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.
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