# Pricing VIX options under the Heston-Hawkes stochastic volatility model

Oriol Zamora Font Department of Mathematics, University of Oslo, P.O. Box 1053
Blindern, N-0316 Oslo, Norway, Email<EMAIL_ADDRESS>
###### Abstract

We derive a semi-analytical pricing formula for European VIX call options
under the Heston-Hawkes stochastic volatility model introduced in [7]. This
arbitrage-free model incorporates the volatility clustering feature by adding
an independent compound Hawkes process to the Heston volatility. Using the
Markov property of the exponential Hawkes an explicit expression of
$\text{VIX}^{2}$ is derived as a linear combination of the variance and the
Hawkes intensity. We apply qualitative ODE theory to study the existence of
some generalized Riccati ODEs. Thereafter, we compute the joint characteristic
function of the variance and the Hawkes intensity exploiting the exponential
affine structure of the model. Finally, the pricing formula is obtained by
applying standard Fourier techniques.

Keywords: VIX options, Volatility with self-exciting jumps, Hawkes process,
Option pricing, Semi-analytical pricing formula.

AMS classification MSC2020: 60G55, 60H10, 91G15, 91G20.

###### Acknowledgements

The author would like to thank David Ruiz Banos and Salvador Ortiz-Latorre for
their help and insightful feedback during the development of this article.

##  1 Introduction

Volatility trading has become remarkably important in finance in recent years
and plays a crucial role in risk management, portfolio diversification, asset
pricing and econometrics. The Chicago Board Options Exchange (CBOE)
established the first volatility index, the VXO, in January 1993. According to
[35], the VXO index represents the implied volatility of a hypothetical at-
the-money option on the S&P 100 (OEX) expiring in 30 days. The CBOE introduced
in September 2003 a new volatility index, the VIX, in a model-free procedure.
The VIX is computed as a weighted sum of out-of-the money S&P 500 (SPX) calls
and puts expiring in 30 days across all available strikes. A convenient
property of the VIX index is that its square approximates the 30-day
conditional risk neutral expectation of the return variance, that is, the
30-day variance swap, see [12]. Many further volatility indices have been
introduced like the VXN, the VXD, the VCAC and others [29].

[9] and [10] described in 1989 and 1993, respectively, the need of volatility
derivatives to allow traders to better manage their portfolio and hedge
against volatility risk. In that regard, the CBOE Futures Exchange (CFE)
introduced VIX futures contracts in March 2004 followed by the launch of VIX
options in February 2006. Thereafter, many new trading opportunities became
available for traders giving them more control on volatility risk. The trading
popularity of volatility derivatives has consistently increased through the
years. This increase is, to some extent, because VIX movements are negatively
correlated with movements in SPX returns.

Due to the importance of volatility trading in finance, a considerable amount
of research has been conducted to develop and expand stochastic volatility
models for pricing VIX derivatives. In particular, including jumps in the
stock and/or the volatility is a common practice. For instance, [29] shows
that a mean reverting logarithmic diffusion with Poisson jumps approximates
the behaviour of the VIX index and an analytical pricing formula for European
VIX options is derived. [31] uses a Heston model with Poisson jumps in the
volatility to derive a pricing formula for volatility derivatives and [32] for
the Heston model with Poisson jumps both in returns and volatility. Also, [39,
24] obtain an exact solution for the pricing of VIX futures in the Heston
model with simultaneous Poisson jumps in the asset price and the volatility.

An important reason to go beyond the Poisson jump diffusion models is that the
volatility clustering feature and the high-volatility effect [3, 5] are not
included due to the constant intensity of the Poisson process. In that regard,
[25, 13, 36, 29] obtain pricing formulas for VIX derivatives under a
stochastic volatility model with jumps in the asset and/or volatility with
state-dependent intensity. Additionally, [20] proposes a Heston model with
Hawkes jumps in the stock price and [36] a Heston model with Hawkes jumps both
in the asset and the volatility, respectively. The main purpose of these
models is to include the jump clustering effect in the stock and/or the
volatility, that is, the occurrence of a jump increases the probability of
future jumps. Other fundamental works about the VIX and volatility derivatives
are [21, 37, 38, 27, 11, 14].

The objective of our work is to obtain a semi-analytical pricing formula for
European VIX call options under the Heston-Hawkes stochastic volatility model
introduced in [7]. Essentially, the Heston-Hawkes model is a Heston model with
a compound Hawkes process in the volatility. In reference to that, the SVCIJ-H
model in [36] and their approach present some similarities but also some
important differences with our paper that are thoroughly described in Section
3. Fundamentally, the SVCIJ-H model in [36] has simultaneous and independent
Hawkes jumps in the stock and volatility and the size of the jumps is
exponentially distributed. In contrast, the self-exciting jumps are only
present in the volatility of the Heston-Hawkes model but their size has a
general distribution, being the exponential distribution a particular case.
This leads to technical challenges in qualitative ODE theory which are
properly tackled in Section 5.

First and foremost, the existence of risk neutral probability measures is
crucial to make sense of the pricing problem. To the best of our knowledge,
the usual procedure in the aforementioned literature is to present the model
directly under $\mathbb{Q}$ after assuming that the model is already
arbitrage-free. Nevertheless, [8, 30] present some stochastic volatility
models where no equivalent local martingale measure exist. Moreover, in terms
of risk management practices, [34, 33] show that exposures computed under the
risk neutral measure are essentially arbitrary and the passage from
$\mathbb{P}$ to $\mathbb{Q}$ is required. Therefore, a proper study about the
existence of risk neutral probability measures is fundamental for the pricing
of VIX derivatives. In that respect, [7] proves that the Heston-Hawkes
stochastic volatility model is arbitrage-free and an explicit family of
equivalent martingale measures is given.

Our approach to derive a semi-analytical pricing formula is based on the use
of the following equality

$\displaystyle\mathbb{E}\left[\max\\{\sqrt{X}-K,0\\}\right]=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{1-\text{erf}\left(K\sqrt{\phi}\right)}{\phi^{3/2}}\mathbb{E}\left[e^{\phi
X}\right]\right]d\phi_{I},$ (1.1)

where erf is the error function and some technical conditions described in
Section 3 must hold. In order to use the previous formula, an explicit
expression of $\text{VIX}^{2}$ is required. Using the Markov property of the
exponential Hawkes, it is possible to derive such an explicit expression as a
linear combination of the variance and the Hawkes intensity. Another
requirement for using the equality is a closed expression of the joint
characteristic function of the variance and the Hawkes intensity. Thanks to
the exponential affine structure of the model, the characteristic function can
be computed after solving some generalized Riccati ODEs. In addition, to apply
(1.1) it is crucial to check that the domain of the characteristic function
intersects the complex numbers with strictly positive real part.

The paper is organized as follows. In Section 2 we outline the Heston-Hakwes
stochastic volatility model introduced in [7]. In Section 3 we present the
problem of pricing European VIX call options and the technical results that
are required to derive a semi-analytical pricing formula. In Section 4 we
summarize the arbitrage-free property of the stochastic volatility model
proven in [7] and some results on the compensators of the Hawkes process under
the risk neutral probability measures given in [6]. In Section 5 we compute
the joint characteristic function of the variance and the Hawkes intensity
using the exponential affine structure of the model. Before, we study the
solution of some generalized Riccati ODEs that appear in the expression of the
characteristic function. In Section 6 we derive an explicit expression of the
VIX index using the Markov property of the exponential Hawkes. Finally, in
Section 7 we present the semi-analytical pricing formula for European VIX call
options under the Heston-Hawkes stochastic volatility model. In the appendix
we give all the technical proofs.

##  2 Heston-Hawkes stochastic volatility model

We outline the Heston-Hawkes stochastic volatility model introduced in [7].
Essentially, this model is an extension of the well-known Heston model that
incorporates the volatility clustering effect by adding an independent
compound Hawkes process to the variance.

Let $T\in{\mathbb{R}}$, $T>0$ be a fixed time horizon. On a complete
probability space $(\Omega,\mathcal{A},\mathbb{P})$, we consider a two-
dimensional standard Brownian motion
$(B,W)=\left\\{\left(B_{t},W_{t}\right),t\in[0,T]\right\\}$ and its minimally
augmented filtration
$\mathcal{F}^{(B,W)}=\\{\mathcal{F}^{(B,W)}_{t},t\in[0,T]\\}$. On
$(\Omega,\mathcal{A},\mathbb{P})$, we also consider a Hawkes process
$N=\\{N_{t},t\in[0,T]\\}$ with stochastic intensity given by

$\displaystyle\lambda_{t}=\lambda_{0}+\alpha\int_{0}^{t}e^{-\beta(t-s)}dN_{s},$

or, equivalently,

$\displaystyle d\lambda_{t}=-\beta(\lambda_{t}-\lambda_{0})dt+\alpha dN_{t},$
(2.1)

where $\lambda_{0}>0$ is the initial intensity, $\beta>0$ is the speed of mean
reversion and $\alpha\in(0,\beta)$ is the self-exciting factor. Note that the
stability condition $\alpha<\beta$ holds. See [4, Section 2] and [19, Section
3.1.1] for further details on Hawkes processes. Then, we consider a sequence
of i.i.d., strictly positive and integrable random variables
$\\{J_{i}\\}_{i\geqslant 1}$ and the compound Hawkes process
$L=\\{L_{t},t\in[0,T]\\}$ given by

$\displaystyle L_{t}=\sum_{i=1}^{N_{t}}J_{i}.$

We make the following assumption on the distribution of the jumps size.

######  Assumption 1.

There exists $\epsilon_{J}>0$ such that the moment generating function
$M_{J}(t)=\mathbb{E}[e^{tJ_{1}}]$ of $J_{1}$ is well defined in
$(-\infty,\epsilon_{J})$. Moreover, $(-\infty,\epsilon_{J})$ is the maximal
domain in the sense that

$\displaystyle\lim_{t\rightarrow\epsilon_{J}^{-}}M_{J}(t)=\infty.$

Since $\epsilon_{J}>0$, all positive moments of $J_{1}$ are finite. Note that
it is a rather mild assumption, and that the exponential distribution belongs
to the class of distributions satisfying the condition.

We assume that $(B,W),N$ and $\\{J_{i}\\}_{i\geqslant 1}$ are independent of
each other. We write
$\mathcal{F}^{L}=\left\\{\mathcal{F}^{L}_{t},t\in[0,T]\right\\}$ for the
minimally augmented filtration generated by $L$ and

$\displaystyle\mathcal{F}=\\{\mathcal{F}_{t}=\mathcal{F}_{t}^{(B,W)}\vee\mathcal{F}_{t}^{L},t\in[0,T]\\},$

for the joint filtration. We assume that $\mathcal{A}=\mathcal{F}_{T}$ and we
will work with $\mathcal{F}$. Since $(B,W)$ and $L$ are independent processes,
$(B,W)$ is also a two-dimensional $(\mathcal{F},\mathbb{P})$-Brownian motion.

Finally, with all these ingredients, we introduce the Heston-Hawkes stochastic
volatility model. We assume that the interest rate is deterministic and
constant equal to $r$. The stock price $S=\\{S_{t},t\in[0,T]\\}$ and its
variance $v=\\{v_{t},t\in[0,T]\\}$ are given by

$\displaystyle\frac{dS_{t}}{S_{t}}$
$\displaystyle=\mu_{t}dt+\sqrt{v_{t}}\left(\sqrt{1-\rho^{2}}dB_{t}+\rho
dW_{t}\right),$ (2.2) $\displaystyle dv_{t}$
$\displaystyle=-\kappa\left(v_{t}-\bar{v}\right)dt+\sigma\sqrt{v_{t}}dW_{t}+\eta
dL_{t},$ (2.3)

where $S_{0}>0$ is the initial price of the stock,
$\mu:[0,T]\rightarrow{\mathbb{R}}$ is a measurable and bounded function,
$\rho\in(-1,1)$ is the correlation factor, $v_{0}>0$ is the initial value of
the variance, $\kappa>0$ is the variance’s mean reversion speed, $\bar{v}>0$
is the long-term variance, $\sigma>0$ is the volatility of the variance and
$\eta>0$ is a scaling factor. We assume that the Feller condition
$2\kappa\bar{v}\geqslant\sigma^{2}$ is satisfied, see [1, Proposition 1.2.15].
For more details and further results on the model see [7].

##  3 Pricing VIX options

The objective of this paper is to derive a semi-analytical pricing formula for
European VIX call options under the Heston-Hawkes stochastic volatility model.
Namely, let $\mathbb{Q}$ be a risk neutral probability measure, the price
$\mathcal{C}^{\mathbb{Q}}(t,T,K)$ at time $t\in[0,T]$ of an European VIX call
option with maturity time $T$ and strike $K$ is given by

$\displaystyle\mathcal{C}^{\mathbb{Q}}(t,T,K)=e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}}\left[\max\\{\text{VIX}^{\mathbb{Q}}_{T}-K,0\\}|\mathcal{F}_{t}\right],$
(3.1)

where the VIX index can be defined (with mathematical simplification) in the
following way

$\displaystyle\left(\text{VIX}_{t}^{\mathbb{Q}}\right)^{2}=-\frac{2}{\Delta}\mathbb{E}^{\mathbb{Q}}\left[\log\left(e^{-r\Delta}\frac{S_{t+\Delta}}{S_{t}}\right)\Big{|}\mathcal{F}_{t}\right]\cdot
100^{2},$

where $\Delta=\frac{30}{365}$ and $S$ follows the dynamics in (2.2). See [23,
25, 16, 39] for further details on the definition of the VIX index. Our
approach to obtain the pricing formula involves the use of Fourier transform
techniques. Precisely, we will employ the following convenient equality

$\displaystyle\mathbb{E}\left[\max\\{\sqrt{X}-K,0\\}\right]=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{\text{erfc}\left(K\sqrt{\phi}\right)}{\phi^{3/2}}\mathbb{E}\left[e^{\phi
X}\right]\right]d\phi_{I},$ (3.2)

where $\phi=\phi_{R}+i\phi_{I}\in\mathbb{C}$ with $\phi_{R}>0$, $X$ is a
positive random variable such that
$\mathbb{E}\left[e^{\phi_{R}X}\right]<\infty$ and erfc is the complementary
error function defined by $\text{erfc}(z)=1-\text{erf}(z)$. For a reference of
this formula see [36, 22, 28]. The usefulness of this formula relies on the
fact that we can derive an explicit expression of $\text{VIX}^{2}$ and its
characteristic function. Several technical issues arise if the formula given
in (3.2) is going to be applied to derive a pricing formula for (3.1):

  1. 1.

The existence of risk neutral probability measures is required to make sense
of (3.1). To the best of our knowledge, the common procedure in the literature
regarding pricing of VIX options under stochastic volatility models with jumps
is to give the model directly under $\mathbb{Q}$ after assuming that the model
is arbitrage-free. It is discussed in [8, 30] that this property is not
obvious in general and a proper study is required. [7] proves that the Heston-
Hawkes stochastic volatility is arbitrage-free and an explicit family of
equivalent martingale measures is given. This is crucial to make sense of this
problem and we summarize the results of [7] in Section 4. Moreover, [34, 33]
show that exposures computed under the risk neutral measure are essentially
arbitrary and the passage from $\mathbb{P}$ to $\mathbb{Q}$ is crucial for
risk management practices.

  2. 2.

An explicit expression of $\text{VIX}^{2}$ is necessary in order to use (3.2).
In Section 6 we obtain such expression by exploiting the Markov property of
the exponential Hawkes [26, Remark 1.22].

  3. 3.

There must exist $\phi=\phi_{R}+i\phi_{I}\in\mathbb{C}$ with $\phi_{R}>0$ such
that $\mathbb{E}\left[e^{\phi_{R}X}\right]<\infty$ to use the formula in
(3.2). In our setting, this condition is reduced to the existence of
$\phi,\psi\in\mathbb{C}$ with $\operatorname{Re}(\phi)>0$ and
$\operatorname{Re}(\psi)>0$ such that $\mathbb{E}^{\mathbb{Q}}\left[e^{\phi
v_{T}+\psi\lambda_{T}}\right]<\infty$ where $\mathbb{Q}$ is a risk neutral
probability measure. This study is conducted in Section 5.

  4. 4.

A closed expression of the characteristic function $\mathbb{E}\left[e^{\phi
X}\right]$ is needed to obtain a tractable pricing formula. In Section 5 we
derive an (almost) explicit expression of the conditional characteristic
function $\mathbb{E}^{\mathbb{Q}}\left[e^{\phi
v_{T}+\psi\lambda_{T}}|\mathcal{F}_{t}\right]$ for $t\in[0,T]$ using the
exponential affine structure of the model [17, 36].

After presenting the model, the problem and the approach, it is pertinent to
thoroughly explain the connection of our paper with [36].

######  Remark 3.1.

It is important to mention the connection of our paper with [36]. While the
objective is the same and the stochastic volatility models are similar, there
is a significant amount of differences that are worth to highlight.

The model SVCIJ-H presented in [36] has simultaneous and independent self-
exciting jumps in the stock and the variance and their size is exponentially
distributed. In our model, the self-exciting jumps occur only in the variance
and their size has a general distribution, being the exponential distribution
a particular case.

The stochastic volatility models presented in [36] are given directly under
$\mathbb{Q}$ while we justify the existence of a family of equivalent
martingale measures for our model. As commented before, the assumption that
the model is arbitrage-free may not be obvious [8, 30] and the passage from
$\mathbb{P}$ to $\mathbb{Q}$ is necessary for risk management procedures like
computing exposures [34, 33].

Since the jumps in our setting can have an arbitrary distribution, the ODEs
that appear when computing the characteristic function take a more general
form than the ones in [36, Proposition 2]. In addition, in Section 5 we prove
that the domain of the characteristic function intersects the complex numbers
with strictly positive real part, which is crucial to use the formula in
(3.2). However, we miss this study in [36, Proposition 2] for the SVCIJ-H
model.

##  4 Arbitrage-free property of the model and compensators

We summarize the results proven in [7] regarding the existence and positivity
of the variance process and the change of measure. In addition, we outline the
results in [6] about the compensator of the Hawkes process under the risk
neutral probability measures.

The existence and positivity results of the variance are given in Proposition
4.1 and the change of measure is compacted in Theorem 4.3. The results about
the compensator of the Hawkes process under the risk neutral measure are given
in Proposition 4.5.

######  Proposition 4.1.

Equation (2.3) has a pathwise unique strong solution and the solution is
strictly positive.

###### Proof.

See [7, Proposition 2.1] and [7, Proposition 2.2]. ∎

In the next result we define the constant $c_{l}$, which depends on the model
parameters. Essentially, it is a strictly positive constant such that for
$c<c_{l}$ the following holds

$\displaystyle\mathbb{E}\left[\exp\left(c\int_{0}^{T}v_{u}du\right)\right]<\infty.$

This constant appears in Theorem 4.3, which is the main result regarding the
change of measure.

######  Proposition 4.2.

For $c\leqslant\frac{\kappa^{2}}{2\sigma^{2}}$, define
$D(c):=\sqrt{\kappa^{2}-2\sigma^{2}c}$, $\Lambda(c):=\frac{2\eta
c\left(e^{D(c)T}-1\right)}{D(c)-\kappa+\left(D(c)+\kappa\right)e^{D(c)T}}$ and

$\displaystyle
c_{l}:=\sup\left\\{c\leqslant\frac{\kappa^{2}}{2\sigma^{2}}:\Lambda(c)<\epsilon_{J}\hskip
8.5359pt\text{and}\hskip
8.5359ptM_{J}\left(\Lambda(c)\right)\leqslant\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)\right\\}.$

Then, $c_{l}>0$ and for $c<c_{l}$

$\displaystyle\mathbb{E}\left[\exp\left(c\int_{0}^{T}v_{u}du\right)\right]<\infty.$

###### Proof.

See [7, Lemma 3.1] and [7, Proposition 3.5]. ∎

######  Theorem 4.3.

Let $a\in\mathbb{R}$ and define
$\theta_{t}^{(a)}:=\frac{1}{\sqrt{1-\rho^{2}}}\left(\frac{\mu_{t}-r}{\sqrt{v_{t}}}-a\rho\sqrt{v_{t}}\right)$,

$\displaystyle Y_{t}^{(a)}$
$\displaystyle:=\exp\left(-\int_{0}^{t}\theta_{u}^{(a)}dB_{u}-\frac{1}{2}\int_{0}^{t}(\theta_{u}^{(a)})^{2}du\right),$
$\displaystyle Z_{t}^{(a)}$
$\displaystyle:=\exp\left(-a\int_{0}^{t}\sqrt{v_{u}}dW_{u}-\frac{1}{2}a^{2}\int_{0}^{t}v_{u}du\right)$

and $X_{t}^{(a)}:=Y_{t}^{(a)}Z_{t}^{(a)}$. Recall the definition of $c_{l}$ in
Proposition 4.2.

  1. 1.

$X^{(a)}$ is a $(\mathcal{F},\mathbb{P})$-martingale for $|a|<\sqrt{2c_{l}}$.

  2. 2.

The set

$\displaystyle\mathcal{E}:=\left\\{\mathbb{Q}(a)\hskip 5.69046pt\text{given
by}\hskip 5.69046pt\frac{d\mathbb{Q}(a)}{d\mathbb{P}}=X_{T}^{(a)}\hskip
5.69046pt\text{with}\hskip 5.69046pt|a|<\sqrt{2c_{l}}\right\\}$

is a set of equivalent local martingale measures.

  3. 3.

Let $\mathbb{Q}(a)\in\mathcal{E}$, the process
$(B^{\mathbb{Q}(a)},W^{\mathbb{Q}(a)})$ defined by

$\displaystyle dB_{t}^{\mathbb{Q}(a)}$
$\displaystyle:=dB_{t}+\theta_{t}^{(a)}dt,$ $\displaystyle
dW_{t}^{\mathbb{Q}(a)}$ $\displaystyle:=dW_{t}+a\sqrt{v_{t}}dt$

is a a two-dimensional standard $(\mathcal{F},\mathbb{Q}(a))$-Brownian motion.

  4. 4.

Let $\mathbb{Q}(a)\in\mathcal{E}$, the dynamics of $S$ and $v$ are given by

$\displaystyle\frac{dS_{t}}{S_{t}}$
$\displaystyle=rdt+\sqrt{v_{t}}\left(\sqrt{1-\rho^{2}}dB_{t}^{\mathbb{Q}(a)}+\rho
dW_{t}^{\mathbb{Q}(a)}\right),$ $\displaystyle dv_{t}$
$\displaystyle=-\kappa^{(a)}(v_{t}-\bar{v}^{(a)})dt+\sigma\sqrt{v_{t}}dW_{t}^{\mathbb{Q}(a)}+\eta
dL_{t}.$ (4.1)

where $\kappa^{(a)}=\kappa+a\sigma>0$ and
$\bar{v}^{(a)}=\frac{k\bar{v}}{k+a\sigma}>0$.

  5. 5.

If $\rho^{2}<c_{l}$,
$\mathbb{E}^{\mathbb{Q}(a)}\left[\exp\left(\frac{\rho^{2}}{2}\int_{0}^{T}v_{u}du\right)\right]<\infty$
and the set

$\displaystyle\mathcal{E}_{m}:=\left\\{\mathbb{Q}(a)\in\mathcal{E}:|a|<\min\left\\{\frac{\sqrt{2c_{l}}}{2},\sqrt{c_{l}-\rho^{2}}\right\\}\right\\}$

is a set of equivalent martingale measures.

###### Proof.

See [7, Theorem 3.6], [7, Observation 3.8] and [7, Theorem 3.9]. Note that
$\kappa^{(a)}=\kappa+a\sigma>0$ because
$|a|<\sqrt{2c_{l}}\leqslant\frac{\kappa}{\sigma}$ since
$c_{l}\leqslant\frac{\kappa^{2}}{2\sigma^{2}}$. ∎

In order to work with the compensators of $N$ and $L$ under the risk neutral
measures we make some assumptions and define a (non-empty) subset of
equivalent martingale measures. For further details see [6].

######  Assumption 2.

Define $D:=\sup_{t\in[0,T]}\left(\mu_{t}-r\right)^{2}<\infty$ and fix
$\varepsilon_{1},\varepsilon_{2}>0$. We assume that

  1. 1.

$\rho^{2}<c_{l}$.

  2. 2.

$\frac{1-\rho^{2}}{D\left[(2+\varepsilon_{1})^{2}-(2+\varepsilon_{1})\right]}\left(\frac{2\kappa\bar{v}-\sigma^{2}}{2\sigma}\right)^{2}>1$.

  3. 3.

$2\kappa\bar{v}>(1+\varepsilon_{2})\sigma^{2}$.

######  Definition 4.4.

Let $q,s>1$, we define a subset of $\mathcal{E}_{m}$ by

$\displaystyle\mathcal{E}_{m}(q,s):=\left\\{\mathbb{Q}(a)\in\mathcal{E}_{m}:|a|<\min\Bigg{\\{}\frac{1}{qs}\sqrt{\frac{c_{l}}{2}},\sqrt{\frac{(1-\rho^{2})c_{l}}{qs\left[2qs(1-\rho^{2})+\rho^{2}s-1\right]}}\Bigg{\\}}\right\\}.$

From now on, we fix $Q_{2}$ such that
$1<Q_{2}<\frac{1-\rho^{2}}{D\left[(2+\varepsilon_{1})^{2}-(2+\varepsilon_{1})\right]}\left(\frac{2\kappa\bar{v}-\sigma^{2}}{2\sigma}\right)^{2}$
and $Q_{1}:=\frac{Q_{2}}{Q_{2}-1}>1$. The subset
$\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ of equivalent martingale measures
is of special interest and will be used throughout the paper thanks to the
properties stated in the next proposition.

######  Proposition 4.5.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$. Then,

  1. 1.

$\mathbb{E}[(X_{t}^{(a)})^{2+\varepsilon_{1}}]<\infty$,
$\mathbb{E}\left[\left(\frac{1}{v_{t}}\right)^{1+\varepsilon_{2}}\right]<\infty$
for all $t\in[0,T]$ and
$\int_{0}^{T}\mathbb{E}\left[\left(\frac{1}{v_{t}}\right)^{1+\varepsilon_{2}}\right]dt<\infty$.

  2. 2.

$N-\Lambda^{N}$ is a $(\mathcal{F},\mathbb{Q}(a))$-martingale.

  3. 3.

$L-\Lambda^{L}$ is a $(\mathcal{F},\mathbb{Q}(a))$-martingale.

###### Proof.

See [6, Lemma 3.7] and [6, Propositon 3.12]. ∎

##  5 Joint characteristic function

The objective of this section is the computation of the conditional
characteristic function of $(v_{T},\lambda_{T})$. Furthermore, we check that
its domain intersects the complex numbers with strictly positive real part,
which is required to use the formula in (3.2). The subsequent results are
based on qualitative ODE theory and the exponential affine structure of the
model [17, 36].

The following result implies that
$\mathbb{E}[f(J_{1})]=\mathbb{E}^{\mathbb{Q}(a)}[f(J_{1})]$ where $f$ is a
deterministic function such that $\mathbb{E}[|f(J_{1})|]<\infty$. Therefore,
whenever there is an expectation under $\mathbb{Q}(a)$ of a random variable of
the type $f(J_{1})$, it can be equivalently computed under $\mathbb{P}$.

######  Lemma 5.1.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, then
$P_{J_{1}}=\mathbb{Q}(a)_{J_{1}}$, that is, the law of $J_{1}$ is the same
under $\mathbb{P}$ and under $\mathbb{Q}(a)$.

###### Proof.

See Lemma A.1 in the appendix. ∎

######  Definition 5.2.

Define the constant
$L_{J}:=\frac{1}{\eta}M_{J}^{-1}\left(\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)\right)$.
Note that $L_{J}>0$ because
$\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)>1$ using
$\alpha<\beta$.

In the following result we apply qualitative ODE theory to study the existence
of some generalized Riccati ODEs. The solutions of such equations appear in
the joint characteristic function of $(v_{T},\lambda_{T})$. A crucial
conclusion of the next lemma is that the ODEs admit as initial conditions
complex numbers with strictly positive real part. This is a requirement to use
the formula in (3.2).

######  Lemma 5.3.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and
$\phi,\psi\in\mathbb{C}$ such that

$\displaystyle
0<\operatorname{Re}(\phi)<\min\Bigg{\\{}\frac{2\kappa^{(a)}}{\sigma^{2}\left(2e^{\kappa^{(a)}T}-1\right)},L_{J}\Bigg{\\}}\hskip
28.45274pt\text{and}\hskip
28.45274pt\operatorname{Re}(\psi)<\frac{\beta-\alpha}{\alpha\beta}.$

Then,

  1. (i)

The ODE

$\displaystyle\frac{d}{dt}G^{(a)}(t;\phi)-\kappa^{(a)}G^{(a)}(t;\phi)+\frac{1}{2}\sigma^{2}G^{(a)}(t;\phi)^{2}=0,\hskip
14.22636ptG^{(a)}(T;\phi)=\phi,$

has a unique solution in the interval $[0,T]$ given by

$\displaystyle
G^{(a)}(t;\phi)=\frac{2\kappa^{(a)}}{\sigma^{2}+e^{\kappa^{(a)}(T-t)}\left(\frac{2\kappa^{(a)}}{\phi}-\sigma^{2}\right)}.$

  2. (ii)

$\sup_{t\in[0,T]}\operatorname{Re}(G^{(a)}(t;\phi))=\operatorname{Re}(\phi)$.

  3. (iii)

$\sup_{t\in[0,T]}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]<\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)$.
In particular, $\Big{|}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]\Big{|}<\infty$ for all $t\in[0,T]$.

  4. (iv)

The ODE

$\displaystyle\frac{d}{dt}H^{(a)}(t;\phi,\psi)-\beta
H^{(a)}(t;\phi,\psi)+e^{\alpha H^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]-1=0,\hskip 14.22636ptH^{(a)}(T;\phi,\psi)=\psi$

has a unique solution in $[0,T]$.

  5. (v)

The ODE

$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\kappa^{(a)}\bar{v}^{(a)}G^{(a)}(t;\phi)+\beta\lambda_{0}H^{(a)}(t;\phi,\psi)=0,\hskip
14.22636ptF^{(a)}(T;\phi,\psi)=0.$

has a unique solution in $[0,T]$.

###### Proof.

See Lemma A.2 in the appendix. ∎

To conclude this section, we give an (almost) closed expression of the joint
characteristic function of $(v_{T},\lambda_{T})$ in terms of the ODEs studied
in the previous result. In addition, a subset of the domain of the
characteristic function is given which intersects the complex numbers with
strictly positive real part.

######  Proposition 5.4.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and
$\phi,\psi\in\mathbb{C}$ such that

$\displaystyle
0<\operatorname{Re}(\phi)<\min\Bigg{\\{}\frac{2\kappa^{(a)}}{\sigma^{2}\left(2e^{\kappa^{(a)}T}-1\right)},L_{J}\Bigg{\\}}\hskip
28.45274pt\text{and}\hskip
28.45274pt\operatorname{Re}(\psi)<\frac{\beta-\alpha}{\alpha\beta}.$

Then,

  1. 1.

$\Big{|}\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}\right]\Big{|}<\infty$.

  2. 2.

The joint conditional characteristic function of $(v_{T},\lambda_{T})$ is
given by

$\displaystyle\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}|\mathcal{F}_{t}\right]=e^{F^{(a)}(t;\phi,\psi)+G^{(a)}(t;\phi)v_{t}+H^{(a)}(t;\phi,\psi)\lambda_{t}}$

where $t\in[0,T]$, and $F^{(a)},G^{(a)}$ and $H^{(a)}$ are solutions to the
following ODEs

$\displaystyle\frac{d}{dt}G^{(a)}(t;\phi)-\kappa^{(a)}G^{(a)}(t;\phi)+\frac{1}{2}\sigma^{2}G^{(a)}(t;\phi)^{2}=0,$
$\displaystyle\hskip 14.22636ptG^{(a)}(T;\phi)=\phi,$
$\displaystyle\frac{d}{dt}H^{(a)}(t;\phi,\psi)-\beta
H^{(a)}(t;\phi,\psi)+e^{\alpha H^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]-1=0,$ $\displaystyle\hskip
14.22636ptH^{(a)}(T;\phi,\psi)=\psi,$
$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\kappa^{(a)}\bar{v}^{(a)}G^{(a)}(t;\phi)+\beta\lambda_{0}H^{(a)}(t;\phi,\psi)=0,$
$\displaystyle\hskip 14.22636ptF^{(a)}(T;\phi,\psi)=0.$ (5.1)

###### Proof.

(i) First, we consider the case
$\operatorname{Im}(\phi)=\operatorname{Im}(\psi)=0$. For $t\in[0,T]$ define
the process $M^{(a)}(t):=g^{(a)}(t,v_{t},\lambda_{t})$ where
$g^{(a)}(t,x,y):=e^{F^{(a)}(t;\phi,\psi)+G^{(a)}(t;\phi)x+H^{(a)}(t;\phi,\psi)y}$
and $F^{(a)},G^{(a)}$ and $H^{(a)}$ are the solutions of the ODEs given in
Lemma 5.3, namely,

$\displaystyle\frac{d}{dt}G^{(a)}(t;\phi)-\kappa^{(a)}G^{(a)}(t;\phi)+\frac{1}{2}\sigma^{2}G^{(a)}(t;\phi)^{2}=0,$
$\displaystyle\hskip 14.22636ptG^{(a)}(T;\phi)=\phi,$
$\displaystyle\frac{d}{dt}H^{(a)}(t;\phi,\psi)-\beta
H^{(a)}(t;\phi,\psi)+e^{\alpha H^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]-1=0,$ $\displaystyle\hskip
14.22636ptH^{(a)}(T;\phi,\psi)=\psi,$
$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\kappa^{(a)}\bar{v}^{(a)}G^{(a)}(t;\phi)+\beta\lambda_{0}H^{(a)}(t;\phi,\psi)=0,$
$\displaystyle\hskip 14.22636ptF^{(a)}(T;\phi,\psi)=0.$ (5.2)

which are well defined for all $t\in[0,T]$. To lighten the notation we omit
the dependence of $g^{(a)}$ on $\phi$ and $\psi$. Note that
$F^{(a)},G^{(a)},H^{(a)}$ take values in $\mathbb{R}$ because we have assumed
that $\operatorname{Im}(\phi)=\operatorname{Im}(\psi)=0$. By convenience we
define $Y_{t}=(t,v_{t},\lambda_{t})$ and we apply Itô formula to the process
$M^{(a)}$

$\displaystyle M^{(a)}(t)=\ $ $\displaystyle
M^{(a)}(0)+\int_{0}^{t}\partial_{t}g^{(a)}(Y_{s})ds+\int_{0}^{t}\partial_{x}g^{(a)}(Y_{s-})dv_{s}+\int_{0}^{t}\partial_{y}g^{(a)}(Y_{s-})d\lambda_{s}$
$\displaystyle+\frac{1}{2}\int_{0}^{t}\partial_{xx}^{2}g^{(a)}(Y_{s-})d[v]^{\text{c}}_{s}+\frac{1}{2}\int_{0}^{t}\partial_{yy}^{2}g^{(a)}(Y_{s-})d[\lambda]_{s}^{\text{c}}+\int_{0}^{t}\partial_{xy}^{2}g^{(a)}(Y_{s-})d[v,\lambda]_{s}^{\text{c}}$
$\displaystyle+\sum_{0<s\leqslant
t}\left[g^{(a)}(Y_{s})-g^{(a)}(Y_{s-})-\partial_{x}g^{(a)}(Y_{s-})\Delta
v_{s}-\partial_{y}g^{(a)}(Y_{s-})\Delta\lambda_{s}\right]$ $\displaystyle=\ $
$\displaystyle
M^{(a)}(0)+\int_{0}^{t}\Big{[}\partial_{t}g^{(a)}(Y_{s})-\kappa^{(a)}\left(v_{s}-\bar{v}^{(a)}\right)\partial_{x}g^{(a)}(Y_{s})-\beta\left(\lambda_{s}-\lambda_{0}\right)\partial_{y}g^{(a)}(Y_{s})$
$\displaystyle+\frac{\sigma^{2}}{2}v_{s}\partial_{xx}^{2}g^{(a)}(Y_{s})\Big{]}ds+\sigma\int_{0}^{t}\sqrt{v_{s}}\partial_{x}g^{(a)}(Y_{s-})dW_{s}^{\mathbb{Q}(a)}$
$\displaystyle+\eta\int_{0}^{t}\partial_{x}g^{(a)}(Y_{s-})dL_{s}+\alpha\int_{0}^{t}\partial_{y}g^{(a)}(Y_{s-})dN_{s}$
$\displaystyle+\sum_{0<s\leqslant
t}\left[g^{(a)}(Y_{s})-g^{(a)}(Y_{s-})-\partial_{x}g^{(a)}(Y_{s-})\Delta
v_{s}-\partial_{y}g^{(a)}(Y_{s-})\Delta\lambda_{s}\right]$ $\displaystyle=\ $
$\displaystyle
M^{(a)}(0)+\int_{0}^{t}\Big{[}\partial_{t}g^{(a)}(Y_{s})-\kappa^{(a)}\left(v_{s}-\bar{v}^{(a)}\right)\partial_{x}g^{(a)}(Y_{s})-\beta\left(\lambda_{s}-\lambda_{0}\right)\partial_{y}g^{(a)}(Y_{s})$
$\displaystyle+\frac{\sigma^{2}}{2}v_{s}\partial_{xx}^{2}g^{(a)}(Y_{s})\Big{]}ds+\sigma\int_{0}^{t}\sqrt{v_{s}}\partial_{x}g^{(a)}(Y_{s-})dW_{s}^{\mathbb{Q}(a)}+\sum_{0<s\leqslant
t}\left[g^{(a)}(Y_{s})-g^{(a)}(Y_{s-})\right].$ (5.3)

We have used the dynamics of $v$ given in (4.1), the dynamics of $\lambda$
given in (2.1) and that

$\displaystyle[v]_{t}$
$\displaystyle=\sigma^{2}\int_{0}^{t}v_{s}ds+\eta^{2}[L]_{t}\implies[v]_{t}^{\text{c}}=\sigma^{2}\int_{0}^{t}v_{s}ds,$
$\displaystyle[\lambda]_{t}$
$\displaystyle=\alpha^{2}[N]_{t}=\alpha^{2}N_{t}\implies[\lambda]_{t}^{\text{c}}=0,$
$\displaystyle[v,\lambda]_{t}$ $\displaystyle=\alpha\eta[L,N]_{t}=\alpha\eta
L_{t}\implies[v,\lambda]_{t}^{\text{c}}=0,$
$\displaystyle\eta\int_{0}^{t}\partial_{x}g^{(a)}(Y_{s-})dL_{s}$
$\displaystyle=\eta\sum_{0<s\leqslant t}\partial_{x}g^{(a)}(Y_{s-})\Delta
L_{s}=\sum_{0<s\leqslant t}\partial_{x}g^{(a)}(Y_{s-})\Delta v_{s},$
$\displaystyle\alpha\int_{0}^{t}\partial_{y}g^{(a)}(Y_{s-})dN_{s}$
$\displaystyle=\alpha\sum_{0<s\leqslant t}\partial_{y}g^{(a)}(Y_{s-})\Delta
N_{s}=\sum_{0<s\leqslant t}\partial_{y}g^{(a)}(Y_{s-})\Delta\lambda_{s}.$

Next, we can write

$\displaystyle\sum_{0<s\leqslant
t}\left[g^{(a)}(Y_{s})-g^{(a)}(Y_{s-})\right]$
$\displaystyle=\sum_{0<s\leqslant t}\left[g^{(a)}(s,v_{s-}+\eta\Delta
L_{s},\lambda_{s-}+\alpha\Delta N_{s})-g^{(a)}(s,v_{s-},\lambda_{s-})\right]$
$\displaystyle=\sum_{0<s\leqslant t}h^{(a)}(s,\Delta L_{s},\Delta N_{s}),$

where

$\displaystyle h^{(a)}(s,u_{1},u_{2}):=g^{(a)}(s,v_{s-}+\eta
u_{1},\lambda_{s-}+\alpha u_{2})-g^{(a)}(s,v_{s-},\lambda_{s-}).$

We now define $R_{s}=(L_{s},N_{s})$ and for $t\in[0,T]$,
$B\in\mathcal{B}\left(\mathbb{R}^{2}\setminus\\{0,0\\}\right)$,

$\displaystyle N^{R}(t,A)=\\#\\{0<s\leqslant t:\Delta R_{s}\in A\\}.$

We add and subtract the compensator of the counting measure $N^{R}$ to split
the expression in (5) into a $(\mathcal{F},\mathbb{Q}(a))$-local martingale
plus a predictable process of finite variation. By Proposition 4.5, the
compensators of $N$ and $L$ under
$\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ are
$\Lambda_{t}^{N}=\int_{0}^{t}\lambda_{u}du$ and
$\Lambda_{t}^{L}=\mathbb{E}[J_{1}]\int_{0}^{t}\lambda_{u}du$ respectively.
Thus,

$\displaystyle\sum_{0<s\leqslant
t}\left[g^{(a)}(Y_{s})-g^{(a)}(Y_{s-})\right]=\ $
$\displaystyle\int_{0}^{t}\int_{(0,\infty)^{2}}h^{(a)}(s,u_{1},u_{2})N^{R}(ds,du)$
$\displaystyle=$
$\displaystyle\int_{0}^{t}\int_{(0,\infty)^{2}}h^{(a)}(s,u_{1},u_{2})\left(N^{R}(ds,du)-\lambda_{s}P_{J_{1}}(du_{1})\delta_{1}(du_{2})ds\right)$
$\displaystyle+\int_{0}^{t}\lambda_{s}\int_{(0,\infty)}h^{(a)}(s,u_{1},1)P_{J_{1}}(du_{1})ds.$

Replacing everything in (5) we get

$\displaystyle M^{(a)}(t)=\ $ $\displaystyle
M^{(a)}(0)+\int_{0}^{t}\Big{[}\partial_{t}g^{(a)}(Y_{s})-\kappa^{(a)}\left(v_{s}-\bar{v}^{(a)}\right)\partial_{x}g^{(a)}(Y_{s})-\beta\left(\lambda_{s}-\lambda_{0}\right)\partial_{y}g^{(a)}(Y_{s})$
$\displaystyle+\frac{\sigma^{2}}{2}v_{s}\partial_{xx}^{2}g^{(a)}(Y_{s})+\lambda_{s}\int_{(0,\infty)}h^{(a)}(s,u_{1},1)P_{J_{1}}(du_{1})\Big{]}ds+\sigma\int_{0}^{t}\sqrt{v_{s}}\partial_{x}g^{(a)}(Y_{s-})dW_{s}^{\mathbb{Q}(a)}$
$\displaystyle+\int_{0}^{t}\int_{(0,\infty)^{2}}h^{(a)}(s,u_{1},u_{2})\left(N^{R}(ds,du)-\lambda_{s}P_{J_{1}}(du_{1})\delta_{1}(du_{2})ds\right),$
(5.4)

where the last two terms are $(\mathcal{F},\mathbb{Q}(a))$-local martingales.
Now, using the definition of $g^{(a)}$ and $h^{(a)}$ we get

$\displaystyle\partial_{t}g^{(a)}(Y_{s})$
$\displaystyle=g^{(a)}(Y_{s})\left(\frac{d}{dt}F^{(a)}(s;\phi,\psi)+\frac{d}{dt}G^{(a)}(s;\phi)v_{s}+\frac{d}{dt}H^{(a)}(s;\phi,\psi)\lambda_{s}\right),$
$\displaystyle\partial_{x}g^{(a)}(Y_{s})$ $\displaystyle=g^{(a)}(Y_{s})G(s),$
$\displaystyle\partial_{xx}^{2}g^{(a)}(Y_{s})$
$\displaystyle=g^{(a)}(Y_{s})G(s)^{2},$
$\displaystyle\partial_{y}g^{(a)}(Y_{s})$ $\displaystyle=g^{(a)}(Y_{s})H(s),$
$\displaystyle\int_{(0,\infty)}h^{(a)}(s,u_{1},1)P_{J_{1}}(du_{1})$
$\displaystyle=g^{(a)}(Y_{s-})\int_{(0,\infty)}\left[e^{\eta u_{1}G(s)+\alpha
H(s)}-1\right]P_{J_{1}}(du_{1})$ $\displaystyle=g^{(a)}(Y_{s-})\left(e^{\alpha
H(s)}\mathbb{E}\left[e^{\eta G(s)J_{1}}\right]-1\right).$

Using that $F^{(a)},G^{(a)}$ and $H^{(a)}$ satisfy the ODEs given in (5) we
see that the drift in (5) is equal to $0$ because

$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\frac{d}{dt}G^{(a)}(t;\phi)v_{t}+\frac{d}{dt}H^{(a)}(t;\phi,\psi)\lambda_{t}$
$\displaystyle-\kappa^{(a)}\left(v_{t}-\bar{v}^{(a)}\right)G^{(a)}(t)-\beta\left(\lambda_{t}-\lambda_{0}\right)H^{(a)}(t)+\frac{\sigma^{2}}{2}v_{t}G(t)^{2}+\lambda_{t}\left(e^{\alpha
H(t)}\mathbb{E}\left[e^{\eta G(t)J_{1}}\right]-1\right)=0,$

We conclude that $M^{(a)}$ is a $(\mathcal{F},\mathbb{Q}(a))$-local
martingale. Moreover, using the final conditions of the ODEs in (5), we see
that $M^{(a)}(T)=e^{\phi v_{T}+\psi\lambda_{T}}$. Since $M^{(a)}$ is bounded
from below it is a supermartingale and the following holds

$\displaystyle\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}\right]=\mathbb{E}^{\mathbb{Q}(a)}\left[M^{(a)}(T)\right]\leqslant
M^{(a)}(0)=\exp\left(F^{(a)}(0)+G^{(a)}(0)v_{0}+H^{(a)}(0)\lambda_{0}\right)<\infty.$

For the general case of arbitrary $\operatorname{Im}(\phi)$ and
$\operatorname{Im}(\psi)$ note that

$\displaystyle\Big{|}\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}\right]\Big{|}\leqslant\mathbb{E}^{\mathbb{Q}(a)}\left[\Big{|}e^{\phi
v_{T}+\psi\lambda_{T}}\Big{|}\right]=\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\operatorname{Re}(\phi)v_{T}+\operatorname{Re}(\psi)\lambda_{T}}\right],$

and we can apply the same argument as before to conclude that the previous
expectation is finite.

(ii) Since $(N,\lambda)$ is a Markov process, see [26, Remark 1.22], the
process $(v,\lambda)$ is also a Markov and we see that there exists a function
$f^{(a)}(\cdot,\cdot,\cdot;\phi,\psi)\colon[0,T]\times(0,\infty)\times[\lambda_{0},\infty)\to\mathbb{R}$
such that

$\displaystyle\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}|\mathcal{F}_{t}\right]=f^{(a)}(t,v_{t},\lambda_{t};\phi,\psi).$

Using that $t\to f^{(a)}(t,v_{t},\lambda_{t};\phi,\psi)$ is a
$(\mathcal{F},\mathbb{Q}(a))$-local martingale and by applying Itô formula
similarly as in the previous step we see that it must satisfy the following
partial integro-differential equation (PIDE)

$\displaystyle\partial_{t}f^{(a)}(t,x,y;\phi,\psi)-\kappa^{(a)}(x-\bar{v}^{(a)})\partial_{x}f^{(a)}(t,x,y;\phi,\psi)-\beta(y-\lambda_{0})\partial_{y}f^{(a)}(t,x,y;\phi,\psi)$
$\displaystyle+\frac{\sigma^{2}}{2}x\partial_{xx}^{2}f^{(a)}(t,x,y;\phi,\psi)+y\int_{(0,\infty)}\left[f^{(a)}(t,x+\eta
u,y+\alpha;\phi,\psi)-f^{(a)}(t,x,y;\phi,\psi)\right]P_{J_{1}}(du)=0$

and final condition $f^{(a)}(T,x,y;\phi,\psi)=e^{\phi x+\psi y}$. Note that
the left-hand side of the previous PIDE is just the drift given in (5). By the
affine structure property of the model, see [17, 36], the solution of the
previous PIDE can be written in exponential affine form, that is
$f^{(a)}(T,x,y;\phi,\psi)=e^{F^{(a)}(t;\phi,\psi)+G^{(a)}(t;\phi)x+H^{(a)}(t;\phi,\psi)y}$.
Substituting that expression in the PIDE we obtain the ODEs given in (2) and
this finishes the proof. ∎

##  6 Explicit expression of the VIX index

In this section we derive an explicit expression of the VIX index under the
Heston-Hawkes model as a linear combination of $v$ and $\lambda$ using the
Markov property of the exponential Hawkes. First, a preliminary result is
required.

######  Lemma 6.1.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, the following
holds

  1. 1.

The process $t\to\int_{0}^{t}\lambda_{u}dX_{u}^{(a)}$ is a square integrable
$(\mathcal{F},\mathbb{P})$-martingale.

  2. 2.

Let $0\leqslant s\leqslant t\leqslant T$, then

$\displaystyle\mathbb{E}[\lambda_{t}X_{t}^{(a)}|\mathcal{F}_{s}]=\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)X_{s}^{(a)}e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}X_{s}^{(a)}}{\beta-\alpha}.$

###### Proof.

See Lemma A.3 in the appendix. ∎

###  6.1 Forward variance

As a starting point, we compute the forward variance under the Heston-Hawkes
stochastic volatility model. Since the computations are rather tedious and
long the proof is postponed to the appendix. In order to avoid a singular case
in the expression of the forward variance we make the following mild
assumption on the family of equivalent martingale measures.

######  Assumption 3.

We assume that $a\neq\frac{\beta-\alpha-\kappa}{\sigma}$. This is equivalent
to $\kappa^{(a)}\neq\beta-\alpha$ because $\kappa^{(a)}=\kappa+a\sigma$ and
ensures that some constants appearing in the next result are well defined.

######  Proposition 6.2.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and define the
forward variance by
$\xi_{s}^{(a)}(t):=\mathbb{E}^{\mathbb{Q}(a)}[v_{t}|\mathcal{F}_{s}]$,
$0\leqslant s\leqslant t\leqslant T$. Then,

  1. 1.

The explicit expression is given by

$\displaystyle\xi_{s}^{(a)}(t)=$ $\displaystyle\
D_{1}^{(a)}(t-s)v_{s}+D_{2}^{(a)}(t-s)\lambda_{s}+D_{3}^{(a)}(t-s),$ (6.1)

where

$\displaystyle D_{1}^{(a)}(h):=e^{-\kappa^{(a)}h},\hskip
28.45274ptD_{2}^{(a)}(h)=C_{1}^{(a)}\left(e^{-(\beta-\alpha)h}-e^{-\kappa^{(a)}h}\right),$
$\displaystyle
D_{3}^{(a)}(h):=\left(\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right)e^{-\kappa^{(a)}h}-\frac{C_{2}^{(a)}}{\beta-\alpha}e^{-(\beta-\alpha)h}+C_{3}^{(a)},$
$\displaystyle
C_{1}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]}{\kappa^{(a)}-(\beta-\alpha)},\hskip
28.45274ptC_{2}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\kappa^{(a)}-(\beta-\alpha)},\hskip
28.45274ptC_{3}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\kappa^{(a)}(\beta-\alpha)}+\bar{v}^{(a)}.$
(6.2)

  2. 2.

The dynamics of the process $t\to\xi_{s}^{(a)}(t)$ is given by

$\displaystyle
d\xi_{s}^{(a)}(t)=\left[-\kappa^{(a)}\left(\xi_{s}^{(a)}(t)-\bar{v}^{(a)}\right)+\eta\mathbb{E}[J_{1}]\left(\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\right]dt.$

  3. 3.

The dynamics of the process $s\to\xi_{s}(t)$ is given by

$\displaystyle d\xi_{s}^{(a)}(t)=$ $\displaystyle\ \sigma
D_{1}^{(a)}(t-s)\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}+\eta
D_{1}^{(a)}(t-s)\left(dL_{s}-\mathbb{E}[J_{1}]\lambda_{s}ds\right)$
$\displaystyle+\alpha D_{2}^{(a)}(t-s)\left(dN_{s}-\lambda_{s}ds\right).$

###### Proof.

See Proposition A.4 in the appendix. ∎

######  Observation 6.3.

Note that all the constants in (6.2) are well defined. By Assumption 3,
$\kappa^{(a)}-(\beta-\alpha)\neq 0$, $\beta-\alpha>0$ due to the stability
condition of the Hawkes process and $\kappa^{(a)}>0$ by Theorem 4.3.

###  6.2 Variance Swap

We now compute an explicit expression of the variance swap using the previous
result on the forward variance. The usefulness of this result relies on the
fact that the VIX index can be written as a particular variance swap.

######  Definition 6.4.

For $k,h>0$, we define
$A_{k}(h):=\frac{1}{h}\int_{0}^{h}e^{-ku}du=\frac{1}{kh}(1-e^{-kh})$.

Let $0\leqslant s\leqslant t\leqslant T$ and
$\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, we define the
variance swap as

$\displaystyle V_{s}^{(a)}(t):=\frac{1}{t-s}\int_{s}^{t}\xi^{(a)}_{s}(u)du.$

######  Proposition 6.5.

Let $0\leqslant s\leqslant t\leqslant T$ and
$\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$. Then, the explicit
expression of $V_{s}^{(a)}(t)$ is given by

$\displaystyle V_{s}^{(a)}(t)=$ $\displaystyle\
K_{1}^{(a)}(t-s)v_{s}+K_{2}^{(a)}(t-s)\lambda_{s}+K_{3}^{(a)}(t-s),$

where

$\displaystyle K_{1}^{(a)}(h):=A_{\kappa^{(a)}}(h),\hskip
28.45274ptK_{2}^{(a)}(h)=C_{1}^{(a)}\left(A_{\beta-\alpha}(h)-A_{\kappa^{(a)}}(h)\right),$
$\displaystyle
K_{3}^{(a)}(h):=\left(\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right)A_{\kappa^{(a)}}(h)-\frac{C_{2}^{(a)}}{\beta-\alpha}A_{\beta-\alpha}(h)+C_{3}^{(a)}$
(6.3)

where $C_{1}^{(a)},C_{2}^{(a)}$ and $C_{3}^{(a)}$ are given in (6.2) and
$A_{k}$ defined in Definition 6.4.

###### Proof.

See Proposition A.5 in the appendix. ∎

######  Observation 6.6.

Note that all the constants in (6.3) are well defined by the same argument as
in Observation 6.3, $\kappa^{(a)}-(\beta-\alpha)\neq 0$, $\beta-\alpha>0$ and
$\kappa^{(a)}>0$.

###  6.3 VIX index

To finish this section, we introduce the mathematical definition of the VIX
index and obtain its explicit expression as a linear combination of the
variance and the Hawkes intensity.

######  Definition 6.7.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, $t\in[0,T]$
and $\Delta=\frac{30}{365}$. The $\text{VIX}^{(a)}$ introduced in the CBOE
(Chicago Board Options Exchange) white paper can be defined, with mathematical
simplification, in the following way

$\displaystyle\left(\text{VIX}_{t}^{(a)}\right)^{2}=-\frac{2}{\Delta}\mathbb{E}^{\mathbb{Q}(a)}\left[\log\left(e^{-r\Delta}\frac{S_{t+\Delta}}{S_{t}}\right)\Big{|}\mathcal{F}_{t}\right]\cdot
100^{2}.$

See [23, 25, 16, 39] for further details on the definition of the VIX index.

The next lemma will be used to deduce that some constants appearing in the
explicit expression of the VIX index are strictly positive.

######  Lemma 6.8.

Let $k_{1},k_{2},h>0$ with $k_{1}\neq k_{2}$. Then,

$\displaystyle\frac{A_{k_{2}}(h)-A_{k_{1}}(h)}{k_{1}-k_{2}}>0.$

###### Proof.

See Lemma A.6 in the appendix. ∎

######  Proposition 6.9.

Let $t\in[0,T]$ and
$\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$. Then,

$\displaystyle\left(\text{VIX}_{t}^{(a)}\right)^{2}=V_{t}^{(a)}(t+\Delta)\cdot
100^{2}=(A^{(a)}v_{t}+B^{(a)}\lambda_{t}+C^{(a)})\cdot 100^{2}.$

where the constants $A^{(a)},B^{(a)}$ and $C^{(a)}$ are independent of $t$ and
are given by

$\displaystyle A^{(a)}$
$\displaystyle=K_{1}^{(a)}(\Delta)=A_{\kappa^{(a)}}(\Delta)>0$ $\displaystyle
B^{(a)}$
$\displaystyle=K_{2}^{(a)}(\Delta)=C_{1}^{(a)}\left(A_{\beta-\alpha}(\Delta)-A_{\kappa^{(a)}}(\Delta)\right)>0,$
$\displaystyle C^{(a)}$
$\displaystyle=K_{3}^{(a)}(\Delta)=\left(\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right)A_{\kappa^{(a)}}(\Delta)-\frac{C_{2}^{(a)}}{\beta-\alpha}A_{\beta-\alpha}(\Delta)+C_{3}^{(a)}$

where $K_{1}^{(a)}$, $K_{2}^{(a)}$ and $K_{3}^{(a)}$ are given in Proposition
6.5, $C_{1}^{(a)}$, $C_{2}^{(a)}$ and $C_{3}^{(a)}$ are given in (6.2) and
$A_{k}$ defined in Definition 6.4.

###### Proof.

Using that

$\displaystyle\frac{S_{t+\Delta}}{S_{t}}=\exp\left(r\Delta-\frac{1}{2}\int_{t}^{t+\Delta}v_{s}ds+\sqrt{1-\rho^{2}}\int_{t}^{t+\Delta}\sqrt{v_{s}}dB_{s}^{\mathbb{Q}(a)}+\rho\int_{t}^{t+\Delta}\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}\right)$

and the fact that both $t\to\int_{0}^{t}\sqrt{v_{u}}dB_{u}^{\mathbb{Q}(a)}$
and $t\to\int_{0}^{t}\sqrt{v_{u}}dW_{u}^{\mathbb{Q}(a)}$ are
$(\mathcal{F},\mathbb{Q}(a))$-martingales by Theorem 4.3, one can see that the
following holds

$\displaystyle\left(\text{VIX}_{t}^{(a)}\right)^{2}=V_{t}^{(a)}(t+\Delta).$

Then, the result follows from Proposition 6.5. Moreover, the fact that
$A^{(a)}>0$ follows directly from
$A_{\kappa^{(a)}}(\Delta)=\frac{1}{\Delta}\int_{0}^{\Delta}e^{-\kappa^{(a)}u}du$
and the fact that $B^{(a)}>0$ follows from the definition of $C_{1}^{(a)}$ in
(6.2) and Lemma 6.8. ∎

##  7 Pricing formula for VIX options

Finally, we have all the results needed to derive a semi-analytical pricing
formula for European VIX call options under the Heston-Hawkes stochastic
volatility model. Essentially, the pricing formula is based on the formula in
(3.2), the explicit expression of $\text{VIX}^{2}$ and the joint
characteristic function of $(v_{T},\lambda_{T})$. By convenience, we define
for $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and $t\in[0,T]$

$\displaystyle\mathcal{C}^{(a)}(t,T,k):=\mathbb{E}^{\mathbb{Q}(a)}\left[\max\\{\text{VIX}^{(a)}_{T}/100-k,0\\}|\mathcal{F}_{t}\right].$

Then, the price at time $t\in[0,T]$ of an European VIX call option with
maturity time $T$ and strike $K=100K^{\prime}$ can be written as

$\displaystyle
e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}(a)}\left[\max\\{\text{VIX}^{(a)}_{T}-K,0\\}|\mathcal{F}_{t}\right]=100e^{-r(T-t)}\mathcal{C}^{(a)}\left(t,T,K^{\prime}\right).$

######  Theorem 7.1.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, the price at
time $t\in[0,T]$ of an European VIX call option with maturity time $T$ and
strike $K=100K^{\prime}$ is given by

$\displaystyle
e^{-r(T-t)}\mathbb{E}^{\mathbb{Q}(a)}\left[\max\\{\text{VIX}^{(a)}_{T}-K,0\\}|\mathcal{F}_{t}\right]=100e^{-r(T-t)}\mathcal{C}^{(a)}\left(t,T,K^{\prime}\right)$

where

$\displaystyle\mathcal{C}^{(a)}(t,T,k)=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{\operatorname{erfc}(k\sqrt{\phi})}{\phi^{3/2}}e^{\phi
C^{(a)}}f^{(a)}(t,v_{t},\lambda_{t};A^{(a)}\phi,B^{(a)}\phi)\right]d\phi_{I},$
(7.1)

erfc is the complementary error function defined by
$\text{erfc}(z)=1-\text{erf}(z)$, $\phi=\phi_{R}+i\phi_{I}\in\mathbb{C}$
satisfies

$\displaystyle
0<\phi_{R}<\min\Bigg{\\{}\frac{2\kappa^{(a)}}{\sigma^{2}A^{(a)}\left(2e^{\kappa^{(a)}T}-1\right)},\frac{L_{J}}{A^{(a)}},\frac{\beta-\alpha}{B^{(a)}\alpha\beta}\Bigg{\\}},$
(7.2)

$A^{(a)},B^{(a)},C^{(a)}$ are given in Proposition 6.9, $L_{J}$ is given in
Definition 5.2 and $f^{(a)}$ is the conditional characteristic function of
$(v,\lambda)$ under $\mathbb{Q}(a)$ given in Proposition 5.4, that is,

$\displaystyle
f^{(a)}(t,v_{t},\lambda_{t};\phi,\psi)=\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi
v_{T}+\psi\lambda_{T}}|\mathcal{F}_{t}\right]=e^{F^{(a)}(t;\phi,\psi)+G^{(a)}(t;\phi)v_{t}+H^{(a)}(t;\phi,\psi)\lambda_{t}},$

where $F^{(a)},G^{(a)}$ and $H^{(a)}$ satisfy the ODEs given in (2), that is,

$\displaystyle\frac{d}{dt}G^{(a)}(t;\phi)-\kappa^{(a)}G^{(a)}(t;\phi)+\frac{1}{2}\sigma^{2}G^{(a)}(t;\phi)^{2}=0,$
$\displaystyle\hskip 14.22636ptG^{(a)}(T;\phi)=\phi,$
$\displaystyle\frac{d}{dt}H^{(a)}(t;\phi,\psi)-\beta
H^{(a)}(t;\phi,\psi)+e^{\alpha H^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]-1=0,$ $\displaystyle\hskip
14.22636ptH^{(a)}(T;\phi,\psi)=\psi,$
$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\kappa^{(a)}\bar{v}^{(a)}G^{(a)}(t;\phi)+\beta\lambda_{0}H^{(a)}(t;\phi,\psi)=0,$
$\displaystyle\hskip 14.22636ptF^{(a)}(T;\phi,\psi)=0.$

###### Proof.

To obtain the pricing formula in (7.1) we use the expression of
$\text{VIX}^{2}$ given in Proposition 6.9, the formula in (3.2) and the joint
conditional characteristic function of $(v_{T},\lambda_{T})$ given in
Proposition 5.4:

$\displaystyle\mathcal{C}^{(a)}(t,T,k)$
$\displaystyle=\mathbb{E}^{\mathbb{Q}(a)}\left[\max\\{\text{VIX}^{(a)}_{T}/100-k,0\\}|\mathcal{F}_{t}\right]$
$\displaystyle=\mathbb{E}^{\mathbb{Q}(a)}\left[\max\\{\sqrt{A^{(a)}v_{T}+B^{(a)}\lambda_{T}+C^{(a)}}-k,0\\}|\mathcal{F}_{t}\right]$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{\text{erfc}\left(k\sqrt{\phi}\right)}{\phi^{3/2}}\mathbb{E}^{\mathbb{Q}(a)}\left[e^{\phi\left(A^{(a)}v_{T}+B^{(a)}\lambda_{T}+C^{(a)}\right)}\big{|}\mathcal{F}_{t}\right]\right]d\phi_{I}$
$\displaystyle=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\operatorname{Re}\left[\frac{\text{erfc}(k\sqrt{\phi})}{\phi^{3/2}}e^{\phi
C^{(a)}}f^{(a)}(t,v_{t},\lambda_{t};A^{(a)}\phi,B^{(a)}\phi)\right]d\phi_{I}.$

Note that the inequality in (7.2) ensures that the characteristic function is
well defined, see Proposition 5.4. Moreover, we have used that
$A^{(a)},B^{(a)}>0$ which is proved in Proposition 6.9 and the fact that
$\phi_{R}>0$ guarantees that we can use the formula in (3.2). ∎

##  Appendix A Appendix: Technical results

We give the proofs that were postponed.

###  A.1 Joint characteristic function

######  Lemma A.1.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, then
$P_{J_{1}}=\mathbb{Q}(a)_{J_{1}}$, that is, the law of $J_{1}$ is the same
under $\mathbb{P}$ and under $\mathbb{Q}(a)$.

###### Proof.

Let $f$ be a measurable and bounded function, we are going to prove that
$\mathbb{E}^{\mathbb{Q}(a)}[f(J_{1})]=\mathbb{E}[f(J_{1})]$ which will imply
that the law of $J_{1}$ is the same under $\mathbb{P}$ and $\mathbb{Q}(a)$. By
Theorem 4.3,
$\frac{d\mathbb{Q}(a)}{d\mathbb{P}}=X_{T}^{(a)}=Y_{T}^{(a)}Z_{T}^{(a)}$. Now,
using that $Z_{T}^{(a)}$ is
$\mathcal{F}_{T}^{W}\vee\mathcal{F}_{T}^{L}$-measurable, that

$\displaystyle
Y_{T}^{(a)}|\mathcal{F}_{T}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}\sim\text{Lognormal}\left(-\frac{1}{2}\int_{0}^{T}\left(\theta_{s}^{(a)}\right)^{2}ds,\int_{0}^{T}\left(\theta_{s}^{(a)}\right)^{2}ds\right)$

because $\theta^{(a)}$ is
$\\{\mathcal{F}_{t}^{W}\vee\mathcal{F}_{t}^{L}\vee\mathcal{F}^{J_{1}}\\}_{t\in[0,T]}$-adapted
and the fact that $Z^{(a)}$ is a
$\\{\mathcal{F}_{t}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}\\}_{t\in[0,T]}$-martingale
we have

$\displaystyle\mathbb{E}^{\mathbb{Q}(a)}[f(J_{1})]$
$\displaystyle=\mathbb{E}[f(J_{1})X_{T}^{(a)}]=\mathbb{E}[f(J_{1})Y_{T}^{(a)}Z_{T}^{(a)}]=\mathbb{E}[\mathbb{E}[f(J_{1})Y_{T}^{(a)}Z_{T}^{(a)}|\mathcal{F}_{T}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}]]$
$\displaystyle=\mathbb{E}[f(J_{1})Z_{T}^{(a)}\mathbb{E}[Y_{T}^{(a)}|\mathcal{F}_{T}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}]]=\mathbb{E}[f(J_{1})Z_{T}^{(a)}]$
$\displaystyle=\mathbb{E}[\mathbb{E}[f(J_{1})Z_{T}^{(a)}|\mathcal{F}_{0}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}]]=\mathbb{E}[f(J_{1})\mathbb{E}[Z_{T}^{(a)}|\mathcal{F}_{0}^{W}\vee\mathcal{F}_{T}^{L}\vee\mathcal{F}^{J_{1}}]]=\mathbb{E}[f(J_{1})].$

∎

######  Lemma A.2.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and
$\phi,\psi\in\mathbb{C}$ such that

$\displaystyle
0<\operatorname{Re}(\phi)<\min\Bigg{\\{}\frac{2\kappa^{(a)}}{\sigma^{2}\left(2e^{\kappa^{(a)}T}-1\right)},L_{J}\Bigg{\\}}\hskip
28.45274pt\text{and}\hskip
28.45274pt\operatorname{Re}(\psi)<\frac{\beta-\alpha}{\alpha\beta}.$

Then,

  1. (i)

The ODE

$\displaystyle\frac{d}{dt}G^{(a)}(t;\phi)-\kappa^{(a)}G^{(a)}(t;\phi)+\frac{1}{2}\sigma^{2}G^{(a)}(t;\phi)^{2}=0,\hskip
14.22636ptG^{(a)}(T;\phi)=\phi,$

has a unique solution in the interval $[0,T]$ given by

$\displaystyle
G^{(a)}(t;\phi)=\frac{2\kappa^{(a)}}{\sigma^{2}+e^{\kappa^{(a)}(T-t)}\left(\frac{2\kappa^{(a)}}{\phi}-\sigma^{2}\right)}.$

  2. (ii)

$\sup_{t\in[0,T]}\operatorname{Re}(G^{(a)}(t;\phi))=\operatorname{Re}(\phi)$.

  3. (iii)

$\sup_{t\in[0,T]}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]<\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)$.
In particular, $\Big{|}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]\Big{|}<\infty$ for all $t\in[0,T]$.

  4. (iv)

The ODE

$\displaystyle\frac{d}{dt}H^{(a)}(t;\phi,\psi)-\beta
H^{(a)}(t;\phi,\psi)+e^{\alpha H^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]-1=0,\hskip 14.22636ptH^{(a)}(T;\phi,\psi)=\psi$
(A.1)

has a unique solution in $[0,T]$.

  5. (v)

The ODE

$\displaystyle\frac{d}{dt}F^{(a)}(t;\phi,\psi)+\kappa^{(a)}\bar{v}^{(a)}G^{(a)}(t;\phi)+\beta\lambda_{0}H^{(a)}(t;\phi,\psi)=0,\hskip
14.22636ptF^{(a)}(T;\phi,\psi)=0.$

has a unique solution in $[0,T]$.

###### Proof.

(i) One can check that the solution of the ODE is

$\displaystyle
G^{(a)}(t;\phi)=\frac{2\kappa^{(a)}}{\sigma^{2}+e^{\kappa^{(a)}(T-t)}\left(\frac{2\kappa^{(a)}}{\phi}-\sigma^{2}\right)}.$

Note that the denominator of $G^{(a)}(t;\phi)$ is $0$ for $t\in[0,T)$ if and
only if

$\displaystyle\phi=\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}.$

A necessary condition for the previous equality to hold is that
$\operatorname{Im}(\phi)=0$. It can be checked that

$\displaystyle\inf_{t\in[0,T]}\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}=\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}T}}{e^{\kappa^{(a)}T}-1},$

and since
$\operatorname{Re}(\phi)<\frac{2\kappa^{(a)}}{\sigma^{2}\left(2e^{\kappa^{(a)}T}-1\right)}<\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}T}}{e^{\kappa^{(a)}T}-1}\leqslant\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}$
for all $t\in[0,T)$, $G^{(a)}$ is well defined in the whole interval $[0,T]$.

(ii) First, we compute the real part of $G^{(a)}$. Writing
$DG^{(a)}(t;\phi)=\sigma^{2}+e^{\kappa^{(a)}(T-t)}\left(\frac{2\kappa^{(a)}}{\phi}-\sigma^{2}\right)$
we have

$\displaystyle\operatorname{Re}(G^{(a)}(t;\phi))=\frac{2\kappa^{(a)}\operatorname{Re}(DG^{(a)}(t;\phi))}{|DG^{(a)}(t;\phi)|^{2}}.$

Then,

$\displaystyle\operatorname{Re}(DG^{(a)}(t;\phi))$
$\displaystyle=\sigma^{2}+e^{\kappa^{(a)}(T-t)}\left(\frac{2\kappa^{(a)}\operatorname{Re}(\phi)}{|\phi|^{2}}-\sigma^{2}\right)$
$\displaystyle=\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)+\frac{2\kappa^{(a)}\operatorname{Re}(\phi)}{|\phi|^{2}}e^{\kappa^{(a)}(T-t)},$
$\displaystyle\operatorname{Im}(DG^{(a)}(t;\phi))$
$\displaystyle=-\frac{2\kappa^{(a)}\operatorname{Im}(\phi)}{|\phi|^{2}}e^{\kappa^{(a)}(T-t)}.$

Note that

$\displaystyle|DG^{(a)}(t;\phi)|^{2}=\ $
$\displaystyle\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}+\frac{4\left(\kappa^{(a)}\right)^{2}\operatorname{Re}(\phi)^{2}}{|\phi|^{4}}e^{2\kappa^{(a)}(T-t)}$
$\displaystyle+\frac{4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)}{|\phi|^{2}}+\frac{4\left(\kappa^{(a)}\right)^{2}\operatorname{Im}(\phi)^{2}}{|\phi|^{4}}e^{2\kappa^{(a)}(T-t)}$
$\displaystyle=\ $
$\displaystyle\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}+\frac{4\left(\kappa^{(a)}\right)^{2}}{|\phi|^{2}}e^{2\kappa^{(a)}(T-t)}+\frac{4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)}{|\phi|^{2}}.$

Thus,

$\displaystyle\operatorname{Re}(G^{(a)}(t;\phi))=\frac{2\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)|\phi|^{2}+4\left(\kappa^{(a)}\right)^{2}e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)}{\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}|\phi|^{2}+4\left(\kappa^{(a)}\right)^{2}e^{2\kappa^{(a)}(T-t)}+4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)}.$

We check that the denominator of $\operatorname{Re}(G^{(a)}(t;\phi))$ in the
previous expression is always strictly positive for $t\in[0,T)$. If
$t\in[0,T)$ we have the following

$\displaystyle\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}|\phi|^{2}+4\left(\kappa^{(a)}\right)^{2}e^{2\kappa^{(a)}(T-t)}+4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)$
$\displaystyle\geqslant\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}\operatorname{Re}(\phi)^{2}+4\left(\kappa^{(a)}\right)^{2}e^{2\kappa^{(a)}(T-t)}+4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}\operatorname{Re}(\phi)$
$\displaystyle=f_{t}(\operatorname{Re}(\phi)),$

where $f_{t}$ is defined by

$\displaystyle
f_{t}(x):=\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}x^{2}+4\left(\kappa^{(a)}\right)^{2}e^{2\kappa^{(a)}(T-t)}+4\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)e^{\kappa^{(a)}(T-t)}x.$

Using that $f_{t}$ is a quadratic function in $x$, one can check that the
minimum of $f_{t}$ is $0$ and it is achieved at
$x_{t}=\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}>0$
and for other values of $x$, $f_{t}(x)$ is strictly positive. Using again that

$\displaystyle\inf_{t\in[0,T)}\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}=\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}T}}{e^{\kappa^{(a)}T}-1},$

and that
$\operatorname{Re}(\phi)<\frac{2\kappa^{(a)}}{\sigma^{2}(2e^{\kappa^{(a)}T}-1)}<\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}T}}{e^{\kappa^{(a)}T}-1}\leqslant\frac{2\kappa^{(a)}}{\sigma^{2}}\frac{e^{\kappa^{(a)}(T-t)}}{e^{\kappa^{(a)}(T-t)}-1}$
for all $t\in[0,T)$, we conclude that the denominator of
$\operatorname{Re}(G^{(a)}(t;\phi))$ is strictly positive for all $t\in[0,T]$.

Finally, we prove that
$0<\operatorname{Re}(\phi)<\frac{2\kappa^{(a)}}{\sigma^{2}(2e^{\kappa^{(a)}T}-1)}$
implies that
$\sup_{t\in[0,T]}\operatorname{Re}(G^{(a)}(t;\phi))=\operatorname{Re}(\phi)$.
Recall that $\operatorname{Re}(G^{(a)}(T;\phi))=\operatorname{Re}(\phi)$ and
we need to prove that
$\operatorname{Re}(G^{(a)}(t;\phi))\leqslant\operatorname{Re}(\phi)$ for
$t\in[0,T)$. Using that the denominator is always positive one can check that
$\operatorname{Re}(G^{(a)}(t;\phi))\leqslant\operatorname{Re}(\phi)$ is
equivalent to

$\displaystyle
L_{1}(t)\operatorname{Re}(\phi)^{3}+L_{2}(t)\operatorname{Re}(\phi)^{2}+L_{3}(t)\operatorname{Re}(\phi)+L_{4}(t)\operatorname{Im}(\phi)^{2}+L_{5}(t)\operatorname{Im}(\phi)^{2}\operatorname{Re}(\phi)\leqslant
0,$ (A.2)

where

$\displaystyle L_{1}(t)$
$\displaystyle=-\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2},$
$\displaystyle L_{2}(t)$
$\displaystyle=2\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right)\left(1-2e^{\kappa^{(a)}(T-t)}\right),$
$\displaystyle L_{3}(t)$
$\displaystyle=4\left(\kappa^{(a)}\right)^{2}e^{\kappa^{(a)}(T-t)}\left(1-e^{\kappa^{(a)}(T-t)}\right),$
$\displaystyle L_{4}(t)$
$\displaystyle=2\kappa^{(a)}\sigma^{2}\left(1-e^{\kappa^{(a)}(T-t)}\right),$
$\displaystyle L_{5}(t)$
$\displaystyle=-\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right)^{2}.$

Using that $L_{4}(t)<0$ and $L_{5}(t)<0$ for all $t\in[0,T)$, we obtain the
following

$\displaystyle
L_{1}(t)\operatorname{Re}(\phi)^{3}+L_{2}(t)\operatorname{Re}(\phi)^{2}+L_{3}(t)\operatorname{Re}(\phi)+L_{4}(t)\operatorname{Im}(\phi)^{2}+L_{5}(t)\operatorname{Im}(\phi)^{2}\operatorname{Re}(\phi)$
$\displaystyle\leqslant
L_{1}(t)\operatorname{Re}(\phi)^{3}+L_{2}(t)\operatorname{Re}(\phi)^{2}+L_{3}(t)\operatorname{Re}(\phi)\leqslant
0.$

Using that $\operatorname{Re}(\phi)>0$ and that
$\left(1-e^{\kappa^{(a)}(T-t)}\right)<0$ for $t\in[0,T)$, the inequality in
(A.2) is equivalent to

$\displaystyle\widetilde{L}_{1}(t)\operatorname{Re}(\phi)^{2}+\widetilde{L}_{2}(t)\operatorname{Re}(\phi)+\widetilde{L}_{3}(t)\leqslant
0,$

where

$\displaystyle\widetilde{L}_{1}(t)$
$\displaystyle=\sigma^{4}\left(1-e^{\kappa^{(a)}(T-t)}\right),$
$\displaystyle\widetilde{L}_{2}(t)$
$\displaystyle=2\kappa^{(a)}\sigma^{2}\left(2e^{\kappa^{(a)}(T-t)}-1\right),$
$\displaystyle\widetilde{L}_{3}(t)$
$\displaystyle=-4\left(\kappa^{(a)}\right)^{2}e^{\kappa^{(a)}(T-t)}.$

Note that

$\displaystyle\widetilde{L}_{1}(t)\operatorname{Re}(\phi)^{2}+\widetilde{L}_{2}(t)\operatorname{Re}(\phi)+\widetilde{L}_{3}(t)$
$\displaystyle\leqslant\sup_{t\in[0,T]}\widetilde{L}_{1}(t)\operatorname{Re}(\phi)^{2}+\sup_{t\in[0,T}\widetilde{L}_{2}(t)\operatorname{Re}(\phi)+\sup_{t\in[0,T]}\widetilde{L}_{3}(t)$
$\displaystyle\leqslant
2\kappa^{(a)}\sigma^{2}\left(2e^{\kappa^{(a)}T}-1\right)\operatorname{Re}(\phi)-4\left(\kappa^{(a)}\right)^{2}\leqslant
0$

which holds because
$0<\operatorname{Re}(\phi)<\frac{2\kappa^{(a)}}{\sigma^{2}(2e^{\kappa^{(a)}T}-1)}$.
This finishes the proof.

(iii) Using that
$L_{J}=\frac{1}{\eta}M_{J}^{-1}\left(\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)\right)$
and that $\operatorname{Re}(\phi)<L_{J}$ we have

$\displaystyle\sup_{t\in[0,T]}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]$
$\displaystyle\leqslant\mathbb{E}\left[\sup_{t\in[0,T]}e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]=\mathbb{E}\left[e^{\eta\sup_{t\in[0,T]}\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]$
$\displaystyle=\mathbb{E}\left[e^{\eta\operatorname{Re}(\phi)J_{1}]}\right]<\mathbb{E}\left[e^{\eta
L_{J}J_{1}}\right]=\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right).$

In particular,

$\displaystyle\Big{|}\mathbb{E}\left[e^{\eta
G^{(a)}(t;\phi)J_{1}}\right]\Big{|}$
$\displaystyle\leqslant\mathbb{E}\left[\Big{|}e^{\eta
G^{(a)}(t;\phi)J_{1}}\Big{|}\right]=\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]$
$\displaystyle\leqslant\sup_{t\in[0,T]}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]<\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)<\infty,$

for all $t\in[0,T]$.

(iv) We introduce the function $h^{(a)}(t;\phi,\psi):=H^{(a)}(T-t;\phi,\psi)$.
Then, the ODE for $h^{(a)}$ is

$\displaystyle\frac{d}{dt}h^{(a)}(t;\phi,\psi)$
$\displaystyle=f^{(a)}(t,h^{(a)}(t;\phi,\psi))$ $\displaystyle=e^{\alpha
h^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(T-t;\phi)J_{1}}\right]-\beta h^{(a)}(t;\phi,\psi)-1,\hskip
28.45274pth^{(a)}(0;\phi,\psi)=\psi,$

where $f^{(a)}(t,x)=e^{\alpha x}\mathbb{E}\left[e^{\eta
G^{(a)}(T-t;\phi)J_{1}}\right]-\beta x-1\in\mathbb{C}$. Since $f^{(a)}$ is a
continuously differentiable function, it is Lipschitz continuous on bounded
intervals and there exists a unique local solution for every initial
condition, see [18, Chapter II, Theorem 1.1]. First, we study the imaginary
part of $h^{(a)}$. Define
$U^{(a)}:=\sup_{t\in[0,T]}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(t;\phi))J_{1}}\right]<\infty$
and note that

$\displaystyle\frac{d}{dt}\operatorname{Im}(h^{(a)}(t;\phi,\psi))$
$\displaystyle\leqslant\Big{|}e^{\alpha
h^{(a)}(t;\phi,\psi)}\mathbb{E}\left[e^{\eta
G^{(a)}(T-t;\phi)J_{1}}\right]\Big{|}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi))$
$\displaystyle\leqslant
e^{\alpha\operatorname{Re}(h^{(a)}(t;\phi,\psi))}\mathbb{E}\left[e^{\eta\operatorname{Re}(G^{(a)}(T-t;\phi))J_{1}}\right]-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi))$
$\displaystyle\leqslant
e^{\alpha\sup_{t\in[0,T]}\operatorname{Re}(h^{(a)}(t;\phi,\psi))}U^{(a)}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi))$
$\displaystyle=C^{(a)}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi)),$

where
$C^{(a)}:=e^{\alpha\sup_{t\in[0,T]}\operatorname{Re}(h^{(a)}(t))}U^{(a)}$.
Similarly, we can get the following lower bound for
$\frac{d}{dt}\operatorname{Im}(h^{(a)}(t;\phi,\psi))$:

$\displaystyle-C^{(a)}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi)))\leqslant\frac{d}{dt}\operatorname{Im}(h^{(a)}(t;\phi,\psi))).$

Therefore,

$\displaystyle-C^{(a)}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi)))\leqslant\frac{d}{dt}\operatorname{Im}(h^{(a)}(t;\phi,\psi)))\leqslant
C^{(a)}-\beta\operatorname{Im}(h^{(a)}(t;\phi,\psi))).$

Define now the two following ODEs

$\displaystyle\frac{d}{dt}h^{(a)}_{IM}(t;\phi,\psi))=C^{(a)}-\beta
h^{(a)}_{IM}(t;\phi,\psi))$ $\displaystyle,\hskip
28.45274pth^{(a)}_{IM}(0;\phi,\psi))=\operatorname{Im}(\psi),$
$\displaystyle\frac{d}{dt}h^{(a)}_{Im}(t;\phi,\psi))=-C^{(a)}-\beta
h^{(a)}_{Im}(t;\phi,\psi))$ $\displaystyle,\hskip
28.45274pth^{(a)}_{Im}(0;\phi,\psi))=\operatorname{Im}(\psi).$

Then, by the comparison theorem

$\displaystyle-\frac{C^{(a)}}{\beta}\left(1-e^{-\beta t}\right)+e^{-\beta
t}\operatorname{Im}(\psi)=h^{(a)}_{Im}(t;\phi,\psi))\leqslant\operatorname{Im}(h^{(a)}(t;\phi,\psi)))$

and

$\displaystyle\operatorname{Im}(h^{(a)}(t;\phi,\psi)))\leqslant
h^{(a)}_{IM}(t;\phi,\psi))=\frac{C^{(a)}}{\beta}\left(1-e^{-\beta
t}\right)+e^{-\beta t}\operatorname{Im}(\psi)$

for all $t\in[0,T]$. This proves that as long as $C^{(a)}<\infty$, the
imaginary part of $h^{(a)}$ does not blow up in $[0,T]$. Since
$C^{(a)}=e^{\alpha\sup_{t\in[0,T]}\operatorname{Re}(h^{(a)}(t))}U^{(a)}$ and
$U^{(a)}<\infty$, it suffices to check that the real part of $h^{(a)}$ does
not explode in $[0,T]$ to ensure existence of $h^{(a)}$ over the whole
interval $[0,T]$.

Similarly as done with the imaginary part we can get the following bounds on
the derivative of the real part of $h^{(a)}$:

$\displaystyle-e^{\alpha\operatorname{Re}(h^{(a)}(t;\phi,\psi))}U^{(a)}-\beta\operatorname{Re}(h^{(a)}(t;\phi,\psi))-1\leqslant\frac{d}{dt}\operatorname{Re}(h^{(a)}(t;\phi,\psi))$

and

$\displaystyle\frac{d}{dt}\operatorname{Re}(h^{(a)}(t;\phi,\psi))\leqslant
e^{\alpha\operatorname{Re}(h^{(a)}(t;\phi,\psi))}U^{(a)}-\beta\operatorname{Re}(h^{(a)}(t;\phi,\psi))-1.$

Define now the two following ODEs

$\displaystyle\frac{d}{dt}h^{(a)}_{RM}(t;\phi,\psi)$
$\displaystyle=f^{(a)}_{M}(h^{(a)}_{RM}(t;\phi,\psi))$
$\displaystyle=U^{(a)}e^{\alpha h^{(a)}_{RM}(t;\phi,\psi)}-\beta
h^{(a)}_{RM}(t;\phi,\psi)-1,\hskip
28.45274pth^{(a)}_{RM}(0;\phi,\psi)=\operatorname{Re}(\psi),$
$\displaystyle\frac{d}{dt}h^{(a)}_{Rm}(t;\phi,\psi)$
$\displaystyle=f^{(a)}_{m}(h^{(a)}_{Rm}(t;\phi,\psi))$
$\displaystyle=-U^{(a)}e^{\alpha h^{(a)}_{Rm}(t;\phi,\psi)}-\beta
h^{(a)}_{Rm}(t;\phi,\psi)-1,\hskip
28.45274pth^{(a)}_{Rm}(0;\phi,\psi)=\operatorname{Re}(\psi),$

where $f^{(a)}_{M}(x)=U^{(a)}e^{\alpha x}-\beta x-1$ and
$f^{(a)}_{m}(x)=-U^{(a)}e^{\alpha x}-\beta x-1$. Then, by the comparison
theorem and as long as $h^{(a)}_{RM}$ and $h^{(a)}_{Rm}$ exist over the whole
interval $[0,T]$ we have

$\displaystyle
h^{(a)}_{Rm}(t;\phi,\psi)\leqslant\operatorname{Re}(h^{(a)}(t;\phi,\psi))\leqslant
h^{(a)}_{RM}(t;\phi,\psi),$ (A.3)

for all $t\in[0,T]$. We study the existence of $h^{(a)}_{RM}$ in the interval
$[0,T]$. One can check that $f^{(a)}_{M}$ has a unique absolute minimum at
$x^{(a)}_{\text{min}}=\frac{1}{\alpha}\log\left(\frac{\beta}{\alpha
U^{(a)}}\right)$ and it is equal to

$\displaystyle
f^{(a)}_{M}(x_{\text{min}})=\frac{\beta}{\alpha}\left(1-\log\left(\frac{\beta}{\alpha
U}\right)\right)-1.$

Note that $f^{(a)}_{M}(x_{\text{min}})<0$ if and only if
$U^{(a)}<\frac{\beta}{\alpha}\exp\left(\frac{\alpha}{\beta}-1\right)$, which
holds by (iii). Moreover,

$\displaystyle
x^{(a)}_{\text{min}}=\frac{1}{\alpha}\log\left(\frac{\beta}{\alpha
U^{(a)}}\right)>\frac{\beta-\alpha}{\alpha\beta}>0.$ (A.4)

Since $\lim_{x\to\pm\infty}f^{(a)}_{M}(x)=\infty$ and there is no local
minimum or maximum, there exist exactly two zeroes $x^{(a)}_{z1}$ and
$x^{(a)}_{z2}$ of $f^{(a)}_{M}$ that satisfy
$x^{(a)}_{z_{1}}<x^{(a)}_{\text{min}}<x^{(a)}_{z2}$. Note that $x^{(a)}_{z1}$
is a stable equilibrium point and for all initial conditions in the interval
$(-\infty,x^{(a)}_{z2})$, $h_{RM}$ exists for all $t\in[0,\infty)$ and it
converges to $x^{(a)}_{z1}$. Therefore, using (A.4) and
$\operatorname{Re}(\psi)<\frac{\beta-\alpha}{\alpha\beta}$ we have

$\displaystyle\operatorname{Re}(\psi)<\frac{\beta-\alpha}{\alpha\beta}<\frac{1}{\alpha}\log\left(\frac{\beta}{\alpha
U^{(a)}}\right)=x^{(a)}_{\text{min}}<x^{(a)}_{z2}$

We conclude that $h^{(a)}_{RM}$ exists for all $t\in[0,\infty)$ and it
converges to $x^{(a)}_{z_{1}}$.

We study now the existence of $h^{(a)}_{Rm}$ in the interval $[0,T]$. One can
check that $f^{(a)}_{m}$ is decreasing in $\mathbb{R}$ and that
$\lim_{x\to-\infty}f^{(a)}_{m}(x)=+\infty$ and
$\lim_{x\to\infty}f^{(a)}_{m}(x)=-\infty$. This implies the existence of a
unique zero which is a stable equilibrium point, and for any initial condition
the solution of the ODE exists for all $t\in[0,\infty)$ and it converges to
that zero of $f_{m}$. Therefore, there is no constraint on
$\operatorname{Re}(\psi)$ to guarantee the existence of $h^{(a)}_{Rm}$ over
the interval $[0,T]$.

Finally, by (A.3) we conclude that $\operatorname{Re}(h^{(a)}(t))$ exists for
all $t\in[0,T]$ and $\operatorname{Im}(h^{(a)}(t))$ as well. Hence, the ODE in
(A.1) has a unique solution in $[0,T]$.

(v) Since $G^{(a)}$ and $H^{(a)}$ exist for all $t\in[0,T]$, $F^{(a)}$ also
does. ∎

###  A.2 Explicit expression of the VIX index

######  Lemma A.3.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$, the following
holds

  1. 1.

The process $t\to\int_{0}^{t}\lambda_{u}dX_{u}^{(a)}$ is a square integrable
$(\mathcal{F},\mathbb{P})$-martingale.

  2. 2.

Let $0\leqslant s\leqslant t\leqslant T$, then

$\displaystyle\mathbb{E}[\lambda_{t}X_{t}^{(a)}|\mathcal{F}_{s}]=\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)X_{s}^{(a)}e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}X_{s}^{(a)}}{\beta-\alpha}.$

###### Proof.

(1) By Theorem 4.3 and Proposition 4.5(1) we know that $X^{(a)}$ is a square
integrable martingale. We check that

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\lambda_{u-}^{2}d[X^{(a)}]_{u}\right]<\infty$

to prove that $t\to\int_{0}^{t}\lambda_{u-}dX_{u}^{(a)}$ is a square
integrable $(\mathcal{F},\mathbb{P})$-martingale. The quadratic variation of
$X^{(a)}$ is given by
$d[X^{(a)}]_{t}=\left[\left(\theta_{t}^{(a)}\right)^{2}+a^{2}v_{t}\right]\left(X_{t}^{(a)}\right)^{2}dt$,
where $\theta^{(a)}$ is defined in Theorem 4.3. Then,

$\displaystyle\mathbb{E}\left[\int_{0}^{T}\lambda_{u-}^{2}d[X^{(a)}]_{u}\right]$
$\displaystyle=\mathbb{E}\left[\int_{0}^{T}\lambda_{u}^{2}\left[\left(\theta_{u}^{(a)}\right)^{2}+a^{2}v_{u}\right]\left(X_{u}^{(a)}\right)^{2}du\right]$
$\displaystyle=\int_{0}^{T}\mathbb{E}\left[\lambda_{u}^{2}\left(\theta_{u}^{(a)}\right)^{2}\left(X_{u}^{(a)}\right)^{2}\right]du+a^{2}\int_{0}^{T}\mathbb{E}\left[\lambda_{u}^{2}v_{u}\left(X_{u}^{(a)}\right)^{2}\right]du.$
(A.5)

We focus on the first term in (A.2). Recall that $\varepsilon_{1}$ and
$\varepsilon_{2}$ where fixed in Assumption 2 and their connection with the
results in Proposition 4.5. Applying Hölder’s inequality with
$p_{1}=\frac{p_{2}p_{3}}{p_{2}p_{3}-p_{2}-p_{3}}>1$,
$p_{2}=1+\varepsilon_{2}>1$, $p_{3}=1+\frac{\varepsilon_{1}}{2}>1$, and then
Doob’s martingale inequality, see [2, Section 2.1.2, Theorem 2.1.5], to the
last expectation we get

$\displaystyle\mathbb{E}\left[\lambda_{u}^{2}\left(\theta_{u}^{(a)}\right)^{2}\left(X_{u}^{(a)}\right)^{2}\right]$
$\displaystyle\leqslant\mathbb{E}[\lambda_{u}^{2p_{1}}]^{\frac{1}{p_{1}}}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2p_{2}}\right]^{\frac{1}{p_{2}}}\mathbb{E}\left[\left(X_{u}^{(a)}\right)^{2p_{3}}\right]^{\frac{1}{p_{3}}}$
$\displaystyle=\mathbb{E}[\lambda_{u}^{2p_{1}}]^{\frac{1}{p_{1}}}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]^{\frac{1}{p_{2}}}\mathbb{E}\left[\left(X_{u}^{(a)}\right)^{2+\varepsilon_{1}}\right]^{\frac{1}{p_{3}}}$
$\displaystyle\leqslant\left(\frac{2+\varepsilon_{1}}{1+\varepsilon_{1}}\right)^{2}\mathbb{E}[\lambda_{u}^{2p_{1}}]^{\frac{1}{p_{1}}}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]^{\frac{1}{p_{2}}}\mathbb{E}\left[\left(X_{T}^{(a)}\right)^{2+\varepsilon_{1}}\right]^{\frac{1}{p_{3}}}.$
(A.6)

By [15, Section 3.1], $\mathbb{E}[\lambda_{u}^{2p_{1}}]<\infty$ for all
$u\in[0,T]$ and it is continuous as a function of $u$. Moreover, by
Proposition 4.5(1) we have
$\mathbb{E}\left[\left(X_{T}^{(a)}\right)^{2+\varepsilon_{1}}\right]<\infty$.
Applying Hölder’s inequality for sums we get

$\displaystyle\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]\leqslant\frac{2^{1+2\varepsilon_{2}}}{(1-\rho^{2})^{1+\varepsilon_{2}}}\left[D^{1+\varepsilon_{2}}\mathbb{E}\left[\left(\frac{1}{v_{u}}\right)^{1+\varepsilon_{2}}\right]+(a\rho)^{2+2\varepsilon_{2}}\mathbb{E}\left[v_{u}^{1+\varepsilon_{2}}\right]\right]$

where $D=\sup_{t\in[0,T]}(\mu_{t}-r)^{2}$ is defined in Assumption 2. By
Proposition 4.5(1) and [6, Lemma 3.5] we have that

$\displaystyle\int_{0}^{T}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]du<\infty.$

Applying Hölder’s inequality with $q_{1}=\frac{p_{2}}{p_{2}-1}>1$ and
$p_{2}>1$ we get

$\displaystyle\int_{0}^{T}\mathbb{E}[\lambda_{u}^{2p_{1}}]^{\frac{1}{p_{1}}}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]^{\frac{1}{p_{2}}}du\leqslant\left(\int_{0}^{T}\mathbb{E}[\lambda_{u}^{2p_{1}}]^{\frac{q_{1}}{p_{1}}}du\right)^{\frac{1}{q_{1}}}\left(\int_{0}^{T}\mathbb{E}\left[\left(\theta_{u}^{(a)}\right)^{2+2\varepsilon_{2}}\right]du\right)^{\frac{1}{p_{2}}}<\infty.$

By (A.2) this implies that

$\displaystyle\int_{0}^{T}\mathbb{E}\left[\lambda_{u}^{2}\left(\theta_{u}^{(a)}\right)^{2}\left(X_{u}^{(a)}\right)^{2}\right]du<\infty.$

Similarly, using again [15, Section 3.1], [6, Lemma 3.5] and Proposition
4.5(1) we can show that the second term in (A.2) is finite. We conclude that
the process $t\to\int_{0}^{t}\lambda_{u-}dX_{u}^{(a)}$ is a square integrable
$(\mathcal{F},\mathbb{P})$-martingale.

(2) By [6, Lemma A.8] we know the process
$t\to\int_{0}^{t}X_{u}^{(a)}d(N-\Lambda^{N})_{u}$ is a square integrable
$(\mathcal{F},\mathbb{P})$-martingale where $d\Lambda^{N}_{t}=\lambda_{t}dt$.
Thus,

$\displaystyle\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}dN_{u}\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}d(N-\Lambda^{N})_{u}\Big{|}\mathcal{F}_{s}\right]+\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}d\Lambda^{N}_{u}\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=\int_{s}^{t}\mathbb{E}\left[X_{u}^{(a)}\lambda_{u}|\mathcal{F}_{s}\right]du.$

Recall that $d\lambda_{t}=-\beta(\lambda_{t}-\lambda_{0})dt+\alpha dN_{t}$.
Then,

$\displaystyle\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}d\lambda_{u}\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=-\beta\int_{s}^{t}\mathbb{E}[X_{u}^{(a)}\lambda_{u}|\mathcal{F}_{s}]du+\beta\lambda_{0}(t-s)X_{s}^{(a)}+\alpha\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}dN_{u}\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=-(\beta-\alpha)\int_{s}^{t}\mathbb{E}[X_{u}^{(a)}\lambda_{u}|\mathcal{F}_{s}]du+\beta\lambda_{0}(t-s)X_{s}^{(a)}.$

Applying Itô formula, using that $\lambda$ is of finite variation and
$X^{(a)}$ is continuous we have

$\displaystyle\lambda_{t}X^{(a)}_{t}=\lambda_{s}X_{s}^{(a)}+\int_{s}^{t}\lambda_{u-}dX_{u}^{(a)}+\int_{s}^{t}X_{u}^{(a)}d\lambda_{u}.$

Using that $t\to\int_{0}^{t}\lambda_{u-}dX_{u}^{(a)}$ is a square integrable
$(\mathcal{F},\mathbb{P})$-martingale we obtain

$\displaystyle\mathbb{E}[\lambda_{t}X_{t}^{(a)}|\mathcal{F}_{s}]$
$\displaystyle=\lambda_{s}X_{s}^{(a)}+\mathbb{E}\left[\int_{s}^{t}\lambda_{u-}dX_{u}^{(a)}\Big{|}\mathcal{F}_{s}\right]+\mathbb{E}\left[\int_{s}^{t}X_{u}^{(a)}d\lambda_{u}\Big{|}\mathcal{F}_{s}\right]$
$\displaystyle=\lambda_{s}X_{s}^{(a)}-(\beta-\alpha)\int_{s}^{t}\mathbb{E}[X_{u}^{(a)}\lambda_{u}|\mathcal{F}_{s}]du+\beta\lambda_{0}(t-s)X_{s}^{(a)}.$

So, writing
$h_{s}^{(a)}(t):=\mathbb{E}[\lambda_{t}X_{t}^{(a)}|\mathcal{F}_{s}]$ we have
$h_{s}^{(a)}(t)=\lambda_{s}X_{s}^{(a)}-(\beta-\alpha)\int_{s}^{t}h_{s}^{(a)}(u)du+\beta\lambda_{0}(t-s)X_{s}^{(a)}$.
Taking the derivative

$\displaystyle
h_{s}^{(a)}(t)^{\prime}=-(\beta-\alpha)h_{s}^{(a)}(t)+\beta\lambda_{0}X_{s}^{(a)}.$

Solving the differential equation with the initial condition
$h_{s}(s)=\lambda_{s}X_{s}^{(a)}$ we obtain

$\displaystyle
h_{s}^{(a)}(t)=\mathbb{E}\left[X_{t}^{(a)}\lambda_{t}|\mathcal{F}_{s}\right]=X_{s}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}X_{s}^{(a)}}{\beta-\alpha}.$

∎

######  Proposition A.4.

Let $\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$ and define the
forward variance by
$\xi_{s}^{(a)}(t):=\mathbb{E}^{\mathbb{Q}(a)}[v_{t}|\mathcal{F}_{s}]$,
$0\leqslant s\leqslant t\leqslant T$. Then,

  1. 1.

The explicit expression is given by

$\displaystyle\xi_{s}^{(a)}(t)=$ $\displaystyle\
D_{1}^{(a)}(t-s)v_{s}+D_{2}^{(a)}(t-s)\lambda_{s}+D_{3}^{(a)}(t-s),$

where

$\displaystyle D_{1}^{(a)}(h):=e^{-\kappa^{(a)}h},\hskip
28.45274ptD_{2}^{(a)}(h)=C_{1}^{(a)}\left(e^{-(\beta-\alpha)h}-e^{-\kappa^{(a)}h}\right),$
$\displaystyle
D_{3}^{(a)}(h):=\left(\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right)e^{-\kappa^{(a)}h}-\frac{C_{2}^{(a)}}{\beta-\alpha}e^{-(\beta-\alpha)h}+C_{3}^{(a)},$
$\displaystyle
C_{1}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]}{\kappa^{(a)}-(\beta-\alpha)},\hskip
28.45274ptC_{2}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\kappa^{(a)}-(\beta-\alpha)},\hskip
28.45274ptC_{3}^{(a)}:=\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\kappa^{(a)}(\beta-\alpha)}+\bar{v}^{(a)}.$

  2. 2.

The dynamics of the process $t\to\xi_{s}^{(a)}(t)$ is given by

$\displaystyle
d\xi_{s}^{(a)}(t)=\left[-\kappa^{(a)}\left(\xi_{s}^{(a)}(t)-\bar{v}^{(a)}\right)+\eta\mathbb{E}[J_{1}]\left(\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\right]dt.$
(A.7)

  3. 3.

The dynamics of the process $s\to\xi_{s}(t)$ is given by

$\displaystyle d\xi_{s}^{(a)}(t)=$ $\displaystyle\ \sigma
D_{1}^{(a)}(t-s)\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}+\eta
D_{1}^{(a)}(t-s)\left(dL_{s}-\mathbb{E}[J_{1}]\lambda_{s}ds\right)$
$\displaystyle+\alpha D_{2}^{(a)}(t-s)\left(dN_{s}-\lambda_{s}ds\right).$

###### Proof.

First we prove (2), then (1) and (3).

(2) We can write

$\displaystyle
v_{t}=v_{s}-\kappa^{(a)}\int_{s}^{t}(v_{u}-\bar{v}^{(a)})du+\sigma\int_{s}^{t}\sqrt{v_{u}}dW_{u}^{\mathbb{Q}(a)}+\eta(L_{t}-L_{s}),$

where $\kappa^{(a)}=\kappa+a\sigma$ and
$\bar{v}^{(a)}=\frac{k\bar{v}}{k+a\sigma}$. By Theorem 4.3.
$\mathbb{E}^{\mathbb{Q}(a)}\left[\exp\left(\frac{\rho^{2}}{2}\int_{0}^{T}v_{u}du\right)\right]<\infty$.
In particular,
$\mathbb{E}^{\mathbb{Q}(a)}\left[\int_{0}^{T}v_{s}ds\right]<\infty$ and the
process $t\to\int_{0}^{t}\sqrt{v_{u}}dW_{u}^{\mathbb{Q}(a)}$ is a
$(\mathcal{F},\mathbb{Q}(a))$-martingale. Therefore,

$\displaystyle\xi_{s}^{(a)}(t)=v_{s}-\kappa^{(a)}\int_{s}^{t}\left(\xi_{s}^{(a)}(u)-\bar{v}^{(a)}\right)du+\sigma\int_{0}^{s}\sqrt{v_{u}}dW_{u}^{\mathbb{Q}(a)}+\eta(\mathbb{E}^{\mathbb{Q}(a)}[L_{t}|\mathcal{F}_{s}]-L_{s}).$
(A.8)

By Theorem 4.3, $X^{(a)}$ is a $(\mathcal{F},\mathbb{P})$-martingale and using
[6, Lemma A.8(2)]

$\displaystyle\mathbb{E}^{\mathbb{Q}(a)}[L_{t}|\mathcal{F}_{s}]$
$\displaystyle=\frac{\mathbb{E}[L_{t}X^{(a)}_{T}|\mathcal{F}_{s}]}{\mathbb{E}[X^{(a)}_{T}|\mathcal{F}_{s}]}$
$\displaystyle=\frac{\mathbb{E}[\mathbb{E}[L_{t}X^{(a)}_{T}|\mathcal{F}_{t}]|\mathcal{F}_{s}]}{X^{(a)}_{s}}$
$\displaystyle=\frac{\mathbb{E}[L_{t}X^{(a)}_{t}|\mathcal{F}_{s}]}{X^{(a)}_{s}}$
$\displaystyle=L_{s}+\mathbb{E}[J_{1}]\int_{s}^{t}\frac{\mathbb{E}[\lambda_{u}X_{u}^{(a)}|\mathcal{F}_{s}]}{X_{s}^{(a)}}du.$

Inserting the last expression in (A.8) and using Lemma 6.1(2) we get

$\displaystyle d\xi_{s}^{(a)}(t)$
$\displaystyle=\left[-\kappa^{(a)}\left(\xi_{s}^{(a)}(t)-\bar{v}^{(a)}\right)+\eta\mathbb{E}[J_{1}]\frac{\mathbb{E}[\lambda_{t}X_{t}^{(a)}|\mathcal{F}_{s}]}{X_{s}^{(a)}}\right]dt$
$\displaystyle=\left[-\kappa^{(a)}\left(\xi_{s}^{(a)}(t)-\bar{v}^{(a)}\right)+\eta\mathbb{E}[J_{1}]\left(\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\right]dt.$

This proves (A.7).

(1) The solution of the previous ODE is given by

$\displaystyle\xi_{s}^{(a)}(t)=e^{-\kappa^{(a)}(t-s)}v_{s}+e^{-\kappa^{(a)}t}\int_{s}^{t}e^{\kappa^{(a)}u}\left(\kappa^{(a)}\bar{v}^{(a)}+f_{s}(u)\right)du$

where

$\displaystyle
f_{s}(t)=\eta\mathbb{E}[J_{1}]\left(\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)e^{-(\beta-\alpha)(t-s)}+\frac{\beta\lambda_{0}}{\beta-\alpha}\right).$

Now

$\displaystyle\int_{s}^{t}e^{\kappa^{(a)}u}\kappa^{(a)}\bar{v}^{(a)}du=\bar{v}^{(a)}\left(e^{\kappa^{(a)}t}-e^{\kappa^{(a)}s}\right)$

and

$\displaystyle\int_{s}^{t}e^{\kappa^{(a)}u}f_{s}(u)du=\eta\mathbb{E}[J_{1}]\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\int_{s}^{t}e^{\kappa^{(a)}u-(\beta-\alpha)(u-s)}du+\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\beta-\alpha}\int_{s}^{t}e^{\kappa^{(a)}u}du$
$\displaystyle=C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left(e^{\kappa^{(a)}t-(\beta-\alpha)(t-s)}-e^{\kappa^{(a)}s}\right)+\left(C_{3}^{(a)}-\bar{v}^{(a)}\right)\left(e^{\kappa^{(a)}t}-e^{\kappa^{(a)}s}\right).$

We conclude that

$\displaystyle\xi_{s}^{(a)}(t)=$ $\displaystyle\
e^{-\kappa^{(a)}(t-s)}v_{s}+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)$
(A.9) $\displaystyle+C_{3}^{(a)}\left(1-e^{-\kappa^{(a)}(t-s)}\right)$
$\displaystyle=$ $\displaystyle\
D_{1}^{(a)}(t-s)v_{s}+D_{2}^{(a)}(t-s)\lambda_{s}+D_{3}^{(a)}(t-s),$

where one can check that

$\displaystyle-C_{1}^{(a)}$
$\displaystyle\frac{\beta\lambda_{0}}{\beta-\alpha}\left(e^{-(\beta-\alpha)h}-e^{-\kappa^{(a)}h}\right)+C_{3}^{(a)}\left(1-e^{-\kappa^{(a)}h}\right)=$
$\displaystyle=\left(\frac{C_{1}^{(a)}\beta\lambda_{0}}{\beta-\alpha}-C_{3}^{(a)}\right)e^{-\kappa^{(a)}h}-\frac{C_{1}^{(a)}\beta\lambda_{0}}{\beta-\alpha}e^{-(\beta-\alpha)h}+C_{3}^{(a)}$
$\displaystyle=\left(\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right)e^{-\kappa^{(a)}h}-\frac{C_{2}^{(a)}}{\beta-\alpha}e^{-(\beta-\alpha)h}+C_{3}^{(a)}=D_{3}^{(a)}(h).$

(3) For convenience we take the explicit expression of $\xi_{s}^{(a)}(t)$
given in (A.9), that is

$\displaystyle\xi_{s}^{(a)}(t)=$ $\displaystyle\
e^{-\kappa^{(a)}(t-s)}v_{s}+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)$
$\displaystyle+C_{3}^{(a)}\left(1-e^{-\kappa^{(a)}(t-s)}\right).$

Then,

$\displaystyle d\xi_{s}^{(a)}(t)=$ $\displaystyle\
\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}v_{s}ds+e^{-\kappa^{(a)}(t-s)}dv_{s}+C_{1}^{(a)}\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)d\lambda_{s}$
$\displaystyle+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left[\left(\beta-\alpha\right)e^{-(\beta-\alpha)(t-s)}-\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}\right]ds$
$\displaystyle-C_{3}^{(a)}\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}ds$
$\displaystyle=$ $\displaystyle\
\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}v_{s}ds+e^{-\kappa^{(a)}(t-s)}\left[-\kappa^{(a)}\left(v_{s}-\bar{v}^{(a)}\right)ds+\sigma\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}+\eta
dL_{s}\right]$
$\displaystyle+C_{1}^{(a)}\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)\left[-\beta(\lambda_{s}-\lambda_{0})ds+\alpha
dN_{s}\right]$
$\displaystyle+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left[\left(\beta-\alpha\right)e^{-(\beta-\alpha)(t-s)}-\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}\right]ds$
$\displaystyle-C_{3}^{(a)}\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}ds.$ (A.10)

Note that the $ds$ coefficients are

$\displaystyle\
\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}v_{s}-e^{-\kappa^{(a)}(t-s)}\kappa^{(a)}\left(v_{s}-\bar{v}^{(a)}\right)+C_{1}^{(a)}\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)\left[-\beta(\lambda_{s}-\lambda_{0})\right]$
$\displaystyle+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\left[\left(\beta-\alpha\right)e^{-(\beta-\alpha)(t-s)}-\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}\right]-C_{3}^{(a)}\kappa^{(a)}e^{-\kappa^{(a)}(t-s)}.$
(A.11)

One can check that the expression in (A.2) is, indeed, equal to,

$\displaystyle-\alpha
C_{1}^{(a)}\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)\lambda_{s}-\eta\mathbb{E}[J_{1}]e^{-\kappa^{(a)}(t-s)}\lambda_{s}.$
(A.12)

Then, the dynamics of $s\to\xi_{s}(t)$ in (A.2) can be written as

$\displaystyle d\xi_{s}^{(a)}(t)=$ $\displaystyle\ \sigma
e^{-\kappa^{(a)}(t-s)}\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}++\eta
e^{-\kappa^{(a)}(t-s)}\left(dL_{s}-\mathbb{E}[J_{1}]\lambda_{s}ds\right)$
$\displaystyle+\alpha
C_{1}^{(a)}\left(e^{-(\beta-\alpha)(t-s)}-e^{-\kappa^{(a)}(t-s)}\right)\left(dN_{s}-\lambda_{s}ds\right)$
$\displaystyle=$ $\displaystyle\ \sigma
D_{1}^{(a)}(t-s)\sqrt{v_{s}}dW_{s}^{\mathbb{Q}(a)}+\eta
D_{1}^{(a)}(t-s)\left(dL_{s}-\mathbb{E}[J_{1}]\lambda_{s}ds\right)$
$\displaystyle+\alpha D_{2}^{(a)}(t-s)\left(dN_{s}-\lambda_{s}ds\right),$

which would finish the proof. We now check that the expressions in (A.2) and
(A.12) are the same. Note that the coefficients of $e^{-(\beta-\alpha)(t-s)}$
in (A.2) are

$\displaystyle\
C_{1}^{(a)}\left[-\beta(\lambda_{s}-\lambda_{0})\right]+C_{1}^{(a)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)(\beta-\alpha)=-\alpha
C_{1}^{(a)}\lambda_{s},$

which is equal to the coefficients of $e^{-(\beta-\alpha)(t-s)}$ in (A.12).
The coefficients of $e^{-\kappa^{(a)}(t-s)}$ in (A.2) are

$\displaystyle\
\kappa^{(a)}\bar{v}^{(a)}+\frac{\eta\mathbb{E}[J_{1}]}{\kappa^{(a)}-(\beta-\alpha)}\beta(\lambda_{s}-\lambda_{0})-\frac{\eta\mathbb{E}[J_{1}]}{\kappa^{(a)}-(\beta-\alpha)}\left(\lambda_{s}-\frac{\beta\lambda_{0}}{\beta-\alpha}\right)\kappa^{(a)}$
$\displaystyle-\left(\frac{\eta\mathbb{E}[J_{1}]\beta\lambda_{0}}{\kappa^{(a)}(\beta-\alpha)}+\bar{v}^{(a)}\right)\kappa^{(a)}$
$\displaystyle=$ $\displaystyle\
\frac{\eta\mathbb{E}[J_{1}]}{\left[\kappa^{(a)}-(\beta-\alpha)\right](\beta-\alpha)}\left[\beta(\lambda_{s}-\lambda_{0})(\beta-\alpha)-\lambda_{s}\kappa^{(a)}(\beta-\alpha)+\beta\lambda_{0}\kappa^{(a)}-\beta\lambda_{0}\left(\kappa^{(a)}-(\beta-\alpha)\right)\right]$
$\displaystyle=$ $\displaystyle\
\frac{C_{1}^{(a)}}{\beta-\alpha}\left[\beta(\lambda_{s}-\lambda_{0})(\beta-\alpha)-\lambda_{s}\kappa^{(a)}(\beta-\alpha)+\beta\lambda_{0}(\beta-\alpha)\right]$
$\displaystyle=$ $\displaystyle\
C_{1}^{(a)}\left[\beta(\lambda_{s}-\lambda_{0})-\lambda_{s}\kappa^{(a)}+\beta\lambda_{0}\right]$
$\displaystyle=$ $\displaystyle\
C_{1}^{(a)}\left[\alpha\lambda_{s}-\alpha\lambda_{s}+\beta(\lambda_{s}-\lambda_{0})-\lambda_{s}\kappa^{(a)}+\beta\lambda_{0}\right]$
$\displaystyle=$ $\displaystyle\ \alpha
C_{1}^{(a)}\lambda_{s}-\frac{\eta\mathbb{E}[J_{1}]}{\kappa^{(a)}-(\beta-\alpha)}\left(\kappa^{(a)}-(\beta-\alpha)\right)\lambda_{s}$
$\displaystyle=$ $\displaystyle\ \alpha
C_{1}^{(a)}\lambda_{s}-\eta\mathbb{E}[J_{1}]\lambda_{s}.$

This proves that the expressions in (A.2) and (A.12) are the same. ∎

######  Proposition A.5.

Let $0\leqslant s\leqslant t\leqslant T$ and
$\mathbb{Q}(a)\in\mathcal{E}_{m}(Q_{1},2+\varepsilon_{1})$. Then, the explicit
expression of $V_{s}^{(a)}(t)$ is given by

$\displaystyle V_{s}^{(a)}(t)=$ $\displaystyle\
K_{1}^{(a)}(t-s)v_{s}+K_{2}^{(a)}(t-s)\lambda_{s}+K_{3}^{(a)}(t-s),$

where

$\displaystyle K_{1}^{(a)}(h):=A_{\kappa^{(a)}}(h),\hskip
28.45274ptK_{2}^{(a)}(h)=C_{1}^{(a)}\left[A_{\beta-\alpha}(h)-A_{\kappa^{(a)}}(h)\right],$
$\displaystyle
K_{3}^{(a)}(h):=\left[\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right]A_{\kappa^{(a)}}(h)-\frac{C_{2}^{(a)}}{\beta-\alpha}A_{\beta-\alpha}(h)+C_{3}^{(a)}$

where $C_{1}^{(a)},C_{2}^{(a)}$ and $C_{3}^{(a)}$ are given in (6.2).

###### Proof.

Using the explicit expression of $\xi_{s}^{(a)}$ given in (6.1) and the
Definition 6.4 we obtain

$\displaystyle V_{s}^{(a)}(t)=$ $\displaystyle\
\frac{1}{t-s}\int_{s}^{t}\xi_{s}^{(a)}(u)du$ $\displaystyle=$ $\displaystyle\
\frac{1}{t-s}\int_{s}^{t}D_{1}^{(a)}(u-s)duv_{s}+\frac{1}{t-s}\int_{s}^{t}D_{2}^{(a)}(u-s)du\lambda_{s}+\frac{1}{t-s}\int_{s}^{t}D_{3}^{(a)}(u-s)du$
$\displaystyle=$ $\displaystyle\
A_{\kappa^{(a)}}(t-s)v_{s}+C_{1}^{(a)}\left[A_{\beta-\alpha}(t-s)-A_{\kappa^{(a)}}(t-s)\right]\lambda_{s}$
$\displaystyle+\left[\frac{C_{2}^{(a)}}{\kappa^{(a)}}-\bar{v}^{(a)}\right]A_{\kappa^{(a)}}(t-s)-\frac{C_{2}^{(a)}}{\beta-\alpha}A_{\beta-\alpha}(t-s)+C_{3}^{(a)}$
$\displaystyle=$ $\displaystyle\
K_{1}^{(a)}(t-s)v_{s}+K_{2}^{(a)}(t-s)\lambda_{s}+K_{3}^{(a)}(t-s).$

∎

######  Lemma A.6.

Let $k_{1},k_{2},h>0$ with $k_{1}\neq k_{2}$. Then,

$\displaystyle\frac{A_{k_{2}}(h)-A_{k_{1}}(h)}{k_{1}-k_{2}}>0.$

###### Proof.

We can assume without loss of generality that $k_{1}>k_{2}$. Now,

$\displaystyle\frac{A_{k_{2}}(h)-A_{k_{1}}(h)}{k_{1}-k_{2}}=\frac{1}{h}\int_{0}^{h}\frac{e^{-k_{2}u}-e^{-k_{1}u}}{k_{1}-k_{2}}du>\frac{1}{h}\int_{0}^{h}\frac{e^{-k_{1}u}-e^{-k_{1}u}}{k_{1}-k_{2}}du=0.$

This finishes the proof. ∎

## References

  * [1] Aurélien Alfonsi  “Affine diffusions and related processes: simulation, theory and applications”  Springer, Cham, 2015
  * [2] David Applebaum  “Lévy processes and stochastic calculus”  Cambridge University Press, 2009
  * [3] Yacine Aït-Sahalia, Julio Cacho-Diaz and Roger J.A. Laeven  “Modeling financial contagion using mutually exciting jump processes”  In _J. Financ. Econ._ 117.3, 2015, pp. 585–606
  * [4] Emmanuel Bacry, Iacopo Mastromatteo and J.F. Muzy  “Hawkes Processes in Finance”  In _Market Microstruct. Liq._ 01.01, 2015, pp. 1550005  DOI: 10.1142/S2382626615500057
  * [5] David S. Bates  “Post-’87 crash fears in the S&P 500 futures option market”  In _J. Econom._ 94.1, 2000, pp. 181–238  DOI: https://doi.org/10.1016/S0304-4076(99)00021-4
  * [6] David Baños, Salvador Ortiz-Latorre and Oriol Zamora  “Thiele’s PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model”, 2023  arXiv:2309.03541
  * [7] David Baños, Salvador Ortiz-Latorre and Oriol Zamora  “Change of measure in a Heston-Hawkes stochastic volatility model”  In _To appear in Adv. in Appl. Probab._ 56.4, 2024  arXiv:2210.15343
  * [8] B. Bibby and M. Sørensen  “A hyperbolic diffusion model for stock prices”  In _Finance Stoch._ 1, 1996, pp. 25–41  DOI: 10.1007/s007800050015
  * [9] Menachem Brenner and Dan Galai  “New Financial Instruments for Hedging Changes in Volatility”  In _Financ. Anal. J._ 45.4 CFA Institute, 1989, pp. 61–65  URL: http://www.jstor.org/stable/4479241
  * [10] Menachem Brenner and Dan Galai  “Hedging volatility in foreign currencies”  In _J. Deriv._ 1, 1993, pp. 53–59
  * [11] Peter Carr and Roger Lee  “Volatility Derivatives”  In _Annu. Rev. Financial Econ._ 1, 2009, pp. 319–339  DOI: 10.1146/annurev.financial.050808.114304
  * [12] Peter Carr and Liuren Wu  “A Tale of Two Indices”  In _J. Deriv._ 13.3, 2006, pp. 13–29
  * [13] Jun Cheng, Meriton Ibraimi, Markus Leippold and Jin E. Zhang  “A remark on Lin and Chang’s paper ‘Consistent modeling of S&P 500 and VIX derivatives”’  In _J. Econ. Dyn. Control_ 36.5, 2012, pp. 708–715  DOI: https://doi.org/10.1016/j.jedc.2012.01.002
  * [14] San-Lin Chung, Wei-Che Tsai, Yaw-Huei Wang and Pei-Shih Weng  “The information content of the S&P 500 index and VIX options on the dynamics of the S&P 500 index”  In _J. Futures Mark._ 31.12, 2011, pp. 1170–1201
  * [15] Lirong Cui, Alan Hawkes and He Yi  “An elementary derivation of moments of Hawkes processes”  In _Adv. in Appl. Probab._ 52.1, 2020, pp. 102–137  DOI: 10.1017/apr.2019.53
  * [16] Jin-Chuan Duan and Chung-Ying Yeh  “Jump and volatility risk premiums implied by VIX”  In _J. Econom. Dynam. Control_ 34.11, 2010, pp. 2232–2244  DOI: 10.1016/j.jedc.2010.05.006
  * [17] Darrell Duffie, Jun Pan and Kenneth Singleton  “Transform Analysis and Asset Pricing for Affine Jump-Diffusions”  In _Econometrica_ 68.6 [Wiley, Econometric Society], 2000, pp. 1343–1376  URL: http://www.jstor.org/stable/3003992
  * [18] Philip Hartman  “Ordinary differential equations”  Birkhäuser, Boston, 1982
  * [19] Thibault Jaisson  “Market impact as anticipation of the order flow imbalance”  In _Quant. Finance_ 15.7, 2015, pp. 1123–1135  DOI: 10.1080/14697688.2015.1013148
  * [20] Bo Jing, Shenghong Li and Yong Ma  “Pricing VIX options with volatility clustering”  In _J. Futures Mark._ 40.6, 2020, pp. 928–944
  * [21] Jan Kallsen, Johannes Muhle-Karbe and Moritz Voß  “Pricing options on variance in affine stochastic volatility models”  In _Math. Finance_ 21.4, 2011, pp. 627–641
  * [22] Alan L. Lewis  “Option valuation under stochastic volatility: With Mathematica code”  Finance Press, Newport Beach, CA, 2000
  * [23] Guang-Hua Lian and Song-Ping Zhu  “Pricing VIX options with stochastic volatility and random jumps”  In _Decis. Econ. Finance_ 36.1, 2013, pp. 71–88  DOI: 10.1007/s10203-011-0124-0
  * [24] Guanghua Lian and Song-Ping Zhu  “Pricing Vix Options with Stochastic Volatility and Random Jumps”  In _Decis. Econ. Finance_ 36, 2013, pp. 71–88
  * [25] Yueh-Neng Lin  “Pricing VIX Futures: Evidence from Integrated Physical and Risk-Neutral Probability Measures”  In _J. Futures Mark._ 27, 2007, pp. 1175 –1217  DOI: 10.1002/fut.20291
  * [26] Thomas Josef Liniger  “Multivariate Hawkes processes”  ETH, 2009  DOI: 10.3929/ethz-a-006037599
  * [27] Zhongjin Lu and Yingzi Zhu  “Volatility components: The term structure dynamics of VIX futures”  In _J. Futures Mark._ 30.3, 2010, pp. 230–256
  * [28] Alexander D. Poularikas  “Transforms and applications handbook”  CRC Press, Boca Raton, FL, 2010
  * [29] Dimitris Psychoyios, George Dotsis and Raphael Markellos  “A jump diffusion model for VIX volatility options and futures”  In _Rev. Quant. Finance Account_ 35, 2010, pp. 245–269  DOI: 10.1007/s11156-009-0153-8
  * [30] Tina Hviid Rydberg  “Generalized hyperbolic diffusion processes with applications in finance”  In _Math. Finance_ 9.2, 1999, pp. 183–201  DOI: 10.1111/1467-9965.00067
  * [31] Artur Sepp  “VIX Option Pricing in a Jump-Diffusion Model”  In _Risk_ 21, 2008
  * [32] Artur Sepp  “Pricing Options on Realized Variance in The Heston Model With Jumps In Returns and Volatility II: An Approximate Distribution of the Discrete Variance”  In _J. Comput. Finance_ 16, 2011  DOI: 10.21314/JCF.2012.240
  * [33] Harvey J. Stein  “Joining Risks and Rewards”  In _SSRN Electronic Journal_ , 2013, pp. Available at SSRN: 2368905
  * [34] Harvey J. Stein  “Fixing risk neutral risk measures”  In _Int. J. Theor. Appl. Finance_ 19.3, 2016, pp. 1650021  DOI: 10.1142/S0219024916500217
  * [35] Robert E. Whaley  “Derivatives on market volatility: hedging tools long overdue”  In _J. Deriv._ 1.1, 1993, pp. 71–84
  * [36] Wuyi Ye, Bin Wu and Pengzhan Chen  “Pricing VIX derivatives using a stochastic volatility model with a flexible jump structure”  In _Probab. Engrg. Inform. Sci._ 37.1, 2023, pp. 245–274  DOI: 10.1017/S0269964821000577
  * [37] Jin Zhang and Yuqin Huang  “The CBOE S&P 500 three-month variance futures”  In _J. Futures Mark._ 30, 2008, pp. 48 –70  DOI: 10.1002/fut.20400
  * [38] Jin E. Zhang, Jinghong Shu and Menachem Brenner  “The new market for volatility trading”  In _J. Futures Mark._ 30.9, 2010, pp. 809–833
  * [39] Song-Ping Zhu and Guang-Hua Lian  “An analytical formula for VIX futures and its applications”  In _J. Futures Mark._ 32.2, 2012, pp. 166–190