The rough Hawkes Heston stochastic volatility model Alessandro BondiSergio PulidoSimone Scotti October 25 2022

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The rough Hawkes Heston stochastic volatility model
Alessandro BondiSergio PulidoSimone Scotti §
October 25, 2022
Abstract
We study an extension of the Heston stochastic volatility model that incorporates rough volatility
and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility
model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the
jump component appearing in the dynamics of the spot variance itself and the log returns. The model
belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the
log returns and the square of the volatility index can be computed explicitly in terms of solutions of
deterministic Riccati-Volterra equations, which can be efficiently approximated using a multi-factor
approximation technique. We calibrate a parsimonious specification of our model characterized by a
power kernel and an exponential law for the jumps. We show that our parsimonious setup is able to
simultaneously capture, with a high precision, the behavior of the implied volatility smile for both
S&P 500 and VIX options. In particular, we observe that in our setting the usual shift in the implied
volatility of VIX options is explained by a very low value of the power in the kernel. Our findings
demonstrate the relevance, under an affine framework, of rough volatility and self-exciting jumps in
order to capture the joint evolution of the S&P 500 and VIX.
JEL code: C63, G12, G13
Keywords: Stochastic volatility, Rough volatility, Hawkes processes, Jump clusters,
Leverage effect, affine Volterra processes, VIX, joint calibration of S&P 500 and VIX
smiles.
1 Introduction
The Black-Scholes model, where volatility is constant, and more generally classical local volatility
models, where volatility is a function of time and spot asset prices, fail to reproduce the dynamics of
implied volatility smiles of options written on the underlying asset. To overcome this limitation, multiple
stochastic, stochastic-local, and path-dependent volatility models have been developed and studied in
recent years. The complexity of volatility modeling, however, has increased with the significant growth
over time of markets on volatility indices, such as the VIX. The rise in popularity of these markets is
explained in part by their relevance to protect portfolios [69]. It has therefore become fundamental to
develop stochastic models able to capture the joint dynamics of the underlying prices and their volatility
index. The task is difficult because classical stochastic models fail to calibrate simultaneously the volatility
smiles of options on the underlying and its volatility index. This modeling challenge, known as the joint
S&P 500/VIX calibration puzzle [49, 50], has inspired the introduction of more sophisticated models, e.g.
[44, 50, 51], that incorporate new features to the joint dynamics of the underlying and the volatility in
order to solve the problem. In this paper we tackle the challenge by proposing a tractable affine model
with rough volatility and volatility jumps that cluster and that have the opposite direction but occur at
the same time as the jumps of the underlying prices. In this introduction we give a brief literature review
to explain the choice of our framework.
The research of Sergio Pulido benefited from the financial support of the chairs “Deep finance & Statistics” and “Machine
Learning & systematic methods in finance” of École Polytechnique. Sergio Pulido and Simone Scotti acknowledge support
by the Europlace Institute of Finance (EIF) and the Labex Louis Bachelier, research project: “The impact of information
on financial markets”.
Classe di Scienze, Scuola Normale Superiore di Pisa, alessandro.bondi@sns.it
Université Paris-Saclay, CNRS, ENSIIE, Univ Evry, Laboratoire de Mathématiques et Modélisation d’Evry (LaMME),
sergio.pulidonino@ensiie.fr
§Università di Pisa, simone.scotti@unipi.it
1
arXiv:2210.12393v1 [q-fin.MF] 22 Oct 2022
The dynamics of the VIX volatility index are highly complex. In particular, they exhibit large and
systematically positive variations over very short periods, with a tendency to form clusters of spikes
during difficult periods like the 2008 financial crisis and the beginning of the COVID-19 pandemic in
2020. This is accompanied by very long periods without any large fluctuation and a less important mean
reversion speed. These observations are in line with an increasing number of studies that indicate the
presence of jumps in the volatility [35, 75], on the underlying [13], and the fact these jumps are common
to the volatility and underlying [73].
The growing interest in volatility indices has driven the standardization of contingent claims written
on the volatility indices themselves. These volatility index markets have very unique features. For VIX
futures and Exchange-Traded products these features are studied in [10]. The complexity of volatility
markets is also exemplified by the difficulty to jointly model the behavior of the volatility smiles of vanilla
options written on the underlying and its volatility index, see for instance [6, 64, 68]. This longstanding
puzzle is known as the S&P 500 (SPX)/VIX calibration puzzle. A growing body of literature explains the
difficulty arguing that “the state-of-the-art stochastic volatility models in the literature cannot capture
the S&P 500 and VIX option prices simultaneously”, see [74]. As pointed out in [49, 50], “all the attempts
at solving the joint S&P 500/VIX smile calibration problem only produced imperfect, approximate fits.”
The problem is that usual stochastic models either fail to reproduce one or both shapes of the implied
volatility for S&P 500 and VIX options or, when both the shapes are coherent, the implied volatility
levels are incorrect.
Access to high frequency data has improved our understanding of the microstructure of financial
markets and the effects on volatility. In particular, recent studies indicate that non-Markovian models
with rough volatility trajectories might be appropriate to better capture long time dependencies due to
meta orders and the large contribution of automatic orders. This is examined in [26] which provides
a general analysis of order-driven markets, the work in [24] which elucidates the memory-features of
volatility, and the studies in [38, 43] which give a micro-structural justification to the newly developed
rough volatility models.
From a modeling point of view, affine models provide a convenient framework because they are
flexible and, thanks to semi-explicit formulas for the Fourier-Laplace transform, fast computations can be
performed using Fourier-based techniques [36, 37, 41]. The most popular affine stochastic volatility model
is the Heston model [52], where the spot variance is a square-root mean-reverting CIR (Cox-Ingersoll-Ross
[28]) process. This model is able to reproduce some stylized features like the mean-reverting property of
the volatility and the leverage effect. It is, however, unable to reproduce other phenomena such as extreme
paths of volatility during crisis periods (even for large values of the volatility of volatility parameter) and
the at the money (ATM) skews of underlying options’ implied volatility simultaneously for short and long
maturities. These limitations, and the micro-structural behavior of markets described in the previous
paragraph, motivated the introduction of the rough Heston model [39, 40]. The rough Heston model
is tractable as it belongs to the class of affine Volterra models [5], and semi-explicit formulas for the
Fourier-Laplace transform are still available. Unfortunately, this model cannot reproduce the features of
options written on the volatility index and the underlying simultaneously.
In order to model the joint behavior of S&P 500 and VIX markets, consistent with empirical evidence,
we add two specific features to the usual Heston model. First, we incorporate rough volatility by adding
a power kernel proportional to tα1, with α(1/2,1], to the dynamics of the spot variance. Second, we
postulate common jumps for the volatility and the underlying with a negative leverage. The presence of
jumps in both underlying and variance helps to reproduce a skewed implied volatility for vanilla options
as in the Barndorff-Nielsen and Shephard model [11, 12]. Inspired by the Hawkes framework, taking
into account jump-clustering and endogeneity of financial markets, we model the spot variance to be
proportional to the intensity process of the jump component appearing in the dynamics of the spot
variance itself and the log returns. For these reasons, we name our model the rough Hawkes Heston
model.
To keep mathematical and numerical tractability, we choose an affine specification of the model.
As such, our model belongs to the class of affine Volterra processes [5], which has been recently ex-
tended to jump processes in [18, 29, 30]. In particular, the Fourier-Laplace transform of the log returns
and the square of the volatility index can be computed explicitly in terms of solutions of deterministic
Riccati-Volterra equations, see Theorems 3 and 10. We approximate the solutions of the Riccati-Volterra
equations via a multi-factor scheme as in [4], see Theorem 11. We leave for future study the implemen-
2
tation and analysis in our framework of other methods such as the Adams method [33, 34], asymptotic
expansions based on Malliavin calculus in the spirit of [7], and hybrid approximation techniques for
Volterra equations similar to those in [20].
The affine property is an advantage of our modeling approach compared to other models proposed to
solve the SPX/VIX calibration problem, such as the quadratic rough Heston model [44], where pricing is
done via Monte Carlo or machine learning techniques [70]. In addition, our affine framework is convenient
because Variance Swap prices and the square VIX index have explicit affine relations to the forward curve,
see Corollary 7 and Remark 3. This is a generalization, to the affine Volterra setting, of the affine relation
already pointed out in [54] within the classical affine exponential framework and empirically confirmed
in [60].
Previous literature on jump-diffusion models focusing on the evolution of S&P 500 and the VIX
proposes either high-dimensional models [27, 63, 73], or models based on hidden Markov chains [46, 65].
These models require a large number of parameters and suffer from the lack of interpretability of the
random factors. Our approach to model the joint SPX/VIX dynamics is different. As in [17], we keep the
number of parameters low by assuming that the jump intensity is proportional to the variance process
itself, and jumps are common to the volatility and underlying with opposite signs. The main new
ingredient of our model, compared to [17], is the addition of a Brownian component and a power kernel
to the variance process. This generates by construction a jump clustering effect and takes into account
related findings in the rough volatility literature [8, 9, 14, 16, 38, 40, 42, 43, 44, 59].
The rough Hawkes Heston model is able to reconcile the shapes and level of the S&P 500 and VIX
volatility smiles. An important role is played by the parameter αcharacterizing the kernel. As is the
case for other rough volatility models, this parameter controls the explosion rate of the term structure
of ATM skews for SPX option smiles as maturity goes to zero. We show that when αis near to 1/2, the
rate of explosion is in the range [0.5,0.6]. This is consistent with similar findings in the rough volatility
literature [9, 14, 16, 38, 42, 43, 44]. In addition, in our framework, the parameter αplays a crucial role
because it controls the level of the implied volatility of VIX options for short maturities. We observe,
that as αapproaches 1/2the levels of S&P 500 and VIX smiles are coherent.
To summarize, the model that we propose in this paper shares many features with other existing
models. These features are mainly: rough volatility [14, 38, 40, 42, 43, 44], jumps [11, 12, 13, 27, 63, 73],
the Hawkes/branching character of volatility [17, 19, 53], and the affine structure [5, 18, 36, 37, 41, 54, 57].
Consequently we take advantage of the low regularity and memory features of rough volatility models, the
large fluctuation of jumps, the clusters of Hawkes processes and the explicit Fourier-Laplace transform
of the affine setup. The specification that we adopt for the joint SPX/VIX calibration is parsimonious
with only five evolution-related parameters. Moreover, all the parameters have a financial interpretation.
The parameter αin the kernel controls the decay of the volatility memory, SPX ATM skews and the
level of VIX smiles. We have in addition the classical parameters controlling the volatility mean reversion
speed and the volatility of volatility, and two parameters related to the leverage effect that specify the
correlation between Brownian motions and between the jumps in the asset and its volatility. Despite its
robustness, the rough Hawkes Heston stochastic volatility model captures remarkably well the implied
volatility surfaces of S&P 500 and VIX at the same time.
The paper is organized as follows. Section 2 lays out the essential hypotheses of our study and
introduces the stochastic model under a general setup, i.e. with a general kernel and law for the jumps.
Section 3 explains the derivation of the Fourier-Laplace transform of the log returns and the application
to undelying’s options pricing. Section 4 focuses on the VIX index characterizing the Fourier-Laplace
transform of the VIX2, and describes the Fourier-based formulas to price options on the VIX. Section
5 studies the multi-factor numerical scheme used in order to approximate the solutions to the Riccati-
Volterra equations arising in Sections 3 and 4. Section 6 details the calibration of our model to S&P
500 and VIX options data. Section 7 presents a complete and detailed sensitivity analysis of implied
volatility curves with respect to the model parameters. Section 8 summarizes the conclusions of our
study. Appendix A contains the proof of the necessary existence, uniqueness and comparison results for
the Riccati-Volterra equations appearing in Section 3. Appendix B presents the proof of the Fourier-
inversion formula used to price options on the underlying. To finish, in Appendix C we prove the
main result related to the convergence of the multi-factor approximation scheme for the Riccati-Volterra
equations.
3
2 The model
We study a stochastic volatility model where the spot variance σ2= (σ2
t)t0is a predictable process,
with trajectories in L2
loc (R+), defined on a stochastic basis (Ω,F,Q,F= (Ft)t0). We assume that the
filtration Fsatisfies the usual conditions and that F0is the trivial σalgebra.
We consider, throughout our study, a kernel Kthat satisfies the next requirement, see [1, 2, 5, 18].
Hypothesis 1. The kernel KL2
loc (R+)is nonnegative, nonincreasing, not identically zero and contin-
uously differentiable on (0,). Furthermore, its resolvent of the first kind Lexists and it is nonnegative
and nonincreasing, i.e. s7→ L[s, s +t]is nonincreasing for every t0.
We recall that, given a kernel KL1
loc (R+;Rd×d), an Rd×dvalued measure Lis called its (measure)
resolvent of the first kind if LK=KL=I, where IRd×dis the identity matrix. The resolvent of
the first kind does not always exist, but if it does then it is unique, see [48, Theorem 5.2, Chapter 5].
We assume that the spot variance σ2is a Qdta.e. nonnegative predictable process which satisfies
the following stochastic affine Volterra equation of convolution type with jumps:
σ2=g0+KdZ, Qdta.e. (1)
Here Z= (Zt)t0is a semimartingale starting at 0with associated jumps-measure µ(dt, dz)and com-
pensated measure ˜µ(dt, dz) = µ(dt, dz)ν(dz)σ2
tdt, with νa nonnegative measure on R+such that
ν({0})=0and RR+|z|2ν(dz)<. Since the intensity of the jumps of σ2is proportional to σ2itself, the
spot variance is a Hawkes-type process, which is coherent with other models that incorporate endogeneity
of financial markets such as [17, 21, 40, 45, 56]. More specifically, Zis given by
dZt=b σ2
tdt+c σtdW2,t +ZR+
z˜µ(dt, dz), Z0= 0,
where bR, c > 0and W2= (W2,t)t0is an FBrownian motion. In the sequel, we denote by
e
Z= ( e
Zt)t0the process e
Zt=c σtdW2,t +RR+z˜µ(dt, dz), t 0. Notice that e
Zis a square-integrable
martingale by [18, Lemma 1]. The function g0is the initial input spot variance curve. By analogy with
the rough Heston model introduced and studied in [39, 40], we consider it of the form
g0(t) = σ2
0+βZt
0
K(s) ds, t 0,(2)
where σ2
0, β 0. According to [18, Appendix A]
σ2=g0RbK g0+Eb,K de
Z, Qdta.e.,(3)
where RbK is the resolvent of the second kind of bK and Eb,K is the canonical resolvent of Kwith
parameter b. We recall that the resolvent of the second kind RKfor a kernel KL1
loc (R+)is the unique
solution RKL1
loc (R+)of the two equations KRK=RKK=KRK. The canonical resolvent
Eλ,K of Kwith parameter λis defined by Eλ,K =λ1RλK for λ6= 0, whereas E0,K =K, see [48,
Theorem 3.1, Chapter 2] and the subsequent definition.
Remark 1. If we assume that Kand the shifted kernels K(·+ 1/n),nN, satisfy Hypothesis 1, the
(weak) existence of the spot variance process σ2, satisfying (1), is ensured by [1, Theorem 2.13] and
[18, Lemma 9]. Assuming weak existence, weak uniqueness is established in [18, Corollary 12] under
Hypothesis 1. We refer to [2] and [18] for more information about stochastic Volterra equations and
stochastic convolution for processes with jumps.
A useful tool for the development of the theory is the adjusted forward process, which we now define.
For every t0, it is denoted by (gt(s))s>t and it is a jointly measurable process on ×(t, )such that
gt(s) = g0(s) + Zt
0
K(sr) dZr,Qa.s., s > t. (4)
Thanks to [62, Theorem 46] and the fact that Fsatisfies the usual conditions, we can consider gt(·)to be
Ft⊗ B(t, )measurable.
4
Analogous arguments provide a version of the conditional expectation process E[σ2|Ft]=(E[σ2
s|Ft])s>t
which is Ft⊗ B(t, )measurable. In particular, from (3),
Ehσ2
sFti=g0RbK g0+Zt
0
Eb,K (sr) d e
Zr,Qa.s., s > t. (5)
We now prescribe the dynamics of the log returns process X= (Xt)t0as follows:
dXt= 1
2+ZR+eΛz1+Λzν(dz)!σ2
tdt+σtp1ρ2dW1,t +ρdW2,t
ΛZR+
z˜µ(dt, dz), X0= 0,(6)
where ρ[1,1] is a correlation parameter, W1= (W1,t)t0is an FBrownian motion independent
from W2and Λ0is a leverage parameter forcing common jumps for volatility and underlying with
opposite signs. This is coherent with empirical findings in [75], stylized features studied in [25], and
the financial/econometric literature with jumps, e.g. [11, 12, 13, 17, 31, 67, 73]. We have assumed, for
the sake of readability and without lost of generality, that interest rates are zero. The price process of
the underlying asset will be S= (St)t0= (S0eXt)t0, where S0>0represents the initial price. An
application of Itô’s formula shows that Sis a local martingale. Indeed,
dSt
St
= 1
2+ZR+eΛz1+Λzν(dz)!σ2
tdt+σtp1ρ2dW1,t +ρdW2,t
ΛZR+
z˜µ(dt, dz) + 1
2σ2
tdt+ZR+eΛz1+Λzµ(dt, dz)
=σtp1ρ2dW1,t +ρdW2,t+ZR+eΛz1˜µ(dt, dz) =:dNt,
where N= (Nt)t0is a local martingale with N0= 0. In particular, since Sstarts at S0, it follows that
S=S0E(N), where Edenotes the Doléans-Dade exponential. In the next section, see Corollary 4, we will
improve on this result by showing that, for every T > 0, the restriction of Sto [0, T ]is a true martingale.
3 The Fourier-Laplace transform of the log returns
In this section we study, for a fixed T0, the conditional Fourier-Laplace transform of XT,
E[ewXT|Ft],t[0, T ]. Here wCis subject to suitable conditions that will be specified in the se-
quel. In particular, we want to find a formula that allow us to compute the prices of options written
on the underlying asset using Fourier-inversion techniques [36, 37, 41, 47]. We will adopt the following
notation: for zCwe denote by Rzand Imzthe real and imaginary parts of z, respectively. We let C+
[resp., C] be the set of complex real numbers with nonnegative [resp., nonpositive] real part.
Let us define the mapping F:C+×CCby
F(u, v) = 1
2u2u+b+ρc uv+c
2v2+ZR+he(vΛu)zueΛz11vziν(dz),(7)
for every (u, v)C+×C. For the development of the theory we need the following result about
deterministic Riccati-Volterra equations, whose proof is postponed to Appendix A.
Theorem 1. Suppose that Ksatisfies Hypothesis 1 and wCis such that Rw[0,1].
(i) There exists a unique continuous solution ψw:R+Cof the Riccati-Volterra equation
ψw(t) = Zt
0
K(ts)F(w, ψw(s)) ds= (KF(w, ψw(·))) (t), t 0.(8)
In particular, ψRwis Rvalued.
5
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TheroughHawkesHestonstochasticvolatilitymodel*AlessandroBondi„SergioPulido…SimoneScottiŸOctober25,2022AbstractWestudyanextensionoftheHestonstochasticvolatilitymodelthatincorporatesroughvolatilityandjumpclusteringphenomena.Inourmodel,namedtheroughHawkesHestonstochasticvolatilitymodel,thespotvariancei...

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