Statistical inference for rough volatility Minimax theory

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arXiv:2210.01214v2 [math.ST] 15 Feb 2024

Annals of Statistics, to appear

STATISTICAL INFERENCE FOR ROUGH VOLATILITY: MINIMAX

THEORY

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

MATHIEU ROSENBAUM4,d,AND GRÉGOIRE SZYMANSKY4,e

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

Science and Technology, acarstenchong@ust.hk

2Ceremade, University 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, rough volatility models have gained consider-

able attention in quantitative ﬁnance. In this paradigm, the stochas-

tic volatility of the price of an asset has quantitative properties sim-

ilar to that of a fractional Brownian motion with small Hurst index

H < 1/2. In this work, we provide the ﬁrst rigorous statistical anal-

ysis of the problem of estimating Hfrom historical observations of

the underlying asset. We establish minimax lower bounds and design

optimal procedures based on adaptive estimation of quadratic func-

tionals based on wavelets. We prove in particular that the optimal rate

of convergence for estimating Hbased on price observations at ntime

points is of order n−1/(4H+2) as ngrows to inﬁnity, extending results

that were known only for H > 1/2. Our study positively answers the

question whether Hcan be inferred, although it is the regularity of a

latent process (the volatility); in rough models, when His close to 0,

we even obtain an accuracy comparable to usual √n-consistent regu-

lar statistical models.

1. Introduction.

1.1. The rough volatility paradigm. Introduced for ﬁnancial engineering purposes

by [35] in 2014, the essence of the rough volatility paradigm is that the volatility pro-

cess (σt), understood as a continuous-time process, exhibits irregular sample paths.

As a prototype model for the log-price of an asset (St), we let

(1) 





St=S0+Rt

0σsdBs,

σ2

t=σ2

0exp(ηW H

t),

where (Bt)is a Brownian motion and (WH

t)is an independent fractional Brownian

motion with Hurst parameter H∈(0,1). The constants σ0and ηare positive and con-

sidered as nuisance parameters. The key feature of rough volatility modelling is that

Hshould be small, of order 10−1. Other rough volatility models involve more com-

plex functionals of the fractional Brownian motion adapted to various applications,

but with the same order of magnitude for H, see for example [23,36]. In particular,

MSC2020 subject classiﬁcations:Primary 60G22, 62C20; secondary 62F12, 62M09, 62P20.

Keywords and phrases: Rough volatility, fractional Brownian motion, wavelets, minimax optimality,

pre-averaging.

for mathematical simplicity, we assume that (Bt)and (WH

t)are independent, and

this rules out leverage effects between price and volatility. The case of leverage effect

is considered in our companion paper [16].

Historically, the rough volatility property was discovered using an empirical ap-

proach, where data sets of interest are time series of daily measurement of historical

volatility over many years and for thousands of assets, see [11,35]. Its daily values

were estimated by ﬁltering high-frequency price observations, using various up-to-

date inference methods for high-frequency data, all of them leading to analogous

results. Several natural empirical statistics were computed on these volatility time

series, in a model-agnostic way. Then it was shown in [35] that strikingly similar

patterns are observed when computing the same statistics in the simple Model (1)

(actually a version of (1) where one considers a piecewise constant approximation

of the volatility). For example, among the statistics advocating for rough volatility,

empirical means of the structure function

(2) ∆7→|log(σt+∆)−log(σt)|q, q > 0,

play an important role, for ∆going from one day to several hundreds of days. For

every value of q, the empirical counterpart of (2) systematically behaves like ∆Aq,

where Aof order 10−1, for the whole range of ∆. This scaling invariance is obvi-

ously reproduced if the volatility dynamics follow (1) with Hof order 10−1, thanks

to the scaling property of fractional Brownian motion. In addition, the fact that this

empirical fact also holds for large ∆somehow discards alternate stationary model

candidates, where the moments of the increments no longer depend on ∆for large

∆. It also kind of rules out the idea that this empirical scaling of the log-volatility

increments could be an artefact due to the estimation error in the volatility process.

1.2. Rough volatility in the literature. At ﬁrst glance, the relevance of the parameter

value in Model (1) may be surprising. It is in stark contrast with the ﬁrst generation

of fractional volatility models (FSV) where H > 1/2in a stationary environment, see

[20]. The goal of FSV models was notably to reproduce long memory properties of

the volatility process and we know that fractional Brownian motion increments ex-

hibit long memory when H > 1/2. However, it turns out that when His very small,

it remains consistent with the behaviour of ﬁnancial data even on very long time

scales, see [35]. In addition to the stylised facts obtained from historical volatility,

the data analysis obtained from implied volatility surfaces also support the rough

volatility paradigm, see [5,8,30,50]. In other words, rough volatility models are, in

ﬁnancial terms, compatible with both historical and risk-neutral measures. Further-

more, rough volatility models can be micro-founded: in fact, only a rough nature

for the volatility can allow ﬁnancial markets to operate under no-statistical arbitrage

conditions at the level of high-frequency trading, see [22,48]. This has paved the way

over the last few years to several new research directions in quantitative ﬁnance.

Among other contributions, we mention risk management of complex derivatives,

as considered for instance in [1,6,23,25,31,36,42,45], numerical issues as addressed

in [2,10,15,34,51,57], asymptotic expansions are provided in [17,24,26–28,46] and

theoretical considerations about the probabilistic structure of rough volatility mod-

els as in [3,9,21,29,33,37,38].

Beyond the popularity of rough volatility models due to their remarkable ability

to mimic data, the domain is certainly mature enough to take a step back with a view

STATISTICAL INFERENCE FOR ROUGH VOLATILITY: MINIMAX THEORY 3

towards a mathematically sound statistical inference program. This is the topic of the

paper. We undertake the rigorous statistical analysis of Model (1), taken as a postu-

late or prototype of a rough volatility model. The intriguing question is obviously

how well can one estimate Hfrom discrete historical data, if Hcan be estimated

at all! How does the postulate of a small Himpact a generic inference program?

Put differently, how well can we distinguish between two values of Hand therefore

overcome the latent nature of the volatility and the noise in its estimation?

These fundamental questions have partially been addressed in the recent litera-

ture. In [32], one postulates the approximation

(3) log c

σ2

t≈log Zt∆

(t−1)∆

σ2

sds+εt,

where (εt)is a Gaussian white noise and c

σ2

tis the quadratic variation computed

from high-frequency observations of the log-price over the interval [(t−1)δ, tδ). Tak-

ing such an approximation for granted, an estimator of His obtained by spectral

methods and a Whittle estimator is proved to be convergent and to satisfy a central

limit theorem in a classical high-frequency framework. Unfortunately, the method-

ology is tightly related to the approximation (3), which is not accurate enough to

apply in Model (1). Another interesting study is that of [13] that requires an ergodic

framework and stationary assumptions. In the present contribution, we refrain from

making such restrictive assumptions. Our companion paper [16] considers a similar

setting to ours, with a slightly different class of models and a practitioner perspec-

tive in mind, focusing on the most useful rough volatility models like rough Heston

and obtaining associated central limit theorems for estimating H.

1.3. Results of the paper.

Piecewise constant volatility. In the spirit of rough volatility models inspired by ﬁ-

nancial engineering, we start our study with a version of Model (1) where the volatil-

ity is piecewise constant at a certain time scale δ. We consider nregularly sampled

observations of the price (St)with 0< H < 1over a ﬁxed time interval [0, T ]. With-

out loss of generality, we take T= 1, hence the time step between observations is

∆ = n−1. We assume ∆≪δ, and this setting is compatible for instance with trading

models that assume a constant volatility over a one day period (for δof the order of

one day). Mathematically, observing equivalently squared increments boils down to

having spot variance observations (at the scale where the volatility is assumed to be

constant) multiplied by noise. Taking logarithm reduces our problem to the setup of

Gloter and Hoffmann [40], where the estimation of the Hurst parameter for a frac-

tional Brownian motion observed with additive measurement error is studied for

H > 1/2. The approach of [40] is based on the scaling properties of wavelet-based

energy levels of fractional Brownian motion

Qj=X

(dj,k)2

where the dj,k are the wavelet coefﬁcients (in an arbitrary wavelet basis) at a dyadic

resolution j. Indeed, we prove that 22jH Qjconverges to some constant with an

explicit rate as j→ ∞, and this yields a strategy for recovering Hbased on an esti-

mation of the ratio Qj+1/Qj, provided it can be estimated accurately enough from

discrete data. Furthermore, the multiresolution nature of wavelet decompositions

enables one to select the optimal resolution jvia an adaptive thresholding proce-

dure. This is somehow simpler than using the scaling properties of p-variations of

the data, although both approaches are similar in spirit. In [40] the energy levels Qj

are estimated from increments of the observations, a strategy that unfortunately can-

not be directly applied here when His small: the roughness of the volatility paths

induces a large bias in the estimation of the energy levels. We mitigate this phe-

nomenon by using a pre-averaging technique introduced in [58]. More precisely, we

show that the energy levels computed from the pre-averaged spot volatility process

still possess nice scaling properties, paving the way to construct a new estimator

that achieves the rate max(n−1/(4H+2), δ1/2). This rate is indeed minimax optimal

for any H∈(0,1). In particular, we obtain – from a rigorous statistical viewpoint –

that estimating Hin this toy model with rough piecewise constant volatility has the

same complexity as classical regular model, the rate n−1/(4H+2) is close to n−1/2in

the rough limit H→0whenever δis small enough.

The general case. In the generic setting of Model (1), the law of the price incre-

ments is more intricate. The piecewise constant volatility part of the previous toy

model is now replaced by local averages of the volatility process. As a matter of fact,

the inference of Hhas been studied in [55] for H > 1/2: energy levels are computed

from price increments and are shown to exhibit a scaling property around a stochas-

tic limit; the same strategy as in [40] can then be undertaken. However, this approach

completely fails in the rough case H < 1/2: over a short time interval, for small H,

local averages of volatility are bad approximations of spot volatility, and this is a cru-

cial element in [55] that requires the condition H > 1/2. We overcome this difﬁculty

as follows: by combining empirical means techniques used in [35] together with our

results in the previous piecewise constant case, we compute energy levels from the

logarithm of the squared price increments and not from price increments. However,

over a small time interval of the form [iδ, (i+ 1)δ), the logarithm of price increments

involves

log 1

δZ(i+1)δ

iδ

exp(ηW H

s)ds

which does not enjoy nice properties when His small: indeed; the roughness of the

trajectories makes this quantity far from a discretized Gaussian process ηW H

iδ . This

creates a bias when computing Hfrom a ratio of energy levels and the scaling law

is no longer exact: additional terms of order 2−2ajH for some a > 1appear. They can

be removed thanks to a suitable bias correction procedure. To that end, we start with

a pilot estimator with no bias correction on the energy levels. We plug in the pilot

estimator to correct the initial energy level estimation and compute a second estima-

tor with an improved rate of convergence. By bootstrapping this procedure O(H−1)

many times, we achieve the minimax optimal rate of convergence n−1/(4H+2) for

every H > 0.

Organisation of the paper. The model with piecewise constant volatility is consid-

ered in Section 2. This assumption is removed in Section 3where we consider the

general model (1). Discussions are gathered in Section 4. Sections 5to 9are devoted

to the proofs. In Section 5, we prove the lower bound for the piecewise constant

STATISTICAL INFERENCE FOR ROUGH VOLATILITY: MINIMAX THEORY 5

volatility model. Note that we formally have a hidden Markov chain, since, condi-

tional on (WH

t), we have a Gaussian Markov chain (actually with independent in-

crements). Yet, due to the absence of ergodicity or stationarity, hidden Markov chain

techniques cannot be applied to handle a likelihood, the key to understanding lower

bounds. Instead, we revisit the hidden pathwise strategy of [41] and [39], later devel-

oped in [40] and for stochastic volatility by [55] for H > 1/2. Here, the case H < 1/2

is substantially more intricate and requires delicate Gaussian calculus. Section 6, es-

tablishes that the estimator for Hconstructed in Section 2in the piecewise constant

volatility model is minimax optimal. This requires us to establish an appropriate

scaling law for an averaged version of the energy levels (namely (25)) that is based

on the Gaussian calculus; see Lemmas 18 and 19. Both Sections 5and 6are relatively

straightforward in terms of testing and estimation strategy, once the techniques of

[40] are well understood. Yet, the computations are more involved. Also, they lay

the necessary ground, both technically and methodologically, for solving the signif-

icantly more difﬁcult case of the general continuous volatility Model (1). Its study

is undertaken in Sections 7to 9, where both the lower bounds and upper bounds

are proved. As far as the lower bound is concerned, only slight modiﬁcations are re-

quired. This is no longer the case for the upper bound, where a new type of scaling

law for the energy levels must be found in order to follow our general approach.

In particular, the scaling law of Proposition 5and its estimation in order to mimick

the results of [40] are the core of the paper. It is based on an appropriate expansion

across scales of the energy levels that strongly uses the representation of the volatil-

ity as an exponential of fractional Brownian motion, and requires intricate Gaussian

calculus, mostly based on Isserlis’ theorem. The tools are presented in Appendix A

and Appendix B, which also contain auxiliary technical results.

2. The piecewise constant rough volatility model.

2.1. Model and notation. We start with a simpliﬁed version of Model 1, as a proto-

type rough volatility model where the volatility is assumed to be piecewise constant

at a given scale δ > 0. (In ﬁnancial terms, δis thought to be of the order of one trading

day, i.e. volatility is assumed to be constant over a day.) Given parameters H∈(0,1)

and η > 0, on a rich enough probability space (Ω,A,PH,η), the price model takes the

form

(4) 









St=S0+Rt

0σsdBs,

σ2

t=σ2

0exp(Xt),

Xt=ηW H

⌊tδ−1⌋δ,

where (Bt)is a Brownian motion and (WH

t)is an independent fractional Brownian

motion with Hurst parameter H∈(0,1) and η, σ0>0are nuisance parameters. With

no loss of generality, we further assume σ2

0= 1.

We observe a sample path of (St)at n+ 1 discrete time points:

S0, S1/n, S2/n, . . . , S1.

To simplify notation and computations, with no loss of generality, we assume that n

and δlive on a dyadic scale in the following sense: for some integer N≥0, we have

n= 2Nand m=nδ = 2Nδis an integer.

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arXiv:2210.01214v2[math.ST]15Feb2024AnnalsofStatistics,toappearSTATISTICALINFERENCEFORROUGHVOLATILITY:MINIMAXTHEORYBYCARSTENH.CHONG1,a,MARCHOFFMANN2,b,YANGHUILIU3,c,MATHIEUROSENBAUM4,d,ANDGRÉGOIRESZYMANSKY4,e1DepartmentofInformationSystems,BusinessStatisticsandOperationsManagement,TheHongKongUnivers...

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Statistical inference for rough volatility Minimax theory

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