IMES DISCUSSION PAPER SERIESA Survey of Rough Volatility
Kazuhiro Hiraki and Yuji Shinozaki
Discussion Paper No. 2024-E-6

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
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Monetary and Economic Studies.

IMES Discussion Paper Series 2024-E-6
July 2024

A Survey of Rough Volatility
Kazuhiro Hiraki* and Yuji Shinozaki**
Abstract
Volatility, which is a measure of asset price uncertainty, is an important subject offinance research. In recent years, a growing number of studies have suggested thatvolatility is rough, namely that volatility fluctuates more wildly than isconventionally recognized, making rough volatility a new trend in volatility research.
This strand of literature includes empirical analysis of the volatility observable inhigh-frequency trading data and derivatives markets, theoretical studies that clarify
the necessity of rough volatility models, and their applications to derivative pricing.
There have also been studies that explore the mechanism giving rise to roughness.
In this paper, we provide an overview of the literature and the history ofdevelopments in the study of volatility from three perspectives—(1) time-series
analysis of asset price fluctuations, (2) derivative pricing and risk management, and(3) the economic mechanism that gives rise to rough volatility—thereby clarifying
the current position and prospects of rough volatility research. In particular, referring
to issues in financial practice and relationship between volatility research andeconomics, we emphasize the importance of cross-disciplinary studies with volatility
as a nexus.
Keywords: Historical volatility; Implied volatility; Market microstructure;
Highfrequency trading; Derivative pricing; Risk management; Market forecast
JEL classification: D53, G12, G13, G17* Director, Institute for Monetary and Economic Studies, Bank of Japan (currently, International
Monetary Fund, E-mail:khiraki@imf.org)** Deputy Director, Institute for Monetary and Economic Studies, Bank of Japan (currently,
Associate Professor, Musashino University, E-mail:y-shino@musashino-u.ac.jp)
The authors appreciate comments from and discussions with Masaaki Fukasawa (Osaka
University), Kazuhiko Ohashi (Hitotsubashi University and Tokyo Institute of Technology), andToshiaki Watanabe (Hitotsubashi University). The authors also thank FORTE Science
Communications (https://www.forte-science.co.jp/) for English language editing. The views
expressed in this paper are those of the authors and do not necessarily reflect the official viewsof the Bank of Japan, the IMF, its Executive Board, or IMF management.

Contents
1

Introduction

1

2

Mathematical preparation: fractional Brownian motion
2.1 Brownian motion and Itô calculus . . . . . . . . . . . . . . . . . . . . . .
2.2 Fractional Brownian motion . . . . . . . . . . . . . . . . . . . . . . . . .

5
5
8

3

Observation of roughness in historical volatility
3.1 History of HV research in ﬁnancial econometrics . . . . . . . . . . . . . .
3.2 Overview of Gatheral et al. [2018] . . . . . . . . . . . . . . . . . . . . . .
3.3 Recent research trends in roughness of HV . . . . . . . . . . . . . . . . . .

12
12
18
20

4

Shape of implied volatility surface and rough volatility models
4.1 History of the development of mathematical ﬁnance and ﬁnancial engineering: IV representation problem . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Progress in using rough volatility . . . . . . . . . . . . . . . . . . . . . . .

25

Market microstructure and origins of roughness
5.1 Research on micro-foundations of asset price ﬂuctuations . . . . . . . . . .
5.2 Framework for analyzing roughness from MM perspective . . . . . . . . .

38
38
39

5

6

Concluding remarks

26
31

42

1 Introduction
One of the key research themes in ﬁnance research is how uncertainty aﬀects various ﬁnancial issues, such as price formation mechanisms of ﬁnancial assets, optimal investment
behaviors, and risk management methods. Consequently, volatility, which is a measure of asset price uncertainty (i.e., magnitude of ﬂuctuations), has been one of the most fundamental
research subjects in ﬁnance, and research on volatility has developed in tandem with various
adjacent ﬁelds, including economics, mathematical science, and computer science.
Volatility can be categorized into two types, depending on how it is measured: historicalvolatility (HV), which is a backward-looking measure of volatility calculated from past priceﬂuctuations, and implied volatility (IV), which is a forward-looking measure of underlying
assets’ volatility calculated from derivative prices. Research on HV focuses on time-series
data and has been developing volatility forecasting models and applications to investment
strategies, whereas research on IV has been providing the theoretical foundation for pricing
and hedging derivatives as well as for volatility indices such as the VIX index.
While the HV research and IV research diﬀer in terms of the motivation for the analysis
and how volatility is measured, they share the common goals of revealing stylized features ofvolatility dynamics (e.g., stochastic ﬂuctuations) and developing realistic volatility models
that express these features. In particular, both strands of research have been utilizing Brownian motions and random walks (the discrete-time counterparts of Brownian motions) as basic
building blocks for modeling the stochastic dynamics of volatility.
From the 2000s onward, however, researchers have been reporting the roughness of volatility observed from IV and HV, which is considered diﬃcult to reconcile with conventional
Brownian motion-based models. Here, roughness refers to the time-series feature that astochastic process ﬂuctuates wildly in a short period of time, and a fractional Brownian motion is known to have this feature.1 In the late 2000s, seminal studies on IV discovered thatthe negative power law, a stylized fact regarding the term structures of IV, can be modeledconsistently by means of fractional Brownian motions (Alòs et al. [2007] and Fukasawa[2011]).2 Subsequently, Gatheral et al. [2018] reported that roughness can be observed fromthe behavior of HVs calculated from high-frequency data for equity indices and bond futures.
These studies heralded one of the recent trends in volatility research that utilizes fractional
Brownian motions to express roughness.3 In particular, volatility models using fractional
The standard deviation of the change in a Brownian motion over a time length ∆ is ∆1/2 , whereas the corresponding standard deviation of a fractional Brownian motion is ∆H , where H ∈ (0, 1) is the Hurst parameter
representing the degree of roughness. When H < 1/2, the standard deviation of the change in a fractional
Brownian motion is large for short time intervals (i.e., small ∆) compared to a Brownian motion, that is, a fractional Brownian motion ﬂuctuates more volatilely than a Brownian motion. Note that roughness is generally
deﬁned based on the Hölder continuity of sample paths (see section 2.2).
2
The negative power law is a stylized feature related to the term structure of IV (i.e., the relationship between
IVs calculated from derivatives of diﬀerent maturities) and is observed mainly in the U.S. equity derivatives
market, where volatility trading is prevalent. For details, see section 4.2.
3
One of the authors of Gatheral et al. [2018], Jim Gatheral (State University of New York), was awarded
Risk Magazine’s Quant of the Year 2021 for his contributions to rough volatility research. This indicates that
his work has attracted a great deal of attention from both researchers and practitioners. He worked at ﬁnancial
1

1

Brownian motion can represent the behaviors of HV and IV mentioned above, and there
have been many applied studies, such as ones on pricing of derivatives including volatility
derivatives (e.g., options whose underlying asset is VIX). In addition, some studies haveanalyzed rough volatility from the market microstructure (MM) perspective and shed lighton what kind of order placing behaviors yield roughness in the volatility. In this way, research on rough volatility has been expanding its coverage from studies on its measurement
and mathematical properties to applied studies on derivatives practices and interdisciplinary
studies on economic rationales for why volatility is rough.
Despite these developments, there are still a number of issues that require further research.
First, discussions over whether volatility is rough or not remain ongoing. On the one hand, as
mentioned above, many studies suggest the existence of roughness and potential applicationsof rough volatility. On the other hand, some studies point out the possibility of bias in theestimated values of roughness indices that may result in spurious detections of roughness.
There are also ongoing discussions about whether fractional Brownian motion is really anappropriate ingredient to model roughness.4 In addition, further technological and theoreticaldevelopments toward eﬃcient numerical computation of rough volatility models are requiredfor practical applications.
Against the above backdrop, this paper surveys cutting-edge research on rough volatility
alongside with the history of existing volatility research from the perspectives of HV, IV,
and MM. Research on HV roughly corresponds to ﬁnancial econometrics, which focuses
mainly on the time-series analysis of asset price ﬂuctuations, while research on IV roughly
corresponds to mathematical ﬁnance, which focuses mainly on derivative pricing. MM is asubset of ﬁnancial economics, which mainly studies asset price formation mechanisms froman economic point of view. We ﬁrst summarize the history of each ﬁeld and its relation topractical issues in the ﬁnancial industry, as well as that to academic issues in the economics,
and then we discuss the position of rough volatility research within each research ﬁeld andits prospects therein. Through these discussions, we describe the possibility and necessity ofcross-disciplinary research with volatility as a nexus, as well as the prospects for practicalapplications of rough volatility.
First, section 2 introduces mathematical deﬁnitions and results for later use. Speciﬁcally,
we introduce Brownian motion and Itô calculus, which have been fundamental tools in the
study of volatility, especially in mathematical ﬁnance. Then, we introduce the deﬁnition andproperties of fractional Brownian motion, focusing on its similarities and diﬀerences withconventional Brownian motion. We also provide an intuitive interpretation of roughness.
Section 3 focuses on HV. After reviewing the history of volatility research in the ﬁnancial
econometrics, we summarize Gatheral et al. [2018], a seminal study on the roughness of HV,and subsequent studies. Research on HV has been developed with the goal of constructing
institutions before becoming a professor at a university, suggesting that practical issues motivated his research
as well.
4
Currently, a majority of the rough volatility literature uses fractional Brownian motion, but this does notmean that it is the only modeling tool for roughness; that other stochastic processes can express observed roughbehaviors has not been ruled out.

2

models that better represent the observed time-series features of HV and have better volatility
forecasting accuracy. In particular, a vast literature has developed models that express theleverage eﬀect and the long-memory properties of HV. It is also important to note that theanalytical methods of HV have evolved along with increases in the availability of asset pricedata. In particular, since the late 1990s, the realized volatility, which is measured based onhigh-frequency asset price data (e.g., intraday data), has become the main variable of interest
in HV research.
In this context, Gatheral et al. [2018], whose working paper version appeared in 2014, is
one of the key papers that have raised the proﬁle of rough volatility. Gatheral et al. [2018]
showed that the time series of realized volatility exhibits roughness and reported that a timeseries model that takes into account roughness could improve the accuracy of volatility forecasting. We distill two main issues by surveying the subsequent studies. The ﬁrst issue is theestimation of roughness (typically measured as the Hurst parameter of fractional Brownian
motion). The realized volatility is subject to measurement errors, as it is an approximationof volatility described by a continuous-time model. Therefore, some studies propose sophisticated estimation procedures for roughness that are robust to measurement errors in therealized volatility. The second issue is about the relationship between roughness and othertraditionally considered features of volatility, especially the long-memory property. The
long-memory property is the fact that volatility is a highly persistent process that exhibits
positive autocorrelation over a long period of time. Since roughness suggests extreme ﬂuctuations in a short period, further theoretical and empirical investigations of the relationshipbetween these two properties are called for.
Section 4 focuses on IV. After reviewing the history of mathematical ﬁnance and related
practical issues in the derivatives business, we provide an overview of rough volatility research in this area. Derivative pricing models are used for pricing complex derivative products and risk management (i.e., derivation of hedging strategies) after calibrating (adjusting)model parameters to represent market prices of highly liquid derivatives. For this reason, it
is a prerequisite for derivative pricing models to have the ability to ﬁt to the market prices ofhighly liquid ﬁnancial instruments. Since IV is calculated from derivative prices, it is necessary to construct a model that can represent IV. Therefore, more sophisticated derivativemodels that can represent the IV observed in the derivatives market have been developed,
especially since the 1990s.
In this context, in the early 2000s, a negative power law for the term structure of IV wasobserved in the U.S. equity derivatives market, and it was pointed out that it is diﬃcult toreconcile this observation with conventional models based on Brownian motion (e.g., Carrand Wu [2003]). In the late 2000s, Alòs et al. [2007], and later in Fukasawa [2011], showedthat a rough volatility model based on fractional Brownian motion could represent the negative power law of IV. These studies are considered to be the pioneers of rough volatility
research. In recent years, research on this issue has progressed, and some studies have provided strong support for the need for rough volatility. Speciﬁcally, Fukasawa [2021] provedthat a rough volatility model with an appropriate roughness parameter (i.e., the roughness
3

consistent with the one implied by the negative power law of IV) is necessary for pricing andhedging derivatives, because otherwise inconsistencies (i.e., arbitrage opportunities) would
arise. In addition to these theoretical studies, studies on practical applications have been
conducted. Unlike conventional Brownian motion, fractional Brownian motion does nothave nice properties such as the Markov property (i.e., future behaviors do not depend on
past behaviors), so it has been considered diﬃcult to apply conventional mathematical ﬁnance theory to fractional Brownian motion. Related to this, recent research has revealed
that the Markov property “recovers” under certain conditions (Bayer et al. [2016]). This
demonstrates how such mathematical diﬃculties are gradually being overcome, increasing
the applicability of rough volatility models to derivatives practices. On the other hand, it
is noteworthy that, at present, the derivatives markets in which roughness is observed (i.e.,markets in which the negative power law is observed) are limited to a few markets, such as
the U.S. equity derivatives market. Therefore, it is still an open question whether roughness
should be taken into account in pricing and risk management for other markets (e.g., othercountries and other asset classes). In addition, the practical application of rough volatility
models requires much more complex numerical calculations than those based on conventional Brownian motion and Itô calculus, so further development of theory and numericalcalculation techniques is needed.
Section 5 focuses on MM. We brieﬂy overview the literature on MM so as to show the
relation between rough volatility research and ﬁnancial economics. The MM literature studies, as its name suggests, how micro-level features, such as the institutional structure of
exchanges and the behaviors of individual market participants, impact asset price formation.
A subset of MM studies are aimed at deriving stochastic processes of asset price dynamics, starting from assumptions on micro-foundations such as the buy and sell order placing
behaviors of individual market participants. In rough volatility research, some studies haveinvestigated the conditions under which market participants’ buy and sell order behaviors
lead to roughness in the volatility of the asset prices. For example, El Euch et al. [2018]
showed that volatility exhibits rough behavior when price updates from a particular periodof trading continue to aﬀect the frequency of future price update occurrences with a certain
degree of persistence. Such studies are considered attempts to identify the source of roughness from a ﬁnancial economic viewpoint, which is an important issue from the perspectiveof the institutional design of markets and its relation to ﬁnancial stability. However, sinceresearch to date has presented only a mathematical framework, it is necessary to develop further research on more realistic modeling of investor behaviors and empirical analysis using
actual data.
In summary, this paper is a cross-disciplinary survey of the ﬁelds of ﬁnancial econometrics, mathematical ﬁnance, and ﬁnancial economics, with volatility research as a nodal point.
Therefore, this paper can be read in various manners, depending on readers’ knowledge andinterest. Readers who wish to get an overview of the study can skip section 2, since it summarizes mathematical results necessary to understand the technical aspects in the later sections.
In addition, readers can read sections 3, 4, and 5 independently, in any order, although these4sections are interrelated.

2 Mathematical preparation: fractional Brownian motion
In this section, we prepare mathematical deﬁnitions and results for their later use. We ﬁrst
introduce Brownian motion and Itô calculus (stochastic diﬀerential equations) in section 2.1,
and then describe the formulation and properties of fractional Brownian motion in section
2.2. Although these concepts are usually formulated based on measure-theoretic probability
theory, we omit rigorous mathematical descriptions and explain only the necessary facts
as intuitively as possible. For details, see Karatzas and Shreve [1998] for a mathematicaltreatment of Brownian motion and Itô calculus, and Nualart [2006] for fractional Brownian
motion.
Note that, as discussed in sections 3 and 4, in rough volatility studies, fractional Brownian
motion is often used to model the logarithm of volatility.5

2.1 Brownian motion and Itô calculus
2.1.1 Deﬁnition and properties of Brownian motion
A mathematical model of a variable that randomly ﬂuctuates along with time is called astochastic process. Mathematically speaking, a collection of random variables {Xt | t ≥ 0}(parameterized by a time parameter t) deﬁned on a probability space (Ω, F , P) is called astochastic process. The value this stochastic process takes at time t can be viewed as arandom variable, which we denote by Xt . If we ﬁx a sample point ω ∈ Ω and consider Xt (ω)
as a function with respect to time, we call it a sample path of the stochastic process for asample point ω.
Brownian motion is one of the most fundamental continuous-time stochastic processes.6 A
Brownian motion is the continuous limit of a random walk in a discrete-time model and hasvarious important properties, such as the fact that its increments (the time variation Xt+∆ −Xt ) follow independent normal distributions. Formally, Brownian motion is deﬁned as astochastic process {Wt | t ≥ 0} satisfying the following properties.7
H

For instance, the instantaneous volatility σt is modeled as σt = eF(Bt ) , where BtH is a fractional Brownian
motion and F is some function. See section 3.2 for more details.
6
Since the beginning of the 20th century, the mathematical formulation and properties of Brownian motion
have been studied, mainly with a focus on modeling various phenomena in physics and economics. In particular, the relationship between Brownian motion and the ﬁeld of ﬁnance goes back to around 1900. Prior
to extensive use in mathematics and physics, Bachelier [1900] discussed option pricing using a model that
could be considered a prototype of Brownian motion. Subsequently, in the early 1900s, Kolmogorov, Wiener,
Kiyoshi Itô, and others developed theories such as measure-theoretic probability theory and Itô calculus. As iswell known, Merton [1973] presented an option pricing method based on Itô calculus, which heralded developments in the ﬁelds of ﬁnancial engineering and mathematical ﬁnance.
7
Note that the existence of stochastic processes satisfying the given conditions is not trivial. For a proof ofthe existence of Brownian motion, see, for example, section 2 of Karatzas and Shreve [1998].
5

5



( i ) Zero initial value :






P (ω | W0 = 0) = 1.






(ii) Continuty of sample path：






P (ω | lim s→t W s (ω) = Wt (ω)) = 1.

(1)



(iii) Increments follow a normal distribution (mean: 0, variance: time-increment)：






Wt − W s ∼ N(0, t − s), for 0 ≤ s < t.






(iv) Increments are independent： For 0 ≤ t0 < t1 < · · · < tk < ∞,





Wt1 − Wt0 , Wt2 − Wt1 , . . . , Wtk − Wtk−1 are indepenedent.
It is well known that Brownian motion has important properties such as the Markov property and martingale property. The Markov property is that the conditional probability distribution of a stochastic process at a future time depends only on the information at thepresent time and not on the past history (information accumulated prior to the present time).
For example, given the value W s of a Brownian motion at time s, the conditional probability
distribution of the Brownian motion at a future time t (> s) satisﬁes the relation
P Wt ∈ B │ F s = P (Wt ∈ B | W s ). Here, both sides are conditional probabilities that Wt
is in some subset B ⊂ R. The left-hand side is a probability distribution conditioned on
the information of Brownian motion up to time s, and the right-hand side is a probability
distribution conditioned only on the information that the value of Brownian motion at times equals W s . The martingale property is the so-called fair betting game property, that is, theconditional expected value of a stochastic process at a future point in time equals the current
value. Mathematically, this property is expressed as E [Wt | F s ] = W s . These two properties
of Brownian motion are derived from the property that a Brownian motion is an independent
incremental process with mean zero.
A Brownian motion has a bounded and nonzero quadratic variation deﬁned up to time T(> 0) as follows:
n−1 
X
2
W i+1n T − W ni T .
⟨W⟩2,T = lim
n→∞

i=0

As shown above, quadratic variation is expressed in terms of sums of squared increments ofa stochastic process for increments representing ﬁnely divided steps within a certain periodof time. Quadratic variation is known to be zero for smooth processes, such as diﬀerentiablesample paths, whereas a Brownian motion, which randomly ﬂuctuates and has non-smoothsample path, has a positive quadratic variation. Speciﬁcally, the above equation converges to
T with probability one; that is,
P ⟨W⟩2,T = T = 1hiholds. This follows from the deﬁnition of Brownian motion (1)(iii), Var Wti − Wt j = ti − t j .

6

2.1.2 Itô calculus
One of the most important features of Brownian motion is that one can deﬁne integration
by Brownian motion, as what is known as an Itô integral. To deﬁne an Itô integral withrespect to an integrand h(s), we proceed as follows. For a time division 0 = s0 < s1 < · · · <sn = t, we multiply the increment of Brownian motion W si+1 − W si by h(si ), then add up allincrements (note that the value of h(s) is the value at the starting time of the increment ofBrownian motion):n−1Xh (si ) W si+1 − W si .

i=0

Then, we can deﬁne the limit of this summation as the time division becomes ﬁner in amathematically rigorous manner. Speciﬁcally, we can deﬁne this limit as the Itô integralRth(s) dW s , that is,
0
Z t
n−1
X

h(s) dW s = limh (si ) W si+1 − W si .
n→∞

0

i=0

Note that the rigorous formulation of this limit and convergence is the core of Itô calculus,
but we omit the mathematical details here since it requires additional preparations for an
accurate description.8
A stochastic process that adds an ordinary integral term to an Itô integral is called an Itô
process. Namely, an Itô process Xt is a stochastic process expressed as follows:
Z t
Z t
Xt = x +µ (s, X s ) ds +σ (s, X s ) dW s .
(3)
0

0

Here, the function µ represents the trend and the function σ represents the magnitude of the
ﬂuctuation of X s . In equation (3), the second and third terms are called the drift and volatility
8

The following is a sketch of this point. First, when the integrand h is an indicator process (i.e.,
R ta piecewiseP
constant stochastic process h(s) = n−1
a
1
),
the
Itô
integralcanbenaturallydeﬁnedash(s)dW s =
i
{s
≤s<s
}
i
i+1
i=0
0
Pn−1

h(s
)
W
−
W
.
For
a
generalstochastic
process
h,
we
rely
on
the
fact
that
h
can
be
approximated
i
si+1
si
i=0
by a sequence of indicator processes. Considering a sequence of Itô integrals for each sequence of indicator
processes approximating h, we can show that this sequence also converges. This limit is deﬁned as the Itôintegral for the more general stochastic process h.
When discussing such approximation and convergence, it is necessary to deﬁne an appropriate functional
space and a norm (distance) on that space. Speciﬁcally,
forhRthe functionalno of integrands, we consider thei spacet 2
2square integrable stochastic process space L = h(s) | E 0 h (s) ds < ∞ equipped with the inner producthR ti⟨h, g⟩ = E 0 h(s)g(s)ds , h, g ∈ L2 . From the deﬁnitions (iii) and (iv) of Brownian motion, we can see thathPP Pn−1
i
n−1
E
= n−1
i=0 h (si ) W si+1 − W si
i=0 g (si ) W si+1 − W si
i=0 h (si ) g (si ) (si+1 − si ). This result proves thatthe destination space of the Itô integral (viewed as a mapping operator) is also a Hilbert space, with the innerproduct
*Z
+
"Z
#
Z
t

0

t

h(s)dW s ,tg(s)dW s = ⟨h, g⟩ = E

0h(s)g(s)ds .

(2)0Thus, the above-mentioned sequence of indicator processes has a limit. Note that equation (2) indicates that aformal calculation dWt · dWt = dt is valid, corresponding to the fact that the quadratic variation of a Brownian
motion is T , that is, P ⟨W⟩2,T = T = 1. This formula is the celebrated Itô lemma.

7terms, respectively. Equation (3) can also be written in diﬀerential formdX s = µ (s, X s ) ds + σ (s, X s ) dW s ,X0 = x,which is called a stochastic diﬀerential equation. Stochastic diﬀerential equations make itpossible to describe more complex models based on Brownian motion (e.g., models in which
drift or the volatility changes). Note that, like a Brownian motion, Itô process Xt is a Markov
process. It is also closely related to the martingale property.9

2.2 Fractional Brownian motion
As will be discussed in section 3, in volatility modeling, there are a number of observations
that cannot be represented by conventional Brownian motion and Itô processes. Therefore,
other models that can represent these stylized facts have been developed. Typical examplesof volatility ﬂuctuations that cannot be represented by Brownian motion and Itô processes
are as follows.
1. While increments of Brownian motion are independent, actual time series of volatility
are autocorrelated.
2. While variances of Brownian motion are proportional to time increments, actual timeseries of volatility do not satisfy this property.10
Fractional Brownian motion, which is an extension of Brownian motion, is known for amodel that can describe these properties, and has been used to describe the stochastic dynamics of volatility.11 Fractional Brownian motion has the property that its increments follow a normal distribution, which is the same as Brownian motion. On the other hand, unlikea Brownian motion, it is a stochastic process that has a parameter called the Hurst parameterand may exhibit nonzero autocorrelationn (past ando future increments are not independent).
H
Fractional Brownian motion B = BtH | t ≥ 0 with the Hurst parameter H ∈ (0, 1) isdeﬁned as a stochastic process that satisﬁes the following conditions.

Rt
Itô integral Yt = 0 σ(s) dW s has the martingale property. Conversely, it is known that any continuous
stochastic process with the martingale property can be represented as an Itô integral. As is well known, themartingale property plays a central role in the formulation of derivative pricing principles such as no-arbitrage
and replication. The Markov property is also an extremely important property for numerical computation ofderivative prices and of associated risk amounts in practical applications.
10
In the case of models based on Brownian motion, the variance of daily data is one-ﬁfth of the variance ofweekly data. On the other hand, empirical studies have revealed that high-frequency data are more volatile thanwhat Brownian motion suggests.
11
Note that Brownian motion and Itô processes are still major modeling tools for the underlying asset priceprocess.
9

8




H


i
)
Zero
initialvalue:
P
ω
|
B
=
0
= 1.
(

0




H
H


(ω
(ii)
Continuityofsamplepaths:
P
lim
B
(ω)
=
B
(ω)
= 1.

s→t
s
t





 (iii) Increments follows a normal
 distribution:



H
H
2H

B
−
B
∼
N
0,
(t
−
s)
, for 0 < s < t.

t
s






(iv) Gaussian increments: For 0 ≤ t0 < t1 < · · · < tk < ∞, increments BH − BH , BH − BH , . . . , BH − BH follow a joint normal distribution.
t1
t0
t2
t1
tk
tk−1

(4)

Namely, the deﬁnitions of Brownian motion (1)(iii)–(iv) are replaced by (4)(iii)–(iv). When
H = 1/2, fractional Brownian motion boils down to Brownian motion, that is, Wt = B1/2
t .
Unlike Brownian motion, when H , 1/2, the increments of fractional Brownian motion
are not independent. In particular, one of the following relationships holds, depending on thevalue of the Hurst parameter.12 For s1 < t1 < s2 < t2 ,
 
h

i

1
H
H
H
H


B
−
B
< 0,
⇒
E
B
−
B
H
∈
0,

t
s
t
s
2
2
2
1
1



h


i


 H=1
⇒ E BtH1 − BHs1 BtH2 − BHs2 = 0,
2



h

i
 



1
H
H
H
H

,
0
⇒
E
B
−
B
B
−
B
> 0.
H
∈

t1
s1
t2
s2

2
This fact indicates that the increments are negatively correlated in the case of H < 1/2 andpositively correlated in the case of H > 1/2. It can be seen that fractional Brownian motion(H , 1/2) is neither a martingale and nor a Markov process. It is also known that fractional
H
Brownian motion has the properties of self-similarity (for a > 0, Bat∼ aH BtH ) and timeHhomogeneity (Bt+s− BHs ∼ BtH ).
Furthermore, from deﬁnitions (4)( i ) and (iii), the following equation holds:qh2 i
Var BtH = t H .
(5)Namely, the time evolution of the standard deviation is time raised to the power of H. Thisis a generalization of the well-known square-root-of-time rule for Brownian motion, whichstates that the standard deviation of a Brownian motion increases proportional to the square 2  2 
h
i
2 


From (4)(iii), for s, t > 0, E BtH BHs = 21 E BtH + BHs − BtH − BHs= 12 t2H + s2H − |t − s|2H holds.
Then we have
12


h

i 1 
E BtH1 − BHs1 BtH2 − BHs2 =(t2 − s1 )2H − (t2 − t1 )2H − (s2 − s1 )2H + (s2 − t1 )2H .
2
If we let a1 = t2 −s1 , a2 = t2 −t1 , b1 = s2 −t1 , b2 = s2 −s1 , then we see that a1 +b1 = a2 +b2 = (t2 −t1 )+(s2 −s1 ) > 0,
and a1 > a2 , b2 > b1 . Setting f (x) = x2H , f is a concave function in the case of H < 1/2 and a convex functionin the case of H > 1/2; thus, the following holds:< 0 (H < 1/2) ,hi f (a1 ) + f (b1 ) f (a2 ) + f (b2 ) 


H
H
H
H
=
0 (H = 1/2) ,E Bt1 − Bs1 Bt2 − Bs2 =
−



2
2
 > 0 (H > 1/2) .

9

root of time t.
This fact implies that a fractional Brownian motion with Hurst parameter H < 1/2 is astochastic process that is highly volatile in a short time (a rough stochastic process). In otherwords, while the value at some time t follows a normal distribution, the same as a Brownian
motion, the standard deviation of the normal distribution at an early time point (when t issmall) is larger than that of a Brownian motion (Figure 1).

Figure 1: Sample paths and their dispersion of fractional Brownian motion
For fractional Brownian motion, when H < 1/2, the short-time ﬂuctuations are more
intense than Brownian motion, that is, the sample path exhibits rough behavior (Figure 2,
upper left), whereas when H > 1/2, the ﬂuctuations are milder, that is, the sample pathexhibits smooth behavior (Figure 2, upper right) relative to Brownian motion.

10

Figure 2: Sample paths of fractional Brownian motion with diﬀerent H
Note that roughness is generally deﬁned as the Hölder continuity of paths (i.e., functions
parameterized by time). The Hölder continuity describes the regularity of the sample path(how extremely the path ﬂuctuates), from which the term rough is derived. Since it is known
that the fractional Brownian motion of the Hurst parameter H is (H − ε)−Hölder continuous,
the Hurst index can be viewed as an index of the Hölder continuity of the fractional Brownian
motion.13
Similarly, while the quadratic variation is ﬁnite in the Brownian motion case, equation(5) indicates that, in the fractional Brownian motion case, 1/H-variation is ﬁnite and thefollowing holds:14⟨B⟩1/H,T = lim

n→∞

n 
X
i=1

BHi T − BHi−1 T
n

n

1/H



, P ⟨B⟩1/H,T = T = 1.

In addition, fractional Brownian motion is represented as an Itô integral with an integrand13
A process {BtH | t ≥ 0} is said to be (H − ε)−Hölder continuous if there exists a continuous modiﬁcationH{ B̃t | t ≥ 0} such that for any ε > 0, there exists a random variable Gε such that B̃tH − B̃Hs ≤ Gε |t − s|H−ε holds.
14

Therefore, if we attempt to deﬁne the integral with respect to fractional Brownian motion following thesame approach as the Itô integral, we cannot apply the formal calculation corresponding to the Itô lemma(dWt · dWt = dt). This means that while the formulation of the Itô integral relies on symmetry and innerproduct structure in the Hilbert space in order to deﬁne the integral, in the case of fractional Brownian motion,
it is necessary to discuss a Banach space which has neither symmetry nor inner product structure. Anotherapproach to formulate integrals with respect to fractional Brownian motion has recently been developed inrough path theory (Lyons [1998]). For example, Bayer et al. [2020] discussed analysis of rough volatility
models based on regularity structure theory, which won the 2014 Fields Prize, and further applications to roughvolatility are expected in the future.

11

that has poles as follows (Levy [1953], Mandelbrot and Van Ness [1968]):
!
Z t
Z 0 

1
H− 12
H− 12
H
H− 21
(t − s)dW s +− (−s)
Bt =(t − s)dW s ,
CH 0
−∞

(6)

R 
 21

1
∞
1 2
1
where C H = 0 (1 + s)H− 2 − sH− 2 ds + 2H
. From equation (6), we can calculate thetime-series correlation of fractional Brownian motion (4)(iii) by considering the Itô integral
including the past history of Brownian motion, as follows:

E BtH − BHs

2 

Z
!2 

1
1
1  ∞
(max{t − u, 0})H− 2 − (max{s − u, 0})H− 2 dWu = 2 E CH−∞
"Z ∞ 
#
2
1
H− 12
H− 21
(max{t − u, 0})= 2E− (max{s − u, 0})duCH−∞= (t − s)2H .

Rt
We can see that, the ﬁrst term on the right-hand side of equation (6), B̃tH = 0 (t − s)H−1/2 dW s ,is a continuously weighted sum of a Brownian motion. Note that B̃tH is called the Riemann–
Liouville-type Itô integral15 and is often used in rough volatility modeling.

3 Observation of roughness in historical volatility
In this section, ﬁrst we summarize the importance of volatility research in section 3.1.
Then, by focusing on HV, we provide an overview of the development of volatility research inthe ﬁeld of ﬁnancial econometrics.16 Next, section 3.2 provides a summary of Gatheral et al.
[2018], which is one of the seminal articles in rough volatility research. This article reports
empirical evidence of the roughness of the time series of realized volatility observed in highfrequency data for various ﬁnancial assets. In section 3.3, we overview studies subsequent
to Gatheral et al. [2018] on rough volatility research.

3.1 History of HV research in ﬁnancial econometrics
3.1.1 Importance of volatility research
In the ﬁeld of ﬁnance, major research topics include the analysis of optimal investment
behavior and asset price formation mechanisms under the presence of uncertainty. For example, in the mean-variance portfolio theory of Markowitz [1952], the investment weights of
15

The Riemann–Liouville-type integral is deﬁned for the usual Riemann integral. These studies have their
origin in the era of Leibniz (15th to 16th century). For a function f , the Riemann–Liouville-type integral oforder λ is deﬁned as
Z t
1
Iλ f =(t − s)λ−1 f (s) ds,
Γ(λ) 0which is an extension of the Reimann integral to the fractional order. This is the origin of the term fractional.
16
For an overview of volatility research in ﬁnancial econometrics, see Takahashi et al. [2023].

12

an optimal portfolio are given as a function of the expected returns and the covariance matrix
of individual stocks. In the Capital Asset Pricing Model (CAPM), one of the prominent asset
pricing models, the expected returns of individual stocks are explained by the covariance
between the return of the stocks and that of the market portfolio under the assumption of arepresentative mean-variance investor (Sharpe [1966], Lintner [1965], Mossin [1966]).
As such, moments of the probability distribution of asset returns play an important rolein ﬁnance research. Accordingly, time-series analysis and modeling of moments of asset
price ﬂuctuations have been a central research theme, especially in ﬁnancial econometrics.
Among these studies, volatility (standard deviation or variance), the second moment of asset
returns, has been attracting more attention than expected return (i.e., the ﬁrst moment).
There are several reasons why research on volatility has progressed further than researchon expected returns. The ﬁrst is that volatility can generally be more accurately estimatedthanks to advances in data availability and the development of quantitative methods, whereas
expected return remains notoriously diﬃcult to estimate precisely. For example, Merton[1980] showed that under the assumption that there are no measurement errors in observed
asset prices, volatility can be more accurately estimated as higher frequency data become
available. Contrarily, he pointed out that accurate estimation of expected return is diﬃcult
even when high-frequency data are available. In addition, expected returns are known to
exhibit large stochastic variations, and estimation from historical data is fraught with diﬃculties (e.g., Martin [2017]). With such theoretical background, vast number of studies havefocused on volatility, which is relatively easy to estimate and forecast.17
The second reason is the importance of volatility in risk management. Knowledge on
the magnitude of price ﬂuctuations during sudden market turmoil is essential in measuring
and analyzing market risks. This implies the importance of the measurement of volatility(and higher-order moments, which capture the shape of the probability distribution of asset
returns) in risk management, as volatility represents the magnitude of asset price ﬂuctuations.
Moreover, in the ﬁelds of mathematical ﬁnance and ﬁnancial engineering, volatility is a key
modeling target for derivative pricing and risk management purposes, whereas estimates of
expected returns are not required. For more details, see section 4.
Volatility and uncertainty in general are also proven as important elements in macroeconomics; a large body of literature has shown that uncertainty has a negative impact on real
economic activity (e.g., Bloom et al. [2007]). For this reason, the VIX index and the Nikkei
Stock Average Volatility Index, which are volatility indicators calculated from option prices,
are widely used in macroeconomic analysis as proxies of uncertainty in ﬁnancial markets andthe macroeconomy, and various studies have proposed alternative indicators of uncertainty(e.g., Bloom [2009]; Baker et al. [2016]; Dew-Becker et al. [2021]).
17

In fact, a majority of empirical studies on optimal portfolio choices focus on minimum variance portfolios
which use only the information on the covariance matrix, rather than mean-variance portfolios (e.g., Michaud[1989], Best and Grauer [1991], Jagannathan and Ma [2003]). This is because the empirical disadvantage of
constructing mean-variance portfolios using estimates of expected returns, which may be contaminated by large
estimation errors, greatly outweighs the theoretical advantage of incorporating the information on expected
returns to construct a mean-variance portfolio.

13

As discussed above, volatility is one of the crucial variables in the various ﬁelds of ﬁnanceand economics. For this reason, in the ﬁeld of ﬁnancial econometrics, vast amounts of eﬀort
have been devoted to analyzing the time-series features of HVs measured from historical
data for actually observed asset prices and developing discrete-time volatility models thathave better ﬁt to the data and better forecast accuracy. The typical features of volatility
ﬂuctuations pointed out by many studies on HVs include persistence of volatility (volatility
clustering), leverage eﬀect, and long memory. In the following, we overview the history of
research on these features.

3.1.2 Persistence and leverage eﬀect of HV
It has long been known that asset returns ﬂuctuate randomly as their autocorrelation isalmost zero. On the other hand, the time series of the absolute values of returns and thesquared returns exhibit very long positive autocorrelations (e.g., Mandelbrot [1963], Fama
[1965], Ding et al. [1993]). Since the absolute value of returns and the squared returns represent the magnitude of the variability of asset prices, this fact indicates that volatility exhibits
strong autocorrelation and persistence. This phenomenon is known as volatility clustering.
Regarding time-series models including persistence of volatility, autoregressive models of
the variance of ﬁnancial assets have been developed since the 1980s. Engle [1982] proposed
the ARCH (autoregressive conditional heteroskedasticity) model, in which volatility is represented by an autoregressive model of squared noise of past asset price changes.18 Bollerslev[1986] extended the ARCH model as a generalized autoregressive conditional heteroskedasticity (GARCH) model, in which volatility depends on autoregressive lag in addition to the
ARCH model terms. These time-series models have been fundamental tools in the analysis
of HV time series, as they are parsimonious yet achieve a good ﬁt to the observed data andgood forecasting accuracy.
A prominent and important feature, along with the persistence of volatility, is the negative
correlation between the return of an equity index and its volatility. This feature means thatvolatility tends to increase as stock prices fall, which is commonly referred to as the leverage eﬀect.19 Under the presence of the leverage eﬀect, the probability distribution of asset
prices has a fat tail in the downside direction. Therefore, it is important to use a model that
can appropriately represent the leverage eﬀect from the perspective of risk management andportfolio selection. Models that extend the GARCH-type model to represent the leverage effect include the exponential GARCH model of Nelson [1991] and the GJR model of Glosten
et al. [1993].

18

For an overview of ARCH-type volatility models, see, for example, Chapter 21 of Hamilton [1994] andPoon and Granger [2003].
19
The term leverage eﬀect comes from the idea that when the stock price falls, the leverage of the ﬁrm, andthus the riskiness of the stock, and the volatility of stock price increases. See Black [1976] and Christie [1982].
Note that, some empirical studies have shown other factors contributing to the negative correlation between
returns and volatility (Wu [2001], Avramov et al. [2006]).

14

3.1.3 Long memory
Among the properties of volatility ﬂuctuations, a vast number of ﬁnancial econometricstudies have noted that asset price dynamics exhibit long memory.20 For example, the aforementioned Ding et al. [1993] reported that time series of absolute values of daily returns
exhibit signiﬁcant positive autocorrelation even for extremely long lags, such as a 10-year
lag. This feature implies that the autocorrelation function (which returns the autocorrelation coeﬃcient as a function of the order of the lag) decreases slowly. On the other hand,
the GARCH-type model described above has the limitation that it could not represent longmemory, because the autocorrelation function decays exponentially. Therefore, research onvolatility modeling based on time-series models with long memory has been actively pursued, especially since the 1990s.
A standard discrete-time model for representing long memory is the autoregressive fractionally integrated moving average (ARFIMA) model (Granger [1980], Granger and Joyeux[1980], Hosking [1981]). The ARFIMA model is an extension of the ARIMA model. For
example, the ARIMA(0, 1, 0) model, the simplest ﬁrst-order integrated model, is equal to
the noise term when the ﬁrst-order diﬀerence is taken.21 Namely, if L is the lag operator(LXt = Xt−1 ), it can be represented as (1 − L)Xt = εt . The ARFIMA model extends thisdiﬀerence order to real numbers: the dth-order ARFIMA model ARFIMA(0, d, 0) is represented as (1 − L)d Xt = εt , where the left-hand side is formally deﬁned as follows:(1 − L)d Xt = Xt − dXt−1 +d(d − 1)
Xt−2 − · · · .
2

The ARFIMA model is a stationary process when −1/2 < d < 1/2. When 0 < d < 1/2,
the autocorrelation function decays with the order of k2d−1 , where k is the lag order. The
ARFIMA model also satisﬁes the following:
lim

n→∞

n
X

ρ j = ∞.

j=−nThis property implies a slow decay of the autocorrelation function, and is one deﬁnition oflong memory. On the other hand, in the autoregressive moving average (ARMA) modeland other models, the sum of the absolute values of the above autocorrelation functions is
ﬁnite. Thus, when 0 < d < 1/2, the ARFIMA model can be regarded as a time-series modelthat lies between a stationary I(0) process without long memory (e.g., ARMA model) and a
20

The existence of long memory has a considerable history in ﬁelds that including hydrology (e.g., the studyof time series of water ﬂow by Hurst [1951], which is the origin of the term Hurst index). In economics,
Granger [1966] and Nerlove [1964] conducted spectral analysis of major macroeconomic variables and observed that they have a large low-frequency component (i.e., the contribution of the long-cycle component islarge). Haubrich and Lo [1991] and Sowell [1992] analyzed macroeconomic variables based on the ARFIMA
model, which is a time-series model with long memory mentioned in the main text. See a survey by Baillie[1996] for more details.
21
In general, the ARIMA(p, d, q) model is a dth-order integrated model with pth-order autoregressive terms
and qth-order moving average terms. The dth-order integrated model becomes a stationary process by diﬀerencing d times, and is called an I(d) process.

15

non-stationary I(1) process (e.g., random walk).
Being related to the subject of this paper, rough volatility and fractional Brownian motion, the relation between the ARFIMA model and FGN (fractional Gaussian noise) model
is worth mentioning. The FGN model is a discrete-time model deﬁned as the diﬀerenceof fractional Brownian motion. The autocorrelation of the ARFIMA(0, d, 0) process decays
with a power exponent similar to the FGN whose Hurst parameter is H = d + 1/2. This relation demonstrates that ARFIMA(0, d, 0) with 0 < d < 1/2, a discrete-time time-series modelwith long memory, is closely related to fractional Brownian motion with Hurst parameter
greater than 1/2.
Early studies that took into account the long memory in the GARCH model include
those of Baillie et al. [1996], who proposed the fractionally integrated generalized autoregressive conditional heteroskedasticity (FIGARCH) model, and Bollerslev and Mikkelsen[1996], who proposed the fractionally integrated exponential generalized autoregressive conditional heteroskedasticity (FIEGARCH) model. Breidt et al. [1998] proposed a discretetime stochastic volatility model with long memory, and continuous-time models by Comteand Renault [1996], Comte and Renault [1998], Cheridito et al. [2003], Comte et al. [2012],
and others have used fractional Brownian motion to express long memory. In these studies,
to represent long memory, the discrete-time ARFIMA model with fractional integration parameter d such that 0 < d < 1/2 and continuous-time fractional Brownian motion with a
Hurst parameter H larger than 1/2 are considered.

3.1.4 Realized volatility
In the early years of ﬁnancial econometric research, when many of the studies mentioned
above were conducted, asset price data with daily or longer frequency were mainly used due
to data constraints. Since the late 1990s, the availability of high-frequency data for intraday asset price ﬂuctuations has expanded. This expansion of data availability has promoted
progress in ﬁnancial econometrics in both theoretical and empirical aspects as described
below.
When intraday price data are available, it is possible to calculate, say, the daily volatility
from time-series data for ﬁve-minute price data within a single business day. The volatility calculated from such high-frequency price data is called the realized volatility and is astandard measure of HV in current ﬁnancial econometrics. Speciﬁcally, realized volatility is
calculated as follows:22
RV (t, t + 1, π) =

n−1
X


log S si+1 − log S si 2 ,i=0where S t is the underling price and π is a partition of [t, t + 1], t = s0 < s1 < … < sn = t + 1.
22This deﬁnition of realized volatility (like integrated volatility, which will be deﬁned later) corresponds tovariance, not volatility (standard deviation), but following the convention in the ﬁeld of ﬁnancial econometrics,
we refer to it as realized (integrated) volatility.

16

For example, realized volatility could be calculated from ﬁve-minute data (i.e., si+1 − si =5 min).
Andersen et al. [2001], Andersen et al. [2003], and Barndorﬀ-Nielsen and Shephard [2002],
among other studies, have revealed that the realized volatility deﬁned in this way has theoretically favorable properties. Since this point is closely related to Gatheral et al. [2018],
which will be discussed in more detail later, we give a brief explanation here.
In a continuous-time model, an underlying asset price with time-varying volatility is generally represented by the following stochastic diﬀerential equation (see section 2 for notation):dS t= µt dt + σt dWt ,St
where σt represents the instantaneous volatility at time t.23 Given the aforementioned observations that volatility ﬂuctuates stochastically, σt (or its logarithm) is often represented
by another stochastic diﬀerential equation. Given a stochastic process for asset prices, wedeﬁne integrated volatility over the period [t, t + ∆] as
Z t+∆
σ2s ds.
t

It is known that realized volatility converges in probability to integrated volatility as the time
split π become ﬁner (i.e., when realized volatility is calculated using higher frequency intraday data). This convergence holds under very mild conditions on σt . For instance, σtneeds be neither an Itô process nor a Markov process, and this convergence holds in caseof fractional Brownian motion as well. As such, realized volatility is not only easily calculated from data without relying on a speciﬁc model, but also has the theoretically desirable
property that it converges to integrated volatility under mild conditions as described above.
These advantageous points have contributed to the widespread use of realized volatility as abenchmark HV measure in ﬁnancial econometrics, especially since the 2000s.
Realized volatility is also known to have long memory, and many analyses have used the
ARFIMA model to describe realized volatility.24 Recently, however, the HAR (Heterogeneous Autoregressive) model proposed by Corsi [2008] and its extensions have been widelyused to describe realized volatility. The HAR model predicts future realized volatility basedon a mixture of realized volatility measured over multiple time horizons. For example, afuture daily realized volatility is predicted based on its lag and historical realized volatility measured at weekly and monthly frequencies. Although the HAR model has a simple
model structure and does not have long-memory property, it is widely used because of its
high predictive ability for daily realized volatility. Similar to the GARCH model, there exist
extensions of the HAR model such as an asymmetric HAR model that takes into account23

In, for example, Andersen et al. [2001], the asset price process is a general semimartingale. We omit thejump term for simplicity in this paper. For the properties of realized volatility and consistent estimators in thepresence of a jump term, see Barndorﬀ-Nielsen and Shephard [2004].
24
There is an alternative approach based on a state-space model called the unobserved components model.
See, for example, Nagakura and Watanabe [2015].

17

leverage eﬀects (e.g., Ubukata and Watanabe [2014], Bekaert and Hoerova [2014]), andHAR-CJ models that consider jumps in the asset pricing process (Andersen et al. [2007]).
Finally, we discuss the empirical properties of the correlation between realized volatility
and asset returns, called the Zumbach eﬀect (Blanc et al. [2017], El Euch et al. [2020]).
First, we denote the cross-correlation function between the daily integrated variance and thesquared asset returns as follows:
h
h i
i
e2 (τ) = E σ2t − E σ2t r2 .
Ct−τ
Using this, we introduce the following quantity called time-reversal asymmetry (TRA):
e2 (τ) − C
e2 (−τ),Z(τ) = C

τ > 0.

In words, TRA measures the diﬀerence between the covariance when volatility precedes thesquared return and the covariance when the squared return precedes volatility. A series ofstudies by Zumbach and his co-authors have revealed that when TRA is calculated using therealized variance as an estimate of the integrated variance, TRA is positive for many ﬁnancial
time series (e.g., Zumbach and Lynch [2001], Lynch and Zumbach [2003], Zumbach [2004],
Zumbach [2009]). On the other hand, one can show theoretically that TRA is (nearly) zerounder standard continuous-time volatility models. This implies that the Zumbach eﬀect cannot be represented by standard models. Recent studies on rough volatility have shown that
the Zumbach eﬀect can be represented by a rough volatility model. This point is discussed
in section 4.

3.2 Overview of Gatheral et al. [2018]
As mentioned above, there has been a long history of theoretical and empirical studies on
the volatility of asset prices. Volatility models have been developed so that they capture thetime-series features of observed data such as long memory as the most prominent example.
A noteworthy trend in recent years is the development of an analytical framework based onrealized volatility calculated from intraday data. Concurrently, the 2000s saw progress inresearch that suggests the need for roughness based on observations about IVs (see section 4for details). Motivated by studies on IVs, Gatheral et al. [2018], the working paper versionof which was published in 2014, analyzed the roughness of realized volatility. In that study,
they revealed that the realized volatility of equity indices and several other asset prices has
a roughness property, which attracted a great deal of attention. In the following, we provide
an overview of Gatheral et al. [2018].
Using data on realized volatility calculated from intraday data for various ﬁnancial assets, Gatheral et al. [2018] estimated the degree of roughness H of their paths (i.e., timeseries of realized volatility). This roughness measure corresponds to the Hurst index of fractional Brownian motion, and thus H < 1/2 suggests that the variation of realized volatility
is rougher than that of a Brownian motion. In the remainder of this subsection, we dis18

cuss roughness with the estimation of the Hurst parameter of fractional Brownian motion inmind.25
As we have seen in section 2.2, when volatility follows a fractional Brownian motion with
Hurst parameter H , 1/2, if the interval ∆ over which the volatility is measured is varied, its
standard deviation does not satisfy the so-called square-root-of-time rule. More speciﬁcally,
when the logarithm of realized volatility follows a fractional Brownian motion with Hurst
parameter H, the following equation holds with respect to the absolute value of the varianceof realized volatility from (4)(iii) in section 2.2, where the left-hand side of the lower equationis the (generalized) moments with q being a non-negative real number representing a power
exponent:hqi(7)
E log σt+∆ − log σt = Kq ∆qH .
 hi
qTaking the logarithm of both sides, log E log σt+∆ − log σt= const + qH log ∆. Provided that this relationship holds, if we ﬁx q and calculate the expectation on the left-handside of equation (7) for various intervals ∆, then we can expect a log-log plot of this left-handside versus ∆ to be a straight line with slope qH.
Gatheral et al. [2018] empirically examined this conjecture based on the daily realized
volatility calculated from ﬁve-minute data for DAX futures, German government bond futures, the S&P 500 index, and the NASDAQ index. They ﬁrst calculated the logarithm of
the left-hand side of equation (7) by replacing the expectation with the sample mean with
ﬁve diﬀerent q (q = 0.5, 1, 1.5, 2, 3) and ∆ = one day to one year. Then, for each ﬁxed q,they regressed the left-hand side on log ∆ to obtain an estimate of qH. Next, they obtained
an estimate of the Hurst parameter H by regressing the estimates of qH obtained above on
q. They reported that the estimated values of the Hurst parameter to be stable at around
0.1 (0.082 − 0.142) regardless of the asset type among the above four, with the ﬁt of theirregressions being extremely good. The fact that the Hurst parameter was estimated stably
below 1/2 suggests that the volatility of these assets exhibits rough stochastic dynamics.
To explain these empirical ﬁndings consistently, Gatheral et al. [2018] proposed the RoughFractional Stochastic Volatility (RFSV) model. In the RFSV model, the instantaneous volatility is given as σt = exp (Xt ), where Xt is a fractional Ornstein-Uhlenbeck (fOU) process,dXt = νdBtH − α (Xt − m) dt,
in which BtH is a fractional Brownian motion with Hurst parameter H, and ν, α, m are constant parameters. The mean reversion parameter α is assumed to be suﬃciently small that thecontribution of the mean reversion term (the dt term in the above equation) is negligible on
short time scales. Under this assumption on α, it has been shown that Xt behaves locally likea fractional Brownian motion, yet it is a stationary process (see Proposition 3.1 in Gatheral
25

As mentioned in section 2.2, roughness is deﬁned based on the Hölder continuity of paths. Therefore, thedegree of roughness can be deﬁned and measured even when volatility follows a stochastic process other thana fractional Brownian motion. Nevertheless, a majority of rough volatility research conducts analyses withfractional Brownian motion in mind. For the sake of the simplicity of exposition, our discussion also has theestimation of the Hurst index of fractional Brownian motion in mind.

19

et al. [2018] for the precise mathematical description). For fractional Brownian motion and
RFSV models to be consistent with the empirical ﬁndings in Gatheral et al. [2018], the Hurst
parameter for fractional Brownian motion in RFSV models must be less than 1/2. Conversely, Gatheral et al. [2018] reported that a Comte and Renault [1998]-type long memory
model with H > 1/2 does not ﬁt to the observations on the behavior of realized volatility
variations.
Gatheral et al. [2018] also compared the accuracy of volatility forecasts based on the RFSV
model with existing models such as AR and HAR models, and reported that the RFSV model
showed better forecasting accuracy than AR and HAR models for almost all stock indicesand volatility measurement horizons. They also showed that only one parameter, the Hurst
index H, needs to be estimated to forecast volatility.26 They argued that this feature of the
RFSV model is a practically appealing point, as it requires estimating much fewer parametersthan AR and HAR models.27
These results of Gatheral et al. [2018] have attracted much attention from researchersand practitioners. One reason for this is the relationship between the long memory property mentioned in section 3.1.3 and roughness. For example, fractional stochastic volatility(FSV) models (e.g., Comte and Renault [1998]) have been used as continuous-time volatility models with long memory. Since FSV models require the Hurst index to be greater than
1/2, strong academic interest has emerged on the relationship between these models and theresults reported by Gatheral et al. [2018]. In addition, practitioners also paid great attention
to rough volatility due to its superior volatility forecasting ability. Progress in research on
the empirical and theoretical properties of IV and its relationship to rough volatility, which
will be discussed in detail in section 4, has also fueled interest in rough volatility.

3.3 Recent research trends in roughness of HV
3.3.1 Measurement error in realized volatility and estimation of the Hurst parameter
Recent research following Gatheral et al. [2018] has mainly dealt with two issues. The
ﬁrst issue is the estimation methods for the Hurst parameter. Gatheral et al. [2018] useddaily realized volatility calculated from ﬁve-minute data as input data and estimated the
Hurst parameter based on a regression analysis based on equation (7). In this estimation, theThe RFSV model has the mean reversion parameter α, as well as other parameters, in addition to the Hurst
parameter. However, as noted above, α does not appear in the volatility forecasting formula, because of theassumption that the mean reversion term can be ignored on short time scales. Gatheral et al. [2018] showedthat the other parameters also do not appear in the forecasting formula, because they cancel each other out inthe derivation process.
27
Note that Gatheral et al. [2018] simply compared the accuracy of predictions of each model without conducting statistical tests regarding the diﬀerences between them. Wang et al. [2023a] compared forecasting
accuracy between rough models and non-rough models (e.g., HAR models) based on the Diebold and Mariano[2002] test and the model conﬁdence set of Hansen et al. [2011], and found that the fOU process, which is arough model, has signiﬁcantly higher predictive ability. In addition, Wang et al. [2023b] analyzed an optimal
way in terms of prediction accuracy to approximate the expected value of fractional Brownian motion with
their discretely sampled values.
26

20

measurement errors of daily realized volatility are not taken into account; it is (implicitly)
assumed that the input data coincide with the true volatility measure.
Note, however, that realized volatility diﬀers from true instantaneous volatility σt and its
Rcumulative value over a given period, integrated volatility σ s ds, because realized volatility
is calculated from discretely sampled asset price data.28 Thus, a naive estimation approach
could lead to bias in the estimated value of the Hurst parameter, due to possible measurement
errors in the input data. In fact, Fukasawa et al. [2022] estimated the Hurst parameter basedon a regression analysis using daily realized volatility calculated from ﬁve-minute data as
input, and found that the Hurst parameter (as Gatheral et al. [2018] reported) takes a value
around 0.1 regardless of the true Hurst parameter for the data-generating processes. They
argued that measurement errors in realized volatility drive this result. Cont and Das [2023]
also reported that estimating the Hurst parameter based on the variation of the path of realized volatility leads to biased estimates. Based on this, they pointed out that obtaining an
estimated Hurst parameter less than 1/2 does not necessarily serve as evidence of roughness.
In response to this issue, several papers have proposed parametric estimation methods for
the Hurst parameter by introducing a model of measurement errors in realized volatility.
The aforementioned Fukasawa et al. [2022] proved a limit theorem on the diﬀerence between realized and integrated volatility (i.e., measurement error of realized volatility) undera mild assumption that the logarithmic process of the underlying asset follows a continuous
semimartingale. Speciﬁcally, they showed that the measurement error of realized volatility
follows a normal distribution with mean 0 and variance 2/m (where m is the number of intraday data per day). They then proposed a quasi-maximum likelihood estimator based on
this theoretical result. When this estimator is applied to the ﬁve-minute realized volatilityof major equity indices such as the S&P 500 and the Nikkei 225 Index, the estimated Hurst
parameters are around 0.02-0.06, which are smaller than the Hurst parameters reported in
Gatheral et al. [2018]. Similar to Fukasawa et al. [2022], Bolko et al. [2023] proposed a
generalized method of moment (GMM) approach to estimate the Hurst parameter under theassumption that the logarithmic process of the underlying asset follows a continuous semimartingale. Their empirical analysis also suggested that the time evolutions of the volatilities
of major equity indices are very rough, with estimated Hurst parameter values below 0.05.
Damian and Frey [2023] proposed a model wherein high-frequency asset price ﬂuctuations are described by a point process and the intensity of the point process is determinedby latent factors that correspond to volatility.29 In this setting, the authors formulated theestimation of volatility as a ﬁltering problem of point process data. However, since fractional Brownian motion is not Markovian and thus the behavior of stochastic processes ispath-dependent, using conventional ﬁltering techniques is not realistic due to computational28

Jumps in the underlying price process also create gaps between realized volatility and integrated volatility,
because realized volatility is aﬀected by jumps, whereas integrated volatility is not. See Andersen et al. [2012]
and Barndorﬀ-Nielsen [2005] for the estimation of realized volatility when the underlying process may jump.
29
A point process is a stochastic process that describes randomly distributed points (events) over time orspace, such as the number of contingent events, and intensity controls the likelihood of the occurrence of
events. See section 5.2 for more details on point processes.

21

burden, including the large memory requirements. To overcome this diﬃculty, Damian andFrey [2023] transformed fractional Brownian motion to superposition of an inﬁnite number
of OU processes. Because an OU process is an Itô process, this transformation renders theapplication of a particle ﬁltering method possible, because it requires the Markov property.
They performed various simulation analyses and reported that their approach is able to distinguish true roughness from the spurious roughness caused by the presence of noise, as
pointed out by Cont and Das [2023].
A topic related to measurement errors is market microstructure noise (MM noise) in observed asset prices. Typical examples of MM noise include rounding errors caused by discrete tick sizes at exchanges (e.g., one cent minimum tick size) and the bid-ask bounce, thatis, mechanical negative autocorrelations due to the existence of bid-ask spreads.30 Whenan underlying asset price contains MM noise, estimation of volatility using high-frequency
data requires caution. As seen in section 3.1, the property that the quadratic variation of theunderlying asset price converges to the integrated volatility as the time split becomes ﬁner,
which is the theoretical basis for using realized volatility, holds under the assumption of no
MM noise in the underlying asset price. However, if the underlying asset price contains MM
noise, it can be shown that realized volatility converges to the variance of the MM noise, notto the integrated volatility. To avoid this problematic result, appropriate treatment is required(Zhou [1996], Aı̈t-Sahalia and Mykland [2009]).
Several approaches exist for dealing with MM noise in high-frequency data. The simplest and most widely used approach recommended in the literature (Andersen et al. [2001],
Gençay et al. [2002], Barndorﬀ-Nielsen and Shephard [2002], among others) is down-sampling,that is, sampling data infrequently (e.g., ﬁve-minute data) so that the eﬀects of MM noise inhigh-frequency data can be mitigated.31 In addition to mitigating MM noise, down-sampling
has the advantage that it is easy to handle both theoretically and empirically because it is estimated from equidistant time-series data (unlike unevenly distanced tick data). Many studies have shown that calculating realized volatility from ﬁve-minute data is very simple andconvenient, yet this method strikes a good balance between having a suﬃciently high data
frequency and obtaining suﬃcient mitigation of MM noise. On these grounds, Fukasawa
et al. [2022] also conducted an analysis using realized volatility calculated from ﬁve-minutedata (Fukasawa et al. [2022], Remark 2.4 (ii)).
30

For an overview of the impact of market microstructure on asset prices, see Campbell et al. [1997].
Bandi and Russell [2008] provided an estimate of the optimal down-sampling data frequency. Note that
down-sampling is in one sense suboptimal, as it discards a portion of high-frequency data, which may containuseful information. For this reason, alternative approaches have been proposed to utilize high-frequency data
as a whole by taking the existence of noise into account. For example, Zhang et al. [2005] proposed a method
called Two Scales Realized Volatility, which simultaneously estimates the integrated volatility and variance of
MM noise in a consistent manner. There are also estimations based on state-space models (e.g., Nagakura and
Watanabe [2015]). Hansen and Lunde [2006] analyzed the eﬀect of MM noise in the measurement of realized
volatility and proposed a realized volatility indicator that corrects for the eﬀect of MM noise. Nevertheless,some studies have suggested that down-sampling is the best approach for practical applications, given the factthat it is a simple method yet it achieves suﬃciently good performance. In fact, Liu et al. [2015] compared morethan 400 estimates of realized volatility and reported that no estimation methods clearly outperform ﬁve-minuterealized volatility.
3122Note that the discussion on MM noise should not be confused with that on measurement
errors in Fukasawa et al. [2022] and others, which analyzed the impact of measurement
errors in realized volatility calculated from discretely sampled asset prices on the estimationof the Hurst parameter, as opposed to integrated volatility deﬁned from a continuous-time
stochastic process of volatility. Such measurement errors can occur even in the absenceof MM noise. Therefore, bias in the estimates of the Hurst parameter from regressions ofrealized volatility, as done by Gatheral et al. [2018], can occur even in the absence of MM
noise.

3.3.2 Relation to the long memory property
The second issue regarding the roughness of realized volatility is the relationship between
the long memory property and roughness. Fractional Brownian motion (and standard Brownian motion) has a property called self-similarity (see section 2) and thus rescaling the timescale does not change its distributional properties once an appropriate scaling coeﬃcient is
attached. In other words, the behavior of fractional Brownian motion over very short time
horizons and that over very long time horizons are essentially identical. This means thatlong memory (positive serial correlation over a long period of time) and roughness (negative
serial correlation due to wild ﬂuctuations over a short period of time) cannot be represented
by a single fractional Brownian motion. There are two directions to trying to address thisissue. One strand of studies has raised the possibility of spurious detection of long memory,while the other strand has aimed to develop a model that can represent roughness and longmemory simultaneously.
The existing literature on long memory has shown that, even when the true data-generatingprocess does not have long memory, estimated long memory parameter (fractional integration parameter) d may be biased in the presence of time trends, structural breaks, regimechanges, and the like, leading to spurious detection of long memory (e.g., Diebold and Inoue[2001]; Granger and Hyung [2004]). Gatheral et al. [2018] also discussed the possibilityof spurious detection of long memory; they reported that when the Andersen et al. [2001]
method for estimating the fractional integration parameter d of the ARFIMA model is applied to the time-series data generated by the RFSV model, the estimation result yields spurious detection of long memory.32 Based on this observation, they argued that the RFSV
model, like those of the previous studies mentioned above, is a time-series model that isrough in nature but can be estimated spuriously to have long memory.
Another strand of literature proposes models that can represent roughness and long memory simultaneously. For example, Bennedsen et al. [2021] proposed a model using a stochastic process called a Brownian semi-stationary process (Barndorﬀ-Nielsen and Schmiegel[2007], Barndorﬀ-Nielsen and Schmiegel [2009]), which is an extension of Brownian mo32

The Andersen et al. [2001] estimation method examines the Fourier transformation of the autocorrelation
function to see whether it satisﬁes a condition corresponding to a hyperbolic decay of the autocorrelation
function in the time domain.

23

tion. In their model, the roughness parameter and the autocorrelation coeﬃcients of longdistance lags are distinct, allowing the degree of roughness and long memory to set independently. Furthermore, they also proposed an empirical method to estimate this model basedon a nonlinear least squares method while taking into account measurement errors in realized volatility. Applying their empirical method to the S&P 500 index futures and individual
stocks data, they obtained results which suggest that the volatility of these assets has both
the rough property and the long memory property.

3.3.3 Summary of this section and future prospects
To summarize the current research trends on the roughness of realized volatility, the following points remain key issues. First, realized volatility contains measurement errors anddiﬀers from its theoretical counterpart, integrated volatility. Therefore, when realized volatility is used as input data as in Gatheral et al. [2018], estimated values of the Hurst parameter
may suﬀer from a downward bias (i.e., bias to the direction of roughness) unless this measurement issue is appropriately taken into account. For this purpose, several recent studies
proposed more sophisticated estimation methods that take this issue in account. These studies reported that the volatility of equity indices and individual stocks exhibit even stronger
roughness according to these improved estimation methods.
Second, due to the self-similarity of fractional Brownian motion, at least simple models
such as a single fractional Brownian motion cannot possess roughness and long memory
simultaneously. There exist two main arguments on this issue. First, some studies have raised
the possibility that the long memory property may be spuriously detected. For example,
Gatheral et al. [2018] demonstrated that a conventional approach spuriously detects longmemory even when it is applied to time series generated by the RFSV model. Anotherstrand of studies proposed stochastic processes that can simultaneously express roughness
and long memory.
Both the above issues exemplify the diﬃculty of analyzing the nature of (instantaneous andintegrated) volatility, which is not directly observable. This calls for continued developmentof less noisy estimates of volatility, more realistic models that can describe the complex
features of volatility, and more robust statistical methods for detecting roughness and longmemory.
Finally, we note that some researchers have expressed skepticism about the need for complex models based on fractional Brownian motion. For example, Rogers [2023] questioned
the rough volatility model based on fractional Brownian motion for several reasons. First,
he pointed out that its non-Markovian nature prevents it from practical applications due toits heavy computational cost, which includes its memory usage cost. Second, he doubted
that there exists a convincing ﬁnancial economic mechanism that yields non-Markovian asset price dynamics. Third, he argued that a conventional Itô process seems to be suﬃcient
to describe the time-series characteristics over typical time scales in empirical research (e.g.,
daily or weekly frequency), suggesting limited empirical beneﬁt from using complicated24fractional Brownian motion-based models. Thinking deeper about these issues may require
holistic judgment, considering not only the perspectives arising from HV research (i.e., observations about realized volatility and estimation methods for the degree of roughness) but
also other perspectives. Speciﬁcally, our subsequent sections provide additional perspectives,
such as the necessity of roughness in derivative pricing and for risk management purposes,
as well as the ﬁnancial economic micro-foundations for explaining why roughness arises.

4 Shape of implied volatility surface and rough volatility
modelsIn this section, we ﬁrst provide an overview of the development of mathematical ﬁnancefrom both practical and academic perspectives, and then discuss the roughness of volatility
indicated by the shape of IV (particularly, its term structure), why we need rough volatility
models for pricing derivatives, and prospects for their practical applications. Mathematicalﬁnance and ﬁnancial engineering33 have mainly focused on pricing and risk management ofderivatives and have provided a theory to calculate fair prices using Itô calculus (stochastic diﬀerential equations) under the principles of no-arbitrage and replication.34 Interrelationship between theory and practice has progressed these ﬁelds; application of theory has
helped solving practical problems and problems arising from practice issues have stimulated
the further development of the theory.
The IV representation problem, on which we focus in this section, is a typical example ofsuch progress. Rough volatility initially attracted attention as a method that could solve thisproblem, especially the expression of the negative power law regarding the term structureof IV, which was considered diﬃcult to solve with existing methods. This discovery led tosubsequent research from the perspective of HV and MM. In recent years, rough volatility
models for derivative pricing have also been developed. In particular, it was shown that,
despite its non-Markovian nature, a rough volatility model can be regarded as a Markov
model by considering additional variables such as forward variances observed in the market.
This result is referred to as the recovery of the Markov property and allows us to use theexisting framework of mathematical ﬁnance theory for rough volatility models. Furthermore,a recent theoretical result showed that under markets where roughness is implied, arbitrage
opportunities will arise if derivatives are not priced and their risk is not managed by roughmodels. This result is of importance as it necessitates the use of rough models. In addition, it
has been found that a rough volatility model can solve the joint calibration problem of SPX
33

Both mathematical ﬁnance and ﬁnancial engineering focus on the pricing of derivatives. The former research ﬁeld is often referred to as a branch of applied mathematics mainly focusing on the fair valuation ofderivatives based on martingale theory, whereas the latter is often referred to as a research ﬁeld oriented towards practical applications, mainly focusing on practical uses of models such as matching model parameters
to reproduce market observations.
34
No-arbitrage is the condition that one cannot make a proﬁt for sure without incurring an initial cost. Replication refers to generating the same cash ﬂow as the derivative by appropriately trading the underlying asset
and other assets.

25

options’ IV and VIX options’ IV, which has been considered diﬃcult to solve with existing
derivatives models.
Section 4.1 summarizes the development of mathematical ﬁnance, by focusing on the IV
representation problem while taking into account practical issues. Section 4.2 introducesrough volatility studies related to IV. Speciﬁcally, we explain that a certain shape of IV
indicates the roughness of volatility and the need for rough volatility models. Then, we summarize how rough volatility models can be used for derivative pricing and risk management,
as well as providing prospects for the further development of rough volatility models.

4.1

History of the development of mathematical ﬁnance and ﬁnancial
engineering: IV representation problem

4.1.1 Implied volatility
As mentioned previously, the Black–Scholes model (Black and Scholes [1973]) is the mostbasic model and has been the key driver of the rise of mathematical ﬁnance and ﬁnancial
engineering, as well as the development of derivatives markets. As will be discussed shortly,
the Black–Scholes model serves as a basic tool in current practice of ﬁnancial institutions,for example, when calculating IVs given the market prices of options.
In the Black–Scholes model, the underlying asset price is assumed to follow a log-normal
distribution, which allows for analytical pricing of vanilla options such as European call options (i.e., the pricing formula is known and the price can be calculated without numericalapproximation). Here, the option price can be calculated as the expected value of the payoﬀ
under a risk-neutral probability measure, a probability measure where the expected returnon the underlying asset price is equal to the risk-free interest rate. As a result, the calculated price is equal to the initial cost required to replicate this option under the no-arbitrage
assumption (for details, see, for example, Björk [2009]). Therefore, by using the Black–
Scholes pricing formula, the option price can be obtained without estimating the expectedunderlying return under the physical probability measure.
Formally, the Black–Scholes pricing formula is formulated as follows. Let WtQ be a Brownian motion under a risk-neutral probability measure and let the underlying asset price S t(BS)follow a geometric Brownian motion (i.e., the price at a certain time follows a log-normal
distribution):!
!
dS t(BS)
σ2
(BS)
(BS)
Q
Q
= r dt + σ dWt , i.e., S t = S 0 exp r −t + σWt ,2
S t(BS)
where t is the current time, r is the risk-free rate, and σ is the volatility parameter. Moreover,let K be the strike and T be the maturity of a European call option. Then, the price of this26

option is given as follows:hi
CBS (r, K, T, σ, S t ) = E Q e−r(T −t) max{S T(BS) − K, 0} | Ft
 (BS)  
 (BS)  






σ2
σ2
 log S t log S t
+
r
+
(T
−
t)
+
r
−
(T
−
t)
K
2
K
2





(BS) 
 − e−r(T −t) KΦ  ,= S t Φ 
√
√




σ T −tσ T −twhere Φ is the standard normal cumulative distribution function.
The parameters (arguments) that appear in the Black–Scholes pricing formula are the current underlying asset price S t , the risk-free rate r, the volatility parameter σ, the strike K, andthe maturity T . Since the underlying asset price and the risk-free rate are observable in themarket35 and the strike and maturity are terms of the instrument being priced, σ is the only
free parameter. This means that there is one-to-one relationship between option prices andthe volatility parameter σ. As a result, the corresponding volatility can be backed out froma given option price using the Black–Scholes formula.36 Speciﬁcally, given a (market) priceC (mkt) (K, T ) of a European call option whose strike is K and maturity is T , the Black–Scholes
implied volatility (BS-IV) is implicitly deﬁned based on the following equation:37
C (mkt) (K, T ) ＝ CBS K, T, σ(mkt)
(K,
T
)
.
BS
In practice, option prices are conventionally quoted in terms of IVs.38 This convention isbased on the idea that the market prices of options reﬂect market expectations on future
uncertainty.

4.1.2 Volatility Surface
If the underlying asset price follows the Black–Scholes model, BS-IVs take a constantvalue equals to the constant volatility parameter σ regardless of maturity or strike. On theother hand, BS-IVs calculated from actual market prices of options usually take diﬀerent
35

The risk-free rate is typically calculated from bond or swap prices.
Since the function CBS in the Black–Scholes formula is a monotonically increasing function with respect
to σ, the implied volatility obtained from the inverse function is uniquely determined.
37
There are other types of IVs. For example, Black (Bachelier) IV is calculated assuming that the underlying
price process follows a Brownian motion, and is used in the options market where the underlying asset price
may become negative (e.g., interest rates). Another prominent type of IV is the so-called model-free IV, whichis calculated without assuming a speciﬁc model for the underlying asset price dynamics. As described below,
the calculation methodology of volatility indices such as VIX is grounded on the theory of model-free IV.
38
Since option prices ﬂuctuate according to the price of the underlying asset, the ﬂuctuation of the option
price is volatile if its underlying asset price is volatile (e.g., exchange rates). Therefore, it would be cumbersome
from a practical standpoint if options were indicated in terms of price, because option prices will change
whenever the price of the underlying asset changes. In addition, since option prices also depend on maturity
and strike, option prices with diﬀerent maturities and strikes are not directly comparable. Therefore, the market
practice of displaying the value of options in terms of (standardized) volatilities that are independent of theunderlying price, maturity, and strike has been widespread. Note that not all derivative instruments display
their value in terms of volatility; for example, foreign exchange options on minor currencies are sometimes
quoted in terms of prices.
36

27

values depending on strike K and maturity T .39 Namely, given the market prices of options
for various strikes and maturities, the BS-IVs depicted as a three-dimensional surface as afunction of strike K and maturity
T is calleda volatility surface. In practice, it is conventional −r(T−t) Keto use log-moneyness κ = log
instead of strike K, and time-to-maturity τ = T − tStinstead of maturity T as arguments. Figure 3 shows a typical volatility surface for, say, an
equity options market.

Figure 3: Volatility surface (conceptual drawing)
In the equity index options markets, which are among the most liquidly traded options
markets, the market prices of options with highly liquid strikes and maturities are often
quoted as volatility surfaces. Looking at the cross-section of the volatility surface at a single
maturity τ, we see that IVs tend to be higher for lower strike prices. This corresponds to a
thicker distribution in the direction of falling stock prices, a phenomenon known as volatility
smile/skew. Looking at the cross-section at a single strike price (cross-section along the Yaxis), we see that IVs tend to be higher for shorter maturities. This corresponds to the factthat the prices of options with shorter maturities are more heavily aﬀected by the risk of
sudden changes in the market.
4.1.3 Representation of IVs
IV representation refers to structuring derivative pricing models40 so that they are consistent with the volatility surfaces quoted in the market. The motivation and details are
39

This means that although the Black–Scholes model is used in calculating BS-IV, the underlying asset pricedoes not follow the Black–Scholes model under a risk-neutral probability measure.
40
Derivative pricing models can be categorized into vanilla models for pricing relatively simple instruments(vanilla options) and exotic models for pricing complex instruments (exotic derivatives). In both cases, themarket prices of highly liquid instruments are used as input to calculate the price of instruments to be priced.
For vanilla options, the main calculation performed in the vanilla model is only the interpolation of the market
prices as input, since similar instruments (e.g., the same derivative products with diﬀerent maturities) havehigh market liquidity. On the other hand, as discussed in the text, the exotic model requires more complex

28

described below. First, note that the general ﬂow of derivative pricing process in practice is
as follows:(1) Build a model to represent the trading prices of highly liquid derivative instruments
quoted in the market.
(2) Using the model built in step (1), price and hedge more complicated derivative instruments.
Here, step (1) is necessary to build a model that accurately represents the prices of highly
liquid derivatives and calculate the prices of more complex derivatives while satisfying the
no-arbitrage condition.41 Since there is one-to-one relationship between option prices and
BS-IVs as mentioned above, it is important to construct a model that can represent IVs ofhighly liquid derivatives (e.g., European options). Accordingly, it has been a central theme ofmathematical ﬁnance to construct models that can represent the features of volatility surfaces
observed in the market (i.e., construct a stochastic diﬀerential equation that the underlying
asset price follows), as well as eﬃcient methods to adjust the parameters of the model toreproduce the observed market prices, which is referred to as calibration. Regarding step (2),the calculation of the prices of complex derivatives and derivation of hedging strategies (i.e.,the derivation of the sensitivity of option prices to changes in underlying asset prices, etc.,
called Greeks) are frequently required in practice. Therefore, it is desirable to perform thesecalculations with high accuracy and speed. In addition, when deriving hedging strategies,
it is important to capture not only the volatility surface at a single time point, but also its
dynamics (time evolution). There have been various studies on these issues as they areimportant themes in theoretical research and practical applications of mathematical ﬁnance.

4.1.4 Existing derivative pricing models
Let us list some representative derivative pricing models: local volatility models that describe volatility as a deterministic function (Dupire [1994]), stochastic volatility models inwhich volatility varies stochastically (Heston [1993], Hagan et al. [2002]), and jump models in which volatility may vary discontinuously (Bates [1996]). Table 1 summarizes thesemodels.

calculations, such as obtaining parameters under risk-neutral probability measures of the underlying asset priceprocess. The concept of IV representation is common to both vanilla and exotic models, but our subsequentdiscussions primarily have the exotic models in mind.
41
Reasons why it is desirable to maintain the no-arbitrage condition include, among others, (1) to avoid being
exploited by arbitrage, (2) the need for reliable mark-to-market valuation for accounting purposes, and (3) theneed to trade other liquid options for risk management purposes.

29Model: BS model (Black and Scholes [1973])
Volatility term: Vt = σ2 (Constant)
Merit: Admits analytical pricing formulae for some options (e.g., European call option)
Drawback: Observed volatilities are not constant
Model: Local Volatility model (Dupire [1994])
Volatility term: Vt = σ (t, S t ) (Deterministic function)
Merit: Can capture volatility surface accurately at a single ﬁxed time
Drawback: Cannot capture the dynamics of volatility surface
Model: Stochastic Volatility model (Heston [1993])
Volatility term: dVt = µV (t, Vt ) dt + σV (t, Vt ) dWtV (Stochastic process)Merit: Can capture the dynamics of volatility surface
Drawback: Cannot fully capture the shape of volatility surface
Model: Jump model (Bates [1996])
Volatility term: dVt = µV (t, Vt ) dt + σV (t, Vt ) dWtV + dJt (Stochastic process with jump dJt )Merit: Can capture the dynamics of volatility surface ﬂexibly thanks to the jump term
Drawback: Hedging involves costly and unstable calculations
Table 1: Summary of representative derivative pricing models
4.1.5 Calibration and asymptotic expansion of IVno
Let us assume that S u(model)
follows the following stochastic diﬀerential equationt≤u≤Tunder the risk-neutral probability measure:dS t(model)= r dt +(model)Stp
Vt dWtQ .

Then, calibration means the calculation of the model parameters so that the following approximation holds:
n
oi


h
CBS r, K, T, σ, S t(model) ≈ E max S T(model) − K, 0 ,(8)where σ̂(mkt)BS (K, T ) is (interpolated) IV observed in the market.
When the expectation on the right-hand side of equation (8) cannot be calculated analytically, it is common to use an approximation method such as asymptotic expansion to address
this calibration problem. Highly accurate asymptotic expansion methods are necessary to obtain suﬃcient accuracy, and have been an important research topic in mathematical ﬁnance(see, for example, Hagan et al. [2002] and Osajima [2007]).

30

4.2 Progress in using rough volatility
Below, we ﬁrst summarize the history of rough volatility studies related to IV, and thendescribe the main technical points of these studies in section 4.2.1 and beyond.42
As already discussed, derivative pricing models have developed toward using more complex volatility functions, motivated by the representation problem of volatility surfaces andrisk management associated with their time evolution. In the early 2000s, empirical studies
on the equity derivatives markets reported a characteristic pattern called the negative powerlaw of volatility surfaces, which is diﬃcult to express by existing derivative pricing models.
In response to this, in the late 2000s, Alòs et al. [2007] and Fukasawa [2011] showed thatthe negative power law can be represented by using a model based on fractional Brownian
motion. This result, together with the subsequent observation of roughness in HVs, triggereda surge of interest in rough volatility research.
In subsequent studies, rough volatility models for pricing derivatives were proposed, suchas the rough Bergomi model (Bayer et al. [2016]) and the rough Heston model (El Euch andRosenbaum [2019]). These models had been considered to have the diﬃculty of being nonMarkovian in nature, due to fractional Brownian motion, which makes the existing framework of mathematical ﬁnance inapplicable. In regard to this, subsequent research has shownthat the Markov property can be recovered by including information on the forward variance(expected value of the variance at a future time). It has been shown that, if such information
is included, the theory of mathematical ﬁnance based on replication and no-arbitrage canbe applied, and thus hedging strategies can be constructed in a practically feasible manner(El Euch and Rosenbaum [2018], Fukasawa et al. [2021]). One of the practical advantagesof a rough volatility model is that it can represent the negative power law of the volatility
surface observed in the market, despite the relatively small number of model parameters. In
addition, Gatheral et al. [2020] showed that the rough volatility model can solve the joint
calibration problem of SPX options’ IVs and VIX options’ IVs (the problem of simultaneously ﬁtting to both the IVs of the S&P 500 index options and that of the VIX options in a
uniﬁed model), which was considered diﬃcult to solve with existing derivatives models.
In addition to these developments, Fukasawa [2021] proved that, in a general setting, arbitrage opportunities arise if we do not use a pricing model with appropriate roughness, as
indicated by the shape of the volatility surface observed in the market. In other words, in markets where the negative power law is observed, pricing and risk management of derivatives
using a conventional non-rough volatility model is undesirable because arbitrage opportunities will arise; that is, volatility has to be rough. This is a very important theoretical resultthat strongly suggests the need to use rough volatility models in the pricing of derivatives.

4.2.1 Negative power law and its representation via fractional Brownian motion
Carr and Wu [2003], Lee [2005], and Fouque et al. [2003] observed the negative power
42

See Alòs and León [2021] and Di Nunno et al. [2023] for surveys.

31

Figure 4: Negative power law (conceptual drawing)
law as a stylized fact about the shape of volatility surfaces based on real data for market
prices of options. The negative power law refers to a speciﬁc shape of the term structure ofIV, which is formulated as follows:ψ(mkt) (τ) =∂ (mkt)1
σBS (κ, τ)≈ cτH− 2 for suﬃciently small τ > 0,∂κκ=0where ψ(mkt) (τ) is the slope of the at-the-money (ATM) volatility skew for options with aﬁxed maturity τ. Namely, the negative power law means that, as a function of maturity, theslope of the volatility surface at each maturity can be described by a two-parameter functionof time-to-maturity to the power of H − 1/2 and a constant coeﬃcient c. It is empiricallyobserved that H < 1/2 holds, namely that the slopes of the volatility surface are steeper for
shorter maturities. The right graph of Figure 4 shows the ATM slopes of a typical volatility
surface at various maturities (black dots) and the curve that ﬁts these ATM slope data (redline, described by two parameters), as observed in a typical example, an equity derivatives
market.
The existing Brownian motion-based models introduced in section 4.1.4 are known tohave diﬃculty in representing the negative power law.43 On the other hand, as mentioned
above, Alòs et al. [2007] and Fukasawa [2011], as well as Fukasawa [2017], proved that thenegative power law can be represented using models based on fractional Brownian motion
with Hurst parameter H < 1/2.44 This fact suggests that the IV observed in the market is43

For example, the stochastic volatility models have diﬃculty in representing the negative power law because
IV is nearly constant at short maturities in these models. The local volatility models can represent the shape of
IV at one point in time, but it is diﬃcult to represent changes in the volatility surface as it evolves over time.
The jump models can represent the behavior of volatility surfaces, but these models tend to be complicated anddiﬃcult to calibrate stably, and hedging operations become diﬃcult in practice.
44
Alòs et al. [2007] used an asymptotic expansion based on Malliavin calculus, and Fukasawa [2011] used a32

rough, which made these studies pioneers in rough volatility research, appearing prior to thework of Gatheral et al. [2018].45
Moreover, Fukasawa [2021] showed that, unless we use a pricing model with an appropriate degree of roughness, namely that implied by the shape of the volatility surface observedin the market (the parameter H in the negative power law), arbitrage opportunities arise, atleast in theory.46 This indicates that no-arbitrage pricing requires a rough volatility model
whose degree of roughness is consistent with the roughness of the volatility implied by theshape of the volatility surface. As mentioned, this result, together with that of research on
HV and market microstructure (MM), establishes a theoretical foundation for adopting arough volatility model.
Below, we provide a technical summary of Fukasawa [2021]. Fukasawa [2021] assumed(model)that the underlying asset price process follows a continuous positive-valued,
D martingale
E St(model)and considered the quadratic variation process of the log asset price log S. This2,tquadratic variation
corresponds
to the integrated variance from time 0 to t, and its timeDEd(model)derivative dt log S, corresponds to the instantaneous variance at time t. Addition2,t
ally, Fukasawa [2021] assumes that the Hölder continuity of the stochastic process of thisinstantaneous variance is equal to H. It is then shown that the following relation holds as an
approximation:
ψ(model) (τ) =∂ (model)1(κ, τ)σ̂BS∝ τ− 2 +H ,∂κκ=0

τ → 0, H ≈ 0.

This result implies that the shape of the volatility surface ψ(model) (τ) exhibits a negative powerlaw as long as the Hölder continuity of volatility satisﬁes H < 1/2. Next, let H be theexponent of the negative power law implied by the shape of the volatility surface, and let H0
be the exponent of the Hölder continuity of the volatility of the stochastic process S t(model)
used as the pricing model. It is shown that if H0 > H, that is, if the roughness of the pricing
model is less than that of the observed IV shape, an arbitrage opportunity can be constructed.
Note that this arbitrage opportunity is constructed using the fact that the roughness of the IV
ampliﬁes the error of the delta hedge based on the Black–Scholes model, and the arbitrage
strategy is composed of a combination of very short-term call and put options.
However, the negative power law of IV described above has been observed mainly in themarket of U.S. equity options with very short maturities, and the appropriateness of the data
used for the observation and the interpolation/extrapolation methods is open to discussion.
In particular, it has been pointed out that ψ(τ) observed from option price data has diﬀerent
curvatures for the long and short maturity portions, so it is necessary to use diﬀerent parammartingale expansion.
45
Gatheral et al. [2018] is a pioneering study in terms of using fractional Brownian motion with H < 1/2 inmathematical ﬁnance. Note that, as mentioned in section 3, there have been previous studies using fractional
Brownian motion with H > 1/2 to represent the long memory property (Comte and Renault [1998]).
46
Fukasawa [2021] assumes an ideal ﬁnancial market, where no market frictions such as transaction costs
exist and vanilla options of any strike and maturity can be traded. Therefore, the results in Fukasawa [2021] do
not necessarily imply that a viable arbitrage strategy could be constructed in a real market.

33eters, depending on the range of IV maturities to be approximated. For example, from theexamination using options market data for the EURO STOXX Index and DAX Index, Guyon
and El Amrani [2022] found that although a two-parameter model ﬁts well when focusing
only on the very short portion (maturities of two weeks or less), when ﬁtting to the entire
IV including the longer maturity portion, three parameters were necessary to ﬁt well to theentire IV. On the other hand, Delemotte et al. [2023] concluded that, consistent with previous
studies, by testing with longer (2007-2015) U.S. stock options market data, a two-parameter
function ﬁts suﬃciently well. Overall, similar to the discussion on long memory (H > 1/2)
and roughness (H < 1/2) in factional Brownian motion mentioned in section 3, this strandof literature suggests that the characteristic patterns of volatility may diﬀer between long andshort time horizons. Other studies that have attempted to model the diﬀerent shapes of IVs
in diﬀerent time horizons include Funahashi and Kijima [2017a] and Funahashi and Kijima[2017b].

4.2.2 Rough volatility model
In recent years, some research has showed that even when the volatility process follows amodel with roughness, the asset price processes can be treated as Markov processes by additionally considering information on forward variances. This ﬁnding, coined as the recoveryof the Markov property, was ﬁrst explicitly shown in the rough Bergomi model (Bayer et al.
[2016]), and since then, the rough Heston model and others have been shown to have similar
properties. We give an overview of these models below.
Bayer et al. [2016] proposed a rough Bergomi model as a pricing model for derivatives.
The rough Bergomi model is a rough stochastic process and is an extension to inﬁnite dimensions of the Bergomi model (Bergomi [2004]), in which the forward variance is described by
an Itô process. Speciﬁcally, it is given by the following equations:

 dS
p
√ 
(1)
t


1 − ρ2 dWt(2) , S t = µt + Vt ρ dW
t +
 √ R

(9)
2


 Vt = E [Vt | Fu ] exp η 2H t (t − s)H− 21 dW s(1) − η (t − u)2H ,u

2t is the current time, u is a certain past time, Vt is the instantaneous variance, andwhere(1)
Wt , Wt(2) is a two-dimensional Brownian motion. E [Vt | Fu ] is the expectation of the future variance conditioned on information up to time u (i.e., Fu ), and is called the forward variance. Bayer et al. [2016] derived the rough Bergomi model, by modeling the spot variancebased on fractional Brownian motion and representing forward variance by a Riemann—
Liouville-type Itô integral.
Given the forward variance {E [Vt+θ | Ft ]}θ>0 , the rough Bergomi model can be regarded asa Markov process. Speciﬁcally, from the general derivative pricing formula Ct = E [F (S T ) | Ft ],the rough Bergomi model admits the following expression:Ct = E [F (S T ) | Ft ] = G T − t, S t , {E [Vt+θ | Ft ]}θ≥0 ,34

where G is a some function. By taking into account the third argument on the right-hand sideof this equation (forward variance), we can adopt existing mathematical ﬁnance theory basedon the Markov property and other desirable properties. Therefore, we can apply existing
notions such as replication, no-arbitrage, and risk-neutral measures to the rough Bergomi
model.
This is a strong advantage of the rough Bergomi model, as it opens the possibility of
application to the U.S. equity volatility market, where forward variances can be estimatedfrom the market price of volatility derivatives.47 In subsequent studies of rough volatility
models, the recovery of the Markov property by forward variances also played an importantrole. Consistent with the discussion in section 4.2.1, Bayer et al. [2016] showed that the SPX
volatility surface, where the negative power law is observed, can be represented well by arough Bergomi model having only a few parameters.48 Extending this argument, asymptotic
expansion formulae for IVs yielded from the rough Bergomi model have also been presented
in El Euch et al. [2019] and Friz et al. [2022], which should lead to practically viable methodsof calibration. In addition, applications to the pricing of volatility products such as VIX
futures have been discussed, for example, in Jacquier et al. [2018]. Note that the roughBergomi model is known to have the martingale property when the parameters are in certain
regions (Gassiat [2019]).
Subsequently, El Euch and Rosenbaum [2019] proposed the rough Heston model, which
is an extension of the Heston model (Heston [1993]), in which the stochastic integration termin the variance dynamics has been replaced with a Riemann–Liouville-type Itô integral:

 dS
p
√ 
(1)
(2)


 S t t = µt + VRt ρ dWt + 1 − ρ2 dWt R ,√ Vt = V0 + 1 t (t − s)α−1 γ(θ − V s ) ds + t (t − s)α−1 γ V s dWt(1) .
Γ(α) 00This model admits a semi-analytical option pricing formula, namely the ordinary diﬀerential
equation satisﬁed by the characteristic function of the solution is a fractional Riccati equation, which renders eﬃcient calibration and pricing. This result parallels the fact that thecorresponding ordinary diﬀerential equation in the Heston model is a Riccati equation. In
addition, El Euch and Rosenbaum [2018] showed that the Markov property is recovered asin the rough Bergomi model by making the autoregressive coeﬃcient θ time-dependent, andreplication using forward variances is possible. Furthermore, Abi Jaber and El Euch [2019]
showed how to approximate the rough Heston model with a family of Markovian models.
This result corresponds to the fact that the rough Bergomi model is an inﬁnite-dimensional
extension of the Bergomi model. Furthermore, it is also known that the rough Heston model
has a micro-foundation from a market microstructure perspective, as discussed below in section 5. The rough Heston model may also exhibit the Zumbach eﬀect described in section
47

Forward variances can be estimated from market prices of variance swaps of various maturities, etc.
The parameters of rough Bergomi model (9) have the following correspondence with observed data: H
corresponds to the curvature of the term structure of the ATM skews, ηρ corresponds to the long-run level of
ATM volatility, and ρ corresponds to the skew of volatility smiles (i.e., the cross-sections of the surface at eachmaturity).
48

35

3 (El Euch et al. [2020]). Finally, Fukasawa and Gatheral [2022] provided the asymptotic
expansion formula of IV for the rough SABR model, which is an extension of the SABR
model.

4.2.3 Joint calibration problem of SPX options’ IV and VIX options’ IV
Besides the representation of the negative power law, rough volatility models are expected
to solve other problems that were considered diﬃcult to solve with existing derivative pricing
models. A prominent example is the joint calibration problem of SPX options’ IV and VIX
options’ IV. Here, SPX is a U.S. equity index (the S&P 500 index) and VIX is the volatility
index of SPX. VIX represents the uncertainty of the SPX index over the next 30 days, andis an example of model-free implied volatility (MFIV). MFIV is an IV that is calculated
based on the fair prices of variance swaps, which can be replicated with a portfolio of vanilla
options in a model-free manner (i.e., replication of variance swaps and hence MFIVs do not
rely on speciﬁc pricing models such as the Black–Scholes model).49
The VIX is published as an index and is used for market analysis and other purposes,
and futures and options written on the VIX as the underlying asset are also actively traded.
Therefore, the volatility surface of the VIX is quoted in the market, and it is a challenging
task to calibrate models to it and the volatility surface of the SPX simultaneously. In otherwords, in order to prevent arbitrage opportunities between the two classes of highly liquid
instruments, an option on the SPX and options on the VIX (which can be regarded as options
on options on the SPX), pricing them in a consistent and uniﬁed manner is an important
issue in the U.S. equity derivatives market. Speciﬁcally, what makes this issue challenging
is calibrating the parameters of the dynamics of the SPX so that the theoretical prices (IVs)of the model ﬁt the two volatility surfaces while preserving the theoretical relationship between the SPX and the VIX. Various studies have been conducted on this problem, startingwith Gatheral [2008], but most of these resorted to jump-type models, because calibration
has been considered to be diﬃcult with conventional Itô process-based models.50 See, forexample, Cont and Kokholm [2013] and Guyon [2020].
Gatheral et al. [2020] showed that this problem can be solved in a continuous stochastic
process model with no jumps, which is an extension of the rough Heston model. This result
is related to the phenomenon that the level of VIX is determined according to the past levelof SPX, and is closely related to the Zumbach eﬀect mentioned in section 3.
Note, however, that whether the rough volatility model is the only solution to this problem
is still an open question. For example, based on about 15 years’ worth of SPX and VIX IV

49

See CBOE [2009] for detailed information on VIX.
One of the reasons why this problem is so diﬃcult is that there are numerous IVs to be ﬁtted (i.e., thecalibration problem has a huge number of constraints). Another is said to be that the short-term portion of the
IV of the SPX tends to have a steeper slope in the strike direction, which requires a large value for the volatility
parameter in the SPX volatility process, while the level of the IV of the VIX is lower than the level of thisparameter.
50

36data, Rømer [2022] argued that the simultaneous calibration problem can be solved with amodel including a conventional Itô process that has neither roughness nor jump terms.

4.2.4 Prospects and challenges of the rough volatility models
As mentioned above, in markets where the negative power law is observed, it is at least
theoretically necessary to use a rough volatility model. In addition, the rough Bergomi modeland the rough Heston model can be used for hedging with forward variance as a hedge
instrument, and these models have the potential to be used in practice as a powerful pricing
model for derivatives. However, as discussed below, further technological development isrequired for practical use.
One of diﬃculties in the practical application of rough volatility models lies in its numerical calculation. As mentioned, in the calibration process, one needs to calculate option
prices repeatedly as the expected payoﬀ value, in order to adjust the model parameters sothat the market quotes and the model prices match. In terms of the cost of calibration, therough Heston and the SABR models are somewhat favorable because these models admit
semi-analytical pricing formulae. However, in general rough volatility models, more computationally expensive calculations are required, compared with the conventional Markovian
models. Horvath et al. [2021a] proposed a deep learning-based technique to solve this problem, called deep calibration. In addition to the calibration, to calculate prices and Greeks for(more complex) derivatives, one needs to calculate the expectations with respect to a general
integrand, which is known as the weak approximation problem. To solve this problem, manymethods and frameworks have been proposed, including a hybrid scheme (Bennedsen et al.
[2017]), a tree method (Horvath et al. [2017]), and a Markovian approximation (Bayer andBreneis [2023], Cuchiero and Teichmann [2019], Hamaguchi [2023]). However, more theoretical studies as well as ones on practical implementations are required in order that roughvolatility models become viable in general derivative pricing operations.51
The aforementioned research works suggest that rough volatility models may be applicableto the U.S. derivatives markets, where the negative power law has been observed. However,if the negative power law or some other empirical patterns indicating roughness are observedin a wider range of markets, the necessity and applicability of the rough volatility model
shall increase. For example, in relation to the discussion on HV in section 3 and on MM
in section 5, if assets that accommodate high-frequency trading exhibit rough volatility, it
is reasonable to expect that the negative power law would be observed in the corresponding
options market. In addition, since there might be some other features of the shape of volatility
surfaces that suggest roughness in volatility, it is possible that roughness of volatility will beobserved in various other markets. It is also noteworthy that the options markets for interest
rate and foreign exchange tend to be more complex than those for equity derivatives due to51

Alòs et al. [2023] attempted to calculate CVA (price adjustment to reﬂect counterparty risk in the valuationof derivatives), which is mandatory in current derivatives practice, with a rough volatility model.

37their market practices.52 Accordingly, there are many problems speciﬁc to these markets, forexample, the joint calibration of the IV of swaptions and the IV of caps.

5 Market microstructure and origins of roughness
As discussed in sections 3 and 4, an increasing number of studies have suggested that
asset price volatility has roughness. Therefore, it is also important to elucidate the ﬁnancialeconomic mechanism that yields roughness in volatility. In this section, we brieﬂy overview
studies related to market microstructure (MM) as such an attempt. In section 5.1, we review
previous studies on MM, focusing on those deriving continuous-time asset price dynamics from discrete individual market participants’ trading order behavior. In section 5.2, wesummarize El Euch et al. [2018], an article that models the behavior of individual market participants and the price formation mechanism, and proves under which circumstances discrete
price movements converge to a continuous-time model with roughness.

5.1 Research on micro-foundations of asset price ﬂuctuations
In theoretical analysis in the ﬁeld of mathematical ﬁnance, as well as in other ﬁelds, asset
prices are often modeled as continuous-time stochastic processes using Brownian motion.
In actual ﬁnancial markets, however, asset prices ﬂuctuate discretely according to discrete
transactions, and thus tick data (price data for each transaction) can be regarded as a discrete
stochastic process. Investors also do not randomly place buy and sell orders.
MM is a research ﬁeld that analyzes the impact of the institutional structure of ﬁnancial
markets (e.g., trading rules on exchanges) on asset price formation as well as on investors’
behaviors at the micro level. One of the issues addressed in MM research is providing a
micro-foundation to the stochastic process of asset price dynamics. As mentioned above,asset price dynamics are ultimately determined by the accumulation of tick-by-tick pricemovements resulting from investors’ buying and selling transactions. This idea has motivateda strand of research that takes microstructural features such as investors’ tick-level trading
behavior (e.g., limit order placements and cancellations, market order placements) and theresulting dynamics of the limit order book as a starting point to derive a model of asset pricedynamics over a longer time horizon. This approach has the advantage of having a microfoundation, that is, trading behaviors of individual investors that are directly modeled. Thisapproach is helpful to better understand the asset price formation mechanism, namely, howinvestors’ behavioral features are related to patterns in asset price ﬂuctuations. These points
are also important from a policymaking perspective, such as the perspective taken when
designing rules and regulations for exchanges.
52

For example, in the case of foreign exchange options, the volatility is quoted with respect to delta andbutterﬂy instead of strike. In the case of interest rate options, the volatility of swaptions (options written ona swap as the underlying asset) is quoted for the maturity of the underlying swap, maturity of the option, andstrike. The volatilities of instruments such as interest rate caps and ﬂoors are also quoted.

38

Research on the limit order book has developed signiﬁcantly since the 1990s, as the availability of individual order-level data began to improve. For example, an early study by
Parlour [1998] analyzed dynamic changes in the limit order book conditions. Cont and
De Larrard [2013] derived continuous-time asset price volatility models starting from themodels of individual buy and sell order behaviors. They analyzed a model in which market
orders, best bid limit orders, best oﬀer limit orders, and order cancellations arrive following independent Poisson processes, and derived the probability distribution of time intervals
between price changes and the probability distribution of the scaling limit of the mid-price(the average of the best bid and best ask prices). In particular, they found that the scaling
limit of the mid-price converges to a Brownian motion, and its coeﬃcient (i.e., volatility)
can be expressed in terms of the arrival intensity of order placements. Lakner et al. [2016]
considered a model in which the state of the limit order book is represented as a probability
distribution. They assumed that this distribution ﬂuctuates stochastically with the arrival oforders, and obtained results regarding its scaling limit.
When a market order is placed, the trade is executed by matching it to the best limit order,
but if the volume of the market order exceeds that of the best limit order, then it is matchedto the second best limit order and if the market order still remains, it will be matched tothe third best limit order, and so on.53 Thus, in the case of large-volume trading, execution
occurs at less favorable prices than the best limit price, and also large changes in the trading
price and mid-price occur. This phenomenon is called market impact. In order to mitigate
such market impact, investors who wish to execute large orders may employ a trading technique called iceberg trading, in which orders are divided into smaller orders and executed
gradually. While iceberg trading reduces market impact, it also requires a longer time to execute the entire order and thus it carries higher price risk (stemming from factors other thanmarket impact). This trade-oﬀ between market impact and increased price risk determines
the optimal execution strategy (see, for example, Almgren and Chriss [2001]). For a survey
on optimal execution strategies, see Donnelly [2022], among others.54 Iceberg trading andother investor behaviors may signiﬁcantly aﬀect the features of high-frequency price dynamics, as well as the long-run asset price dynamics (i.e., the continuous-time stochastic process
obtained as a scaling limit).

5.2 Framework for analyzing roughness from MM perspective
El Euch et al. [2018] provided an example where a rough volatility model emerges as ascaling limit of one of the micro-founded models described above. In particular, El Euchet al. [2018] proved that the degree of the persistence of the impact of a price update on
future price update intensity is a crucial determinant of whether the volatility of the scaling53

For more information on limit order book models, including those incorporating non-best limit orderbooks, see, for example, Cont et al. [2023] and the references therein.
54
Moreover, a vast literature studies market impact from the perspective of market liquidity. See Amihud[2002] and Goyenko et al. [2009], among others.

39limit process exhibits the rough property. Speciﬁcally, they analyzed the scaling limit ofa model in which tick-level upward and downward price changes are described by a point
process called the Hawkes process and the persistence of the eﬀect of price update on future
price updates is expressed through how the intensity of the Hawkes process reacts to theoccurrence of price update. The result of their analysis was that whether the scaling limit
process poses roughness depends on the parameters related to the intensity of the Hawkes
process. Subsequent studies have discussed generalizations of the El Euch et al. [2018]
model, as well as the applications of the model to the analysis of the rough Heston model.
5.2.1 Hawkes process
To formally deﬁne the Hawkes process, we ﬁrst describe the Poisson distribution andPoisson process. The Poisson distribution is a discrete probability distribution that expresses
the number of occurrences of certain contingent events. It has a parameter µ that controls
the likelihood of the occurrence of the events (i.e., the average number of occurrences overa unit time interval). Formally, a discrete random variable X follows a Poisson distribution
with parameter µ if it satisﬁes the following equation:µk −µP (X = k) = e .
k!
Note that both the expected value and variance of the Poisson distribution equal µ (i.e.,
E[X] = Var[X] = µ).
The Poisson process Lt is the most basic point process that counts the occurrence of certain
contingent events whose occurrence follows a Poisson distribution. Speciﬁcally, the numberof occurrences of events between time t and t + s, Lt+s − Lt , follows a Poisson distribution
with parameter µ = λs. Namely, the following holds:
P (Lt+s − Lt = k) =(λs)k −λse ,k!

where λ is a parameter called the intensity of the Poisson process. As can be seen from thefact that the average number of events over time duration s is λs, the intensity λ represents
the instantaneous likelihood of the occurrence of an event, that is, P (Lt+dt − Lt = 1) = λdt
holds.55
As such, the intensity of the (stationary) Poisson process is constant. However, there exist
a variety of real-life contingent events, where the occurrence of an event stimulates following
events, such as aftershocks following a main earthquake, chain reaction bankruptcies after
the bankruptcy of a large company, or sell-oﬀ after a sharp market decline. In other words,
the occurrence of an event increases the intensity of following events in these cases, andthis property is called the self-exciting property. The Hawkes process is an extension of the
Poisson process that has the self-exciting property. The intensity λt of the Hawkes process
55

Mathematically, a Poisson process is deﬁned to be an independent increments process that satisﬁes
P (Lt+dt − Lt = 1) = λdt and P (Lt+dt − Lt ≥ 2) = 0.

40

Nt at a certain point in time depends on the number of the events that have occurred up to thatpoint (that is, Nt ). Speciﬁcally, the intensity of the Hawkes process is given as the following
equation:
Z
t

λt =

h(t − s) dN s ,0where the function h is called a kernel function of the Hawkes process. For example, if thekernel function is positive (i.e., h(·) > 0), then the intensity will increase with the occurrence of events and the process exhibits the self-exciting property. If the kernel function h
is a positive-valued decreasing function, the eﬀect of the occurrence of past events on theintensity will decay over time.

5.2.2 Framework of El Euch et al. [2018]
El Euch et al. [2018] modeled tick-by-tick price movements using a Hawkes process andproved that the convergence property of the intensity determines whether roughness arisesin the volatility of the price process obtained as a scaling limit. Below, we give a brief
explanation of this article.

Let Nt+ and Nt− be the numbers of price increases and decreases, respectively, and Nt+ , Nt−
be a two-dimensional Hawkes process. Namely, let λ+ and λ−t be the intensities of Nt+ and
Nt− , respectively, given by the following equation:
 +  + Z t  λt  ν ϕ1 (t − s) ϕ2 (t − s) dN s+  −  =  −  +  − ,
λt
ν
dN s
0 ϕ3 (t − s) ϕ4 (t − s)(10)where the kernel functions ϕ1 , ϕ2 , ϕ3 , ϕ4 represent how past price increases and decreases
aﬀect the intensity of price increases and decreases. For example, the ﬁrst row of (10) is theintensity of the price increase and is written as
Z t
Z t
+
+
+
λt = ν +ϕ1 (t − s) dN s +ϕ2 (t − s) dN s− .
00The second term on the right-hand side represents the impact of past price increases on theintensity of current price increases, and the third term on the right-hand side represents theimpact of past price decreases on the intensity of current price increases. Similarly, theintensity of price decrease is aﬀected by past price increases and decreases. Thus, the model
represents the so-called mutually exciting property, with which price increases and pricedecreases interact with each other to aﬀect the intensity of future price changes.
Under this framework, the article puts additional conditions on the self-exciting parameters and discusses whether roughness arises in the stochastic process followed by the scaling
limit Pt formed under high-frequency trading.56 As a result, if the eﬀect of price changes on

56

The scaling limit is the price Pt = limT →∞

+
−
NtT
−NtT
formed under a hypothetical trade in inﬁnitesimal time.
T

41

the intensity lingers over a long period of time, that is, in the case that the kernel function,ϕ1 (u) ϕ2 (u)
Φ(u) =  ,ϕ3 (u) ϕ4 (u)decays slowly as u → ∞, roughness arises.57
El Euch et al. [2018] interpreted this condition for roughness as the market trading behavior of iceberg trading described above, where a large order is split into consecutive smaller
trade orders over a longer period of time so that an order is followed by many other orders.
However, this is one of the interpretations of the convergence condition on the intensity of the
Hawkes process described above; it has not been excluded that other patterns of investors’
behaviors may result in the same mathematical condition. Therefore, the result by El Euchet al. [2018] does not immediately lead to the conclusion that iceberg trading is the source ofroughness in high-frequency trading.

5.2.3 Prospects for rough volatility research from MM perspective
Subsequent studies have extended the framework of El Euch et al. [2018]. For example,
Jusselin and Rosenbaum [2020] clariﬁed the correspondence between the degree of impactof a transaction on market prices and roughness (i.e., Hurst parameter H) in a more general
setting of the market. In addition, Rosenbaum and Tomas [2021] extended the discussionof two-dimensional Hawkes processes to multidimensional Hawkes processes and presented
a framework for analyzing micro-structures in markets where multiple assets are traded.
Further research from the MM perspective, including more realistic modeling of investors’
behaviors and empirical analysis using real data, would be important to elucidate the sourceof the roughness of volatility.

6 Concluding remarks
In this paper, we have surveyed empirical and theoretical studies on rough volatility anddiscussed its potential applications from various perspectives. In particular, as evidence forthe roughness of volatility, we have introduced statistical observations on time-series data forrealized volatility (section 3) and the shape of IV observed in derivatives markets (section 4).
R∞
For example, if there exists a constant 1/2 < α < 1 such that αxα x ϕ1 (s) ds → C (x → ∞), thenthe scaling limit converges to the rough Heston model. Here, α appearing in the convergence condition corresponds to the Hurst index of the rough Heston model obtained as the scale limit through H = α − 1/2. The
technical gist of this result is as follows.
Under high-frequency trading, the kernel function satisﬁes certainR∞
integrability conditions (such as 0 ϕ1 (s) ds converging in an appropriate sense, a condition called being anearly unstable Hawkes! process). Under this condition, they discussed the convergence of the eigenvalue of
ϕ1 (u) ϕ2 (u)Φ(u) =. The degree of decay of these eigenvalues determines whether the convergent scale
ϕ3 (u) ϕ4 (u)limit process has roughness.
57

42

Both of these observations can be expressed consistently using fractional Brownian motion,a model that has a rough feature.
Furthermore, in section 4, we have explained an important theoretical result that indicatesthe necessity of rough volatility models. Namely, for derivatives markets where roughness
is observed, a pricing model with appropriate roughness is necessary because otherwise arbitrage opportunities arise. Section 5 introduced a study from the MM research strand that
attempted to shed light on the mechanism by which roughness arises. In addition, as applications, roughness could improve volatility forecasting accuracy (section 3) and derivative
pricing models (section 4). As such, the necessity and usefulness of roughness have been
discussed extensively in various ﬁelds, including ﬁnancial econometrics and mathematicalﬁnance, which originally developed around diﬀerent issues. Rough volatility has become anew trend in volatility research.
It should be noted, however, that although a growing number of studies have suggested
the existence of roughness in volatility, there is an ongoing discussion as to whether roughvolatility modeling is really necessary. Most of the studies to date have suggested that theassumption of rough volatility is just a suﬃcient condition to explain empirical observations
consistently. In addition, roughness has only been empirically conﬁrmed in a few markets,
such as the U.S. equity market. Therefore, it is important to deepen our understanding ofwhether the roughness of volatility is a universal property that appears regardless of asset
class, as well as other issues. It is also important to uncover ﬁnancial economic mechanisms
behind the stylized features revealed by the existing empirical studies, especially what kindof trading behaviors of market participants give rise to the roughness. Therefore, further research from the perspective of MM is desirable, as described in section 5. Even if the roughness of volatility is conﬁrmed, there are various challenges for application. For example, it isdebatable whether the models based on fractional Brownian motion and Riemann–Liouvilletype Itô integral formulations presented in this paper are desirable in terms of representability
and implementability. Various other issues remain to be addressed, including parameter estimation for time-series models, stabilization of numerical methods for derivative pricing
models, and analysis of hedging strategies.
Rough volatility research is expected to develop further, expanding its related ﬁelds in bothpractical and academic directions. For example, application to high-frequency (algorithmic)
trading (Guasoni et al. [2021]), integration with machine learning and ﬁnancial time-series
data generation (Rosenbaum and Zhang [2022], Horvath et al. [2021b]), and application tobroad risk management have already begun. It is expected that research will continue toprogress, inﬂuenced by changes in the issues of interest in ﬁnancial practice, while also intersecting with the ﬁelds of artiﬁcial intelligence, economics, information science, and mathematical science. Therefore, it is important to keep abreast of the leading-edge discussions.

43

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