H− 1

2 ds 1s<t

where VH(s, t) = γ2∫ tt0

[(t − u)H−12 + 1

2 (s − u)H−12 1s<t]

2du . Cancelling terms in the exponent, we see that δIsimplifies to

δI = e12γ

2∫ tt0

(s−u)H−12 (t−u)H−

12 du− 1

8γ2∫ tt0

(s−u)2H−11s<tdu)(t− s)H− 1

2 ds 1s<t

= e12γ

2Covt0 (ZHs ZHt )− 1

8 Vart0 (ZHs ))(t− s)H− 12 ds 1s<t

Then

Et0((XHT −XH

t0 )3) = 3Et0∫ T

t0

∫ T

t0

ρ(s)ξ12t0(s) ξt0(t)δIdt

and (6) and (7) follow. Finally we recall that a general stochastic integral∫ t

0φsdMs with respect to a martingale

M is defined as an L2- limit of∫ t

0φ 1n [ns]dMs; using this construction we can rigourize the formal argument above

with δI (we omit the tedious details for the sake of brevity).

C Monotonicity properties of gH(s, t) for the Riemann-Liouville case

The covariance of the RL process for s < t < 1 is

R(s, t) =

∫ s

0

(s− u)H−12 (t− u)H−

12 du =

∫ s

0

uH−12 (t− s+ u)H−

12 du . (C-1)

12

Differentiating this expression using the Leibniz rule we see that

Rs(s, t) = sH−12 tH−

12 + (

1

2−H)

∫ s

0

uH−12 (t− s+ u)H−

32 du

and recall that gH(s, t) = 1Rs(s,t)

. Then we can infer monotonicity properties of g from Rs:

• By inspection Rs is a decreasing function of t, so g is increasing in t.

• For 0 < s < t, (t− s+ u)H−12 is a smooth function of u on [0, s] so the integral term in our expression for Rs

is finite ∀t > 0. Thus Rs(s, t) tends to +∞ as s→ 0 so gH(0, t) = 0 for t > 0.

• For s = t > 0 the first term in (3) is finite but the integral diverges, so we also have gH(t, t) = 0.

• For s, t ∈ (0, 1]2, (st)H−12 , 1

2 −H and uH−12 (t− s+ u)H−

32 are non-negative and decreasing in H, so gH(s, t)

is increasing in H.

• By inspection, gH(s, t) is continuous for s ∈ [0, t], and performing a Taylor series expansion of ∂∂sgH(s, t)(s, t)

we can show that ∂∂sgH(s, t)→ −∞ as s 0 and s t.

These properties can be seen in Figure 3.

13