# Pricing and delta computation in jump-diffusion models with stochastic
intensity by Malliavin calculus

Ayub Ahmadi, Mahdieh Tahmasebi111Corresponding author. E-mail addresses:
<EMAIL_ADDRESS>(M.Tahmasebi<EMAIL_ADDRESS>(A.Ahmadi).
${}^{\mbox{}}$Department of Applied Mathematics, Faculty of Mathematics,
Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

Abstract: In this paper, the pricing of financial derivatives and the
calculation of their delta Greek are investigated as the underlying asset is a
jump-diffusion process in which the stochastic intensity component follows the
CIR process. Utilizing Malliavin derivatives for pricing financial derivatives
and challenging to find the Malliavin weight for accurately calculating delta
will be established in such models. Due to the dependence of asset price on
the information of the intensity process, conditional expectation attribute to
show boundedness of moments of Malliavin weights and the underlying asset is
essential. Our approach is validated through numerical experiments,
highlighting its effectiveness and potential for risk management and hedging
strategies in markets characterized by jump and stochastic intensity dynamics.

keywords:Malliavin calculus, stochastic intensity, delta computing, pricing of
derivatives, Bismuth-Elworthy-Li formula.

##  1 Introduction

Stochastic intensity is the most important attribute of financial markets
which represent the realism of the arrival rate of events in the models
better. Flexibility in randomness of intensity, in spite of constant
intensity, can remarkably capture the arrival of new information, changes in
the behavior of investors in the market and the arrival rate of jumps such as
market crashes, large price movements, or sudden changes in volatility.
Financial institutions, portfolio managers, and investors can use these models
to assess the likelihood and potential consequences of extreme events, leading
to more informed decision-making and the development of effective risk
mitigation strategies, see [Brigo and Alfonsi(2005)] and [Leung and
Kwok(2009)] respectively. In addition, the stochastic jump intensity to the
rate of default of firms for risk assessments and portfolio managements have
been exposed in [Feng(2017)] and [Lévy dit Véhel and Lévy Véhel(2018)].
The self-exciting point process in which the current intensity of events is
determined by events in the past is firstly introduced by [Hawkes(1971)]. The
crucial role of jumps with stochastic intensity in option pricing are
supported by the empirical results in [Fang(2000)] and its role in modeling
the jump intensity risk is supported in [Santa-Clara and Yan(2010)],
empirically. In Markov intensity models with discrete state, which the model
called Markov-modulated jump model, pricing of the risky underlying assets is
considered by [Elliott et al.(2007)], [Bo et al.(2010)], [Chang et al.(2013)]
and recently by [Shan et al.(2023)] in which the Markov-modulated jump
diffusion process is used to model the discrete dividend process in financial
markets. In the continuous framework, [Brigo and Alfonsi(2005)] have derived
an analytical formula for pricing of credit derivatives under CIR stochastic
intensity models. Later, [Brigo and El-Bachir(2006)] have considered a smile-
extended jump stochastic intensity to price credit default swaptions. Non-
Gaussian intensity models have been investigated by [Bianchi and
Fabozzi(2015)]. In 2019, the authors [Yang et al.(2019)] proposed these models
for variance exchange rate to price the variance swaps. This subject in [Huang
et al.(2014)] and in [Chang and Wang(2020)] and recently in [Ma et al.(2023)]
was dealt with option pricing under double exponential jump model with
stochastic volatility and stochastic intensity using Fourier transform.
On the other hand, Malliavin calculus is a sophisticated mathematical tool
that extends the traditional calculus to differentiate random variables and
quantify their sensitivities, the accurate calculation of delta and pricing of
financial derivatives, hedging strategies, and investment decision-making, see
for instance, [Alos and Ewald(2008)], [Hillairet et al.(2018)],
[Yilmaz(2018)], [Kuchuk-Iatsenko et al.(2016)], [Fournié et al.(2001)]. In
2004, [El-Khatib and Privault(2004)] have computed Greeks in a market driven
by a discontinuous process with Poisson jump times and random jump sizes using
the Malliavin calculus on Poisson space. Numerical simulations are presented
for the delta and gamma of Asian options, and confirm the efficiency of this
approach over classical finite difference Monte-Carlo approximations of
derivative. In [Huehne(2005)] the stochastic weights for the fast and accurate
computation of Greeks for options whose underlying is driven by a pure-jump
Levy process have been derived. Later, Bavouset and Messaoud discuss this
subject [Bavouzet and Messaoud(2006)] by both the Malliavin derivative with
respect to the jump amplitudes and to the Wiener process. Also, The
computation of delta with Malliavin calculus for options on the underlying
asset modeled by Levy processes are stated in [Mhlanga(2011)],
[Khedher(2012)], [Matchie(2009)], [Coffie et al.(2021)], as we refer the
readers to [Nunno et al.(2009)], [Nualart and Nualart(2018)] for more details
about the Malliavin calculus on Levy processes. Recently, sensitivity analysis
with respect to the stock price for singular SDEs is considered in [Coffie et
al.(2021)] and regularity of distribution-dependent SDEs with jump processes
is proved in [Song and Wang(2022)] by using Malliavin calculus. [Hudde and
Rüschendorf(2023)] have represented a closed-form expression for Asian Greeks
in an exponential Levy process model.
In general, in this article, we are interested in the jump-diffusion models
with stochastic intensity as follows in the CIR model, called self-exciting
Cox process. We will investigate the pricing of financial derivatives and will
derive an expression for the delta calculation by a Malliavin weight. In the
presence of the Malliavin derivative of the intensity in the Malliavin
derivative of the underlying process, some Wiener-direction which belongs to
the domain of Skorokhod operator in the Gaussian case is found, Theorem 3.3,
to be used in the duality formula appeared in calculating the delta and the
price of financial derivatives. Meanwhile, the use of conditional expectation
with respect to the information of the intensity is unavoidable. We should
point out here that there are two different approaches to define the Malliavin
derivative with respect to jump processes {chapters 10, 11 of [Nualart and
Nualart(2018)]}. So that we will find two different Skorokhod integrals as the
Malliavin weight associated to each approach in computation of delta.
This article is organized as follows: In section 2, we recall Malliavin
concepts on Wiener spaces and Poisson spaces. In section 3, we will introduce
the main model with a stochastic intensity process and check the necessities
to exist the solution of the model and to be bounded of its moments, We will
demonstrate the Malliavin derivative of the solution and find some direction
which its derivative is invertible in. In section 4, we will calculate the
delta and price of the European option . In section 5, We express the
numerical results and compare them with the finite difference method. Finally,
we introduce the fund that supported us and we present the conclusion.

##  2 A review on Malliavin calculus

Let us review some concepts of Malliavin calculus on Wiener space and in the
Poisson framework, See standard reference [Nualart and Nualart(2018)].

###  2.1 Malliavin calculus concepts on Wiener space

For a positive real number $T$, suppose that $\Omega:=C_{0}(\left[0,T\right])$
is the space of real continuous functions $w$ on $\left[0,T\right]$ with
$w(0)=0$ equipped with the uniform norm

$\rVert{w}\rVert_{\infty}=\sup_{t\in\left[0,T\right]}\lvert{w(t)}\rvert.$
(2.1)

Consider $\Big{(}\Omega,\mathcal{F},\mathcal{F}_{t},P\Big{)}$ as a filtered
probability space, with coordinate map $t\to{W(t,w)}$ for Brownian motion
$B(t)$ corresponding to the filtration $\\{\mathcal{F}_{t}\\}$. For every
$\gamma\in\Omega$ of the Cameron-Martin space, the set of the functions in the
form $\gamma(t)=\int_{0}^{t}g(s)ds$ for some $g\in{L^{2}}(\left[0,T\right])$,
and a random variable $F:\Omega\to{\mathbb{R}}$, the directional derivative of
$F$ in the $\gamma$ direction, have defined as the following form, if the
limit exists. In fact,

$D^{W}_{\gamma}{F}(w)=\frac{d}{d\epsilon}[F(w+\epsilon\gamma)]_{\epsilon=0}.$

If there exists some $\psi\in{L^{2}(\left[0,T\right]\times\Omega)}$ satisfying
the following equation

$D^{W}_{\gamma}{F}(w)=\int_{0}^{T}\psi(t,w).g(t)dt.$

the variable $F$ is Malliavin differentiable in Wiener space and
$D^{W}F=(D^{W}_{t}{F})_{0\leq{t}\leq{T}}:=(\psi(t,w))_{0\leq{t}\leq{T}}$. We
define the set of all $F:\Omega\to\mathbb{R}$ such that $F$ is differentiable
by $\mathbb{D}_{W}^{1,2}$. In fact, if we denote by $\mathcal{S}$ the set of
all functionals $F=\varphi(\theta_{1},\theta_{2},...,\theta_{n})$ where $\phi$
is a smooth function with bounded derivatives of any order and
$\theta_{i}=\int_{0}^{T}f_{i}(t)dB_{t}$ with
$f_{i}\in{L^{2}(\left[0,T\right])}$, Then $F\in\mathbb{D}_{W}^{1,2}$ and the
derivative of $F$ is

$D^{W}_{t}{F}(w)=\sum_{i=1}^{n}\frac{\partial{\varphi}}{\partial
x_{i}}(\theta_{1},....,\theta_{n})f_{i}(t).$

For every integer $n$ and $p\geq 2$, the space $\mathbb{D}_{W}^{n,p}$ is the
closure of $\mathcal{S}$ with respect to the norm defined by

$\lVert{F}\rVert_{n,p}=\lVert{F}\rVert_{L^{p}(\Omega)}+\lVert{(D^{W})^{n}{F}}\rVert_{L^{p}(\left[0,T\right]^{n}\times\Omega)}.$

The Skorohod operator is the adjoint operator of $D_{W}$ from
$L^{2}([0,T]\times\Omega)$ to $\mathbb{D}_{W}^{1,2}$. Later, we will use the
following duality relation, which states that for given $F\in\mathbb{D}^{1,2}$
and $u\in Dom(\delta^{W})$

$\mathbb{E}\Big{(}\left\langle{D^{W}F,u}\right\rangle_{L^{2}[0,T]}\Big{)}:=\mathbb{E}\Big{(}\int_{0}^{T}(D^{W}_{t}F)u_{t}dt\Big{)}=\mathbb{E}\Big{(}F\delta^{W}(u)\Big{)}.$

For every adapted process $u$, $\delta^{W}(u)$ can be represented by the
stochastic integral $\int_{0}^{T}u(s)dW_{s}$.

###  2.2 The Malliavin calculus on Poisson space

There are two different approach to introduce the Malliavin derivative of Levy
processes. One is introduced by the chaos expansion criteria which does not
satisfy in the rule chain, and the other is introduced on the closure of the
set of Poisson functionals that satisfies the chain rule. We recall some
concepts and for more details, we refer to [Nualart and Nualart(2018)].

####  2.2.1 First approach

Consider a Levy process $N$ with the Levy measure $v$ on a complete separable
metric space $(\mathbb{R}_{0},\mathcal{B})$. Let
$L^{2}([0,T]\times\mathbb{R}_{0}^{n})$ be the space of symmetric square
integrable functions on the $([0,T]\times\mathbb{R}_{0}^{n},m\times
v\times\cdots\times v)$, where $m$ is an atomless measure on $[0,T]$. Given
$h\in L^{2}([0,T]\times\mathbb{R}_{0}^{n})$ and fixed $z\in\mathbb{R}_{0}$, we
write $h(t,.,z)$ to indicate the function on $\mathbb{R}_{0}^{n-1}$ given by
$(z_{1},...,z_{n-1})\to h(t,z_{1},...,z_{n-1},z)$. Denote the set of random
variables $F$ in $L^{2}(\Omega)$ with a chaotic decomposition
$F=\sum_{n=0}^{\infty}I_{n}(h_{n})$ by $\mathbb{D}_{N}^{1,2}$, that $h_{n}\in
L_{s}^{2}([0,T]\times\mathbb{R}_{0}^{n})$, satisfying

$\sum_{n\geq
1}nn!\lVert{h_{n}}\rVert_{L^{2}([0,T]\times\mathbb{R}_{0}^{n})}^{2}<\infty.$

Then, if $F\in\mathbb{D}_{N}^{1,2}$ we define the Malliavin derivative $D^{N}$
of $F$ as the $L^{2}([0,T]\times\mathbb{R}_{0})$-valued random variable given
by

$D_{t,z}^{N}F=\sum_{n\geq 1}nI_{n-1}(h_{n}(t,.,z)),\;\;z\in\mathbb{R}_{0}.$

The operator $D^{N}$ is a closed operator from $\mathbb{D}_{N}^{1,2}\subset
L^{2}(\Omega)$ into $L^{2}(\Omega\times[0,T]\times\mathbb{R}_{0})$ and satisfy
the following rules.

######  Lemma 2.1.

[Nualart and Nualart(2018)] Let $F,G\in\mathbb{D}_{N}^{1,2}$ Suppose that
$FG\in L^{2}(\Omega)$ and $(F+D^{N}F)(G+D^{N}G)\in
L^{2}(\Omega\times[0,T]\times\mathbb{R}_{0})$.Then the product FG also belongs
to $\mathbb{D}_{N}^{1,2}$ and

$D_{t,z}^{N}(FG)=FD_{t,z}^{N}G+GD_{t,z}^{N}F+D_{t,z}^{N}FD_{t,z}^{N}G.$

######  proposition 2.2.

[Nualart and Nualart(2018)] Let $F$ be a random variable in $\mathbb{D}^{1,2}$
and let $\varphi$ be a real continuous function such that $\varphi(F)$ belongs
to $L^{2}(\Omega)$ and $\varphi(F+D^{N}F)$ belongs to $L^{2}(\Omega\times Z)$.
Then $\varphi(F)$ belongs to $\mathbb{D}^{1,2}$ and

$D_{t,z}^{N}\varphi(F)=\varphi(F+D_{t,z}^{N}F)-\varphi(F).$

Now, given stochastic process $u$ in
$L^{2}(\Omega\times[0,T]\times\mathbb{R}_{0})$ admits a unique representation
of the following form that for each $(t,z)\in[0,T]\times\mathbb{R}_{0}$

$u(t,z)=\sum_{n\geq 0}I_{n}(h_{n}(t,.,z)),$

where the function $h_{n}\in L^{2}([0,T]\times\mathbb{R}_{0}^{n})$. If

$\sum_{n\geq
0}(n+1)!\lVert{h_{n}}\rVert_{L(\mathbb{R}_{0}^{n+1})}^{2}<\infty,$

we say $u$ is in the domain of the divergence operator $\delta^{N}$, denoted
by $Dom\delta^{N}$ and

$\delta^{N}(u)=\sum_{n\geq 0}I_{n+1}(\tilde{h}_{n}),$

where $\tilde{h}_{n}$ stands for the symmetrization of $h$ as a function in
the last $n+1$ variables. For instance, if $u(z)=h(z)$ is a deterministic
function in $L^{2}(\mathbb{R}_{0})$ then $\delta(u)=I_{1}(h)$. If
$u(z)=I_{1}(h(.,z))$, with $h\in L^{2}(\mathbb{R}_{0})$, then
$\delta(u)=I_{2}(h)$.
The following result characterizes $\delta^{N}$ as the adjoint operator of
$D^{N}$.

######  proposition 2.3.

[Nualart and Nualart(2018)] If $u\in Dom\delta^{N}$, then $\delta^{N}(u)$ is
the unique element of $L^{2}(\Omega)$ such that, for all
$F\in\mathbb{D}_{N}^{1,2}$,

$\mathbb{E}(\left\langle{D^{N}F,u}\right\rangle_{L^{2}([0,T]\times\mathbb{R}_{0})})=\mathbb{E}(F\delta^{N}(u)).$

Conversely, if $u$ is a stochastic process in
$L^{2}(\Omega\times[0,T]\times\mathbb{R}_{0})$ such that, for some $G\in
L^{2}(\Omega)$ and for all $F\in\mathbb{D}_{N}^{1,2}$,

$\mathbb{E}(\left\langle{D^{N}F,u}\right\rangle_{L^{2}([0,T]\times\mathbb{R}_{0})})=\mathbb{E}(FG),$

then $u$ belongs to $Dom\delta^{N}$ and $\delta^{N}(u)=G$.

The divergence operator $\delta$ satisfies the following product rule.

######  proposition 2.4.

[Nualart and Nualart(2018)] Let $F\in\mathbb{D}_{N}^{1,2}$ and $u\in
Dom\delta$ such that the product $uDF$ belongs to $Dom\delta^{N}$ and the
right-hand side of (2.2) below belongs to $L^{2}(\Omega)$. Then $Fu\in
Dom\delta$ and

$\delta^{N}(Fu)=F\delta^{N}(u)-\left\langle{D^{N}F,u}\right\rangle_{L^{2}([0,T]\times\mathbb{R}_{0})}-\delta^{N}(uDF).$
(2.2)

####  2.2.2 Second approach

We make use of the notation

$N(h):=\int_{[0,T]}\int_{\mathbb{R}_{0}}h(t,z)N(dt.dz)$

for every $h\in L^{1}([0,T]\times\mathbb{R}_{0},m\times v)$. Denote by
$C_{0}^{0,2}([0,T]\times\mathbb{R}_{0})$ the set of continuous functions
$h:[0,T]\times\mathbb{R}_{0}\to\mathbb{R}$ that have compact support and are
twice differentiable on $\mathbb{R}_{0}$. We consider the set $\mathcal{S}$ of
cylindrical random variables of the form

$F=\varphi(N(h_{1}),...,N(h_{n})),$ (2.3)

where $\varphi\in C_{0}^{2}(\mathbb{R}^{n})$ and $h_{i}\in
C_{0}^{0,2}([0,T]\times\mathbb{R}_{0})$ for $1\leq i\leq n$. It is easy to
show that the set $\mathcal{S}$ is dense in $L^{2}(\Omega)$. The Malliavin
derivative of a simple random variable $F$ in $\mathcal{S}$ of the form (2.3)
is defined as the two parameter process

$D_{t,z}^{N_{P}}F=\sum_{k=1}^{n}\frac{\partial\varphi}{\partial
x_{k}}(N(h_{1}),...,N(h_{n}))\partial_{z}h_{k}(t,z),\;\quad(t,z)\in[0,T]\times\mathbb{R}_{0}.$

In particular, $D_{t,z}^{N_{P}}(N(h))=\partial_{z}h$. Define the scalar
product $<.,.>$ for every $u,\tilde{u}\in L^{2}(\Omega)$ as

$<u,\tilde{u}>_{N}:=\int_{0}^{T}\int_{\mathbb{R}_{0}}u(s,z)\tilde{u}(s,z)N(ds,dz),$

and denote $\|.\|_{N}$ as its associated norm. Also, let
$\mathbb{D}_{N_{p}}^{1,p}$, for every $p\geq 1$, the closure of $\mathcal{S}$,
as the domain of the operator $D_{t,z}^{N_{P}}$, with respect to the seminorm

$\|F\|_{1,N}^{p}:=\mathbb{E}(|F|^{p})+\mathbb{E}(\|D^{N_{P}}F\|^{p}_{N}).$

The next result is the chain rule for the Malliavin derivative in the Poisson
framework.

######  proposition 2.5.

[Nualart and Nualart(2018)] Let $\varphi$ be a function in $C^{1}(\mathbb{R})$
with bounded derivative, and let $F$ be a random variable in
$\mathbb{D}_{N_{p}}^{1,2}$. Then, $\varphi(F)$ belongs to
$\mathbb{D}_{N_{p}}^{1,2}$ and

$D_{t,z}^{N_{P}}(\varphi(F))=\varphi^{\prime}(F)D_{t,z}^{N_{P}}(F).$

The authors in [Song and Wang(2022)] have stated a powerful tool called
integration by parts formula for this type of derivative in the following form
in some Sobolev spaces we recall here. For every $p\geq 1$, denote by
$\mathbb{L}_{p}$ the set of all predictable processes $\psi$ on
$[0,T]\times\mathbb{R}_{0}$ with finite norm

$\|\psi\|_{\mathbb{L}_{p}}=\Big{[}\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{0}^{T}\psi(s,z)m(ds)\nu(dz)\Big{)}^{p}\Big{]}^{\frac{1}{p}}+\Big{[}\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{0}^{T}\psi^{p}(s,z)m(ds)\nu(dz)\Big{)}\Big{]}^{\frac{1}{p}},$

and denote by $\mathbb{V}_{p}$ the set of all predictable processes $\psi$ on
$[0,T]\times\mathbb{R}_{0}$ with finite norm

$\|\psi\|_{\mathbb{V}_{p}}=\|\frac{\partial\psi}{\partial
z}\|_{\mathbb{L}_{p}}+\|\rho\psi\|_{\mathbb{L}_{p}},$

where $\rho(z)=|z|^{-1}$. We shall write $\mathbb{V}_{\infty}:=\bigcap_{p\geq
1}\mathbb{V}_{p}$.

######  proposition 2.6.

Given $F\in\mathbb{D}_{N_{p}}^{1,p}$, for $p\geq 2$, and
$w_{0}\in\mathbb{V}_{\infty}$ we have

$\mathbb{E}\Big{(}<D^{N_{P}}(F),w_{0}>_{N}\Big{)}=\mathbb{E}\Big{(}F\int_{\mathbb{R}_{0}}\int_{0}^{T}\frac{1}{\theta}\frac{\partial(\theta
w_{0})}{\partial z}(t,z)\tilde{N}(dt,dz)\Big{)},$

where $\nu(dz)=\theta(z)dz$.

##  3 Stochastic jump processes with stochastic intensity

In this section, we recall the concept of stochastic intensity desired by
Bérmaud in Chapter 5 of [Brèmaud(2020)] and introduce the model and state the
assumptions and some lemmas we need in the main results.
Let $(\Omega,\mathcal{F},\mathit{P})$ be a Wiener-Poisson space with a risk
neutral probability $\mathit{P}$. Assume that $N_{t}$ is a Poisson process and
$\mathcal{F}_{t}^{N}$ is an $\sigma$-field generated by $N$ with the density
of jumps sizes $C_{z}$, as $z\in\mathbb{R}_{0}$ and stochastic intensity
process $\lambda$. For given $\sigma$-field $\mathcal{F}_{t}$, the process
$\lambda_{t}$ is an $\mathcal{F}_{t}$-intensity of $N_{t}$ if for every
$s,t\in[0,T]$

$\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{t}^{s}N(du,dz)|\mathcal{F}_{t}\Big{)}=\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{t}^{s}C_{z}\lambda_{u}dudz|\mathcal{F}_{t}\Big{)},$

and so that
$\tilde{N}(t,z)=N(t.z)-\int_{\mathbb{R}_{0}}\int_{0}^{t}C_{z}\lambda_{s}dsdz$
is an $\mathcal{F}_{t}$-martingale. Also, obviously, for every $0\leq t,s\leq
T$ and for every $\mathcal{F}_{t}$-predictable function $k$

$\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{t}^{s}k(u,z)N(du,dz)|\mathcal{F}_{t}\Big{)}=\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{t}^{s}k(u,z)C_{z}\lambda_{u}dudz|\mathcal{F}_{t}\Big{)}.$

We refer the reader to Chapter 5 of [Brèmaud(2020)] for more details. It is
worth mention that one can easily show [Brèmaud(2020)] that if $\lambda$ is
$\mathcal{G}$-measurable, for every measurable function $k$ such that
$\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{0}^{t}(k(t,s))^{2}\lambda_{s}dsC_{z}dz\Big{)}<\infty$,

$\mathbb{E}\Big{(}exp\\{iu\int_{\mathbb{R}_{0}}\int_{0}^{t}k(s,z)N(ds,dz)\\}\Big{|}\mathcal{G}\Big{)}=exp\Big{\\{}\int_{\mathbb{R}_{0}}\int_{0}^{t}(e^{iuk(t,z)}-1)\lambda_{s}C_{z}dsdz\Big{\\}}.$
(3.1)

In this manuscript, we assume that the underlying asset price
$S=(S_{t})_{t\in[0,T]}$ with the jump stochastic intensity process
$\lambda=(\lambda_{t})_{t\in[0,T]}$ of Poisson process $N_{t}$ can be governed
by the following system of SDEs:

$\begin{cases}dS_{t}&=\mu
S_{t}dt+\sigma_{1}S_{t}dW_{t}^{S}+\int_{\mathbb{R}_{0}}(e^{J_{t,z}}-1)S_{t}\tilde{N}(dt,dz),\\\
d\lambda_{t}&=\kappa(\Theta-\lambda_{t})dt+\sigma_{2}\sqrt{\lambda_{t}}dW_{t},\end{cases}$
(3.2)

where $(W_{t})_{t\in[0,T]}$ and $(W_{t}^{S})_{t\in[0,T]}$ are independent
Brownian motions, $N_{t}$ is independent of $W_{t}^{S}$, $\mu$ denotes the
riskless interest rate, $J$ is a cadlag function, the mean-reverting speed
parameter $\kappa,\sigma_{2}$ and $\sigma_{1}$ are positive constants and the
long term mean $\Theta$ satisfying $2\kappa\Theta\;>\sigma_{2}^{2}$.
For $\mathcal{F}_{t}^{\lambda}$, $\sigma$-field generated by $\lambda$, let
$\mathcal{F}_{t}=\mathcal{F}_{t}^{N}\vee\mathcal{F}_{t}^{\lambda}$ and
$\mathcal{G}=\mathcal{F}_{t}^{\lambda}$. In this case, obviously,
$\mathcal{F}_{t}=\mathcal{F}_{t}^{N}$ and for every $0\leq t,s\leq T$ and for
every $\mathcal{F}_{t}$-predictable function $k$

$\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{t}^{s}k(u,z)N(du,dz)|\mathcal{F}_{t}^{\lambda}\Big{)}=\int_{\mathbb{R}_{0}}\int_{t}^{s}k(u,z)C_{z}\lambda_{u}dudz.$

We also assume the following conditions throughout the paper.
Condition H1:

  * •

For every $p\geq 1$ and for almost everywhere $0\leq t\leq T$

$\int_{\mathbb{R}_{0}}(e^{J_{t,z}}-1)^{p}C_{z}dz\leq u_{p}<\infty,\qquad
v_{t}:=\int_{\mathbb{R}_{0}}(e^{J_{t,z}}-1)C_{z}dz\neq 0.$ (3.3)

  * •

There exists some $p_{0}\geq 2$ such that $2u_{p_{0}}\sigma_{2}^{2}\leq k$.

  * •

$\int_{0}^{T}\frac{1}{v_{u}^{2}}du<\infty.$

we know that the solution to the stochastic differential (3.2) is as follows,
see [Øksendal and Sulem(2019)].

$\displaystyle S_{t}$
$\displaystyle=S_{0}\exp\Big{\\{}(\mu-\frac{\sigma_{1}^{2}}{2})t+\sigma_{1}W_{t}^{S}+\int_{0}^{t}\int_{\mathbb{R}_{0}}(J_{s,z}-e^{J_{s,z}}+1)C_{z}\lambda_{s}dzds$
$\displaystyle+\int_{\mathbb{R}_{0}}\int_{0}^{t}J_{s,z}\tilde{N}(ds,dz)\Big{\\}}$
$\displaystyle=:S_{0}\exp\Big{\\{}X_{t}\Big{\\}}=:S_{0}Y_{t}\exp\Big{\\{}(\mu-\frac{\sigma_{1}^{2}}{2})t+\sigma_{1}W_{t}^{S}\Big{\\}},$

where $Y_{t}$ satisfying

$dY_{t}=Y_{t}(e^{J_{t,z}}-1)\tilde{N}(dt,dz),\qquad Y_{0}=1.$ (3.4)

It follows that this solution is in $L^{p}$-space for every $2\leq p\leq
p_{0}$, as we see in the following lemma.

######  Lemma 3.1.

The solution $S_{t}$ of (3.2) is unique and uniformly is in $\bigcap_{2\leq
p\leq p_{0}}L^{p}(\Omega)$, i.e., for every $2\leq p\leq p_{0}$,

$\mathbb{E}\Big{(}\sup_{0\leq t\leq T}|S_{t}|^{p}\Big{)}<\infty.$

###### Proof.

We know that for any $p\geq 2$,

$\mathbb{E}\Big{(}\sup_{t\in\left[0,T\right]}\exp\Big{\\{}p(\mu-\frac{\sigma_{1}^{2}}{2})t+p\sigma_{1}W_{t}^{S}\Big{\\}})<\infty.$

So, it is sufficient to show that equation (3.4) has a unique solution. To do
this, with the same proof of Lemma 2.3. in [Song and Zhang(2015)] and Section
5.1.1 of [Menaldi(2008)], we derive that for any $p\geq 2$ and every step time
$h$, there exists a constant $C_{p}>0$ such that:

$\displaystyle\mathbb{E}\biggl{(}\sup_{s\in\left[t,t+h\right]}\bigg{|}\int_{\mathbb{R}_{0}}\int_{t}^{s}$
$\displaystyle(e^{J_{u,z}}-1)Y_{u}\tilde{N}(du,dz)\bigg{|}^{p}|\mathcal{F}_{t}^{\lambda}\biggr{)}$
$\displaystyle\leq{C_{p}}\mathbb{E}\biggl{(}\bigg{[}\int_{t}^{t+h}\int_{\mathbb{R}_{0}}Y_{u}^{2}(e^{J_{u,z}}-1)^{2}C_{z}\lambda_{u}dzdu\bigg{]}^{\frac{p}{2}}|\mathcal{F}_{t}^{\lambda}\biggr{)}$
$\displaystyle+{C_{p}}\mathbb{E}\biggl{(}\bigg{[}\int_{t}^{t+h}\int_{\mathbb{R}_{0}}Y_{u}^{p}(e^{J_{u,z}}-1)^{p}C_{z}\lambda_{u}dzdu|\mathcal{F}_{t}^{\lambda}\biggr{)}.$

Define the new probability measure
$p_{1}(A)=\lambda_{s}1_{A}(s)/\int_{t}^{t+h}\lambda_{s}ds$, for every
$A\subset[t,t+h]$ as $1_{A}$ is the indicator function, and applying Young
inequality to result

$\displaystyle\mathbb{E}\biggl{(}\sup_{s\in\left[t,t+h\right]}Y_{s}^{p}|\mathcal{F}_{t}^{\lambda}\biggl{)}$
$\displaystyle\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}$
$\displaystyle+{C_{p}}\frac{1}{\int_{t}^{t+h}\lambda_{s}ds}\mathbb{E}\biggl{(}\bigg{[}\int_{t}^{t+h}\int_{\mathbb{R}_{0}}Y_{u}^{p}(e^{J_{u,z}}-1)^{p}C_{z}\lambda_{u}dzdu|\mathcal{F}_{t}^{\lambda}\biggr{)}$
$\displaystyle+{C_{p}}\mathbb{E}\biggl{(}\bigg{[}\int_{t}^{t+h}\int_{\mathbb{R}_{0}}Y_{u}^{p}(e^{J_{u,z}}-1)^{p}C_{z}\lambda_{u}dzdu|\mathcal{F}_{t}^{\lambda}\biggr{)}$
$\displaystyle\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}+{C_{p}}\frac{u_{p}}{\int_{t}^{t+h}\lambda_{s}ds}\int_{t}^{t+h}\mathbb{E}\biggl{(}\sup_{t\leq
s\leq u}Y_{s}^{p}|\mathcal{F}_{t}^{\lambda}\biggr{)}\lambda_{u}du$
$\displaystyle+{C_{p}}u_{p}\int_{t}^{t+h}\mathbb{E}\biggl{(}\sup_{t\leq s\leq
u}Y_{s}^{p}|\mathcal{F}_{t}^{\lambda}\biggr{)}\lambda_{u}du.$ (3.5)

Now apply Gronwall inequality for
$m(s)=\int_{t}^{s}\mathbb{E}\big{(}\sup_{t\leq s\leq
u}Y_{s}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}\lambda_{u}du$, $0\leq s\leq T$,
and derive that

$\displaystyle m(s)$
$\displaystyle\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}\int_{t}^{s}\lambda_{u}exp\Big{\\{}u_{p}\int_{u}^{s}\lambda_{r}(1+\frac{1}{\int_{t}^{s}\lambda_{v}dv})dr\Big{\\}}du$
$\displaystyle\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}e^{u_{p}}\int_{t}^{s}\lambda_{u}exp\Big{\\{}u_{p}\int_{u}^{s}\lambda_{r}dr\Big{\\}}du$
$\displaystyle\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}\frac{e^{u_{p}}}{u_{p}}exp\Big{\\{}u_{p}\int_{t}^{s}\lambda_{r}dr\Big{\\}}.$
(3.6)

Substituting (3.6) into (3.5) obtain

$\mathbb{E}\biggl{(}\sup_{s\in\left[t,t+h\right]}Y_{s}^{p}|\mathcal{F}_{t}^{\lambda}\biggl{)}\leq\mathbb{E}\big{(}Y_{t}^{p}|\mathcal{F}_{t}^{\lambda}\big{)}\biggl{(}1+C_{p}(1+\frac{1}{\int_{t}^{t+h}\lambda_{s}ds})exp\Big{\\{}u_{p}+u_{p}\int_{t}^{s}\lambda_{r}dr\Big{\\}})\biggr{)}.$
(3.7)

Therefore, in the sequence, we should show that the expectation of the right
hand side of equation (3.6), as $t=0$ and $2\leq p\leq p_{0}$, is bounded. For
this purpose, we note that using Ito formula, for a positive constant
$\gamma$, we have

$\displaystyle de^{\gamma\lambda_{t}}$
$\displaystyle=\gamma\kappa(\Theta-\lambda_{t})e^{\gamma\lambda_{t}}+\sigma_{2}\gamma
e^{\gamma\lambda_{t}}\sqrt{\lambda_{t}}dW_{t}+\frac{\sigma_{2}^{2}}{2}\gamma^{2}e^{\gamma\lambda_{t}}\lambda_{t}dt$
$\displaystyle=\kappa\Theta\gamma
e^{\gamma\lambda_{t}}dt+(\frac{\gamma^{2}\sigma_{2}^{2}}{2}-\kappa\gamma)\lambda_{t}e^{\gamma\lambda_{t}}dt+\sigma_{2}\gamma
e^{\gamma\lambda_{t}}\sqrt{\lambda_{t}}dW_{t}.$

Taking the expectation on both sides and applying Gronwall inequality to
deduce

$\mathbb{E}(e^{\gamma\lambda_{t}})\leq
e^{\gamma\lambda_{0}}+\kappa\Theta\gamma
e^{\gamma\lambda_{0}}e^{\kappa\Theta\gamma
t}=e^{\gamma\lambda_{0}}(1+\kappa\Theta\gamma e^{\kappa\Theta\gamma t}),$

Now, set $\gamma=u_{p_{0}}$ and $\gamma=2u_{p_{0}}$ in above inequality to use
in the expectation of the following equation.

$\int_{0}^{t}\kappa\Theta\gamma
e^{\gamma\lambda_{s}}ds=e^{\gamma\lambda_{t}}-e^{\gamma\lambda_{0}}+\int_{0}^{t}\kappa\gamma\lambda_{s}e^{\gamma\lambda_{s}}ds-\frac{\gamma^{2}\sigma_{2}^{2}}{2}\int_{0}^{t}\lambda_{s}e^{\gamma\lambda_{s}}ds-\gamma\sigma_{2}\int_{0}^{t}\sqrt{\lambda_{s}}e^{\gamma\lambda_{s}}dW_{s}.$

Thus,

$\displaystyle\kappa\Theta\gamma\mathbb{E}\big{(}\int_{0}^{t}e^{\gamma\lambda_{s}}ds\big{)}$
$\displaystyle\leq\mathbb{E}(e^{\gamma\lambda_{t}})-e^{\gamma\lambda_{0}}+\kappa\gamma\mathbb{E}\big{(}\int_{0}^{t}\lambda_{s}e^{\gamma\lambda_{s}}ds\big{)}$
$\displaystyle\leq\mathbb{E}(e^{\gamma\lambda_{t}})-e^{\gamma\lambda_{0}}+\kappa\gamma\int_{0}^{t}[2\mathbb{E}(\lambda_{s}^{2})+2\mathbb{E}(e^{2\gamma\lambda_{s}})]ds<\infty,$

which results that

$\mathbb{E}\big{(}exp\\{u_{p_{0}}\int_{0}^{t}\lambda_{s}ds\\}\big{)}\leq\int_{0}^{t}\mathbb{E}(exp\\{u_{p_{0}}\lambda_{s}\\})ds<\infty.$

This fact and taking the expectation from (3.7) complete the proof. The
inequality (3.7) will obviously prove the uniqueness of the solution. ∎

######  remark 3.2.

We mention that if $sup_{p\geq 2}u_{p}<\infty$, then Lemma 3.1 is held for
every $p\geq 2$.

In the last part of this section note that getting the partial derivatives of
$S_{t}$ with respect to $S_{0}$ show that the stochastic flow of $S_{t}$
exists and it is

$\displaystyle\frac{\partial{S_{t}}}{\partial{S_{0}}}=\frac{S_{t}}{S_{0}}=\exp\\{X_{t}\\}=Y_{t}\exp\\{(\mu-\frac{\sigma_{1}^{2}}{2})t+\sigma_{1}W_{t}\\}.$
(3.8)

Therefore, this flow is in $L^{p}$-space for every $2\leq p\leq p_{0}$.

###  3.1 Malliavin derivative of the solution on Wiener space

In this section, we obtain Malliavin derivative of the solution $S_{t}$ and
also we will consider some Skorokhod integrable directions in which the
inverse of directional derivatives are in $L^{p}(\Omega)$, for all $2\leq
p\leq p_{0}$.
Due to the representation of the solution with respect to $X_{t}$, it needs to
find its derivative. Gaussian Malliavin derivative of $X_{t}$ comes as
follows:

$D_{u}^{W}X_{t}=-\int_{u}^{t}D_{u}^{W}\lambda_{s}\int_{\mathbb{R}_{0}}(e^{J_{s,z}}-1)C_{z}dzds=:-\int_{u}^{t}v_{u}D_{u}^{W}\lambda_{s}ds,$
(3.9)

In [Altmayer and Neuenkirch(2015)], the authors have shown that using the Ito
formula and taking the Malliavin derivative with respect to the Brownian
motion, for every $s\leq t$

$D_{s}^{W}\lambda_{t}=\sigma_{2}\sqrt{\lambda_{t}}1_{0\leq s\leq
t}\exp\Big{\\{}-{\int_{s}^{t}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{r}})dr}\Big{\\}},$
(3.10) $\mathbb{E}(\sup_{0\leq t\leq T}\lambda_{t}^{p})<\infty\quad\forall
p\geq 1,\quad and\quad\sup_{0\leq t\leq
T}\mathbb{E}(\lambda_{t}^{-p})<\infty,\quad\forall p\geq
1~{}s.t.~{}2\kappa\theta>p\sigma_{2}^{2},$ (3.11)

where $C_{\sigma}=\frac{\kappa\Theta}{2}-\frac{\sigma_{2}^{2}}{8}$ is a
positive number. Here, we represent that the inverse of directed Malliavin
derivative of $X_{t}$ in some directions, which are also in the domain of
Skorokhod operator, can belong to all $L^{p}$ spaces for any $p\geq 2$.

######  Theorem 3.3.

When $2\kappa\theta>3\sigma_{2}^{2}$, there exists a direction $h(.)\in
dom(\delta^{W})$, defined as the following

$h(u)=\frac{1}{v_{u}}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{u}}),\qquad
0\leq u\leq T,$

such that $\mathcal{B}_{T}=\left\langle{D_{.}^{W}X_{T},h(.)}\right\rangle$ is
almost surely invertible and

$\Big{(}\left\langle
D_{.}^{W}X_{T},h(.)\right\rangle\Big{)}^{-1}\in\bigcap_{2\leq p}L^{p}.$

###### Proof.

From (3.9), Fubini theorem and the expression (3.10), we derive

$\displaystyle\mathcal{B}_{T}$ $\displaystyle=-\int_{0}^{T}\left\langle
v_{.}{D_{.}^{W}\lambda_{s},h(.)}\right\rangle ds$
$\displaystyle=-\int_{0}^{T}\Big{[}\sigma_{2}\sqrt{\lambda_{s}}exp\\{-\int_{0}^{s}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{r}})dr\\}\int_{0}^{s}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{u}})exp\\{\int_{0}^{u}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{r}})dr\\}du\Big{]}ds$
$\displaystyle=-\int_{0}^{T}\Big{[}\sigma_{2}\sqrt{\lambda_{s}}(1-exp\\{-\int_{0}^{s}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{r}})dr\\})\Big{]}ds.$
(3.12)

Applying the Gamma function results

$\displaystyle\mathbb{E}(\frac{1}{|\mathcal{B}_{T}|^{p}})=$
$\displaystyle\frac{1}{\Gamma(p)}\mathbb{E}\big{(}\int_{0}^{\infty}z^{p-1}e^{-z\int_{0}^{T}\Big{[}\sigma_{2}\sqrt{\lambda_{s}}(1-exp\\{-\int_{0}^{s}(\frac{\kappa}{2}+\frac{C_{\sigma}}{\lambda_{r}})dr\\})\Big{]}ds}dz\big{)}$
$\displaystyle\leq\frac{1}{\Gamma(p)}\mathbb{E}\big{(}\int_{0}^{\infty}z^{p-1}e^{-z(1-e^{-\frac{T\kappa}{2}})\int_{0}^{T}\sigma_{2}\sqrt{\lambda_{s}}ds}dz\big{)}$
$\displaystyle=\frac{1}{\sigma_{2}^{p}(1-e^{-\frac{T\kappa}{2}})^{p}}\mathbb{E}\big{(}\frac{1}{(\int_{0}^{T}\sqrt{\lambda_{s}}ds)^{p}}\big{)}<\infty,$

thanks to Lemma 5.2. of [Altmayer and Neuenkirch(2015)] in the last
inequality. So, we conclude that for every $p\geq 2$, $\mathcal{B}_{T}\in
L^{P}(\Omega)$. In the sequel, we will show that $h(.)\in Dom(\delta^{W})$.
According to Proposition 1.3.1 in [Nualart(2000)], it is sufficient to show
that $h(.)\in\mathbb{D}_{W}^{1,2}$. To do that, from these facts that for
every $x,y\in\mathbb{R}$, $(x+y)^{2}\leq 2x^{2}+2y^{2}$ we result

$\displaystyle\mathbb{E}(\int_{0}^{T}$ $\displaystyle
h^{2}(t)dt)+\mathbb{E}(\int_{0}^{T}(D^{W}h)^{2}(t)dt)$
$\displaystyle\leq\kappa^{2}\int_{0}^{T}\frac{1}{2v_{u}^{2}}du+2C_{\sigma}^{2}\int_{0}^{T}\frac{1}{v_{u}^{2}}\mathbb{E}(\frac{1}{\lambda_{u}^{2}})du+\sigma_{2}^{2}C_{\sigma}^{2}\int_{0}^{T}\frac{1}{v_{u}^{2}}\mathbb{E}(\frac{1}{\lambda_{u}^{3}})du<\infty,$
(3.13)

where we used form (3.11) in the last inequality. ∎

##  4 Pricing and Delta calculation

In this section, we discuss the pricing of the payoff function by weighted
Malliavin described in the previous section.we also present an explicit
formula to calculate the delta Greek. To do this, we state a representation of
the delta as a combination of the Wiener-Malliavin weight and the Poisson-
Malliavin weight. We assume the following condition on the payoff functions.
Condition H2: The payoff function $\mathit{f}:\mathbb{R^{+}}\to\mathbb{R^{+}}$
is a measurable function with at most polynomial growth $\frac{p_{0}}{2}$, and
locally Riemann integrable, possibly, having discontinuities of the first
kind.
Let us introduce the following notations presented in [Bezborodov et
al.(2019)]: for every $x\geq 0$,

$F(x)=\int_{0}^{x}\mathit{f}(z)dz,\qquad\mathit{g(y)}=\mathit{f(e^{y})},\qquad
G(y)=\int_{0}^{y}\mathit{g}(z)dz.$

In this notation, we have
$\mathbb{E}\mathit{f}(S_{T})=\mathbb{E}\mathit{g}(X_{T})$ and

$G(x)=\frac{F(e^{x})}{e^{x}}+\int_{0}^{x}\frac{F(e^{y})}{e^{y}}dy-F(1).$

######  Theorem 4.1.

Under condition H2, the price of a simple derivative can be represented as

$\mathbb{E}\Big{(}\mathit{f}(S_{T})\Big{)}=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}(1+{Z_{T}})\Big{)}=\mathbb{E}\Big{(}G(X_{T})Z_{T}\Big{)},$
(4.1)

where $Z_{T}=\delta^{W}(\frac{h(.)}{\mathcal{B}_{T}})$.

###### Proof.

Suppose that the function $\mathcal{K}$ is a locally Lipschitz function with
$\mathcal{K}^{\prime}(x)=k(x)$ almost everywhere with respect to the Lebesgue
measure. Assume additionally that $k$ is of exponential growth and
$\mathcal{K}(X_{T})\in\mathbb{D}_{W}^{1,2}$. Namely, the Skorokhod integral is
the adjoint operator to the Malliavin derivative, therefore

$\displaystyle\mathbb{E}\Big{(}k(X_{T})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\int_{0}^{T}k(X_{T})\frac{D_{u}^{W}X_{T}h(u)}{\left\langle{D_{.}^{W}X_{T},h(.)}\right\rangle}du\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\int_{0}^{T}\frac{D_{u}^{W}\mathcal{K}(X_{T})h(u)}{\left\langle{D_{.}^{W}X_{T},h(.)}\right\rangle}du\Big{)}=\frac{1}{T}\mathbb{E}\Big{(}\int_{0}^{T}\mathcal{K}(X_{T})\frac{h(.)}{\left\langle{D_{.}^{W}X_{T},h(.)}\right\rangle}dW_{u}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\mathcal{K}(X_{T})\int_{0}^{T}\frac{h(u)}{\left\langle{D_{.}^{W}X_{T},h(.)}\right\rangle}dW_{u}\Big{)}=\mathbb{E}\Big{(}\mathcal{K}(X_{T})Z_{T}\Big{)}.$
(4.2)

In particular, for the function $G$ which is a locally Lipschitz function and
$g$ is of exponential growth, we rewrite (4.1) for $k=g$ as follows:

$\displaystyle\mathbb{E}(f(S_{T}))$
$\displaystyle=\mathbb{E}(g(X_{T}))=\mathbb{E}(G(X_{T})Z_{T})$
$\displaystyle=\mathbb{E}\Big{(}\Big{(}\frac{F(S_{T})}{S_{T}}+\int_{0}^{X_{T}}\frac{F(e^{y})}{e^{y}}dy-F(1)\Big{)}Z_{T}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}Z_{T}\Big{)}+\mathbb{E}\Big{(}Z_{T}\int_{0}^{X_{T}}\frac{F(e^{y})}{e^{y}}dy\Big{)}-\mathbb{E}\Big{(}F(1)Z_{T}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}Z_{T}\Big{)}+\mathbb{E}\Big{(}\int_{0}^{X_{T}}\frac{F(e^{y})}{e^{y}}dyZ_{T}\Big{)},$

Applying equation (4.1) to $k(x)=\frac{F(e^{x})}{e^{x}}$, we get that

$\mathbb{E}\Big{(}\int_{0}^{X_{T}}\frac{F(e^{y})}{e^{y}}dyZ_{T}\Big{)}=\mathbb{E}\Big{(}\frac{F(e^{X_{T}})}{e^{X_{T}}}\Big{)}=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}\Big{)}.$

Hence

$\mathbb{E}\Big{(}f(S_{T})\Big{)}=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}Z_{T}\Big{)}+\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}\Big{)}=\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}(1+{Z_{T}})\Big{)}.$

∎

####  4.0.1 Delta with Wiener-Malliavin weight

Now, we are ready to present an explicit formula to calculate the Delta Greek.
To do this, we state a representation of the delta as a combination of the
Wiener-Malliavin weight and the Poisson-Malliavin weight.

######  Theorem 4.2.

Under condition H2, the delta display with respect to the Wiener -Malliavin
weight as

$\displaystyle\Delta^{W}=$
$\displaystyle\frac{\partial}{\partial{s}}\mathbb{E}\Big{(}f(S_{T})\Big{)}=\mathbb{E}\Big{(}f(S_{T})\frac{Z_{T}}{S_{0}}\Big{)}.$

###### Proof.

From the fact that $\frac{\partial{Z_{T}}}{\partial{S_{0}}}=0$, we derive

$\displaystyle\Delta^{W}$
$\displaystyle=\frac{\partial}{\partial{S_{0}}}\mathbb{E}\Big{(}f(S_{T})\Big{)}=\frac{\partial}{\partial{S_{0}}}\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}(1+{Z_{T}})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{\frac{\partial{F(S_{T})}}{\partial{S_{0}}}S_{T}-F(S_{T})\frac{\partial{S_{T}}}{\partial{S_{0}}}}{S_{T}^{2}}(1+{Z_{T}})+\frac{F(S_{T})}{S_{T}}\frac{\partial{Z_{T}}}{\partial{S_{0}}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}(\frac{F^{\prime}(S_{T})}{S_{T}}-\frac{F(S_{T})}{{S_{T}}^{2}})\frac{\partial{S_{T}}}{\partial{S_{0}}}(1+{Z_{T}})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{f(S_{T})}{S_{T}}\frac{{S_{T}}}{{S_{0}}}(1+{Z_{T}})\Big{)}-\mathbb{E}\Big{(}\frac{F(S_{T})}{{S_{T}}^{2}}\frac{{S_{T}}}{{S_{0}}}(1+{Z_{T}})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{f(S_{T})}{S_{0}}(1+{Z_{T}})\Big{)}-\mathbb{E}\Big{(}\frac{F(S_{T})}{S_{T}}\frac{1}{{S_{0}}}(1+{Z_{T}})\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{f(S_{T})}{S_{0}}{Z_{T}}\Big{)},$

where we used Theorem 4.1 in the last equality. ∎

####  4.0.2 Delta with Poisson-Malliavin weight

We will use the literature in [Coffie et al.(2021)] and calculate the delta
with a Malliavin weight regarding the Poisson random measure in two
approaches.
In the first approach.
Due to Proposition 2.2, we know that $D_{u,z}^{N}X_{t}=J_{u,z}1_{u\leq t}$ and
then $D_{u,z}^{N}S_{t}=S_{t}(exp\\{D_{u,z}^{N}X_{t}\\}-1)$ satisfying

$\displaystyle D_{s,z}^{{N}}S_{t}$
$\displaystyle=S_{s}(e^{J_{s,z}}-1)+\int_{s}^{t}\mu
D_{s,z}^{N}S_{u}du+\int_{s}^{t}\sigma_{1}D_{s,z}^{N}S_{u}dW_{u}$
$\displaystyle+\int_{s}^{t}\int_{\mathbb{R}_{0}}(e^{J_{u,z}}-1)D_{s,z}^{N}S_{u}\tilde{N}(du,dz).$
(4.3)

It is remarkable that in this approach of Malliavin derivative with respect ot
the Poisson random measure, the following lemma will be held.

######  Lemma 4.3.

Let $F=1_{A}\in\mathbb{D}_{N}^{1,2}$, where $1_{A}$ is the indicator function
of the set $A\in\mathcal{F}$. Then $D^{N}F=0$ almost everywhere.

###### Proof.

From Proposition 2.2 and Lemma 2.1, we have

$\displaystyle D^{N}1_{A}$
$\displaystyle=D^{N}(1_{A})^{2}=(1_{A}+D^{N}1_{A})^{2}-1_{A}^{2}=21_{A}D^{N}1_{A}+(D^{N}1_{A})^{2},$
$\displaystyle D^{N}1_{A}$
$\displaystyle=D^{N}(1_{A})^{3}=(1_{A}+D^{N}1_{A})^{3}-1_{A}^{3}=31_{A}D^{N}1_{A}+31_{A}(D^{N}1_{A})^{2}.$

Then, $D^{N}1_{A}=0$. ∎

Thanks to Theorem 5.6.1 in [Mhlanga(2011)] and Proposition 2.3, if there
exists a random variable $u(.,.)\in Dom(\delta^{N})$ such that

$\displaystyle\mathbb{E}\Big{(}f^{\prime}(S_{T})\frac{\partial{S_{T}}}{\partial{S_{0}}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}u(t,z)(f(S_{T}+D_{t,z}^{{N}}S_{T})-f(S_{T}))C_{z}\lambda_{t}dzdt\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}u(t,z)(f(S_{T}e^{J_{t,z}})-f(S_{T}))C_{z}\lambda_{t}dzdt\Big{)},$
(4.4)

then $\Delta^{N}:=\frac{\partial}{\partial
S_{0}}\mathbb{E}\Big{(}f(S_{T})\Big{)}=\mathbb{E}\Big{(}f(S_{T})\delta^{N}(u)\Big{)}$.
Now we calculate the delta with respect to the Poisson process in the
following examples desired in [Huehne(2005)].
Example: Consider the European call option with the payoff function
$f(S_{T})=\max(S_{T}-K,0)$. In fact, one can define the function $u$ of the
form

$\displaystyle u(t,z)=$
$\displaystyle\begin{cases}\frac{{\frac{\partial{S_{T}}}{\partial{S_{0}}}H_{K}(S_{T})}}{\int_{0}^{T}\int_{\mathbb{R}_{0}}D_{t,z}S_{T}C_{z}\lambda_{t}dzdt}\;&ifD_{t,z}S_{T}+{S_{T}-K}\geq
0\\\ \\\
\frac{{\frac{\partial{S_{T}}}{\partial{S_{0}}}}H_{K}(S_{T})}{\int_{0}^{T}\int_{\mathbb{R}_{0}}(K-S_{T})C_{z}\lambda_{t}dzdt}\;&ifD_{t,z}S_{T}+{S_{T}-K}<0,\end{cases}$
(4.5)

where $H_{y}(x)=1_{x\geq y}$ is the Heaviside function and $1_{A}$ is the
indicator function of the set $A$. Obviously, the equality (4.0.2) will be
held for this function. Also, it is in the domain of $\delta^{N}$, due to the
similar proof of Lemma 5.1 in [Alos and Ewald(2008)] for every $p\geq 2$
instead of $\frac{1}{2}$, we have
$\mathbb{E}\Big{(}(\int_{0}^{T}\lambda_{t}dt)^{-p}\Big{)}<\infty$. Rewrite the
definition of the function $u$ in (4.5) in the following form.

$S_{0}u(t,z)=\frac{H_{K}(S_{T})}{\int_{0}^{T}v_{t}\lambda_{t}dt}1_{S_{T}e^{J_{t,z}}-K\geq
0}+\frac{H_{K}(S_{T})}{\int_{0}^{T}\lambda_{t}dt}\frac{S_{T}}{K-S_{T}}1_{S_{T}e^{J_{t,z}}-K<0}.$

Therefore,

$\displaystyle\Delta^{N}$
$\displaystyle=\frac{\partial}{\partial{S_{0}}}\mathbb{E}(f(S_{T}))=\mathbb{E}(f^{\prime}(S_{T})\frac{\partial{S_{T}}}{\partial{S_{0}}})=\mathbb{E}\Big{(}f(S_{T})\delta^{N}(u)\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}f(S_{T})\frac{1}{S_{0}}\delta^{N}\Big{(}\frac{H_{K}(S_{T})}{\int_{0}^{T}v_{t}\lambda_{t}dt}1_{S_{T}e^{J_{t,z}}-K\geq
0}+\frac{H_{K}(S_{T})}{\int_{0}^{T}\lambda_{t}dt}\frac{S_{T}}{K-S_{T}}1_{S_{T}e^{J_{t,z}}-K<0}\Big{)}\Big{)}.$

According to (2.2), Proposition 2.4 and Lemma 4.3,

$\displaystyle\delta^{N}\Big{(}H_{K}(S_{T})1_{S_{T}e^{J_{t,z}}-K\geq
0}\Big{)}=H_{K}(S_{T})\delta^{N}\Big{(}1_{S_{T}e^{J_{t,z}}-K\geq 0}\Big{)},$

and

$\displaystyle\delta^{N}\Big{(}H_{K}(S_{T})\frac{S_{T}}{K-S_{T}}1_{S_{T}e^{J_{t,z}}-K<0}\Big{)}$
$\displaystyle=H_{K}(S_{T})\delta^{N}\Big{(}\frac{S_{T}}{K-S_{T}}1_{S_{T}e^{J_{t,z}}-K<0}\Big{)}.$

Thus,

$\displaystyle\Delta^{N}$
$\displaystyle=\mathbb{E}\Big{(}\frac{S_{T}-K}{S_{0}\int_{0}^{T}v_{t}\lambda_{t}dt}H_{K}(S_{T})\delta^{N}\Big{(}1_{S_{T}e^{J_{t,z}}-K\geq
0}\Big{)}\Big{)}$
$\displaystyle+\mathbb{E}\Big{(}\frac{S_{T}-K}{S_{0}\int_{0}^{T}\lambda_{t}dt}H_{K}(S_{T})\delta^{N}\Big{(}\frac{S_{T}}{(K-S_{T})}1_{S_{T}e^{J_{t,z}}-K<0}\Big{)}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}\frac{S_{T}H_{K}(S_{T})}{S_{0}\int_{0}^{T}v_{t}\lambda_{t}dt}\int_{0}^{T}\int_{\mathbb{R}_{0}}1_{S_{T}e^{J_{t,z}}-K\geq
0}(e^{J_{t,z}}-1)C_{z}\lambda_{t}dzdt\Big{)}$
$\displaystyle-\mathbb{E}\Big{(}\frac{S_{T}^{2}H_{K}(S_{T})}{S_{0}(S_{T}-K)\int_{0}^{T}\lambda_{t}dt}\int_{0}^{T}\int_{\mathbb{R}_{0}}1_{S_{T}e^{J_{t,z}}-K<0}(e^{J_{t,z}}-1)C_{z}\lambda_{t}dzdt\Big{)}.$

In the second approach.
In this part, we need the following assumption.

######  Assumption 4.4.

For $\alpha\in(0,2)$ and some constants $c_{0}$ and $c$,

$C_{.}\in C^{1}(\mathbb{R}_{0}),\qquad|\frac{\partial}{\partial
z}logC_{z}|\leq c_{0}\rho(z),$

and

$\lim_{\epsilon\rightarrow
0}\epsilon^{\alpha-2}\int_{|z|\leq\epsilon}|z|^{2}C_{z}dz=c.$ (4.6)

As a result of the assumption (4.6), shown in [Song and Wang(2022)] Lemma 2.5,
for any $p\geq 2$, there exist some constants $c_{0,p}$ and $c_{1,p}$ such
that

$c_{0,p}\epsilon^{p-\alpha}\leq\int_{|z|\leq\epsilon}|z|^{p}C_{z}dz\leq
c_{1,p}\epsilon^{p-\alpha}.$ (4.7)

Condition K1: First and second derivatives of the function $J$ with respect to
$z$ is bounded, i.e., there exists some non-negative constant $\gamma$ and
$c_{J}>0$ such that

$\sup_{0\leq t\leq T,z\in\mathbb{R}_{0}}|\frac{\partial J_{t,z}}{\partial
z}|^{-1}\leq c_{J}|z|^{-\gamma},\qquad\sup_{0\leq t\leq
T,z\in\mathbb{R}_{0}}|\frac{\partial^{2}J_{t,z}}{\partial z^{2}}|\leq
c_{J}|z|^{\gamma-1}.$

In the same way as the proof of Lemma 4.1 in [Song and Wang(2022)], one can
show the following lemma.

######  Lemma 4.5.

Under Assumption 4.4, for every $p\geq 2$ and $\theta\geq 2$, there exists
some constant $c_{p}$ such that for every $t\in[0,T]$ and $\epsilon\in(0,1)$,

$\mathbb{E}\Big{(}\Big{[}\int_{0<|z|\leq\epsilon}\int_{0}^{t}|z|^{\theta}N(ds,dz)\Big{]}^{-p}|\mathcal{F}_{t}^{\lambda}\Big{)}\leq
c_{p}\Big{(}\epsilon^{\theta-\alpha}\int_{0}^{t}\lambda_{s}ds\Big{)}^{-p}+\Big{(}\int_{0}^{t}\lambda_{s}ds\Big{)}^{-\frac{\theta
p}{\alpha}}.$

###### Proof.

According to (3.1) and the proof of Lemma 4.1 in [Song and Wang(2022)], we
have

$\displaystyle\mathbb{E}\Big{(}\Big{[}\int_{0<|z|\leq\epsilon}$
$\displaystyle\int_{0}^{t}|z|^{\theta}N(ds,dz)\Big{]}^{-p}|\mathcal{F}_{t}^{\lambda}\Big{)}$
$\displaystyle\leq\frac{1}{\Gamma(p)}\int_{0}^{\infty}r^{p-1}exp\\{\int_{\\{0<|z|\leq\epsilon\\}}\int_{0}^{t}(e^{-r|z|^{\theta}}-1)\lambda_{t}C_{z}dtdz\\}dr$
$\displaystyle\leq\frac{1}{\Gamma(p)}\int_{0}^{\infty}r^{p-1}exp\\{\int_{\\{0<|z|\leq\epsilon\wedge
r^{-\frac{1}{\theta}}\\}}\int_{0}^{t}c_{0}r|z|^{\theta}\lambda_{t}C_{z}dtdz\\}dr$
$\displaystyle\leq
c_{p}\Big{(}\epsilon^{\theta-\alpha}\int_{0}^{t}\lambda_{s}ds\Big{)}^{-p}+\Big{(}\int_{0}^{t}\lambda_{s}ds\Big{)}^{-\frac{\theta
p}{\alpha}}.$

for some $c_{0}>0$ that $1-e^{-x}\geq c_{0}x$ as $|x|\leq 1$. ∎

Now, we calculate the delta with respect to the Poisson process when the
Malliavin derivative is defined in the second approach. To do this, we note
that from the definition of Malliavin derivative in this approach and (3.8),
we know

$D_{r,z}^{N_{p}}S_{T}=\frac{\partial J_{r,z}}{\partial z}S_{T},$ (4.8)

satisfying the following equation for every $0\leq s\leq t$

$\displaystyle D_{s,z}^{{N_{p}}}S_{t}$ $\displaystyle=S_{s}\frac{\partial
J_{s,z}}{\partial z}e^{J_{s,z}}+\int_{s}^{t}\mu
D_{s,z}^{N_{p}}S_{u}du+\int_{s}^{t}\sigma_{1}D_{s,z}^{N_{p}}S_{u}dW_{u}$
$\displaystyle+\int_{\mathbb{R}_{0}}\int_{s}^{t}(e^{J_{u,z}}-1)D_{s,z}^{N_{p}}S_{u}\tilde{N}(du,dz).$

With a similar way to [Song and Wang(2022)], set
$\mathcal{A}(t,z):=\frac{1}{S_{0}}\Big{(}\frac{\partial J_{t,z}}{\partial
z}\Big{)}^{-1}\xi(z)$ where $\xi$ is a non-negative smooth function that

$\xi(z)=|z|^{3+\gamma}~{}~{}if~{}|z|\leq\frac{1}{4}\Big{(}\int_{0}^{T}\mathbb{E}(\lambda_{s})ds\Big{)}^{\frac{1}{\alpha}},\qquad\xi(z)=0~{}~{}if~{}|z|\geq\frac{1}{2}\Big{(}\int_{0}^{T}\mathbb{E}(\lambda_{s})ds\Big{)}^{\frac{1}{\alpha}},$

and $|\frac{\partial}{\partial z}\xi(z)|\leq c_{1}|z|^{2+\gamma}$ and
$|\xi(z)|\leq c_{1}|z|^{3+\gamma}$, for some constant $c_{1}$. Then, according
to Lemma 4.5, under Assumption 4.4 and condition $K1$, one can arrive at

$\mathbb{E}\Big{(}\mathcal{N}_{\xi}\Big{)}^{-p}:=\mathbb{E}\Big{(}\int_{\mathbb{R}_{0}}\int_{0}^{T}\xi(z){N}(dr,dz)\Big{)}^{-p}\leq
2c_{p}\Big{(}\int_{0}^{T}\mathbb{E}(\lambda_{s})ds\Big{)}^{-\frac{(3+\gamma)p}{\alpha}},$

and for some constant $c_{Jp}$, in connection with (4.7),

$\displaystyle\|\mathcal{A}\|^{p}_{\mathbb{V}_{p}}$ $\displaystyle\leq
2^{p-1}(\|\frac{\partial\mathcal{A}}{\partial
z}\|^{p}_{\mathbb{L}_{p}}+\|\rho\mathcal{A}\|^{p}_{\mathbb{L}_{p}})$
$\displaystyle\leq
c_{Jp}\Big{[}\mathbb{E}\Big{(}\int_{0<|z|\leq(\int_{0}^{T}\mathbb{E}(\lambda_{s})ds)^{\frac{1}{\alpha}}}\int_{0}^{T}|z|^{2}\lambda_{s}dsC_{z}dz\Big{)}^{p}$
$\displaystyle+\mathbb{E}\Big{(}\int_{0<|z|\leq(\int_{0}^{T}\mathbb{E}(\lambda_{s})ds)^{\frac{1}{\alpha}}}\int_{0}^{T}|z|^{2p}\lambda_{s}dsC_{z}dz\Big{)}\Big{]}$
$\displaystyle\leq
c_{1,p}c_{Jp}\Big{[}\Big{(}\int_{0}^{T}\mathbb{E}(\lambda_{s})ds\Big{)}^{\frac{p(2-\alpha)}{\alpha}}\mathbb{E}\Big{(}\int_{0}^{T}\lambda_{s}ds\Big{)}^{p}+\Big{(}\int_{0}^{T}\mathbb{E}(\lambda_{s})ds\Big{)}^{\frac{2p}{\alpha}}\Big{]}<\infty.$

Now, multiply (4.8) in $\mathcal{A}$ and get integration to derive the
Poisson-Malliavin weight of the computation of delta.

$<D^{N_{p}}S_{T},\mathcal{A}>_{N}=\int_{\mathbb{R}_{0}}\int_{0}^{T}D_{r,z}^{N_{p}}S_{T}\Big{(}\frac{\partial
J_{r,z}}{\partial z}\Big{)}^{-1}\xi(z){N}(dr,dz)=S_{T}\mathcal{N}_{\xi},$

and then, in connection with Propositions 2.5 and 2.6,

$\displaystyle\Delta_{p}^{N}:=\frac{\partial}{\partial
S_{0}}\mathbb{E}(f(S_{T}))$
$\displaystyle=\mathbb{E}\Big{(}f^{\prime}(S_{T})\frac{\partial{S_{T}}}{\partial{S_{0}}}\Big{)}=\mathbb{E}\Big{(}<D^{N_{p}}f(S_{T}),\mathcal{A}>_{N}\frac{1}{\mathcal{N}_{\xi}}\Big{)}$
$\displaystyle=\mathbb{E}\Big{(}f(S_{T})\frac{1}{\mathcal{N}_{\xi}}\int_{\mathbb{R}_{0}}\int_{0}^{T}\frac{1}{C_{z}}\frac{\partial(C_{.}\mathcal{A})(s,z))}{\partial
z}\tilde{N}(ds,dz)\Big{)}$
$\displaystyle=:\mathbb{E}\Big{(}f(S_{T})\frac{1}{\mathcal{N}_{\xi}}\delta^{N_{p}}(\mathcal{A})\Big{)}.$

######  Lemma 4.6.

Under Assumption 4.4 and Condition $K1$, for every $p\geq 2$,

$\mathbb{E}\Big{(}\delta^{N_{p}}(\mathcal{A})\Big{)}^{p}<\infty.$

###### Proof.

From Assumption 4.4 and Section 5.1.1 of [Menaldi(2008)], there exist
constants $c_{jp}$ and $a$ such that

$\displaystyle\mathbb{E}\Big{(}\delta^{N_{p}}(\mathcal{A})\Big{)}^{p}$
$\displaystyle\leq
c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{1}{C_{z}^{2}}[\frac{\partial(C_{.}\mathcal{A})(s,z))}{\partial
z}]^{2}\lambda_{s}dsC_{z}dz\Big{)}^{\frac{p}{2}}$
$\displaystyle+c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}\frac{1}{C_{z}^{p}}[\frac{\partial(C_{.}\mathcal{A})(s,z))}{\partial
z}]^{p}\lambda_{s}dsC_{z}dz\Big{)}$ $\displaystyle\leq
2^{p}c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}(|\frac{\partial}{\partial
z}logC_{z}|^{2}\mathcal{A}^{2}(s,z)\lambda_{s}dsC_{z}dz\Big{)}^{\frac{p}{2}}$
$\displaystyle+2^{p}c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}(|\frac{\partial}{\partial
z}logC_{z}|^{p}\mathcal{A}^{p}(s,z)\lambda_{s}dsC_{z}dz\Big{)}$
$\displaystyle+2^{p}c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}\mathcal{A}^{2}(s,z)\lambda_{s}dsC_{z}dz\Big{)}^{\frac{p}{2}}+2^{p}c_{jp}\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}\mathcal{A}^{p}(s,z)\lambda_{s}dsC_{z}dz\Big{)}$
$\displaystyle\leq
a\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}|z|^{2}\lambda_{s}dsC_{z}dz\Big{)}^{\frac{p}{2}}+a\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}|z|^{2p}\lambda_{s}dsC_{z}dz\Big{)}$
$\displaystyle+a\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}|z|^{6}\lambda_{s}dsC_{z}dz\Big{)}^{\frac{p}{2}}+a\mathbb{E}\Big{(}\int_{0}^{T}\int_{\mathbb{R}_{0}}|z|^{3p}\lambda_{s}dsC_{z}dz\Big{)}<\infty.$

∎

##  5 Numerical Example

In this section, we calculate the delta in both cases of the Malliavin
derivative for the European option and show the results.
Let $J_{t,z}=z$ for $z\in\mathbb{R}$ and consider an European call option with
the expiration date $T$ and the strike price $K$, as

$f(S_{T})=\max(S_{T}-K,0).$

The exact expression for $\Delta$ is

$\Delta=\mathbb{E}[H_{K}(S_{T})\frac{S_{T}}{S_{0}}],$

whereas for the symmetric Finite Difference approach gives

$\Delta=\frac{\partial}{\partial
S_{0}}\mathbb{E}[\max(S_{T}-K,0)]=\frac{F(S_{0}+h)-F(S_{0}-h)}{2h},$

where $F(S_{0})=\mathbb{E}[f(S_{T})|S_{0}]$, and $h$ is an arbitrary small
constant.
Figure 1 shows the pricing of a European option for parameters
$\sigma_{1}=0.40$, $\sigma_{2}=0.10$, $\Theta=1$, $\kappa=0.15$, $\mu=1$,
$T=1$, $\lambda_{0}=0.10$, $S_{0}=5$, the number of simulated paths is $1000$
and $K=S_{0}\times u$ which $u=0.3,0.45,1.00,\ldots,6.45,7$. In figures 3 and
3, sensitivity of the price of call option are presented with respect to the
parameters of the stochastic intensity model; $\kappa$ and $\sigma$.

Figure 1: Pricing of European call option for $T=1,S_{0}=5$ and the function
$J_{t,z}=z$ with Guassian jump distribution.

Figure 2: sensitivity of price with respect to k

Figure 3: sensitivity of price with respect to sigma2

###  5.1 Delta in the first and second approach

Figures 5 and 5 show the behaviour of these four expressions $\Delta$,
$\Delta^{W}$, $\Delta^{N}$ and $\Delta^{W}/2+\Delta^{N}/2$ for
$\sigma_{1}=0.10$, $\sigma_{2}=0.05$, $\Theta=0.30$, $\kappa=0.05$,
$\mu=0.01$, $T=1$, $\lambda_{0}=0.10$, $S_{0}=5$, $K=S_{0}\times 1.2$ and time
discretization $h=0.0001$. The jumps are generated by a normal distribution
with a mean of $-0.10$ and a standard deviation of $0.50$ which satisfies
Condition H1. The exact solution is $0.0481509$. The execution time of the
program code in Malliavin method and finite difference method in the first
approach are $2.2104\times 10^{4}$ and $4.4111\times 10^{4}$, and in the
second approach are $2.2817\times 10^{4}$ and $4.5626\times 10^{4}$
respectively. The specifications of the computer system with which the program
is implemented are Intel(R) Core i$7-9700$K CPU and $64$ GB Memory.

Figure 4: Greek Delta for European call option in the first approach for
$J_{t,z}=z$ and Gaussian jump distribution.

Figure 5: Greek Delta for European call option in the second approach for
$J_{t,z}=z$ and Gaussian jump distribution. Table 1: The mean square error of
four methods

$The\ Method$ | $MSE\ of\ the\ first\ approach$ | $MSE\ of\ the\ second\ approach$
---|---|---
$Winner-Malliavin(W_{M})\ Weight$ | $0.0004$ | $0.0003$
$Poisson-Malliavin(P_{M})\ Weight$ | $0.0023$ | $0.0021$
$Mean\ W_{M}\ and\ P_{M}\ Weight$ | $0.0003$ | $0.0002$
$Symmetric\ Finite\ Difference$ | $0.0254$ | $0.0190$

##  6 Funding

This work is based upon research funded by Iran National Science
Foundation(INSF) under project No.4022879.

##  7 Conclusions

The main purpose of this article is to study the pricing of financial
derivatives and to calculate the delta of them in a stochastic model with
stochastic jump and intensity by using the Mallivain calculation. In the
presence of the Malliavin derivative of the intensity, some Wiener-direction
is found to be used in the duality formula of the Gaussian case appeared in
calculating the delta and the price of financial derivatives. This subject,
delta computation, is also considered in Poisson space with two different
approaches. Finally, through the numerical results, we compare the price
sensitivity computation in two methods, the finite difference method and the
Malliavin method, with the exact solution in the models with jumps and
stochastic intensity on asset prices and financial derivatives. The method
developed in this paper can be extended to other pricing problems and Greeks
associated with stochastic volatility processes and fractional Brownian
motion. We leave these problems for our future work.

##  8 Acknowledgments

We wish to thank the referees for their valuable suggestions and comments
which will have improved the paper.

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# Generation of hyperentangled states and two-dimensional quantum walks using
$J$\- (q)- plates and polarization beamsplitters

P. A. Ameen Yasir<EMAIL_ADDRESS>Department of Instrumentation &
Applied Physics, Indian Institute of Science, Bengaluru 560012, India  C. M.
Chandrashekar<EMAIL_ADDRESS>Department of Instrumentation & Applied
Physics, Indian Institute of Science, Bengaluru 560012, India  The Institute
of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India
Homi Bhabha National Institute, Training School Complex, Anushakti Nagar,
Mumbai 400094, India

###### Abstract

A single photon can be made to entangle simultaneously in its different
internal degrees of freedom (DoF) – polarization, orbital angular momentum
(OAM), and frequency – as well as in its external DoF – path. Such
entanglement in multiple DoF is known as hyperentanglement and provide
additional advantage for quantum information processing. We propose a passive
optical setup using $q$-plates and polarization beamsplitters to hyperentangle
an incoming single photon in polarization, OAM, and path DoF. By mapping
polarization DoF to a two-dimensional coin state, and path and OAM DoF to two
spatial dimensions, $x$ and $y$, we present a scheme for realization of two-
dimensional discrete-time quantum walk using only polarization beamsplitters
and $q$-plates ensuing the generation of hyperentangled states. The amount of
hyperentanglement generated is quantified by measuring the entanglement
negativity between any two DoF. We further show the hyperentanglement
generation can be controlled by using an additional coin operation or by
replacing the $q$-plate with a $J$-plate.

##  I Introduction

It is possible to entangle photons in more than one degree of freedom (DoF)
such as polarization, time-energy, path, orbital angular momentum (OAM), and
so on Kwiat (1997); Barreiro  _et al._ (2005); Zhao  _et al._ (2019). Such
states are called hyperentangled states Kwiat (1997). Due to extension in the
dimension of the Hilbert space of such paired photons, increase in the channel
capacity has been demonstrated Barreiro  _et al._ (2008) and as a consequence
hyperentanglement is poised to offer additional quantum advantage. Single
photons can also be simultaneously entangled in polarization, path, and OAM
DoF. While polarization and OAM correspond to internal DoF of the photon, path
DoF corresponds to external DoF. The amount of entanglement between these
three DoF can be, for instance, generated and controlled using devices such as
waveplates (both quarter and half-waveplates), polarization beamsplitters
(PBS), and $q$-plates Devlin  _et al._ (2017) or $J$-plates Marrucci  _et al._
(2006). Single photon entangled in these three DoF can also be thought of a
quantum walker in higher dimensions. For example, in a discrete-time quantum
walk in two spatial dimensions, a coin degree of freedom can be mapped to
polarization DoF and the two spatial dimensions can be mapped to path and OAM
DoF. Therefore, controlled engineering of interactions between different DoF
of single photon to generate and control hyperentanglement can be directly
mapped to the controlled realization of higher dimensional quantum walks.

Quantum walks, the quantum analog of classical random walks, are broadly
classified into two categories : discrete-time quantum walk (DTQW) and
continuous-time quantum walk (CTQW) Venegas-Andraca (2012). In the former
case, the quantum coin dictates the direction in which the walker moves and
the walk evolves in the Hilbert space $\mathcal{H}_{c}\otimes\mathcal{H}_{p}$,
where $\mathcal{H}_{c}$ denotes the Hilbert space corresponding to the coin
space and $\mathcal{H}_{p}$ denotes the position space in which the walker
moves. In the case of CTQW, no coin operation is necessary, and the state
evolves only in the position Hilbert space.

For one-dimensional DTQW the Hilbert space $\mathcal{H}_{c}$ is spanned by
two-dimensional (2D) basis vectors $|H\rangle=[1,0]^{T}$ and
$|V\rangle=[0,1]^{T}$. It can be noted that $|H\rangle$ and $|V\rangle$ denote
Jones vectors corresponding to horizontal and vertical polarization states of
photons, respectively. The Hilbert space $\mathcal{H}_{p}$ is spanned by the
position basis $\\{|x\rangle\\}$, where $x\in\mathbb{Z}$. Each step of DTQW
can be described using a composition of quantum coin operation
$\hat{C}_{\boldsymbol{\sigma}}$ in SU(2),

$\displaystyle\hat{C}_{\boldsymbol{\sigma}}=\begin{bmatrix}e^{i\xi}\cos\theta&e^{i\zeta}\sin\theta\\\
-e^{-i\zeta}\sin\theta&e^{-i\xi}\cos\theta\end{bmatrix}$ (1)

on $\mathcal{H}_{c}$, followed by a position shift operation,

$\displaystyle\hat{S}=\sum_{x=-\infty}^{\infty}[|H\rangle\langle
H|\otimes|x-1\rangle\langle x|+|V\rangle\langle V|\otimes|x+1\rangle\langle
x|]$ (2)

on the combined Hilbert space Chandrashekar  _et al._ (2008). After each step
of walk operation the walker will evolve in superposition of position space
entangling the two Hilbert spaces. In an optical setting with polarization
DoF, any $\hat{C}_{\boldsymbol{\sigma}}$ in SU(2) can be realized using two
quarter waveplates and a half waveplate Simon and Mukunda (1990) and
$\hat{S}_{x}$ can be realized using PBS. In general, the 1D DTQW evolution
after $n$ steps can be given by

$\displaystyle|\Psi_{n}\rangle$
$\displaystyle=[\hat{S}_{x}(\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{x})]^{n}|\Psi_{\rm
in}\rangle$
$\displaystyle=\sum_{x=-\infty}^{\infty}[a_{x}^{(n)}|H\rangle+b_{x}^{(n)}|V\rangle]\otimes|x\rangle,$
(3)

where $|\Psi_{\rm in}\rangle$ is the initial state, $\mathds{1}_{x}$ refers to
identity operator in the position space, and $a_{x}^{(n)}$ and $b_{x}^{(n)}$
are normalized complex coefficients. The evolved state is evidently entangled
in coin and spatial DoF.

For extension of DTQW to the 2D space, the Hilbert space will be a composition
of $\mathcal{H}_{c}\otimes\mathcal{H}_{p_{x}}\otimes\mathcal{H}_{p_{y}}$,
where $\mathcal{H}_{c}$ corresponds to the coin space and
$\mathcal{H}_{p_{x}}$ and $\mathcal{H}_{p_{y}}$ refer to the Hilbert spaces
corresponding to the position spaces in $x$ and $y$-directions, respectively.
Since the state has to simultaneously evolve in both $x$ and $y$\- spaces, it
is natural to expect the use of 4-dimensional coin space and a corresponding
coin operation. Two well-known examples of such coin choices are Grover coin
and 4-dimensional discrete-Fourier transform coin Tregenna  _et al._ (2003).
However, it was shown that such 2D DTQW can as well be implemented using just
2D coin operation Di Franco  _et al._ (2011); Chandrashekar  _et al._ (2010);
Chandrashekar and Busch (2013). For instance, the Grover walk with an initial
state
$(\frac{1}{2})(|0\rangle-|1\rangle-|2\rangle+|3\rangle)\otimes|x=0\rangle\otimes|y=0\rangle$
can be implemented using a two-state alternate walk – in which a two
dimensional coin operation is used and each step of walk is split into
evolution in one dimension followed by an evolution in the other dimension. It
has also been demonstrated that the alternate walk can be implemented in the
form of Pauli walk, where Pauli operators' bases are used as conditions in the
shift operators and no coin operation is therefore necessary Chandrashekar and
Busch (2013).

There has been a continued interest for efficient implementation of quantum
walks (both, in 1D and 2D space) in various quantum systems. For example, in
1D, quantum walk has been realized using physical systems such as NMR Ryan
_et al._ (2005), optical lattice Karski  _et al._ (2009), linear optical
devices Schreiber  _et al._ (2010); Crespi  _et al._ (2013); Broome  _et al._
(2010); Zhang  _et al._ (2007), ion traps Schmitz  _et al._ (2009); Zähringer
_et al._ (2010), and $q$-plates (single photons Cardano  _et al._ (2015) as
well as bright classical light Goyal  _et al._ (2013); Sephton  _et al._
(2019)) to name a few. In 2D, the quantum walk has been realized using
photonic waveguide arrays Tang  _et al._ (2018), liquid-crystal devices
D’Errico  _et al._ (2020), etc. Various new schemes have been proposed for the
realization of 1D quantum walk which include $q$-plates and waveplates Zhang
_et al._ (2010), passive optical devices Jeong  _et al._ (2004); Do  _et al._
(2005), cross-Kerr nonlinearity Gao  _et al._ (2019).

Inspired by the Pauli walk where different bases are used for evolution in
different spatial dimensions, in this paper, we propose a passive optical
setup – using $J$-plates Devlin  _et al._ (2017) or $q$-plates Marrucci  _et
al._ (2006) and polarization beamsplitters (PBS) – to generate
hyperentanglement in polarization, path, and OAM DoF of a single photon. Here,
$J$-plates or $q$-plates will be used to control the OAM Allen  _et al._
(1992) and polarization DoF, while PBS will be used to control the path DoF.
Upon evolution, we show that the photon will be hyperentangled in these three
DoF. This setup can also effectively simulate a 2D modified form of Pauli walk
in OAM and position DoF where coin operation is not required. Due to basis
change that $J$\- ($q-$) plate and PBS introduce, the effect of coin operation
in the path dimension is absorbed into the $J$\- ($q$-) plate and the effect
of coin operation in OAM dimension is absorbed into PBS. By mapping the path
and OAM DoF to $x$ and $y$-dimensions we can recover the DTQW in two
dimensional position space.

This paper is organized as follows. In Section II we briefly review schemes
for realizing 2D DTQW such as Pauli and alternate walks and explain how the
evolved state is hyperentangled in the associated Hilbert spaces. In Section
III we propose a passive optical setup to hyperentangle the incoming single
photon in the three DoF (polarization, path, and OAM). This hyperentanglement
is quantified by measuring the entanglement negativity between any two of the
three DoF. We then present our numerical results by simulating the two-
dimensional modified Pauli walk – which does not require an explicit coin
operator. Finally, in Section IV we conclude with some remarks.

##  II Two-dimensional quantum walk

In this section we show the equivalence between the alternate and Pauli walks
for any arbitrary choice of coin operator in SU(2). We also propose modified
Pauli walk and discuss its implementation in optical setting using $J$-plates
and PBS for a particular choice of SU(2) parameters. To quantify
hyperentanglement, we use entanglement negativity which measures entanglement
between any two of three DoF.

Mathematical framework : Quantum walk in 2D can be implemented using 2D coin
operator and shift operators in $x$ and $y$-directions (Chandrashekar  _et
al._ , 2010; Chandrashekar and Busch, 2013; Di Franco  _et al._ , 2011). We
define coin operator as $\hat{C}_{\boldsymbol{\sigma}}$ [Eq. (1)], and shift
operators can be defined as

$\displaystyle\hat{S}_{x}$
$\displaystyle=\sum_{x=-\infty}^{\infty}[|H\rangle\langle
H|\otimes|x-1\rangle\langle x|\otimes\mathds{1}_{y}$
$\displaystyle\,\,\,+|V\rangle\langle V|\otimes|x+1\rangle\langle
x|\otimes\mathds{1}_{y}],$ (4) $\displaystyle\hat{S}_{y}$
$\displaystyle=\sum_{y=-\infty}^{\infty}[|H\rangle\langle
H|\otimes\mathds{1}_{x}\otimes|y-1\rangle\langle y|$
$\displaystyle\,\,\,+|V\rangle\langle
V|\otimes\mathds{1}_{x}\otimes|y+1\rangle\langle y|],$ (5)

where $\mathds{1}_{x}$ and $\mathds{1}_{y}$ are identity operators in $x$ and
$y$ spaces, respectively. If $|\Psi_{\rm in}\rangle$ represents initial state,
the evolution operator $\hat{\mathcal{O}}$ corresponding to alternate walk is
(Chandrashekar and Busch, 2013)

$\displaystyle|\Psi_{1}\rangle=\hat{S}_{x}[\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{xy}]\hat{S}_{y}[C_{\boldsymbol{\sigma}}^{\dagger}\otimes\mathds{1}_{xy}]|\Psi_{\rm
in}\rangle=\hat{\mathcal{O}}|\Psi_{\rm in}\rangle,$ (6)

where $\mathds{1}_{xy}=\mathds{1}_{x}\otimes\mathds{1}_{y}$. This alternate
walk evolution operator can also be implemented using just two shift
operators, $\hat{S}_{x}$ and $\hat{S}_{\boldsymbol{\sigma}}$, where

$\displaystyle\hat{S}_{\boldsymbol{\sigma}}$
$\displaystyle=[\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{xy}]\hat{S}_{y}[\hat{C}_{\boldsymbol{\sigma}}^{\dagger}\otimes\mathds{1}_{xy}]$
$\displaystyle=\sum_{y}[|u_{1}\rangle\langle
u_{1}|\otimes\mathds{1}_{x}\otimes|y-1\rangle\langle y|$
$\displaystyle\,\,\,+|u_{2}\rangle\langle
u_{2}|\otimes\mathds{1}_{x}\otimes|y+1\rangle\langle y|].$ (7)

The states $|u_{1}\rangle$ and $|u_{2}\rangle$ denote the first and second
column vectors of $\hat{C}_{\boldsymbol{\sigma}}$ [see Eq. (1)], respectively.
In any physical system with provision to directly realize $\hat{S}_{x}$ and
$\hat{S}_{\boldsymbol{\sigma}}$, without explicit use of coin operation, we
can realize a 2D DTQW. When $\\{|u_{1}\rangle,|u_{2}\rangle\\}$ are the
eigenvectors of the Pauli matrices,

$\displaystyle\sigma_{1}=\begin{bmatrix}0&1\\\ 1&0\end{bmatrix}\,\,\,{\rm
or}\,\,\,\sigma_{2}=\begin{bmatrix}0&-i\\\ i&0\end{bmatrix},$ (8)

the evolution operator $\hat{S}_{x}\hat{S}_{\boldsymbol{\sigma}}$ readily
implements the Pauli walk Chandrashekar and Busch (2013). Therefore, the
operator $\hat{S}_{x}\hat{S}_{\boldsymbol{\sigma}}$ can be thought of as a
generalized Pauli walk and its evolution can be given by

$\displaystyle|\Psi_{1}\rangle=\hat{S}_{x}\hat{S}_{\boldsymbol{\sigma}}|\Psi_{\rm
in}\rangle=\hat{\mathcal{O}}|\Psi_{\rm in}\rangle.$ (9)

Figure 1: Probability distribution (Prob. dist.) of the 2D DTQW without
explicit coin operation in position and OAM DoF – beginning with an initial
state $[(|H\rangle+|V\rangle)/\sqrt{2}]\otimes|x=0\rangle\otimes|y=0\rangle$ –
after 50 steps. (a) Modified Pauli walk using $q$-plates and PBS [see Eq.
(II)]. Note that the probability distribution is identical for the Pauli walk
too [see Eqs. (9) and (II)] which is realized using $q$-plates, PBS, and HWP.
(b) Modified Pauli walk using $J$-plates and PBS for the choice
$|u_{1}\rangle=[1,-1]^{T}/\sqrt{2}$ and $|u_{2}\rangle=[1,1]^{T}/\sqrt{2}$ in
Eq. (II). (c) Modified Pauli walk using $J$-plates and PBS for the choice
$|u_{1}\rangle=[1,\sqrt{3}i]^{T}/2$ and $|u_{2}\rangle=[\sqrt{3},-i]^{T}/2$ in
Eq. (II).

Shift operators $\hat{S}_{x}$ and $\hat{S}_{y}$ given in Eqs. (4) and (5)
shifts the position vector without changing the coin state vectors. However,
we can also define a modified shift operator which induces a flip in the coin
state vector along with the shift in position vector. For example, we can
define

$\displaystyle\hat{S}^{\prime}_{y}=[\sigma_{1}\otimes\mathds{1}_{xy}]\hat{S}_{y}$
$\displaystyle=\sum_{y=-\infty}^{\infty}[|V\rangle\langle
H|\otimes\mathds{1}_{x}\otimes|y-1\rangle\langle y|$
$\displaystyle\,\,\,+|H\rangle\langle
V|\otimes\mathds{1}_{x}\otimes|y+1\rangle\langle y|],$ (10)

and $\hat{S}^{\prime}_{x}$ defined likewise. Because of the bit-flip symmetry
Chandrashekar  _et al._ (2007), it can be shown that both $\hat{\mathcal{O}}$
and

$\displaystyle\hat{\mathcal{O}}^{\prime}\equiv\hat{S}_{x}[\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{xy}]\hat{S}^{\prime}_{y}[C_{\boldsymbol{\sigma}}^{\dagger}\otimes\mathds{1}_{xy}]|\Psi_{\rm
in}\rangle$ (11)

lead to the equivalent evolution of the initial state $|\Psi_{\rm in}\rangle$
in Eq. (6). Thus, we can also define modified versions of both alternate and
generalized Pauli walks as

$\displaystyle|\Psi_{1}\rangle$
$\displaystyle=\hat{S}_{x}[\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{xy}]\hat{S}^{\prime}_{y}[\hat{C}_{\boldsymbol{\sigma}}^{\dagger}\otimes\mathds{1}_{xy}]|\Psi_{\rm
in}\rangle,$ (12) $\displaystyle{\rm and}\,\,\,|\Psi_{1}\rangle$
$\displaystyle=\hat{S}_{x}\hat{S}^{\prime}_{\boldsymbol{\sigma}}|\Psi_{\rm
in}\rangle,$ (13)

respectively, with

$\displaystyle\hat{S}^{\prime}_{\boldsymbol{\sigma}}$
$\displaystyle=[\hat{C}_{\boldsymbol{\sigma}}\otimes\mathds{1}_{xy}]\hat{S}^{\prime}_{y}[\hat{C}_{\boldsymbol{\sigma}}^{\dagger}\otimes\mathds{1}_{xy}]$
$\displaystyle=\sum_{y}[|u_{2}\rangle\langle
u_{1}|\otimes\mathds{1}_{x}\otimes|y-1\rangle\langle y|$
$\displaystyle\,\,\,+|u_{1}\rangle\langle
u_{2}|\otimes\mathds{1}_{x}\otimes|y+1\rangle\langle y|].$ (14)

If we begin with the initial state

$\displaystyle|\Psi_{\rm
in}\rangle=(\cos\alpha|H\rangle+e^{i\beta}\sin\alpha|V\rangle)\otimes|x=0\rangle\otimes|y=0\rangle,$
(15)

then, after $n$ steps, the state will be of the form

$\displaystyle|\Psi_{n}\rangle$
$\displaystyle=(\hat{S}_{x}\hat{S}^{\prime}_{\boldsymbol{\sigma}})^{n}|\Psi_{\rm
in}\rangle$
$\displaystyle=\sum_{x,y}\left[(a_{x,y}^{(n)}|H\rangle+b_{x,y}^{(n)}|V\rangle)\otimes|x\rangle\otimes|y\rangle\right],$
(16)

where $a_{x,y}^{(n)}$ and $b_{x,y}^{(n)}$ are normalized complex coefficients.
The iterative relations between $a_{x,y}^{(n)}$ and $b_{x,y}^{(n)}$ are

$\displaystyle a_{x,y}^{(n)}$
$\displaystyle=a_{x+1,y+1}^{(n-1)}(e^{i\zeta}\sin\theta)(e^{-i\xi}\cos\theta)$
$\displaystyle\,\,\,+a_{x+1,y-1}^{(n-1)}(e^{i\xi}\cos\theta)(e^{-i\zeta}\sin\theta)$
$\displaystyle\,\,\,+b_{x-1,y+1}^{(n-1)}(e^{-i\xi}\cos\theta)(e^{-i\xi}\cos\theta)$
$\displaystyle\,\,\,+b_{x-1,y-1}^{(n-1)}(-e^{-i\zeta}\sin\theta)(e^{-i\zeta}\sin\theta),$
(17)

and

$\displaystyle b_{x,y}^{(n)}$
$\displaystyle=a_{x+1,y+1}^{(n-1)}(e^{i\zeta}\sin\theta)(-e^{i\zeta}\sin\theta)$
$\displaystyle\,\,\,+a_{x+1,y-1}^{(n-1)}(e^{i\xi}\cos\theta)(e^{i\xi}\cos\theta)$
$\displaystyle\,\,\,+b_{x-1,y+1}^{(n-1)}(e^{-i\xi}\cos\theta)(-e^{i\zeta}\sin\theta)$
$\displaystyle\,\,\,+b_{x-1,y-1}^{(n-1)}(-e^{-i\zeta}\sin\theta)(e^{i\xi}\cos\theta).$
(18)

The above described state evolution after $n$ steps is in superposition of the
tensor products of the three Hilbert spaces, namely coin Hilbert space
$\mathcal{H}_{c}$, and two position Hilbert spaces $\mathcal{H}_{p_{x}}$ and
$\mathcal{H}_{p_{y}}$ associated with the dynamics. The interwinding
coefficients of the state vectors after evolution clearly indicate that the
Eq. (II) is hyperentangled Kwiat (1997).

Optical realization : It is possible to realize the shift operators in Eq.
(13) without an explicit coin operation using passive optical devices PBS and
$J$-plates on polarization and OAM DoF, respectively. Operator $\hat{S}_{x}$
($\hat{S}_{x,\rm pos}$ from now on, with `pos' referring to the position DoF)
can be readily realized using the PBS – which reflect horizontal polarization
and transmit vertical polarization. Operator $\hat{S}^{\prime}_{\sigma}$
($\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$) can be realized using a
$J$-plate.

A single photon carrying an OAM of $m\hbar$ per photon is represented by the
state $|m\rangle$ passes through a $J$-plate Devlin  _et al._ (2017). It is
known that the $J$-plate can decrease (increase) the OAM of the incoming
photon with Jones vector $|u_{1}\rangle$ ($|u_{2}\rangle$) by $\hbar$ per
photon while simultaneously transforming the Jones vector of the photon to
$|u_{2}\rangle$ ($|u_{1}\rangle$). Thus, the Jones matrix corresponding to the
$J$-plate is

$\displaystyle J(\phi)=e^{-i\phi}|u_{2}\rangle\langle
u_{1}|+e^{i\phi}|u_{1}\rangle\langle u_{2}|,$ (19)

where $\phi=\tan^{-1}(y^{\prime}/x^{\prime})$, with $(x^{\prime},y^{\prime})$
being the spatial coordinates of the $J$-plate placed in the transverse plane.
Since $|u_{1}\rangle$ and $|u_{2}\rangle$ are themselves functions of
$(\xi,\zeta,\theta)$ [Eqs. (1) and (II)], $J(\phi)$ can also be written as

$\displaystyle J(\phi)\equiv J(\phi,\xi,\zeta,\theta).$ (20)

Thus, the shift operator realizing this transformation will be [cf. Eq. (II)],

$\displaystyle\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$
$\displaystyle=\sum_{m}[|u_{2}\rangle\langle u_{1}|\otimes\mathds{1}_{x,{\rm
pos}}\otimes|m-1\rangle\langle m|$ $\displaystyle\,\,\,+|u_{1}\rangle\langle
u_{2}|\otimes\mathds{1}_{x,{\rm pos}}\otimes|m+1\rangle\langle m|].$ (21)

For the special case when $|u_{1}\rangle=|R\rangle=[1,-i]^{T}/\sqrt{2}$ and
$|u_{2}\rangle=|L\rangle=[1,i]^{T}/\sqrt{2}$, or equivalently for the choice
$J(\phi,0,-\pi/2,\pi/4)$, $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$
is realized using a $q$-plate Marrucci  _et al._ (2006). With these, we find
that a single photon in the initial state $|\Psi_{\rm in}\rangle$, under the
action of PBS and $J$-plates, will evolve as

$\displaystyle|\Psi_{n}\rangle$ $\displaystyle=(\hat{S}_{x,{\rm
pos}}\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}})^{n}|\Psi_{\rm
in}\rangle$
$\displaystyle=\sum_{x,m}\left[(a_{x,m}^{(n)}|H\rangle+b_{x,m}^{(n)}|V\rangle)\otimes|x\rangle\otimes|m\rangle\right],$
(22)

where the normalized complex coefficients are iteratively related as in Eqs.
(II) and (II) with the position label $y$ being replaced by the OAM label $m$.

We can also realize

$\displaystyle\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$
$\displaystyle=\sum_{m}[|u_{1}\rangle\langle u_{1}|\otimes\mathds{1}_{x,{\rm
pos}}\otimes|m-1\rangle\langle m|$ $\displaystyle\,\,\,+|u_{2}\rangle\langle
u_{2}|\otimes\mathds{1}_{x,{\rm pos}}\otimes|m+1\rangle\langle m|]$ (23)

[see Eq. (II)] using a $J$-plate with Jones matrix

$\displaystyle\tilde{J}(\phi)$ $\displaystyle=e^{-i\phi}|u_{1}\rangle\langle
u_{1}|+e^{i\phi}|u_{2}\rangle\langle u_{2}|$
$\displaystyle\equiv\tilde{J}(\phi,\xi,\zeta,\theta).$ (24)

However, in order to realize $\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$ with
$|u_{1}\rangle=|R\rangle$ and $|u_{2}\rangle=|L\rangle$ using a $q$-plate
instead of a $J$-plate, we will require an additional half-waveplate (HWP).

Figure 2: Optical implementation of Pauli and modified Pauli walks. Both Pauli
and modified Pauli walks with any orthogonal set of vectors
$\\{|u_{1}\rangle,|u_{2}\rangle\\}$ can be realized using $J$-plates and PBS
[see Eqs. (9), (II), and (II)]. When
$\\{|u_{1}\rangle,|u_{2}\rangle\\}=\\{|R\rangle,|L\rangle\\}$, every $J$-plate
can be replaced with a $q$-plate ($q$-plate and HWP) to realize the modified
Pauli walk (Pauli walk). $D_{i}$ denotes a detector unit placed at the
position $x=i$. Each detector unit consists of a spatial light modulator
(SLM), a single mode fiber (SMF), and a single photon detector (SPD). A single
photon is sent through a $q$-plate (or $J$-plate) to the PBS placed at $x=0$
(shown with an arrow). Here, $J$-plate implements either one of the shift
operators $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$ [Eq. (II)] or
$\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$ [Eq. (II)], and PBS implements the
shift operator $\hat{S}_{x,{\rm pos}}$ [see Eq. (II)]. Evidently, no explicit
2D coin operation is necessary to implement these type of quantum walks.

The probability distribution of both Pauli and modified Pauli walks for $n=50$
steps beginning with an initial state $|\Psi_{\rm
in}\rangle=[(|H\rangle+|V\rangle)/\sqrt{2}]\otimes|x=0\rangle\otimes|m=0\rangle$
[that is, by substituting $\alpha=\pi/4$ and $\beta=0$ in Eq. (15)] has been
shown in Fig. 1. In (a) we have considered modified Pauli walk using
$q$-plates and PBS [see Eq. (II)]. Owing to bit-flip symmetry Chandrashekar
_et al._ (2007), the probability distribution for the Pauli walk – realized
using $q$-plates, PBS, and HWP – will also be identical to that of (a) [Eqs.
(9) and (II)]. In (b) and (c) we have considered modified Pauli walk –
realized using $J$-plates and PBS. The orthogonal state vectors
$\\{|u_{1}\rangle,|u_{2}\rangle\\}$ for (b) and (c) were chosen to be

$\displaystyle\\{[1,-1]^{T}/\sqrt{2},[1,1]^{T}/\sqrt{2}\\},\,\,\,{\rm and}$
$\displaystyle\\{[1,\sqrt{3}i]^{T}/2,[\sqrt{3},-i]^{T}/2\\},$

respectively.

##  III Generation of Hyperentanglement

Figure 3: Entanglement negativity $\mathcal{N}$ [see Eq. (25)] plotted against
increasing number of steps between two DoF for various $J$-plate parameters
$(\xi,\zeta,\theta)$ [see (20)]. Here, $J(\phi,0,-\pi/2,\pi/4)$ represents a
$q$-plate, and the initial state is
$[(|H\rangle+|V\rangle)/\sqrt{2}]\otimes|x=0\rangle\otimes|y=0\rangle$. (a)
$\mathcal{N}$ between path and OAM DoF. (b) $\mathcal{N}$ between polarization
and OAM DoF. Note that $\mathcal{N}$ between polarization and path DoF is
identical as that of (b).

In this section we present an outline of the optical setup which can
hyperentangle the incoming single photon in polarization, path, and OAM DoF
and realize 2D DTQW. The hyperentanglement between these three DoF is then
quantified using entanglement negativity between the combination of the the
Hilbert spaces.

In Fig. 2 we present the schematic representation of the setup for optical
implementation of both Pauli and modified Pauli walks. In the case of modified
Pauli walk, the shift operator $\hat{S}_{x,{\rm pos}}$ is realized using a
PBS, $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$ is realized using a
$J$-plate for any given orthogonal set of vectors
$\\{|u_{1}\rangle,|u_{2}\rangle\\}$ [see Eq. (II)]. To realize the Pauli walk,
we just have to replace $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$ in
Eq. (II) with $\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$ in Eq. (II). Clearly,
$\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$ is also realized using a $J$-plate
with the Jones matrix given in Eq. (II). When
$\\{|u_{1}\rangle,|u_{2}\rangle\\}=\\{|R\rangle,|L\rangle\\}$, every $J$-plate
can be replaced with a $q$-plate in the modified Pauli walk. On the other
hand, every $J$-plate has to be replaced with a $q$-plate and a HWP to realize
the Pauli walk.

While $\hat{S}_{x,{\rm pos}}$ controls the path DoF of the single photon,
$\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$ (or
$\hat{S}_{\boldsymbol{\sigma},{\rm OAM}}$) controls both polarization and OAM
DoF of the same. Therefore, we don't need to explicitly use a coin operation
to control the polarization DoF. This setup requires $n(n+1)/2$ PBS and
$n(n-1)+1$ $J$-plates to realize the modified Pauli walk (or Pauli walk) for
$n$ steps. Here we have two remarks to make. First, the number of PBS and
$J$-plates required to implement this type of quantum walk will scale
quadratically with the number of steps. Second, when
$\\{|u_{1}\rangle,|u_{2}\rangle\\}=\\{|H\rangle,|V\rangle\\}$, the evolved
state is localized at the center and no spread is therefore observed.

The single photon, after $n$-steps, would have evolved in superposition of
position and OAM space. Upon measurement it will collapse at any one of the
detector units $D_{0}$, $D_{1}$, $D_{-1}$, $\ldots$, placed as shown in the
Fig. 2. Each detector unit contains a spatial light modulator (SLM), a single
mode fiber (SMF), and a single photon detector (SPD). The measurement of the
OAM DoF requires all three of these components Zhang  _et al._ (2010); Cardano
_et al._ (2015), whereas the measurement of the path DoF requires just a SPD
Do  _et al._ (2005).

After $n$ steps, the single photon will be entangled in polarization, path,
and OAM DoF. To quantify the amount of entanglement between any two DoF, we
adopt a measure known as the entanglement negativity Vidal and Werner (2002).
Here, we first partial trace out the density matrix
$|\Psi_{n}\rangle\langle\Psi_{n}|$ with respect to the third DoF. After
partial transpose of the resulting reduced density matrix we compute

$\displaystyle\mathcal{N}=\sum_{i}\frac{(|\lambda_{i}|-\lambda_{i})}{2},$ (25)

where $\lambda_{i}$'s are the eigenvalues of the partial transposed reduced
density matrix. Evidently, $\mathcal{N}=0$ implies the reduced system is
unentangled.

Figure 4: Entanglement negativity $\mathcal{N}$ [see Eq. (25)] between any two
of three DoF plotted against any one of the three $J$-plate parameters
$(\xi,\zeta,\theta)$ [see (20)] after 25 steps. The initial state was taken to
be $[(|H\rangle+|V\rangle)/\sqrt{2}]\otimes|x=0\rangle\otimes|y=0\rangle$. In
(a), $\mathcal{N}$ between path and OAM DoF is considered, whereas in (b) that
between polarization and OAM DoF is considered. In both (a) and (b) the black
curve was obtained for $J(\phi,\pi/2,\pi/2,\underline{\theta})$ and PBS
combination, where $\underline{\theta}$ denotes that $\theta$ is varied from 0
through $\pi/2$ in steps of $\pi/180$. And the red curve was obtained for
$J(\phi,\underline{\xi},-\pi/2,\pi/4)$ and PBS combination, where
$\underline{\xi}$ denotes that $\xi$ is varied from 0 through $\pi/2$ in steps
of $\pi/180$. Note that the choice $J(\phi,0,-\pi/2,\pi/4)$ represents a
$q$-plate and is encircled in both (a) and (b).

Now we present our numerically simulated results of a single photon passing
through the optical setup schematically outlined in the Fig. 2. The
probability distribution of the evolved state after 50 steps for three
different $J$-plate parameters $(\xi,\zeta,\theta)$ [see (20)] has been shown
in the Fig. 1. Note that $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$
[Eq. (II)] with any $(\xi,\zeta,\theta)$ (or equivalently,
$\\{|u_{1}\rangle,|u_{2}\rangle\\}$) can be realized using a $J$-plate.
Nevertheless, $\hat{S}^{\prime}_{\boldsymbol{\sigma},{\rm OAM}}$ with the
choice $(\xi,\zeta,\theta)=(0,-\pi/2,\pi/4)$ can be implemented using a
$q$-plate. In order to demonstrate that the three DoF are entangled, we first
trace out the polarization DoF from the density matrix corresponding to
$|\Psi_{n}\rangle$ (see Eq. (II)) and compute the entanglement negativity
$\mathcal{N}$ [see Eq. (25)] corresponding to the partial transposed reduced
density matrix. We then plot $\mathcal{N}$ against increasing number of steps
$n$ in Fig. 3.

In frame (a) of Fig. 3 we have shown $\mathcal{N}$ between the path and OAM
DoF as a function of number of steps. As we increase the number of steps,
$\mathcal{N}$ also increases linearly. The value $\mathcal{N}$ can be
controlled using the $J$-plate parameters $(\xi,\zeta,\theta)$. If we partial
trace any one of the spatial DoF (path or OAM DoF), $\mathcal{N}$
corresponding to the reduced density matrix between the polarization and OAM
(or polarization and path) DoF reaches a steady value as we increase the
number of steps (see frame (b) of Fig. 3) for various choices of the $J$-plate
parameters $(\xi,\zeta,\theta)$. For instance, for the choice
$(\xi,\zeta,\theta)=(0,-\pi/2,\pi/4)$, i.e., a $q$-plate, $\mathcal{N}$
between polarization and OAM (or path) DoF reaches a steady state value
$0.17927$ after 25 steps, provided we begin with an initial state
$[(|H\rangle+|V\rangle)/\sqrt{2}]\otimes|x=0\rangle\otimes|y=0\rangle$.

We now demonstrate how the $J$-plate parameters $(\xi,\zeta,\theta)$,
beginning with an initial state, can be used to control the amount of
entanglement between three DoF : polarization, path, and OAM. In other words,
we demonstrate numerically how $\mathcal{N}$ can be controlled by tuning the
$J$-plate parameters. In Fig. 4 we have shown how the negativity $\mathcal{N}$
between any two DoF varies with respect to the $J$-plate parameters
$(\xi,\zeta,\theta)$. Here, we have allowed one of the three $J$-plate
parameters $(\xi,\zeta,\theta)$ to vary while keeping the remaining two
constant and plotted the respective $\mathcal{N}$ values. Furthermore,
$\mathcal{N}$ between any two DoF has been computed after evolving the quantum
walk for 25 steps. In frame (a) of Fig. 4 $\mathcal{N}$ between path and OAM
DoF has been plotted as a function of one of the three $J$-plate parameters.
And in frame (b) of Fig. 4 $\mathcal{N}$ between polarization and OAM (or
path) DoF has been obtained as a function of the same. By keeping
$\xi=\zeta=\pi/2$ and varying $\theta$ in steps of $\pi/180$ from 0 through
$\pi/2$, we obtain a black curve as shown in Fig. 4. Likewise, keeping
$\zeta=-\pi/2$ and $\theta=\pi/4$ and varying $\xi$ in steps of $\pi/180$ from
0 through $\pi/2$, we obtain a red curve as shown in Fig. 4. The entanglement
negativity $\mathcal{N}$ corresponding to the $q$-plate is encircled in both
(a) and (b).

##  IV Conclusion

To summarize, we have proposed a passive optical setup – using $J$-plates or
$q$-plates, and PBS – to hyperentangle an incoming single photon in
polarization, path, and OAM DoF. We have shown that this optical setup can be
efficiently used to simulate the 2D DTQW with variable evolution parameters
(modified Pauli walk) without explicitly using a quantum coin operation. The
evolved state has been numerically shown to be hyperentangled in polarization,
path, and OAM DoF. The amount of entanglement between any two of the three DoF
has been computed using entanglement negativity. It was observed that the
entanglement negativity increased linearly between the path and OAM DoF,
whereas the same between the polarization and path (or OAM) DoF remained
constant after few number of steps due to the bound on the dimension of the
coin space. The amount of entanglement between any two of the three DoF and
the hyperentanglement in the system can be controlled by varying the $J$-plate
parameters. Ability to control and engineer the dynamics of quantum walks
using optical components can also play an important role in realization of
non-Markovian quantum channels Naikoo  _et al._ (2020) and study of open
quantum systems.

## Acknowledgment

Yasir would like to thank Abhaya S. Hegde for useful discussions on
Hyperentangled states. Yasir and CMC acknowledge the support from the Office
of Principal Scientific Advisor to Government of India, project no.
Prn.SA/QSim/2020 and Interdisciplinary Cyber Physical Systems (ICPS) program
of the Department of Science and Technology, India, Grant No.:
DST/ICPS/QuST/Theme-1/2019/1 for the support,

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