Statistical inference for rough volatility Central limit theorems

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arXiv:2210.01216v3 [math.ST] 14 Jun 2024

STATISTICAL INFERENCE FOR ROUGH VOLATILITY:

CENTRAL LIMIT THEOREMS

BYCARSTEN H. CHONG1,a, MARC HOFFMANN2,b, YANGHUI LIU3,c,

MATHIEU ROSENBAUM4,dAND GRÉGOIRE SZYMANSKI4,e

1Department of Information Systems, Business Statistics and Operations Management, The Hong Kong University of Science

and Technology, acarstenchong@ust.hk

2CEREMADE, Université Paris Dauphine-PSL, bhoffmann@ceremade.dauphine.fr

3Department of Mathematics, Baruch College CUNY, cyanghui.liu@baruch.cuny.edu

4CMAP, École Polytechnique, dmathieu.rosenbaum@polytechnique.edu;egregoire.szymanski@polytechnique.edu

In recent years, there has been a substantive interest in rough volatility

models. In this class of models, the local behavior of stochastic volatility is

much more irregular than semimartingales and resembles that of a fractional

Brownian motion with Hurst parameter H < 0.5. In this paper, we derive a

consistent and asymptotically mixed normal estimator of Hbased on high-

frequency price observations. In contrast to previous works, we work in a

semiparametric setting and do not assume any a priori relationship between

volatility estimators and true volatility. Furthermore, our estimator attains a

rate of convergence that is known to be optimal in a minimax sense in para-

metric rough volatility models.

1. Introduction. For many years, continuous-time stochastic volatility models were pre-

dominantly based on stochastic differential equations driven by Brownian motion or Lévy

processes. But more recently, [29] found empirical evidence that stochastic volatility is ac-

tually much rougher than semimartingales, in the sense that it locally resembles a fractional

Brownian motion with Hurst index H < 0.5, a statement that was further supported by other

empirical work based on both return data [13,28,30] and options data [11,27,41].

The data-driven approach of [29] to uncover rough volatility starts by considering high-

frequency log-price data {xiδn:i= 0,...,[T/δn]}, where for example δn= 5 min and T=

1year. In a next step, daily spot variance estimates are calculated from the formula

(1.1) ˆcj=1

knδn

i=1

(δn

(j−1)kn+ix)2, j = 1,...,[T/(knδn)],

where δn

ix=xiδn−x(i−1)δnand kn= 78 is the number of 5 min increments during one

trading day. Afterwards, realized power variations of log ˆcj, that is,

m(q, ∆) = 1

[T/∆]

j=1 |log ˆcj∆−log ˆc(j−1)∆|q

are computed for different values of q > 0and ∆∈ {1day,2days,...}. If log ˆcjwere discrete

observations of a continuous Itô semimartingale, then one would expect that m(q, ∆) scales

as ∆q/2, implying that the slope ζqin a regression of log m(q, ∆) on log ∆ satisﬁes

ζq/q ≈1

MSC2020 subject classiﬁcations:Primary 62G15, 62G20, 62M09; secondary 60F05, 62P20.

Keywords and phrases: Central limit theorem, fractional Brownian motion, Hurst parameter, nonparametric

estimation, rough volatility, spot volatility, volatility of volatility.

However, for large set of high-frequency data, [29] consistently found values of ζq/q < 1

indicating that stochastic volatility locally behaves as a fractional Brownian motion with

Hurst parameter H < 1

2. (To be precise, [29] actually computed the realized q-variations of

daily realized variance, that is, of log RVj=knδnˆcj. But since log RVj∆−log RV(j−1)∆ =

log ˆcj∆−log ˆc(j−1)∆, this amounts to the same as computing m(q, ∆).)

As was pointed out by [13,28], the above approach rests on the assumption that realized

variances have the same scaling behavior as the true unobserved volatility. At the same time,

it is well known (see e.g., [4, Chapter 8]) that in the absence of jumps and if volatility is a

semimartingale, spot variance estimators of the type (1.1) converge to true spot variance plus

a small modulated white noise. In a ﬁrst attempt to take estimation errors for spot variance

into account, [13,28] assume that

(1.2) log ˆcj= log true spot variancej+εj,

where εjis a zero-mean iid sequence that is independent of everything else. Under assump-

tion (1.2), [13,28] derive consistent estimators of the roughness parameter Hin parametric

rough volatility models and uphold the conclusion of [29] that volatility is rough in a large

set of ﬁnancial time series. We also refer to [14], where the authors assume (1.2) with slightly

different assumptions on (log ˆcj, εj), and to [46], where a central limit theorem (CLT) for H

is established under (1.2) (see also [50]).

This paper aims to substantially generalize the aforementioned results in two directions:

ﬁrst, we establish consistent and asymptotically mixed normal estimators of Hin a semipara-

metric setting, where except for Hall other model ingredients are fully nonparametric; and

second, we shall do so without assuming any relationship (such as (1.2)) between volatility

proxies and true volatility. The rate of convergence of our best estimator is

(1.3) δ−1/(4H+2)

which as our companion paper [18] shows is optimal in a minimax sense in parametric rough

volatility models.

The remaining paper is structured as follows: in Section 2, after introducing the model

assumptions, we state the main technical result of this paper, Theorem 2.2, a CLT for volatility

of volatility (VoV) estimators in a rough volatility framework. Section 3discusses how we turn

Theorem 2.2 into rate-optimal and feasible estimators of H. In addition to a usual application

of the delta method, the rough volatility setting requires us overcome two distinct challenges:

• eliminating a nonnegligible asymptotic bias term for which we do not have a sufﬁciently

fast estimator;

• constructing an optimal sequence knfor spot variance estimation that depends on the un-

known parameter Hwithout losing a marginal bit of convergence rate.

Our ﬁnal estimator Hnfor His given in Equation (3.34). As Theorems 3.3 and 3.5 show,

Hnis a feasible and rate-optimal estimator of Hif H∈(0,1

2)and is equal to 1

2with high

probability if volatility is a continuous Itô semimartingale. In Section 4, we report the results

of a short simulation study. Section 5contains the main steps of the proof of Theorem 2.2,

with certain technical details postponed to Appendices A–C.

In what follows, we write A.Bif there is a constant C∈(0,∞)that does not depend on

any important parameter such that A≤CB. Furthermore, if An(t)and Bn(t)are stochas-

tic processes, we write An≈Bnif E[supt∈[0,T ]|An(t)−Bn(t)|]→0as n→ ∞. For two

sequences anand bnwe write an∼bnif an/bn→1as n→ ∞. If x∈Rn, we denote its

Euclidean norm by |x|. For any α∈R, we write xα

+=xαif x > 0and xα

+= 0 otherwise. We

also use the notation N={1,2,...}and N0={0,1,2,...}.

STATISTICAL INFERENCE FOR ROUGH VOLATILITY 3

2. Model and CLT for VoV estimators. On a ﬁltered probability space (Ω,F,F=

(Ft)t≥0,P)satisfying the usual conditions, we assume that the log-price xof an asset is

given by a continuous Itô semimartingale of the form

(2.1) xt=x0+Zt

bsds +Zt

σsdWs, t ≥0.

We assume that the squared volatility process c=σ2satisﬁes

(2.2) ct=c0+Zt

asds +Zt

˜g(t−s)˜ηsds +Zt

g(t−s)(ηsdWs+ ˆηsdˆ

Ws),

where

η2

t=η2

0+Zt

aη

sds +Zt

˜gη(t−s)˜

θsds +Zt

gη(t−s)θsd¯

Ws,

ˆη2

t= ˆη0+Zt

aˆη

sds +Zt

˜gˆη(t−s)˜

ϑsds +Zt

gˆη(t−s)ϑsd¯

Ws.

(2.3)

The ingredients of (2.1)–(2.3) are assumed to satisfy the following conditions.

ASSUMPTION CLT. Suppose that the log-price process xis given by (2.1)with the fol-

lowing speciﬁcations:

1. There is H∈(0,1

2]such that the squared volatility process ct=σ2

tsatisﬁes (2.2)with η

and ˆηgiven by (2.3). The variables x0,c0,η2

0and ˆη2

0are F0-measurable.

2. The processes a,b,aηand aˆη(resp., θand ϑ) are adapted and locally bounded real-

valued (resp., R1×4-dimensional) processes. Moreover, for all T > 0, we assume that

(2.4) lim

h→0sup

s,t∈[0,T ],|s−t|≤hE[1 ∧|bt−bs|] + E[1 ∧|at−as|]= 0.

3. The processes ˜η,˜

θand ˜

ϑare adapted, locally bounded and for all T > 0, there is KT∈

(0,∞)such that

(2.5) sup

s,t∈[0,T ]E[1 ∧|˜ηt−˜ηs|] + E[1 ∧|˜

θt−˜

θs|] + E[1 ∧|˜

ϑt−˜

ϑs|]≤KT|t−s|H.

4. The processes Wand ˆ

Ware independent standard F-Brownian motions and ¯

Wis a four-

dimensional F-Brownian motion that is jointly Gaussian with (W, ˆ

W). The components

of ¯

Wmay depend on each other and on (W, ˆ

W).

5. We have

(2.6) g(t) = gH(t) + g0(t), gη(t) = gHη(t) + gη

0(t), gˆη(t) = gHˆη(t) + gˆη

0(t),

˜g(t) = g˜

H(t) + ˜g0(t),˜gη(t) = g˜

Hη(t) + ˜gη

0(t),˜gˆη(t) = g˜

Hˆη(t) + ˜gˆη

0(t),

where

(2.7) gH(t) = K−1

HtH−1/2

+, KH=Γ(H+1

psin(πH)Γ(2H+ 1) ,

and Hη, Hˆη∈(0,1

2],˜

H, ˜

Hη,˜

Hˆη∈[H, 1

2]and g0, gη

0, gˆη

0,˜g0,˜gη

0,˜gˆη

0∈C1([0,∞)) are func-

tions vanishing at t= 0.

Let us comment on the conditions imposed in Assumption CLT. Except for the parameter

H, the assumptions on x,c,ηand ˆηare fully nonparametric and designed in such a way that it

contains the rough Heston model [25,26] as an example, which is a particular important one

as it is founded in the microstructure of ﬁnancial markets [24,36]. Note that we allow c,ηand

ˆηto have both a usual (differentiable) and a rough (non-differentiable) drift. Moreover, by

considering W,ˆ

Wand ¯

W, we allow for the most general dependence between the Brownian

motions driving x,c,η,ˆη. Also note that Hηand Hˆηare not coupled with H, so the VoV

processes ηand ˆηcan be much rougher than the volatility process citself.

The structural assumptions in (2.3) for η2and ˆη2are important and cannot be replaced by

just merely assuming that η2and ˆη2are H-Hölder regular (see the paragraph after (5.27) for

the reason). At the same time, (2.3) is a mild assumption: it is not only satisﬁed by the rough

Heston model but by any model in which VoV is assumed to be continuous and stationary,

thanks to the Wold–Karhunen representation theorem. (Due to the coefﬁcients θsand ϑs,

the processes η2and ˆη2do not have to be stationary.) Also, in the estimation of VoV in a

semimartingale context, (2.3) with Hη=Hˆη=1

2is typically assumed; see [39,48].

REMARK 2.1 (Rough volatility vs. long memory vs. jumps).

1. There is long-standing debate in the literature whether volatility has long memory or not

(see, e.g., [6,20,22,23]). Because we include various g0-functions in (2.6), the kernels in

(2.2) and (2.3) are only speciﬁed around t= 0. In particular, H,Hηand Hˆηare parameters

of roughness and are not related to long-range dependence / long-memory / persistence. In

particular, in our model, volatility can be rough and have long memory at the same time,

which is important as [13,44,45] point out.

2. In some parts of the previous literature, the notion of “roughness” is used as a synonym

for jumps or discontinuities (see, e.g., [5,15]). Since our model neither includes price nor

volatility jumps, one may ask whether rough volatility can be explained through jumps,

which are a well-documented feature of high-frequency price series [3]. In this respect,

we ﬁrst note that [19] shows that rough volatility (in the sense that H < 1

2in (2.2)) can

be statistically distinguished from both price and volatility jumps. In order to make the

estimators developed in this paper robust to jumps, a natural idea is to include truncation in

(2.8) and (2.9) below. We leave it to future work to study the details of such an extension.

3. While this paper focuses on the rough case H≤1/2, we conjecture that the results of this

paper can be extended to H < 3

4without major obstacles. Since noncentral limit theorems

are known to appear for variation statistics of directly observable processes if H≥3

4(see,

e.g., [42]), we do not know whether our results extend to H≥3

If cwas directly observable, a classical way to feasibly estimate Hwould be to prove a

joint CLT for realized autocovariances δ1−2H

nP[T/δn]−ℓ

i=1 δn

ic δn

i+ℓcwith different values of

ℓ∈N0and then to obtain an estimator of Hfrom the ratio of two such functionals; see [9,

16,17,21,32,34,40]. Since we do not observe c, we ﬁrst consider spot variance estimators

ˆcn

t,s =1

knδn

t,s,ˆ

t,s =

[(t+s)/δn]−1

i=[t/δn]

(δn

ix)2, δn

ix=xiδn−x(i−1)δn,(2.8)

where kn∈Nand kn∼θδ−κ

nfor some κ, θ > 0. Then we form realized autocovariances of

these spot variance estimators by deﬁning

Vn,ℓ,kn

t= (knδn)1−2H1

[t/δn]−(ℓ+2)kn+1

i=1 ˆcn

(i+kn)δn,knδn−ˆcn

iδn,knδn

×ˆcn

(i+(ℓ+1)kn)δn,knδn−ˆcn

(i+ℓkn)δn,knδn

(2.9)

STATISTICAL INFERENCE FOR ROUGH VOLATILITY 5

for ℓ≥0. Note that we write [x]and {x}for the integer and fractional part of x, respec-

tively. The normalization in the last line is chosen in such a way that ˜

Vn,ℓ,kn

tconverges in

probability. In the semimartingale context (with H=1

2and ℓ= 0), the functional ˜

Vn,0,kn

was used in [48] (see also [31,39]) to estimate the integrated VoV process Rt

0(η2

s+ ˆη2

s)ds (to

be very precise, this is actually the integrated variance of variance). Still in the semimartin-

gale framework, functionals similar to (2.9) have also been investigated in the literature to

estimate the leverage effect; see [2,7,8,37,47,49].

To state a CLT for ˜

Vn,ℓ,knfor H < 1

2, we have to introduce some additional notation:

for n∈N,h > 0and a function f:R→R, we deﬁne the forward and central difference

operators by

∆n

hf(t) =

i=0

(−1)n−in

if(t+ih), δn

hf(t) =

i=0

(−1)in

if(x+ (n

2−i)h),

respectively. For n= 1, we simply write ∆hf(t) = ∆1

hf(t) = f(t+h)−f(t)and δhf(t) =

δ1

hf(t) = f(t+h

2)−f(t−h

2). Moreover, given α∈R, we use the shorthand notation ∆n

htα

or ∆n

h|t|αfor ∆n

hf(t)where f(t) = tα

+or f(t) = |t|α(δn

htα

+and δn

h|t|αare used similarly).

Finally, for any d∈N, we use st

=⇒to denote functional stable convergence in law in the space

of càdlàg functions [0,∞)→Rdequipped with the local uniform topology. The following

CLT is the main technical result of this paper.

THEOREM 2.2. Let d∈Nand ℓ1,...,ℓd≥2be integers. Furthermore, consider deter-

ministic integer sequences (k(1)

n)n∈N,...,(k(d)

n)n∈Nsuch that for some κ∈[2H

2H+1 ,1

2]and

θ1,...,θd∈(0,∞)we have k(j)

n∼θjδ−κ

nfor all j= 1,...,d. For each j= 1,...,d, let

(2.10) Zn,j

t=δ−(1−κ)/2

n(˜

Vn,ℓj,k(j)

t−Vℓj

t−An,ℓj,k(j)

t),

where for ℓ≥2, we deﬁne

(2.11) Vℓ

t= ΦH

ℓZt

(η2

s+ ˆη2

s)ds

with

ΦH

ℓ=δ4

1|ℓ|2H+2

2(2H+ 1)(2H+ 2)

=(ℓ+ 2)2H+2 −4(ℓ+ 1)2H+2 + 6ℓ2H+2 −4(ℓ−1)2H+2 + (ℓ−2)2H+2

2(2H+ 1)(2H+ 2)

(2.12)

and for a general integer sequence kn,

An,ℓ,kn

t=−2K−1

H+1

(knδn)−1/2−HZt

kn−1

i=0

∆3

1(ℓ−1−i+{u/δn}

kn)H+1/2

×Zu

[u/δn]δn

σvdWv(σuηu−σ[u/δn]δnη[u/δn]δn)du.

(2.13)

Under Assumption CLT, the process Zn

t= (Zn,1

t,...,Zn,d

t)Tsatisﬁes the joint CLT

(2.14) Znst

=⇒Z,

where Z= ((Z1

t,...,Zd

t)T)t≥0is a continuous Rd-valued process that is deﬁned on a very

good ﬁltered extension (¯

Ω,¯

F,¯

F= ( ¯

Ft)t≥0,¯

P)of the original probability space (see e.g. [35,

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arXiv:2210.01216v3[math.ST]14Jun2024STATISTICALINFERENCEFORROUGHVOLATILITY:CENTRALLIMITTHEOREMSBYCARSTENH.CHONG1,a,MARCHOFFMANN2,b,YANGHUILIU3,c,MATHIEUROSENBAUM4,dANDGRÉGOIRESZYMANSKI4,e1DepartmentofInformationSystems,BusinessStatisticsandOperationsManagement,TheHongKongUniversityofScienceandTechno...

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Statistical inference for rough volatility Central limit theorems

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