Power‐type derivatives for rough volatility with jumps

• Published date: 01 July 2022
• Author: Liang Wang,Weixuan Xia
• Date: 01 July 2022
• DOI: http://doi.org/10.1002/fut.22337

Received: 2 December 2021

|

Accepted: 19 April 2022

DOI: 10.1002/fut.22337

RESEARCH ARTICLE

Power‐type derivatives for rough volatility with jumps

Liang Wang

1

|Weixuan Xia

2

1

Department of Mathematics and

Statistics, Boston University Graduate

School of Arts and Sciences, Boston,

Massachusetts, USA

2

Department of Finance, Boston

University Questrom School of Business,

Boston, MA, USA

Correspondence

Weixuan Xia, Department of Finance,

Boston University Questrom School of

Business, Boston, MA, USA.

Email: gabxia@bu.edu

Abstract

This paper proposes a novel analytical pricing–hedging framework for

volatility derivatives which simultaneously takes into account rough volatility

and volatility jumps. Directly targeting the instantaneous variance of a risky

asset, our model consists of a generalized fractional Ornstein–Uhlenbeck

process driven by a Lévy subordinator and an independent sinusoidal‐

composite Lévy process, and allows the characteristic function of average

forward variance to be obtainable in semiclosed form, without having to

invoke any geometric‐mean approximations. Pricing–hedging formulae are

proposed for a general class of power‐type derivatives, in the spirit of

numerical Fourier transform. A comparative empirical study is conducted on

two independent recent data sets on Volatility Index options, before and

during the COVID‐19 pandemic, to demonstrate that the proposed framework

is highly amenable to efficient model calibration under various choices of

kernels. The price dynamics of the underlying asset can be readily considered

and the possibility of studying rough volatility of volatility is given as well.

KEYWORDS

Lévy subordinators, rough volatility, sinusoidal processes, VIX options, volatility jumps

MSC 2020 CLASSIFICATIONS

60E10, 60G22, 60J76

JEL CLASSIFICATION

C65, G13

1|INTRODUCTION

In the present paper we are interested in the efficient pricing and hedging of volatility derivatives—in particular

European‐style swaps and options written on average‐forward volatility (refer to Carr & Lee, 2009 for a résumé), such as

the Volatility Index (VIX) index—in the presence of short‐termdependence (or rough volatility) as well as suitable jumps

in the instantaneousvolatility. The main idea is largely motivated by BNS‐type modelsproposed in Barndorff‐Nielsen and

Shephard (2001) and Barndorff‐Nielsen et al. (2002), aiming to coalesce memory effects intopure‐jump volatility. Analytic

tractability will be the focus in establishing our model framework, which also permits considerationsof comovements,

especially largecojumps, between the associated asset price and the instantaneous volatility.

On the implementation side, our study objects are a more general class ofderivatives raising the underlying volatility

or the standard option payoff to a nonnegative power. Derivatives with power payoff functions written on equity have

been thoroughly examined in the literature; see Tompkins (1999), Raible (2000), Macovschi and Quittard‐Pinon (2006),

J Futures Markets. 2022;42:1369–1406. wileyonlinelibrary.com/journal/fut © 2022 Wiley Periodicals LLC.

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and Xia (2017) on single‐asset options and Blenman and Clark (2005), Wang(2016), and Xia (2019) on exchange options.

Similar exchange options on zero‐coupon bonds have recently been studied in Blenman et al. (2020). Thus, along these

lines consideration of power‐type derivatives in the volatility market will also have some appealing consequences in

generating nonlinear leverage effects on the investor's risk exposure, and will be conducive to hedging volatility‐of‐

volatility risks as well. Noteworthily, there are alsovolatility derivatives with payoffs involving the underlying risky asset

prices, including—but not limited to—future‐style log contracts on volatility (Neuberger, 1994), covariance swaps

(Habtemicael & SenGupta, 2016) embodying correlation risks, and target volatility options(Cao, Badescu, et al., 2020)

based on a combination of the asset priceand forward volatility.

1.1 |Balancing rough volatility and volatility jumps

“Rough volatility”is an already‐familiar jargon that has flourished since the pioneering research work of Gatheral et al. (2018),

which suggested rough sample paths of volatility observed in high‐frequency financial time series.

1

Overthepast4 years,a

good number of works have been devoted to empirical justifications of rough volatility in various asset types. To name a few,

Livieri et al. (2018) confirmed the existence of rough volatility by studying spot volatility of the S&P500 index, Takaishi (2020)

collected further evidence supporting volatility roughness in the cryptocurrency (in particular Bitcoin) market, and Da Fonseca

and Zhang (2019) even showed that rough volatility is also present in the VIX index.

Although rough volatility has successfully reproduced stylized facts of historical volatility derived from asset prices,

some difficulties have arisen due to the loss of Markov properties. When developing pricing–hedging techniques

accounting for rough volatility one will most likely sojourn at Monte‐Carlo simulation methods, whereas the

inaccessibility of infinitesimal generators has disabled methods based on the Feynman–Kac formula. So far, simulation‐

based pricing–hedging methods have already been studied in depth; for example, Jacquier et al. (2018)adoptedahybrid

simulation schemefor the calibration of the rough Bergomi model initially proposed in Bayer et al. (2016) on VIX futures

and options. On the other hand, under the so‐called “rough Heston model”with a stationary power‐type kernel that

belongs to the family of affine Volterra processes discussed in Jaber et al. (2019) (see also Gatheral & Keller‐Ressel, 2019),

characteristic function‐based pricing methods were developed in El Euch and Rosenbaum (2019), which depend,

partially, on solving a fractional Riccati equation and whose applicability was also demonstrated by calibrating the

S&P500 implied volatility surfaces; the paper El Euch and Rosenbaum (2018) considered from a theoretical standpoint

similar hedging problems, after being able to write the characteristic function of the log‐assetprice in terms of a function

of its corresponding forward variance curve. We alsonotice the up‐to‐date work of Horvath et al.(2020), which adopted a

martingale framework using forward variance curveswith the goal of studying volatility options, Xi and Wong (2021),

which investigated the valuation of discrete variance swaps, and Alfeus etal. (2019), which advanced the rough Heston

model to include regime switching. It is worth mentioning that these recent works have universally emphasized the role

of a Brownian motion, having paid little attention to jumps in asset prices and their volatility.

All relevant models notwithstanding, one should however bear in mind that the key feature of rough volatility is a short‐

term dependence structure, rather than inherent reliance on Brownian sample paths (or path continuity), which

characteristic is mostly an estimation assumption imposed in Gatheral et al. (2018) and extensive use of the Brownian motion

in the cited literature is more or less an act of simplicity. On the other hand, disregarding the exclusive use of the Brownian

motion highlights another important aspect—volatility jumps. In a semimartingale setting, this would send us back to the

work of Todorov and Tauchen (2011), which, by analyzing from high‐frequency VIX index data the activity level of presumed

mean‐reverting instantaneous variance models, showed that stock market volatility should be most suitably depicted as a

purely discontinuous process without a Brownian component. On the surface, this concern may seem inconsequential in a

non‐semimartingale model with frictions: In short, the activity level of the process can be flexibly adjusted according to the

controlling fraction parameter. For this reason, inclusion of volatility jumps in a model that is already fractional has

seemingly been ignored for investigation. However, since increased activity levels are an ineluctable consequence of increased

path roughness, using a fractional Brownian motion with a fraction parameter less than 1 will only increase the activity level

of the resultant variance process, which to a degree neglects the empirical findings of Todorov and Tauchen (2011); see also

Bollerslev and Todorov (2011), Bardgett et al. (2019), and Cao, Ruan et al. (2020) (concerning VIX options as well) for similar

1

Meanwhile, the idea of introducing frictions into volatility quantities goes back to the much earlier work of Alòs et al. (2007), motivated by

observations in option price‐implied volatility surfaces.

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WANG AND XIA

confirmations of the necessity of volatility jumps. In connection with this, we expect that replacing the Brownian motion with

a purely discontinuous process whose sample paths are less active, alongside suitable modifications, is able to strike a balance

between these two important aspects and eventually yield desirable modeling outcomes.

We are hence inspired to take on a new path deviating from the use of the fractional Brownian motion in

establishing rough volatility and switch to purely discontinuous square‐integrable Lévy processes of infinite activity.

We aim for the instantaneous variance based on a generalized fractional Ornstein–Uhlenbeck process subject to an

integrable kernel. A key feature of this formulation is that it is not derived from logarithms but is yet capable of

capturing short‐term dependence and possible jumps, as well as achieving an arbitrary suitable activity level of the

corresponding forward variance curve. Besides Barndorff‐Nielsen and Shephard (2001) as aforementioned, we note that

models of this nonexponential type have also been widely applied in volatility analysis, some recent developments

including Hofmann and Schulz (2016), Issaka and SenGupta (2017), and Cao, Ruan et al. (2020). More importantly

introducing jumps into the instantaneous variance gives rise to an analytically tractable form for the characteristic

function of the average‐forward volatility, thus facilitating the pricing and hedging of its derivatives. This structure

requires no inexact transformations, such as the geometric‐mean approximation adopted in, for example, Horvath et al.

(2020), for the average‐forward volatility, which cannot be avoided under exponential models.

Despite the intended generality of our model framework, attention will be drawn to three particular types of

stationary kernels. While the first type is recognized for its incommensurable simplicity, the second is compatible with

the transformation of the instantaneous variance dynamics into a usual Ornstein–Uhlenbeck process but driven by a

fractional Lévy process. The third type is a result of reverse engineering and is designed specifically to avoid certain

transcendental functions that are costly to implement. We will also compare the overall suitability of various types of

kernels, in spite of their considerable similarity to each other in shape.

1.2 |Structure of paper

The remainder of this paper is organized as follows. In Section 2we establish our model framework starting from the

instantaneous variance dynamics and provide a comprehensive analysis of its properties, including covariance function and

path regularity, and then build up to an integral representation for the characteristic function of the average‐forward variance.

Some simulation techniques are discussed in Section 3.Section4contains our new pricing–hedging formulae for power‐type

derivatives, and subsequently initiates a comparative empirical study in Section 5focusedaroundVIXoptionsutilizingtwo

independent data sets and two of the proposed kernels. Lastly, we discuss how the model framework can be connected to the

associated risky asset price dynamics and even further extended to accommodate rough volatility of volatility in Section 6.

Conclusions and future research directions are outlined in Section 7and all mathematical proofs are presented at the end.

2|CONSTRUCTION OF ROUGH VOLATILITY WITH JUMPS

2.1 |Fractional lévy processes

We begin with some crucial ingredients of a non‐Gaussian fractional Lévy process. Allowing for positivity of the

instantaneous variance the background‐driving Lévy process must be nonnegative (i.e., a subordinator).

Consider a continuous‐time stochastic basis(Ω,,; {}

)

tt 0

¤≔≡

≥

S

, where the filtration



is assumed to satisfy

the usual conditions. Let XX(

)

t

≡be an adapted and square‐integrable Lévy subordinator supported on

S

,

exclusively characterized by a Poisson random measure

NX

defined on

(

,

)

++ +

¦¦

. According to the Lévy–Khintchine

representation, X

1

has the characteristic exponent



[]

ϕle eνzl

l

og( )log=(−1)(d ),

,

XlXlz X

i

0+

i

1

1

¦≔∈

∞

where

i

denotes the imaginary unit and ν

X

is the intensity measure associated with

NX

. For practicality we impose the

assumption that

ν

is nonatomic so that the distribution of X

1

is absolutely continuous with respect to the Lebesgue

measure (see, e.g., Kohatsu‐Higa & Takeuchi, 2019) and we denote by

ξ

X[]>0

11

≔and

ξ

XVar[] > 0

21

≔. Since

X

has independent andstationary increments,it hasthe familiarcovariance function,for any u>0

,XX ξt

C

ov[,] =

ttu+2

.

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