# Option pricing in Volterra sandwiched volatility model

Giulia Di Nunno1,2
<EMAIL_ADDRESS>Yuliya Mishura3
<EMAIL_ADDRESS>Anton Yurchenko-Tytarenko1,111Corresponding author.
<EMAIL_ADDRESS>
(1Department of Mathematics, University of Oslo
2Department of Business and Management Science, NHH Norwegian School of
Economics, Bergen
3Department of Probability Theory, Statistics and Actuarial Mathematics, Taras
Shevchenko National University of Kyiv
)

###### Abstract

We introduce a new model of financial market with stochastic volatility driven
by an arbitrary Hölder continuous Gaussian Volterra process. The
distinguishing feature of the model is the form of the volatility equation
which ensures the solution to be “sandwiched” between two arbitrary Hölder
continuous functions chosen in advance. We discuss the structure of local
martingale measures on this market, investigate integrability and Malliavin
differentiability of prices and volatilities as well as study absolute
continuity of the corresponding probability laws. Additionally, we utilize
Malliavin calculus to develop an algorithm of pricing options with
discontinuous payoffs.

Keywords: stochastic volatility, sandwiched process, Hölder continuous noise,
option pricing, Malliavin calculus
MSC 2020: 91G30; 60H10; 60H35; 60G22
Acknowledgements. The present research is carried out within the frame and
support of the ToppForsk project nr. 274410 of the Research Council of Norway
with title STORM: Stochastics for Time-Space Risk Models. The second author
was supported by Japan Science and Technology Agency CREST JPMJCR2115.

##  1 Introduction

Despite its fundamental role in the development of mathematical finance, the
classic Black–Scholes approach does not reflect numerous phenomena observed on
real financial markets. In particular, numerous empirical arguments clearly
indicate that the volatility is far from being constant: it is well known that
time series of realized volatility are prone to clustering [17, 18] and the
surface profile of implied volatility is not flat [19, 20, 30]. One of common
ways to modify the vanilla Black–Scholes framework to account for such effects
is to model the instantaneous volatility as a stochastic process itself.
Initially, such solution was proposed in [34] and [32], and has evolved into a
separate full-fledged area of research since then, see e.g. books [5, 27, 35,
36] for an overview of classical approaches.

A separate line of research on stochastic volatility is associated with the
inclusion of market memory. Here, we mention the paper [12], where
fractionally integrated GARCH and EGARCH models were considered, as well as
the pioneering work [16], where the idea to model the log-volatility by
fractional Brownian motion (fBm) with $H>1/2$ was proposed. The latter “fBm
approach” turned out to be very fruitful and gave rise to numerous studies
among which we note [13, 15] where the volatility is modeled by a fractional
Ornstein-Uhlenbeck process. For a long time, the Hurst index $H>1/2$ prevailed
in fractional models due to the stylized fact stating presence of long memory
in volatility [3, 4, 24]. Some of more recent empirical studies such as [31]
argue that it is reasonable to model volatility with Hölder regularity
strictly less than $1/2$. This observation became a core of the so-called
“rough volatility” models such as [9, 26] which, in addition, prove to be
consistent with the shape of the realized volatility surface [25, 29]. Another
degree of freedom can be introduced by assuming that local roughness of
volatility depends on time. Such an extension was justified in e.g. [7, 21]
with a suggestion to use multifractional Brownian motion. In this regard, we
also mention [37] where volatility is modeled by a function of a general
Gaussian Volterra process and, in a sense, the latter approach is related to
the model proposed in the present paper, since we also choose a Gaussian
Volterra process as the source of noise in volatility. Namely, our model has
the form

$\displaystyle
S_{i}(t)=S_{i}(0)+\int_{0}^{t}\mu_{i}(s)S_{i}(s)ds+\int_{0}^{t}Y_{i}(s)S_{i}(s)dB^{S}_{i}(s),$
(1.1)
$Y_{i}(t)=Y_{i}(0)+\int_{0}^{t}b_{i}(s,Y_{i}(s))ds+\int_{0}^{t}\mathcal{K}_{i}(t,s)dB_{i}^{Y}(s),$
(1.2)

$i=1,...,d$, where $S_{i}$ and $Y_{i}$ are price and volatility processes
respectively, $\mu_{i}$ are deterministic continuous functions, $B^{S}_{i}$,
$B^{Y}_{i}$ are correlated standard Brownian motions, $\mathcal{K}_{i}$ are
arbitrary square integrable kernels such that each process
$Z_{i}(t):=\int_{0}^{t}\mathcal{K}_{i}(t,s)dB_{i}^{Y}(s)$ is Hölder continuous
up to the order $H_{i}\in(0,1)$. The main distinguishing property of the model
is related to the drift functions $b_{i}$: for each $i=1,...,d$ there exist
two deterministic continuous functions $0<\varphi_{i}<\psi_{i}$ and constants
$c>0$, $\gamma>\frac{1}{H_{i}}-1$ and $y_{*}>0$ such that

$\displaystyle b_{i}(t,y)\geq\frac{c}{(y-\varphi_{i}(t))^{\gamma}},\quad
y\in(\varphi_{i}(t),\varphi_{i}(t)+y_{*}),$ $\displaystyle
b_{i}(t,y)\leq-\frac{c}{(\psi_{i}(t)-y)^{\gamma}},\quad
y\in(\psi_{i}(t)-y_{*},\psi_{i}(t)).$

Processes of the type (1.2) were extensively studied in [23] and will be
referred to as _sandwiched processes_ due to the fact that the abovementioned
shape of the drift ensures that

$\varphi_{i}(t)<Y_{i}(t)<\psi_{i}(t),$ (1.3)

i.e. the volatility process $Y_{i}$ is “sandwiched” between $\varphi_{i}$ and
$\psi_{i}$.

The model (1.1)–(1.2) that will be referred to as the Volterra Sandwiched
Volatility (VSV) model has several substantial advantages. First of all, it is
very flexible in terms of the choice of noises: the class of Hölder continuous
Volterra Gaussian processes is very broad and includes e.g. fractional
Brownian motion. In fact, in order to ensure existence and uniqueness of the
solution to equation (1.2), it is sufficient to demand only Hölder continuity
of the additive noise without any assumptions on distribution. Nevertheless,
we still prefer to use Gaussian Volterra processes because, on the one hand,
they allow to apply various Malliavin calculus techniques and, on the other
hand, enjoy adaptedness to the filtration generated by $B_{i}^{S}$,
$B_{i}^{Y}$, $i=1,...,d$. The second advantage of the VSV model is its
tractability. As it is noted in [23, Section 4], sandwiched processes can be
used to define generalizations of some processes that are common for
stochastic volatility modeling such as Cox-Ingersoll-Ross (CIR) or
Chan–Karolyi–Longstaff–Sanders (CKLS) processes. Here we also mention [33] as
well as the series of papers [38, 39, 40], where a process of the sandwiched
type with $\varphi_{i}\equiv 0$, $\psi_{i}\equiv\infty$ was used in the
context of CIR model driven by fractional Brownian motion with Hurst index
$H>1/2$. The third benefit of the VSV model is related to the functions
$\varphi_{i}$, $\psi_{i}$. Leaving aside the fact that they can be regarded as
another calibration parameters and a thoughtful choice of them can improve the
performance of the model, the “sandwichness” (1.3) substantially improves
analytical properties of both price and volatility. For example, since we
choose $\varphi_{i}>0$, the process $Y_{i}$ is bounded away from zero and
hence inverse volatilities $\frac{1}{Y_{i}}$ are integrable w.r.t. Brownian
motion generating the model. This will allow us to provide a clear description
of equivalent local martingale measures on the market – a feature which is
often left aside in many popular rough volatility models. Moreover, the
boundedness of $Y_{i}$ from above ensures that for any $r\in\mathbb{R}$

$\mathbb{E}\left[\sup_{t\in[0,T]}S^{r}_{i}(t)\right]<\infty$ (1.4)

– a property that is often absent even in standard Brownian stochastic
volatility models (see e.g. [2]).

In the first half of the paper, we introduce the VSV model and present a
detailed discussion of its properties. In particular, we prove (1.4), provide
description of the set of martingale measures and obtain Malliavin
differentiability of both volatility and price using the method similar to
[33, Theorem 3.3] based on characterization of Sobolev spaces over an abstract
Wiener space from [45]. Results on Malliavin differentiability and absolute
continuity of the probability law of price pave the way for some advanced
numerical methods and the second part of the paper is devoted to one of them:
the Malliavin integration-by-parts quadrature of discontinuous option payoffs.
Namely, consider a European option with a discontinuous payoff $f(S(T))$.
Usually, it is impossible to compute the expected payoff $\mathbb{E}f(S(T))$
analytically, so one must apply numerical methods for that: for instance, one
can take an approximation $\widehat{S}(T)$ of $S(T)$ using some numerical
scheme and utilize one of Monte Carlo techniques to compute
$\mathbb{E}f(\widehat{S}(T))$. However, [6] states that discontinuities in $f$
lead to a substantial deterioration in convergence speed: for instance, the
rate of convergence of $\mathbb{E}[|f({S}(T))-f(\widehat{S}(T))|^{2}]$ to zero
is halved in comparison to the order of the scheme for $\widehat{S}$. This
aggravation is additionally worsened by typically low convergence rates of
numerical schemes for models with rough volatility and it significantly limits
application of some advanced Monte Carlo methods such as multi-level Monte
Carlo. In order to overcome this problem, [1] suggested to use Malliavin
integration-by-parts to replace the discontinuous $f$ under the expectation
with some Lipschitz functional and, as a result, the initial convergence rate
was preserved. Similar technique was applied in [10, 40] for models driven by
a fractional Brownian motion and, in the present paper, we utilize the same
approach for the VSV model (1.1)–(1.2) providing some useful extensions. In
particular, all [1, 10, 40] take $\mathbb{E}f(\widehat{S}(T))$ with respect to
the physical measure (i.e. it is is an expected payoff rather than a price)
whereas we provide a quadrature formula under the change of measure as well.

The paper is organised as follows. In Section 2, we specify the VSV market
model, provide some moment bounds and describe the set of martingale measures.
In Section 3, we prove Malliavin differentiability of volatility and price
processes as well as present results on absolute continuity of probability
laws of the latter. Section 4 adapts the Malliavin integration-by-parts
quadrature method from [1] to the VSV setting. In Section 5, we give error
estimates the mentioned method provided that the volatility is discretized
using the drift-implicit Euler scheme from [22]. Section 6 contains simulation
results.

##  2 Model description

In this Section, we define the Volterra sandwiched volatility (VSV) model as
well as provide some basic results regarding its properties.

######  Notation 2.1.

Throughout this paper, $C$ denotes any positive deterministic constant the
exact value of which is not relevant. Note that $C$ may change from line to
line (or even within one line).

###  2.1 Preliminaries and assumptions

#### Probability space, filtration and correlation structure of Brownian
motions.

Let $(\Omega,\mathcal{F},\mathbb{P})$ be the canonical $2d$-dimensional Wiener
space, i.e. $\Omega:=C_{0}([0,T];\mathbb{R}^{2d})$, $\mathbb{P}$ is the
classical Wiener measure, $\mathcal{F}$ is the $\mathbb{P}$-augmented Borel
sigma-algebra. Let also $W$ be the corresponding $2d$-dimensional Brownian
motion, i.e. $W$: $[0,T]\times\Omega\to\mathbb{R}^{2d}$,
$W(t,\omega)=\omega(t)$, and $\mathbb{F}=\\{\mathcal{F}_{t},~{}t\in[0,T]\\}$
be the $\mathbb{P}$-augmented filtration generated by $W$.

Now, let $\Sigma=(\sigma_{i,j})_{i,j=1}^{2d}$ be an arbitrary non-degenerate
$2d\times 2d$ real correlation matrix, i.e. $\Sigma$ is symmetric, positive
definite, $\sigma_{i,j}\in[-1,1]$ and $\sigma_{i,i}=1$ for all $i=1,...,2d$.
Since $\Sigma$ is symmetric and positive definite, it admits a Cholesky
decomposition of the form
$\Sigma=\operatorname{\mathcal{L}}\operatorname{\mathcal{L}}^{T}$, where
$\operatorname{\mathcal{L}}=(\ell_{i,j})_{i,j=1}^{2d}$ is a lower triangular
matrix with strictly positive diagonal entries. Define
$B(t)=(B_{1}(t),...,B_{2d}(t))^{T}:=\operatorname{\mathcal{L}}W(t)$ and note
that components of the $2d$-dimensional stochastic process
$\\{B(t),~{}t\in[0,T]\\}$ are $1$-dimensional Brownian motions with

$\operatorname{\mathbb{E}}\left[B_{i}(t)B_{j}(s)\right]=(t\wedge
s)\sigma_{i,j},\quad t,s\in[0,T],~{}i,j=1,...,2d.$

In what follows, we will need to distinguish the first $d$ components of $B$
(that will generate the “outer” noise for the price) from the other $d$
components (that will generate the noise for the volatility) and thus we
introduce the notation

$\displaystyle
B^{S}(t)=\left(B^{S}_{1}(t),...,B^{S}_{d}(t)\right)^{T}:=(B_{1}(t),...,B_{d}(t))^{T},$
$\displaystyle
B^{Y}(t)=\left(B^{Y}_{1}(t),...,B^{Y}_{d}(t)\right)^{T}:=(B_{d+1}(t),...,B_{2d}(t))^{T}.$

Clearly, since the matrix $\operatorname{\mathcal{L}}$ is not degenerate, the
filtration generated by the $2d$-dimensional process $B$ coincides with
$\mathbb{F}$.

#### Stochastic volatility process.

Consider now $d$ measurable functions $\mathcal{K}_{i}$:
$[0,T]^{2}\to\mathbb{R}$, $i=1,...,d$, that satisfy the following assumption.

######  Assumption A.

For any $i=1,...,d$

  * (i)

$\mathcal{K}_{i}$ is a square integrable Volterra kernel, i.e.
$\mathcal{K}_{i}(t,s)=0$ whenever $t<s$,

$\int_{0}^{T}\int_{0}^{T}\mathcal{K}^{2}_{i}(t,s)dsdt=\int_{0}^{T}\int_{0}^{t}\mathcal{K}^{2}_{i}(t,s)dsdt<\infty$

and

$\int_{0}^{T}\mathcal{K}^{2}_{i}(t,s)ds<\infty,\quad\int_{0}^{T}\mathcal{K}^{2}_{i}(t,s)dt<\infty,\quad\forall
s,t\in[0,T];$

  * (ii)

there exists a constant $H_{i}\in(0,1)$ such that for any
$\lambda\in(0,H_{i})$

$\int_{0}^{t}(\mathcal{K}_{i}(t,u)-\mathcal{K}_{i}(s,u))^{2}du\leq
C_{\lambda}|t-s|^{2\lambda},\quad 0\leq s\leq t\leq T,$ (2.1)

where $C_{\lambda}>0$ is a constant, possibly dependent on $\lambda$.

Assumption A(i) allows to define a $d$-dimensional Gaussian Volterra process
$Z=\left(Z_{1},...,Z_{d}\right)$ such that

$Z_{i}(t):=\int_{0}^{t}\mathcal{K}_{i}(t,s)dB^{Y}_{i}(s),\quad t\in[0,T].$
(2.2)

Moreover, Assumption A(ii) ensures that the paths of each $Z_{i}$ are regular
a.s., which is summarized in the following proposition.

######  Proposition 2.2 ([8], Theorem 1 and Corollary 4).

The Gaussian process $\\{Z_{i}(t),~{}t\in[0,T]\\}$ defined by (2.2) has a
modification that is a.s. Hölder continuous up to the order $H_{i}\in(0,1)$ if
and only if (2.1) holds for any $\lambda\in(0,H_{i})$. Moreover, for any
$\lambda\in(0,H_{i})$, the random variable

$\Lambda_{\lambda,i}:=\sup_{0\leq s<t\leq
T}\frac{|Z_{i}(t)-Z_{i}(s)|}{|t-s|^{\lambda}}$ (2.3)

has moments of all orders.

######  Remark 2.3.

By Proposition 2.2, it is evident that one can choose the modification of
$Z=(Z_{1},...,Z_{d})$ such that for _all_ $\omega\in\Omega$

$|Z_{i}(\omega,t)-Z_{i}(\omega,s)|\leq\Lambda_{\lambda,i}(\omega)|t-s|^{\lambda},$

where $\lambda\in(0,H_{i})$. In what follows, this Hölder continuous
modification will be used.

Now, let $\varphi_{i}$, $\psi_{i}$: $[0,T]\to\mathbb{R}$, $i=1,...,d$, be $d$
pairs of functions such that

  1. 1)

$\varphi_{i}$, $\psi_{i}$ are Hölder continuous up to the order $H_{i}$, where
$H_{i}$ is from Assumption A(ii);

  2. 2)

for each $i=1,...,d$ and $t\in[0,T]$:

$0<\varphi_{i}(t)<\psi_{i}(t).$ (2.4)

For any
$a_{1},a_{2}\in\left[0,\min_{i=1,...,d}\frac{1}{2}\lVert\psi_{i}-\varphi_{i}\rVert_{\infty}\right)$,
where $\lVert\cdot\rVert_{\infty}$ is the standard $\sup$-norm, denote

$\mathcal{D}^{i}_{a_{1},a_{2}}:=\\{(t,y)\in[0,T]\times\mathbb{R}_{+},~{}y\in(\varphi_{i}(t)+a_{1},\psi_{i}(t)-a_{2})\\}$
(2.5)

and consider $d$ functions $b_{i}$: $\mathcal{D}^{i}_{0,0}\to\mathbb{R}$,
$i=1,...,d$, that satisfy the following assumptions.

######  Assumption B.

  * (i)

The functions $b_{i}\in C(\mathcal{D}^{i}_{0,0})$, $i=1,...,d$;

and there exist constants $c>0$, $p>1$,
$y_{*}\in\left(0,\frac{1}{2}\min_{i=1,...,d}\lVert\psi_{i}-\varphi_{i}\rVert_{\infty}\right)$
such that for any $i=1,...,d$ the following conditions hold:

  * (ii)

for any
$\varepsilon\in\left(0,\min\left\\{1,\min_{i=1,...,d}\frac{1}{2}\lVert\psi_{i}-\varphi_{i}\rVert_{\infty}\right\\}\right)$

$|b_{i}(t_{1},y_{1})-b_{i}(t_{2},y_{2})|\leq\frac{c}{\varepsilon^{p}}\left(|y_{1}-y_{2}|+|t_{1}-t_{2}|^{\lambda}\right),\quad(t_{1},y_{1}),(t_{2},y_{2})\in\mathcal{D}^{i}_{\varepsilon,\varepsilon};$

  * (iii)

for some constant $\gamma_{i}>\frac{1}{H_{i}}-1$, where $H_{i}$ is from
Assumption A(ii),

$b_{i}(t,y)\geq\frac{c}{(y-\varphi_{i}(t))^{\gamma_{i}}},\quad(t,y)\in\mathcal{D}^{i}_{0,0}\setminus\mathcal{D}^{i}_{y_{*},0},$
$b_{i}(t,y)\leq-\frac{c}{(\psi_{i}(t)-y)^{\gamma_{i}}},\quad(t,y)\in\mathcal{D}^{i}_{0,0}\setminus\mathcal{D}^{i}_{0,y_{*}};$

  * (iv)

there exists a continuous partial derivative $\frac{\partial b_{i}}{\partial
y}$ with respect to the spatial variable and

$\frac{\partial b_{i}}{\partial y}(t,y)<c,\quad(t,y)\in\mathcal{D}^{i}_{0,0}.$

Next, consider the $d$-dimensional _sandwiched process_ $(Y_{1},...,Y_{d})$
defined by

$Y_{i}(t)=Y_{i}(0)+\int_{0}^{t}b_{i}(s,Y_{i}(s))ds+Z_{i}(t),\quad
t\in[0,T],\quad i=1,...,d,$ (2.6)

where

  * –

$Y_{i}(0)\in\left(\varphi_{i}(0),\psi_{i}(0)\right)$ are deterministic
constants;

  * –

$Z_{i}$, $i=1,...,d$, are defined by (2.2) with
$\operatorname{\mathcal{K}}_{i}$ satisfying Assumption A;

  * –

$b_{i}$, $i=1,...,d$, satisfy Assumption B.

######  Remark 2.4.

Each equation in (2.6) is treated pathwisely and, according to [23], has a
unique solution for the given $\omega\in\Omega$ provided that the
corresponding path $Z_{i}(\cdot,\omega)$ of the noise is Hölder continuous up
to the order $H_{i}$ (and this happens a.s. by Remark 2.3). Moreover, in this
case, according to [23, Theorem 3.2], for each $i=1,...,d$ and
$\lambda\in\left(\frac{1}{\gamma_{i}+1},H_{i}\right)$, where $\gamma_{i}$ is
from Assumption B(iii), one can find deterministic constants $L_{1,i}$,
$L_{2,i}>0$ and $\alpha_{i}>0$ that depend only on the shape of $b_{i}$ and
$\lambda$ such that for all $t\in[0,T]$:

$\varphi(t)+\frac{L_{1,i}}{(L_{2,i}+\Lambda_{\lambda,i})^{\alpha_{i}}}\leq
Y_{i}(t)\leq\psi(t)-\frac{L^{1}_{i}}{(L^{2,i}+\Lambda_{\lambda,i})^{\alpha_{i}}},$
(2.7)

where $\Lambda_{\lambda,i}$ is a random variable defined by (2.3). In
particular, this means that $0<\varphi_{i}(t)<Y_{i}(t)<\psi_{i}(t)$ a.s.,
$t\in[0,T]$, $i=1,...,d$ (which justifies the name “ _sandwiched_ ”). Note
that $Z_{i}$ does not have to be Gaussian for (2.7) to hold: it is sufficient
to assume $\lambda$-Hölder continuity with $\lambda>\frac{1}{\gamma_{i}+1}$.

Since $\varphi_{i}$, $i=1,...,d$, are continuous positive functions, it is
clear that for any $r>0$:

$\operatorname{\mathbb{E}}\left[\sup_{t\in[0,T]}\frac{1}{Y^{r}_{i}(t)}\right]\leq\frac{1}{\min_{t\in[0,T]}\varphi_{i}^{r}(t)}<\infty.$

Furthermore, since $\Lambda_{\lambda,i}$ has moments of all orders, it is easy
to see that for any $r>0$:

$\operatorname{\mathbb{E}}\left[\sup_{t\in[0,T]}\frac{1}{\left(Y_{i}(t)-\varphi_{i}(t)\right)^{r}}\right]<\infty,\quad\operatorname{\mathbb{E}}\left[\sup_{t\in[0,T]}\frac{1}{\left(\psi_{i}(t)-Y_{i}(t)\right)^{r}}\right]<\infty.$

######  Remark 2.5.

Note that item (iv) of Assumption B are not required for existence and
uniqueness of the solution to (2.6). It will be exploited later for the
numerical scheme.

#### Price process.

The components of the $d$-dimensional price process $S=(S_{1},...,S_{d})$ will
be defined as solutions to the SDEs of the form

$S_{i}(t)=S_{i}(0)+\int_{0}^{t}\mu_{i}(s)S_{i}(s)ds+\int_{0}^{t}Y_{i}(s)S_{i}(s)dB^{S}_{i}(s),\quad
t\in[0,T],\quad i=1,...,d,$ (2.8)

where

  * –

$S_{i}(0)>0$, $i=1,...,d$, are deterministic constants;

  * –

$\mu_{i}$: $[0,T]\to\mathbb{R}$, $i=1,...,d$, are $H_{i}$-Hölder continuous
functions, where each $H_{i}$ is from Assumption A(ii);

  * –

$Y_{i}$, $i=1,...,d$ are sandwiched volatility processes defined above.

We now list some simple properties of processes $S_{i}$, $i=1,...,d$.

######  Theorem 2.6.

For any $i=1,...,d$, equation (2.8) has a unique solution of the form

$S_{i}(t)=S_{i}(0)\exp\left\\{\int_{0}^{t}\left(\mu_{i}(s)-\frac{Y_{i}^{2}(s)}{2}\right)ds+\int_{0}^{t}Y_{i}(s)dB^{S}_{i}(s)\right\\}.$
(2.9)

Furthermore, for any $r\in\mathbb{R}$:

$\mathbb{E}\left[\sup_{t\in[0,T]}S_{i}^{r}(t)\right]<\infty.$

###### Proof.

Each $Y_{i}$, $i=1,...,d$, is a bounded process and thus existence and
uniqueness of solution to the corresponding equation follows from e.g. [14,
Theorem 16.1.2] while the explicit form of the solution can be checked
straightforwardly via Itô’s formula.

Next, using the explicit form of $S_{i}$ as well as boundedness of $Y_{i}$ one
can see that there exists a constant $C>0$ such that

$\displaystyle S_{i}^{r}(t)$
$\displaystyle=S_{i}^{r}(0)\exp\left\\{r\int_{0}^{t}\mu_{i}(s)ds-\frac{r}{2}\int_{0}^{t}Y_{i}^{2}(s)ds+r\int_{0}^{t}Y_{i}(s)dB^{S}_{i}(s)\right\\}$
(2.10) $\displaystyle\leq
C\exp\left\\{r\int_{0}^{t}Y_{i}(s)dB^{S}_{i}(s)\right\\}$ $\displaystyle\leq
C\exp\left\\{\frac{r^{2}}{2}\int_{0}^{t}Y_{i}^{2}(s)ds\right\\}\times$
$\displaystyle\qquad\times\exp\left\\{\int_{0}^{t}rY(s)dB_{i}^{S}(s)-\frac{1}{2}\int_{0}^{t}(rY(s))^{2}ds\right\\}$
$\displaystyle\leq
C\exp\left\\{\int_{0}^{t}rY_{i}(s)dB_{i}^{S}(s)-\frac{1}{2}\int_{0}^{t}(rY_{i}(s))^{2}ds\right\\}.$

Note that the Novikov’s criterion immediately yields that the process

$\mathcal{E}_{t}\\{rY_{i}\cdot
B^{S}_{i}\\}:=\exp\left\\{\int_{0}^{t}rY_{i}(s)dB^{S}_{i}(s)-\frac{1}{2}\int_{0}^{t}(rY_{i}(s))^{2}ds\right\\},\quad
t\in[0,T],$

is a uniformly integrable martingale such that

$\mathbb{E}\left[\mathcal{E}_{t}\\{rY_{i}\cdot B^{S}_{i}\\}\right]=1,\quad
t\in[0,T],$

and, moreover,

$\mathcal{E}_{t}\\{rY_{i}\cdot
B^{S}_{i}\\}=\int_{0}^{t}\mathcal{E}_{s}\\{rY_{i}\cdot
B^{S}_{i}\\}rY_{i}(s)dB^{S}_{i}(s),\quad t\in[0,T].$

By the Burkholder–Davis–Gundy inequality, there exists a constant $C>0$ such
that

$\displaystyle\mathbb{E}\left[\sup_{t\in[0,T]}\mathcal{E}_{t}\\{rY_{i}\cdot
B^{S}_{i}\\}\right]$
$\displaystyle=\mathbb{E}\left[\sup_{t\in[0,T]}\int_{0}^{t}\mathcal{E}_{s}\\{rY_{i}\cdot
B^{S}_{i}\\}rY_{i}(s)dB^{S}_{i}(s)\right]$ (2.11) $\displaystyle\leq
C\mathbb{E}\left[\left(\int_{0}^{T}\mathcal{E}^{2}_{s}\\{rY_{i}\cdot
B^{S}_{i}\\}Y_{i}^{2}(s)ds\right)^{\frac{1}{2}}\right]$ $\displaystyle\leq
C\sup_{s\in[0,T]}\psi(s)\left(\int_{0}^{T}\mathbb{E}\left[\mathcal{E}^{2}_{s}\\{rY_{i}\cdot
B^{S}_{i}\\}\right]ds\right)^{\frac{1}{2}}.$

By Novikov’s criterion, $\mathbb{E}\left[\mathcal{E}_{s}(2rY_{i}\cdot
B^{S}_{i})\right]=1$, so

$\displaystyle\mathbb{E}\left[\mathcal{E}^{2}_{s}\\{rY_{i}\cdot
B^{S}_{i}\\}\right]$
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{s}\\{2rY_{i}\cdot
B^{S}_{i}\\}\exp\left\\{\int_{0}^{s}(rY_{i}(u))^{2}du\right\\}\right]$
$\displaystyle\leq C\mathbb{E}\left[\mathcal{E}_{s}\\{2rY_{i}\cdot
B^{S}_{i}\\}\right]<\infty$

and hence, taking into account (2.10), we obtain the required result. ∎

#### Numéraire and discounted price process.

As a numéraire, we will use the function $e^{\int_{0}^{t}\nu(s)ds}$,
$t\in[0,T]$, where $\nu$: $[0,T]\to\mathbb{R}_{+}$ denotes a Hölder continuous
function of order $\min_{i=1,...,d}H_{i}$ representing an instantaneous
interest rate. The discounted price process denoted as

$\widetilde{S}_{i}(t):=e^{-\int_{0}^{t}\nu(s)ds}S_{i}(t),\quad t\in[0,T],\quad
i=1,...,d,$

has thus dynamics of the form

$\widetilde{S}_{i}(t)=S_{i}(0)+\int_{0}^{t}\widetilde{\mu}_{i}(s)\widetilde{S}_{i}(s)ds+\int_{0}^{t}Y_{i}(s)\widetilde{S}_{i}(s)dB^{S}_{i}(s),\quad
t\in[0,T],\quad i=1,...,d,$

where $\widetilde{\mu}_{i}:=\mu_{i}-\nu$.

###  2.2 Incompleteness of the market and martingale measures

Recall that our model is driven by the $2d$-dimensional Gaussian process

$\left(\begin{matrix}B^{S}(t)\\\
B^{Y}(t)\end{matrix}\right)=\operatorname{\mathcal{L}}W(t),\quad t\in[0,T],$

where $W=\\{W(t),~{}t\in[0,T]\\}$ is a standard $2d$-dimensional Brownian
motion and $\operatorname{\mathcal{L}}=(\ell_{i,j})_{i,j=1}^{2d}$ is
nondegenerate lower triangular $2d\times 2d$ real matrix with positive entries
on a diagonal such that
$\operatorname{\mathcal{L}}\operatorname{\mathcal{L}}^{T}=\Sigma$. Let also

$\operatorname{\mathcal{L}}=\left(\begin{matrix}\operatorname{\mathcal{L}}_{11}&\operatorname{\mathcal{L}}_{12}\\\
\operatorname{\mathcal{L}}_{21}&\operatorname{\mathcal{L}}_{22}\end{matrix}\right),$

where $\operatorname{\mathcal{L}}_{ij}$ we denote the $d\times d$ blocks of
matrix $\operatorname{\mathcal{L}}$. Note that
$\operatorname{\mathcal{L}}_{12}$ is a zero matrix and
$\operatorname{\mathcal{L}}_{11}$ is a lower triangular matrix with positive
elements on the diagonal. In particular, $\operatorname{\mathcal{L}}_{11}$ is
invertible, so we can define its inverse
$\operatorname{\mathcal{L}}_{11}^{-1}=\left(\ell^{(-1)}_{j,k}\right)_{j,k=1}^{d}$.
Finally, for a given semimartingale $X$, we denote the corresponding Doléans-
Dade exponential by

$\mathcal{E}_{t}\left\\{X\right\\}:=\exp\left\\{X_{t}-X_{0}-\frac{1}{2}[X]_{t}\right\\}.$

The next lemma is in the spirit of Lemma 2.10 in [11].

######  Lemma 2.7.

Let $M$ be a positive local $\mathbb{P}$-martingale w.r.t. $\mathbb{F}$ with
$M(0)=1$ a.s. Then $M\widetilde{S}_{i}$, $i=1,...,d$, are all local
martingales if and only if

$\displaystyle M(t)$
$\displaystyle=\mathcal{E}_{t}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\}\mathcal{E}_{t}\left\\{\sum_{l=d+1}^{2d}\int_{0}^{\cdot}\xi_{l}(s)dW_{l}(s)\right\\}$
(2.12)
$\displaystyle=\mathcal{E}_{t}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)+\sum_{l=d+1}^{2d}\int_{0}^{\cdot}\xi_{l}(s)dW_{l}(s)\right\\},$

where $\left(\ell^{(-1)}_{j,k}\right)_{j,k=1}^{d}$ are the elements of
$\operatorname{\mathcal{L}}_{11}^{-1}$ and $\xi_{l}$, $l=d+1,...,2d$, are
predictable processes such that $\int_{0}^{\cdot}\xi_{l}(s)dW_{l}(s)$ are
well-defined local martingales.

###### Proof.

By the virtue of the local martingale representation theorem, $M$ is a local
$\mathbb{P}$-martingale w.r.t. $\mathbb{F}$ if and only if there exist
predictable processes $\zeta_{j}=\\{\zeta_{j}(t),~{}t\in[0,T]\\}$,
$j=1,...,2d$, such that

$\mathbb{P}\left(\int_{0}^{T}\zeta_{j}^{2}(s)ds<\infty\right)=1,\quad
j=1,...,2d,$

and

$M(t)=1+\sum_{j=1}^{2d}\int_{0}^{t}\zeta_{j}(s)dW_{j}(s).$ (2.13)

By Itô’s formula,

$\displaystyle d(M(t)\widetilde{S}_{i}(t))$
$\displaystyle=\left(\tilde{\mu}_{i}(t)M(t)+Y_{i}(t)\sum_{j=1}^{2d}\ell_{i,j}\zeta_{j}(t)\right)\widetilde{S}_{i}(t)dt$
$\displaystyle\quad+\sum_{j=1}^{2d}\left(\ell_{i,j}M(t)Y_{i}(t)+\zeta_{j}(t)\right)\widetilde{S}_{i}(t)dW_{j}(t),$

and thus $M\widetilde{S}_{i}$ is a local martingale if and only if

$\int_{0}^{t}\left(\tilde{\mu}_{i}(s)M(s)+Y_{i}(s)\sum_{j=1}^{2d}\ell_{i,j}\zeta_{j}(s)\right)\widetilde{S}_{i}(s)ds=0,\quad
t\in[0,T].$

This, in turn, implies that

$\sum_{j=1}^{2d}\ell_{i,j}\zeta_{j}(t)=-M(t)\frac{\tilde{\mu}_{i}(t)}{Y_{i}(t)},\quad
i=1,...,d,$

almost everywhere w.r.t. $dt\otimes d\mathbb{P}$. Since
$\operatorname{\mathcal{L}}$ is lower-triangular and each $Y_{i}(t)$ is
positive a.s., we have that

$\operatorname{\mathcal{L}}_{11}\left(\begin{matrix}\zeta_{1}(t)\\\ \vdots\\\
\zeta_{d}(t)\end{matrix}\right)=-M(t)\left(\begin{matrix}\frac{\tilde{\mu}_{1}(t)}{Y_{1}(t)}\\\
\vdots\\\ \frac{\tilde{\mu}_{d}(t)}{Y_{d}(t)}\end{matrix}\right)$ (2.14)

almost everywhere w.r.t. $dt\otimes d\mathbb{P}$ and hence

$\left(\begin{matrix}\zeta_{1}(t)\\\ \vdots\\\
\zeta_{d}(t)\end{matrix}\right)=\operatorname{\mathcal{L}}^{-1}_{11}\left(\begin{matrix}-M(t)\frac{\tilde{\mu}_{1}(t)}{Y_{1}(t)}\\\
\vdots\\\ -M(t)\frac{\tilde{\mu}_{d}(t)}{Y_{d}(t)}\end{matrix}\right)$ (2.15)

almost everywhere w.r.t. $dt\otimes d\mathbb{P}$. Using notation
$\operatorname{\mathcal{L}}_{11}^{-1}=\left(\ell^{(-1)}_{j,k}\right)_{j,k=1}^{d}$
and substituting (2.15) to (2.13), we see that the process $M$ should satisfy
the SDE of the form

$dM(t)=-M(t)\sum_{j=1}^{d}\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(t)}{Y_{k}(t)}dW_{j}(t)+\sum_{l=d+1}^{2d}\zeta_{l}(t)dW_{l}(t).$

This SDE has a unique solution (see e.g. [14]) of the form

$\displaystyle M(t)$
$\displaystyle=\mathcal{E}_{t}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\}\times$
(2.16)
$\displaystyle\qquad\times\left(1+\sum_{l=d+1}^{2d}\int_{0}^{t}\mathcal{E}^{-1}_{u}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\}\zeta_{l}(u)dW_{l}(u)\right)$
$\displaystyle=:\mathcal{E}_{t}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\}\left(1+\sum_{l=d+1}^{2d}\int_{0}^{t}\widetilde{\zeta}_{l}(s)dW_{l}(u)\right).$

Furthermore, since the process $M$ is positive, the continuous process
$\widetilde{M}$ defined by

$\widetilde{M}(t):=1+\sum_{l=d+1}^{2d}\int_{0}^{t}\widetilde{\zeta}_{l}(s)dW_{l}(s)$

must also be positive, thus it can be represented as

$\displaystyle\widetilde{M}(t)=\mathcal{E}_{t}\left(\sum_{l=d+1}^{2d}\int_{0}^{\cdot}\xi_{l}(s)dW_{l}(s)\right)$
(2.17)

with $\xi_{l}$, $l=d+1,...,2d$, being some predictable processes such that the
corresponding stochastic integrals are well-defined. It remains to notice that
$\widetilde{\zeta}_{l}$, $l=d+1,...,2d$, can be arbitrary predictable
stochastic processes such that the corresponding stochastic integrals are
well-defined, thus $\xi_{l}$, $l=d+1,...,2d$, can also be chosen arbitrarily.
Taking into account (2.16) and (2.17), we get the representation (2.12). ∎

In the literature, positive densities $M(T)$ associated with local
$\mathbb{P}$-martingales $M$ such that $M\widetilde{S}_{i}$ all become local
martingales are commonly called _strict local martingale densities_ (see [43]
for more detail) and Lemma 2.7 thus gives the description of the set of all
strict local martingale densities on the market. However, note that the
measure defined by $d\mathbb{Q}:=M(T)d\mathbb{P}$ is not necessarily a
probability measure and one has to check that
$\mathbb{Q}(\Omega)=\mathbb{E}[M(T)]=1$ (or, equivalently, that $M$ is a
uniformly integrable martingale) separately. Moreover, exploiting the Bayes’
theorem for conditional expectations, it is easy to see that if $\mathbb{Q}$
defined above is a probability measure, then $\widetilde{S}_{i}$, $i=1,...,d$,
are local $\mathbb{Q}$-martingales, which would imply the no-arbitrage
property of the market. It turns out that the market model in consideration
indeed has this property which is formulated in the next theorem.

######  Theorem 2.8.

The market defined in Subsection 2.1 is arbitrage-free and incomplete.

###### Proof.

Each $Y_{k}$, $k=1,...,d$, is bounded away from zero by the corresponding
$\min_{t\in[0,T]}\varphi_{k}(t)>0$, thus, by Novikov’s condition,

$M(t)=\mathcal{E}_{t}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\}$

is a uniformly integrable martingale (this case corresponds to $\xi_{l}\equiv
0$ in (2.12), $l=d+1,...,2d$). Therefore the associated $\mathbb{Q}$ is a
probability measure, i.e. the market is arbitrage free.

Furthermore, again by Novikov’s condition, any $\xi_{l}$ such that

$\operatorname{\mathbb{E}}\left[\exp\left\\{\frac{1}{2}\sum_{l=d+1}^{2d}\int_{0}^{T}\xi_{l}^{2}(t)dt\right\\}\right]<\infty$

leads to $M$ being a uniformly integrable martingale, thus the market is
incomplete. ∎

##  3 Malliavin differentiability and of volatility and price processes and
absolute continuity of their laws

###  3.1 Malliavin differentiability of volatility and price w.r.t. the
initial Brownian motion $W$

In order to introduce the concept of the Malliavin differentiability, one must
first fix the underlying isonormal Gaussian process and an obvious choice is
the initial Brownian motion $W$. Recall that $(\Omega,\mathcal{F},\mathbb{P})$
is the canonical $2d$-dimensional Wiener space, $W=(W_{1},...,W_{2d})$ is the
associated $2d$-dimensional Brownian motion and stochastic volatility
processes $Y_{i}=\\{Y_{i}(t),~{}t\in[0,T]\\}$, $i=1,...,d$, are given by

$Y_{i}(t)=Y_{i}(0)+\int_{0}^{t}b_{i}(s,Y_{i}(s))ds+Z_{i}(t),\quad t\in[0,T],$

where $Y_{i}(0)\in(\varphi_{i}(0),\psi_{i}(0))$, each $b_{i}$ satisfies
Assumption B, $Z_{i}(t)=\int_{0}^{t}\mathcal{K}_{i}(t,s)dB^{Y}_{i}$ is a
Gaussian Volterra process with the kernel $\mathcal{K}_{i}$ satisfying
Assumption A and

$B^{Y}_{i}:=\sum_{j=1}^{2d}\ell_{d+i,j}W_{j}.$

Denote also by $\mathcal{H}$ the associated Cameron-Martin space, i.e. the
space of all continuous functions $F=(F_{1},...,F_{2d})\in
C_{0}([0,T];\mathbb{R}^{2d})$ such that each $F_{i}$, $i=1,...,2d$, can be
represented as

$F_{i}(\cdot)=\int_{0}^{\cdot}f_{i}(s)ds,$

where $f=(f_{1},...,f_{2d})\in L^{2}([0,T];\mathbb{R}^{2d})$. For the relation
between $F$ and $f$ specified above, we will use the notation
$F=\int_{0}^{\cdot}f(s)ds$. By $\mathbb{D}^{1,2}$, we denote the space of
Malliavin differentiable functions (w.r.t. the $L^{2}$-closure).

Before proceeding to the main results of this Subsection, let us give an
important auxiliary theorem.

######  Theorem 3.1 ([45], Theorem 3.1).

A random variable $\eta\in L^{2}(\Omega,\mathcal{F}_{T},\mathbb{P})$ belongs
to $\mathbb{D}^{1,2}$ w.r.t. $W$ if and only if the following two conditions
are satisfied:

  1. (i)

$\eta$ is _ray absolutely continuous_ , i.e. for any $F\in\mathcal{H}$ there
exists a version of the process $\\{\eta(\omega+\varepsilon
F),~{}\varepsilon\geq 0\\}$ that is absolutely continuous;

  2. (ii)

$\eta$ is _stochastically Gateaux differentiable_ , i.e. there exists a random
vector

$D\eta\in L^{2}\left(\Omega;L^{2}([0,T];\mathbb{R}^{2d})\right)$

such that for any $F=\int_{0}^{\cdot}f(s)ds\in\mathcal{H}$, $f\in
L^{2}([0,T];\mathbb{R}^{2d})$,

$\frac{1}{\varepsilon}\left(\eta\left(\omega+\varepsilon
F\right)-\eta(\omega)\right)\xrightarrow{\mathbb{P}}\langle
D\eta,f\rangle_{L^{2}\left([0,T];\mathbb{R}^{2d}\right)},\quad\varepsilon\to
0.$

In this case the random vector $D\eta$ from (ii) is the Malliavin derivative
of $\eta$.

In order to apply Theorem 3.1 to obtain the Malliavin differentiability of
$Y_{i}(t)$, $i=1,...,d$, $t\in[0,T]$, we first need to formally identify

$Y_{i}\left(\omega+\varepsilon F,t\right),\quad i=1,...,d,\quad t\in[0,T],$

for any $F=\int_{0}^{\cdot}f(s)ds\in\mathcal{H}$. For the given $\omega$, it
is reasonable to associate these expressions with random variables
$Y^{\varepsilon,F}_{i}(\omega,t)$ given by

$Y^{\varepsilon,F}_{i}(\omega,t)=Y(0)+\int_{0}^{t}b_{i}\left(s,Y^{\varepsilon,F}_{i}(\omega,s)\right)ds+Z_{i}^{\varepsilon,F}\left(\omega,t\right),\quad
t\in[0,T],$

where
$Z_{i}^{\varepsilon,F}\left(\omega,t\right):=Z_{i}\left(\omega+\varepsilon
F,t\right)$. Indeed, equations (2.6) are treated pathwisely and each
$Z_{i}^{\varepsilon,F}\left(\omega,t\right)$ is Hölder continuous w.r.t. $t$
for all $\omega\in\Omega$. This means that the equation above has a unique
solution for all $\omega\in\Omega$ and

$Y^{\varepsilon,F}_{i}(\omega,t)=Y_{i}\left(\omega+\varepsilon
F,t\right),\quad\omega\in\Omega,\quad t\in[0,T].$

However, in order to apply Theorem 3.1, one needs to get a more convenient
representation of $Z_{i}^{\varepsilon,F}\left(\omega,t\right)$ that “splits”
the pieces corresponding to $\omega$ and $F$. Intuitively, it is clear that a
natural candidate for such a representation is

$Z_{i}^{\varepsilon,F}\left(\cdot,t\right)=Z_{i}(\cdot,t)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds,\quad
t\in[0,T],$

and this guess is formalized in the next simple proposition.

######  Proposition 3.2.

Let $\varepsilon\geq 0$ and $F=\int_{0}^{\cdot}f(s)ds\in\mathcal{H}$. Then
there exists $\Omega_{\varepsilon,F}$, $\mathbb{P}(\Omega_{\varepsilon,F})=1$,
such that for any $i=1,...,d$ and $\omega\in\Omega_{\varepsilon,F}$

$Z_{i}^{\varepsilon,F}\left(\omega,t\right)=Z_{i}(\omega,t)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds,\quad\varepsilon\in\mathbb{R}_{+},\quad
t\in[0,T].$

###### Proof.

By definition, for any $t\in[0,T]$

$\displaystyle Z_{i}(\cdot,t)$
$\displaystyle=L^{2}\text{-}\lim_{n\to\infty}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(B^{Y}_{i}(\cdot,t_{n,l+1})-B^{Y}_{i}(\cdot,t_{n,l})\right)=$
$\displaystyle=L^{2}\text{-}\lim_{n\to\infty}\sum_{j=1}^{2d}\ell_{d+i,j}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(W_{j}(\cdot,t_{n,l+1})-W_{j}(\cdot,t_{n,l})\right)$
$\displaystyle=:L^{2}\text{-}\lim_{n\to\infty}Z^{n}_{i}(\cdot,t),$

where $\pi^{n}=\\{0=t_{n,0}<t_{n,1}<...<t_{n,k_{n}}=t\\}$, $n\geq 1$, is a
system of partitions such that $\sup_{l=0,...,k_{n}-1}(t_{n,l+1}-t_{n,l})\to
0$ as $n\to\infty$ and
$\mathcal{K}^{n}_{i}(t,s):=\sum_{l=0}^{k_{n}-1}K^{n,l}_{i}(t)\mathbbm{1}_{[t_{n,l},t_{n,l+1})}(s)$
is such that
$\int_{0}^{t}\left(\mathcal{K}_{i}(t,s)-\mathcal{K}^{n}_{i}(t,s)\right)^{2}ds\to
0$, $n\to\infty$.

By the Riesz theorem, there exist $\Omega^{\prime}_{t}\subset\Omega$,
$\mathbb{P}(\Omega^{\prime}_{t})=1$, and a subsequence of partitions that will
again be denoted by $\\{\pi^{n},~{}n\geq 1\\}$ such that for all $i=1,...,d$
and $\omega\in\Omega^{\prime}_{t}$

$Z_{i}(\omega,t)=\lim_{n\to\infty}Z^{n}_{i}(\omega,t).$ (3.1)

Moreover, note that, by the definition of the probability space and Brownian
motion $W$, for any
$\omega=(\omega_{1},...,\omega_{2d})\in\Omega=C_{0}([0,T];\mathbb{R}^{2d})$

$\displaystyle Z^{n}_{i}(\omega,t)$
$\displaystyle=\sum_{j=1}^{2d}\ell_{d+i,j}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(W_{j}(\omega,t_{n,l+1})-W_{j}(\omega,t_{n,l})\right)$
$\displaystyle=\sum_{j=1}^{2d}\ell_{d+i,j}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(\omega_{j}(t_{n,l+1})-\omega_{j}(t_{n,l})\right),$

whence, for all $\omega\in\Omega$,

$Z^{n}_{i}(\omega+\varepsilon
F,t)=Z^{n}_{i}(\omega)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(F_{j}(t_{n,l+1})-F_{j}(t_{n,l})\right)$

and thus, for all $\omega\in\Omega^{\prime}_{t}$,

$\displaystyle\lim_{n\to\infty}Z^{n}_{i}(\omega+\varepsilon F,t)$
$\displaystyle=\lim_{n\to\infty}Z^{n}_{i}(\omega)+\lim_{n\to\infty}\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\sum_{l=0}^{k_{n}-1}K_{i}^{n,l}(t)\left(F_{j}(t_{n,l+1})-F_{j}(t_{n,l})\right)$
(3.2)
$\displaystyle=Z_{i}(\omega,t)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds.$

Now, denote
$\Omega_{\varepsilon,F,t}:=\\{\omega\in\Omega^{\prime}_{t}:~{}\omega+\varepsilon
F\in\Omega^{\prime}_{t}\\}$. Then $\mathbb{P}(\Omega_{\varepsilon,F,t})=1$
since, by the Cameron-Martin theorem,

$\displaystyle\mathbb{P}(\Omega_{\varepsilon,F,t})$
$\displaystyle=\int_{\Omega}\mathbbm{1}_{\Omega^{\prime}_{t}}(\omega+\varepsilon
F)d\mathbb{P}(\omega)=\operatorname{\mathbb{E}}\left[\mathbbm{1}_{\Omega^{\prime}_{t}}\exp\left\\{\varepsilon\mathcal{I}_{f}-\frac{\varepsilon^{2}}{2}\lVert\mathcal{I}_{f}\rVert^{2}_{L^{2}(\Omega)}\right\\}\right]=1,$

where $\mathcal{I}_{f}:=\sum_{j=1}^{2d}\int_{0}^{T}f_{j}(s)dW_{j}(s)$, and
therefore, taking into account (3.1) and (3.2), for any
$\omega\in\Omega_{\varepsilon,F,t}$

$Z_{i}(\omega+\varepsilon
F,t)=Z_{i}(\omega,t)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds.$
(3.3)

Finally, it remains to notice that, for the fixed $\varepsilon\geq 0$ and
$F\in\mathcal{H}$, both left- and right-hand sides of (3.3) are continuous
w.r.t. $t$, whence we can put

$\Omega_{\varepsilon,F}:=\bigcap_{t\in\mathbb{Q}\cap[0,T]}\Omega_{\varepsilon,F,t}.$

∎

Now we are ready to prove the key result of the Subsection. We will use the
same approach as in [33, Theorem 3.3] and simply check the conditions of
Theorem 3.1.

######  Theorem 3.3.

Let Assumptions A and B hold. Then, for all $i=1,...,d$ and $t\in[0,T]$,
$Y_{i}(t)\in\mathbb{D}^{1,2}$ and

$\begin{gathered}DY_{i}(t)=(D^{1}Y_{i}(t),...,D^{2d}Y_{i}(t))\in
L^{2}\left(\Omega;L^{2}([0,T];\mathbb{R}^{2d})\right),\\\
D^{j}_{u}Y_{i}(t):=\ell_{d+i,j}\left(\mathcal{K}_{i}(t,u)+\int_{u}^{t}\mathcal{K}_{i}(s,u)\frac{\partial
b_{i}}{\partial y}(s,Y_{i}(s))e^{\int_{s}^{t}\frac{\partial b_{i}}{\partial
y}(v,Y_{i}(v))dv}ds\right)\mathbbm{1}_{[0,t]}(u),\end{gathered}$ (3.4)

with $\ell_{d+i,j}$, $i=1,...,d$, $j=1,...,2d$ being the corresponding
elements of the matrix $\operatorname{\mathcal{L}}$.

###### Proof.

We will split the proof into three steps. First, we verify that $DY_{i}(t)$
given by (3.4) indeed satisfies

$\sum_{j=1}^{2d}\mathbb{E}\int_{0}^{T}\left(D^{j}_{u}Y_{i}(t)\right)^{2}du<\infty$

and then check the conditions of Theorem 3.1.

Step 1: correctness of the form of derivative. By Remark 2.4, for all
$s\in[0,T]$:
$(s,Y_{i}(s))\in\mathcal{D}^{i}_{\frac{1}{\xi_{i}(0)},\frac{1}{\xi_{i}(0)}}$,
where

$\xi_{i}(0):=\frac{(L_{2,i}+\Lambda_{\lambda,i})^{\alpha_{i}}}{L_{1,i}}$

with $\Lambda_{\lambda,i}$ being defined by (2.3) and $L_{1,i}$, $L_{2,i}$,
$\alpha_{i}$ being some deterministic positive constants that depend only on
$\lambda\in\left(\frac{1}{\gamma_{i}+1},H_{i}\right)$ and the shape of the
drift $b_{i}$. Whence, due to Assumption B(ii),

$\left|\frac{\partial b_{i}}{\partial y}(s,Y_{i}(s))\right|\leq
c\xi_{i}^{p}(0),\quad s\in[0,T].$

Moreover, by Assumption B(iv),

$e^{\int_{s}^{t}\frac{\partial b_{i}}{\partial y}(v,Y_{i}(v))dv}\leq e^{cT}.$

Therefore

$\displaystyle\mathbb{E}\int_{0}^{T}\left(D^{j}_{u}Y_{i}(t)\right)^{2}du$
$\displaystyle\leq 2\ell_{d+i,j}^{2}\int_{0}^{t}\mathcal{K}^{2}_{i}(t,u)du$
$\displaystyle\quad+2Tc^{2}e^{2cT}\ell_{d+i,j}^{2}\mathbb{E}\left[\xi_{i}^{2p}(0)\right]\int_{0}^{t}\int_{u}^{t}\mathcal{K}_{i}^{2}(s,u)dsdu$
$\displaystyle<\infty,$

because $\int_{0}^{t}\mathcal{K}_{i}^{2}(t,s)ds<\infty$ by Assumption A(i) and
the random variable $\xi_{i}(0)$ has moments of all orders.

Step 2: ray absolute continuity. Let $F\in\mathcal{H}$ be fixed. Then, as
explained above, the random process
$\\{Y^{\varepsilon,F}_{i}(t),~{}\varepsilon\geq 0\\}$ defined for every
$\varepsilon\geq 0$ by the equation

$Y^{\varepsilon,F}_{i}(t)=Y_{0}+\int_{0}^{t}b_{i}(s,Y^{\varepsilon,F}_{i}(s))ds+Z_{i}(t)+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds$
(3.5)

is a version of the process $\\{Y_{i}(\omega+\varepsilon
F,t),~{}\varepsilon\geq 0\\}$. Let us prove that this version is Lipschitz
w.r.t. $\varepsilon$ and hence absolutely continuous. First, fix
$\lambda\in\left(\frac{1}{\gamma_{i}+1},H_{i}\right)$, where $H_{i}$ is from
Assumption A(ii) and $\gamma_{i}$ is from Assumption B(iii), and take an
arbitrary $\varepsilon^{*}>0$. Then, by Remark 2.3 and Assumption A(ii), for
any $\varepsilon\in[0,\varepsilon^{*}]$

$\displaystyle|Z_{i}(t_{1})-Z_{i}(t_{2})|+\varepsilon\left|\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t_{1}}\mathcal{K}_{i}(t_{1},s)f_{j}(s)ds-\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t_{2}}\mathcal{K}_{i}(t_{2},s)f_{j}(s)ds\right|$
$\displaystyle\qquad\leq\Lambda_{\lambda,i}|t_{1}-t_{2}|^{\lambda}+\varepsilon\left(\int_{0}^{t_{1}\vee
t_{2}}(\mathcal{K}_{i}(t_{1},s)-\mathcal{K}_{i}(t_{2},s))^{2}ds\right)^{\frac{1}{2}}\Big{\lVert}\sum_{j=1}^{2d}\ell_{d+i,j}f_{j}\Big{\rVert}_{L^{2}([0,T])}$
$\displaystyle\qquad\leq\left(\Lambda_{\lambda,i}+\varepsilon^{*}C_{\lambda}\Big{\lVert}\sum_{j=1}^{2d}\ell_{d+i,j}f_{j}\Big{\rVert}_{L^{2}([0,T])}\right)|t_{1}-t_{2}|^{\lambda},$

which implies, by Remark 2.4, that
$(t,Y^{\varepsilon,F}_{i}(t))\in\mathcal{D}^{i}_{\frac{1}{\xi_{i}(\varepsilon^{*})},\frac{1}{\xi_{i}(\varepsilon^{*})}}$
for any $\varepsilon\in[0,\varepsilon^{*}]$ and $t\in[0,T]$, where
$\mathcal{D}^{i}_{\cdot,\cdot}$ is defined by (2.5) and

$\xi_{i}(\varepsilon^{*}):=\frac{\left(L_{2,i}+\Lambda_{\lambda,i}+\varepsilon^{*}C_{\lambda}\Big{\lVert}\sum_{j=1}^{2d}\ell_{d+i,j}f_{j}\Big{\rVert}_{L^{2}([0,T])}\right)^{\alpha_{i}}}{L_{1,i}}$
(3.6)

for some deterministic positive constants $L_{1,i}$, $L_{2,i}$ and
$\alpha_{i}$ that depend only on $\lambda$ and the shape of the drift $b_{i}$.
This implies, by Assumption B(ii), that for all $s\in[0,T]$ and
$0\leq\varepsilon_{1}\leq\varepsilon_{2}\leq\varepsilon^{*}$,

$\left|b_{i}(s,Y^{\varepsilon_{2},F}_{i}(s))-b_{i}(s,Y^{\varepsilon_{1},F}_{i}(s))\right|\leq
c\xi_{i}^{p}(\varepsilon^{*})\left|Y^{\varepsilon_{2},F}_{i}(s)-Y^{\varepsilon_{1},F}_{i}(s)\right|,$

whence

$\displaystyle|Y^{\varepsilon_{2},F}_{i}(t)-Y^{\varepsilon_{1},F}_{i}(t)|$
$\displaystyle\leq\int_{0}^{t}\left|b_{i}(s,Y^{\varepsilon_{2},F}_{i}(s))-b_{i}(s,Y^{\varepsilon_{1},F}_{i}(s))\right|ds$
$\displaystyle\quad+|\varepsilon_{2}-\varepsilon_{1}|\left|\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds\right|$
$\displaystyle\leq
c\xi_{i}^{p}(\varepsilon^{*})\int_{0}^{t}\left|Y^{\varepsilon_{2},F}_{i}(s)-Y^{\varepsilon_{1},F}_{i}(s)\right|ds$
$\displaystyle\quad+|\varepsilon_{2}-\varepsilon_{1}|\left(\int_{0}^{T}\mathcal{K}_{i}^{2}(t,s)ds\right)^{\frac{1}{2}}\Big{\lVert}\sum_{j=1}^{2d}\ell_{d+i,j}f_{j}\Big{\rVert}_{L^{2}([0,T])}.$

where $\int_{0}^{T}\mathcal{K}_{i}^{2}(t,s)ds<\infty$ by Assumption A(i). Now,
it follows from Gronwall’s inequality that for all
$0\leq\varepsilon_{1}\leq\varepsilon_{2}\leq\varepsilon^{*}$

$|Y^{\varepsilon_{2},F}_{i}(t)-Y^{\varepsilon_{1},F}_{i}(t)|\leq\left(\int_{0}^{T}\mathcal{K}_{i}^{2}(t,s)ds\right)^{\frac{1}{2}}\Big{\lVert}\sum_{j=1}^{2d}\ell_{d+i,j}f_{j}\Big{\rVert}_{L^{2}([0,T])}e^{cT\xi_{i}^{p}(\varepsilon^{*})}|\varepsilon_{2}-\varepsilon_{1}|,$

which implies the desired absolute continuity w.r.t. $\varepsilon$.

Step 3: stochastic Gateaux differentiability. Again, it is sufficient to
compute the limit in probability of the form

$\mathbb{P}\text{-}\lim_{\varepsilon\to
0}\frac{Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)}{\varepsilon},$

where $Y_{i}^{\varepsilon,F}(t)$ is defined by (3.5).

Using the mean-value theorem, we obtain that

$\displaystyle Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)$
$\displaystyle=\int_{0}^{t}(b_{i}(s,Y_{i}^{\varepsilon,F}(s))-b_{i}(s,Y_{i}^{0,F}(s))ds$
(3.7)
$\displaystyle\quad+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds$
$\displaystyle=\int_{0}^{t}\Theta_{i,\varepsilon}(s)(Y_{i}^{\varepsilon,F}(s)-Y_{i}^{0,F}(s))ds$
$\displaystyle\quad+\varepsilon\sum_{j=1}^{2d}\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds,$

where $\Theta_{i,\varepsilon}(s):=\frac{\partial b_{i}}{\partial
y}\left(s,Y_{i}^{0,F}(s)+\theta_{i,\varepsilon}(s)(Y_{i}^{\varepsilon,F}(s)-Y_{i}^{0,F}(s))\right)$
with $\theta_{i,\varepsilon}(s)$ being some value between 0 and 1. It is easy
to verify that (3.7) implies the representation of the form

$Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)=\varepsilon\sum_{j=1}^{2d}\int_{0}^{t}\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}dG_{i,j}(s),$

where

$G_{i,j}(t):=\ell_{d+i,j}\int_{0}^{t}\mathcal{K}_{i}(t,s)f_{j}(s)ds.$

Note that, just like in Step 1, it follows from Remark 2.4 that for all
$\varepsilon\in[0,1]$ and $s\in[0,T]$
$(s,Y^{\varepsilon,F}_{i}(s))\in\mathcal{D}^{i}_{\frac{1}{\xi_{i}(1)},\frac{1}{\xi_{i}(1)}}$,
where $\xi_{i}(1)$ is defined via (3.6). This, together with continuity of
$\frac{\partial b_{i}}{\partial y}$ due to Assumption B(iv) and compactness of
$\overline{\mathcal{D}^{i}_{\frac{1}{\xi_{i}(1)},\frac{1}{\xi_{i}(1)}}}$,
implies that there exists a finite positive random variable $\eta_{i}$ such
that $|\Theta_{i,\varepsilon}(s)|\leq\eta_{i}$ for all $\varepsilon\in[0,1]$
and $s\in[0,T]$. Moreover, the mapping
$s\mapsto\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}$,
$s\in[0,t]$, is Lipschitz and whence the integrals
$\int_{0}^{t}\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}dG_{i,j}(s)$,
$j=1,...,2d$, are well-defined as pathwise limits of Riemann-Stieltjes
integral sums due to the classical results of Young ([47, Section 10], see
also [28, Chapter 6] for a more recent overview on the subject).

Now, applying the integration by parts formula, we obtain

$\displaystyle\frac{Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)}{\varepsilon}$
$\displaystyle=\sum_{j=1}^{2d}\int_{0}^{t}\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}dG_{i,j}(s)$
$\displaystyle=\sum_{j=1}^{2d}\left(G_{i,j}(t)-\int_{0}^{t}G_{i,j}(s)d\left(\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}\right)\right)$
$\displaystyle=\sum_{j=1}^{2d}\left(G_{i,j}(t)+\int_{0}^{t}G_{i,j}(s)\Theta_{i,\varepsilon}(s)\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}ds\right),$

and, taking into account the definition of $G_{i,j}$,

$\displaystyle\frac{Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)}{\varepsilon}$
$\displaystyle=\sum_{j=1}^{2d}\int_{0}^{T}D_{i,j}(\varepsilon,t,u)f_{j}(u)du$

with

$D_{i,j}(\varepsilon,t,u):=\ell_{d+i,j}\left(\mathcal{K}_{i}(t,u)+\int_{u}^{t}\mathcal{K}_{i}(s,u)\Theta_{i,\varepsilon}(s)\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}ds\right)\mathbbm{1}_{[0,t]}(u).$

Now, note that for all $s\in[0,T]$ and $\varepsilon\in[0,1]$

$|\Theta_{i,\varepsilon}(s)|\exp\left\\{\int_{s}^{t}\Theta_{i,\varepsilon}(v)dv\right\\}<\eta_{i}\exp\\{cT\\},$

where $c$ is from Assumption B(iv), and therefore we can apply the dominated
convergence theorem to obtain that

$\frac{Y_{i}^{\varepsilon,F}(t)-Y_{i}^{0,F}(t)}{\varepsilon}\to\langle
DY_{i}(t),f\rangle_{L^{2}([0,T];\mathbb{R}^{2d})}\quad
a.s.,\quad\varepsilon\to 0,$

where $DY_{i}(t)$ has exactly the form (3.4), which finishes the proof. ∎

An obvious way to obtain the Malliavin differentiability of the price
processes $(S_{1},...,S_{d})^{T}$ would be to notice that each $S_{i}$ has the
form

$S_{i}(t)=e^{X_{i}(t)},\quad t\in[0,T],$

where $X_{i}$ is the log-price process that, due to Theorem 2.6, can be
written as

$X_{i}(t):=\log
S_{i}(t)=X_{i}(0)+\int_{0}^{t}\left(\mu_{i}(s)-\frac{Y_{i}^{2}(s)}{2}\right)ds+\int_{0}^{t}Y_{i}(s)dB_{i}^{S}(s).$
(3.8)

Then, in order to prove that $S_{i}(t)\in\mathbb{D}^{1,2}$, one can first
establish that $X_{i}(t)\in\mathbb{D}^{1,2}$ and then apply an appropriate
chain rule. This approach will be implemented in the rest of this Subsection.

We first begin with a version of a chain rule for the Malliavin derivative.
Note that classical chain rule results (see e.g. [41, Proposition 1.2.3,
Proposition 1.2.4]) require either all partial derivatives $\frac{\partial
f}{\partial x_{j}}$ to be bounded (which is not the case for such functions as
$f(x)=x^{2}$ or $f(x)=e^{x}$) or the law of $\eta$ to be absolutely continuous
(which we have not established yet for the log-price $X_{i}(t)$). Note that
our version of the chain rule is essentially based on [42, Lemma A.1], but
proof of the latter contains a mistake in the argument. For reader’s
convenience, we give our own proof where the mentioned inaccuracy is treated.

######  Proposition 3.4.

Let $\eta=(\eta_{1},...,\eta_{k})\in(\mathbb{D}^{1,2})^{k}$ and $f\in
C^{(1)}(\mathbb{R}^{k};\mathbb{R})$ be a real-valued function such that

  * 1)

if $\frac{\partial f}{\partial x_{j}}$ is unbounded, then there exist
sequences $\\{r_{n},~{}n\geq 1\\}$, $\\{s_{j,n},~{}n\geq 1\\}$,
$r_{n},s_{j,n}\to\infty$ as $n\to\infty$, such that

$\left|\frac{\partial f}{\partial x_{j}}(x_{1},...,x_{k})\right|\geq s_{j,n}$

implies that $|f(x_{1},...,x_{k})|\geq r_{n}$;

  * 2)

$\mathbb{E}\left[|f(\eta)|^{2}+\left\lVert\sum_{j=1}^{k}\frac{\partial
f}{\partial
x_{j}}(\eta)D\eta_{j}\right\rVert^{2}_{L^{2}([0,T];\mathbb{R}^{2d})}\right]<\infty$.

Then $f(\eta)\in\mathbb{D}^{1,2}$ and $Df(\eta)=\sum_{j=1}^{k}\frac{\partial
f}{\partial x_{j}}(\eta)D\eta_{j}$.

###### Proof.

According to [41, Proposition 1.2.3], if all partial derivatives
$\frac{\partial f}{\partial x_{j}}$ are bounded, then the claim of our
Proposition is true and the chain rule holds. For the general case, let
$\phi\in C^{\infty}(\mathbb{R})$ be compactly supported function such that
$\phi(x)=x$ whenever $|x|\leq 1$ and $|\phi(x)|\leq|x|$ for all
$x\in\mathbb{R}$.

For any $m\geq 1$ define $f_{m}(x):=m\phi\left(\frac{f(x)}{m}\right)$ and
observe that $f_{m}$ has bounded partial derivatives. Indeed, if
$\frac{\partial f}{\partial x_{j}}$ is bounded, so is $\frac{\partial
f_{m}}{\partial x_{j}}$. Otherwise, let $n^{*}$ be such that
$\frac{r_{n^{*}}}{m}>\max\left\\{-\inf\operatorname{supp}\phi,\sup\operatorname{supp}\phi\right\\}$.
Then, for all $(x_{1},...,x_{k})\in\mathbb{R}^{k}$ such that $|\frac{\partial
f}{\partial x_{j}}(x_{1},...,x_{k})|\geq s_{j,n^{*}}$, it holds that
$\frac{f(x_{1},...,x_{k})}{m}\notin\operatorname{supp}\phi$ and therefore

$\displaystyle\frac{\partial f_{m}}{\partial
x_{j}}(x_{1},...,x_{k})=\frac{\partial f}{\partial
x_{j}}(x_{1},...,x_{k})\phi^{\prime}\left(\frac{f(x_{1},...,x_{k})}{m}\right)=0.$

Whence

$\displaystyle\left|\frac{\partial f_{m}}{\partial
x_{j}}(x_{1},...,x_{k})\right|=\left|\frac{\partial f}{\partial
x_{j}}(x_{1},...,x_{k})\right|\left|\phi^{\prime}\left(\frac{f(x_{1},...,x_{k})}{m}\right)\right|\leq
s_{j,n^{*}}\max_{z\in\mathbb{R}}|\phi^{\prime}(z)|.$

Note that $|f_{m}(\eta)|\leq|f(\eta)|$ for all $m\geq 1$ and
$\lim_{m\to\infty}f_{m}(\eta)=f(\eta)$ a.s. Moreover,

$\displaystyle|Df_{m}(\eta)|$
$\displaystyle=\left|\sum_{j=1}^{k}\frac{\partial f_{m}}{\partial
x_{j}}(\eta)D\eta_{j}\right|\leq\max_{z\in\mathbb{R}}|\phi^{\prime}(z)|\left|\sum_{j=1}^{k}\frac{\partial
f}{\partial x_{j}}(\eta)D\eta_{j}\right|$

for all $m\geq 1$ and

$\lim_{m\to\infty}Df_{m}(\eta)=\sum_{j=1}^{k}\frac{\partial f}{\partial
x_{j}}(\eta)D\eta_{j}\quad a.s.$

Therefore, by the dominated convergence theorem,

$\lim_{m\to\infty}\operatorname{\mathbb{E}}\left[|f_{m}(\eta)-f(\eta)|^{2}+\left\lVert
Df_{m}(\eta)-\sum_{j=1}^{k}\frac{\partial f}{\partial
x_{j}}(\eta)D\eta_{j}\right\rVert^{2}_{L^{2}([0,T];\mathbb{R}^{2d})}\right]=0,$

and the result follows by closedness of the operator $D$. ∎

Now we can get the following simple corollary.

######  Corollary 3.5.

For any $i=1,...,d$ and $t\in[0,T]$, $X_{i}(t)\in\mathbb{D}^{1,2}$ and its
Malliavin derivative $DX_{i}(t)=(D^{1}X_{i}(t),...,D^{2d}X_{i}(t))$ has the
form

$D^{j}_{u}X_{i}(t)=\left(-\int_{0}^{t}Y_{i}(s)D^{j}_{u}Y_{i}(s)ds+\ell_{i,j}Y_{i}(u)+\int_{0}^{t}D^{j}_{u}Y_{i}(s)dB_{i}^{S}(s)\right)\mathbbm{1}_{[0,t]}(u).$

###### Proof.

1) First, note that if $u>t$, then the claim is true due to [41, Corollary
1.2.1]. Let now $u\leq t$. By the definition of the Malliavin derivative,

$D^{j}_{u}\left[\int_{0}^{t}\mu_{i}(s)ds\right]=0.$ (3.9)

Next, note that for any $s\in[0,T]$ $Y_{i}^{2}(s)\in\mathbb{D}^{1,2}$. Indeed,
by Proposition 3.4, it is sufficient to check that

$\operatorname{\mathbb{E}}\left[Y_{i}^{2}(s)+\lVert
Y_{i}(s)DY_{i}(s)\rVert^{2}_{L^{2}([0,T];\mathbb{R}^{2d})}\right]<\infty$

which is true since $Y_{i}(s)$ is bounded. Whence
$\frac{1}{2}\int_{0}^{t}D_{u}Y_{i}^{2}(s)ds=\int_{0}^{t}Y_{i}(s)D^{j}_{u}Y_{i}(s)ds$,
$j=1,...,2d$, these integrals are well defined and so, using the chain rule
and closedness of the operator $D$, we can write (see e.g. [46, Theorem
1.2.2])

$D^{j}_{u}\left[\frac{1}{2}\int_{0}^{t}Y_{i}^{2}(s)ds\right]=\int_{0}^{t}Y_{i}(s)D^{j}_{u}Y_{i}(s)ds,\quad
j=1,...,2d.$ (3.10)

Finally, by adaptedness and boundedness of $Y_{i}$ and [41, Proposition
1.3.2],

$\displaystyle D^{j}_{u}\left[\int_{0}^{t}Y_{i}(s)dB_{i}^{S}(s)\right]$
$\displaystyle=D^{j}_{u}\left[\sum_{k=1}^{2d}\int_{0}^{t}\ell_{i,k}Y_{i}(s)dW_{k}(s)\right]$
(3.11)
$\displaystyle=\ell_{i,j}Y_{i}(u)+\sum_{k=1}^{2d}\ell_{i,k}\int_{0}^{t}D^{j}_{u}Y_{i}(s)dW_{k}(s)$
$\displaystyle=\ell_{i,j}Y_{i}(u)+\int_{0}^{t}D^{j}_{u}Y_{i}(s)dB_{i}^{S}(s),$

as required. ∎

###  3.2 Malliavin differentiability w.r.t. the transformed Brownian motion.
Absolute continuity of probability laws

In order to proceed, we will require absolute continuity of the law of
$X_{i}(t)$ w.r.t. the Lebesgue measure. It is well known (see e.g. [41,
Chapter 2]) that a sufficient condition for that is

$\lVert D_{\cdot}X_{i}(t)\rVert_{L^{2}([0,T];\mathbb{R}^{2d})}>0\quad a.s.,$
(3.12)

thus it would be sufficient to check (3.12). However, the form of $DX_{i}(t)$
obtained in Subsection 3.1 is not very convenient for this matter. In order to
get the desired properties, we will transform the original Brownian motion $W$
and consider the Malliavin derivative w.r.t. this transformation.

Namely, recall that $B=(B_{1},...,B_{2d})^{T}$ is a $2d$-dimensional Gaussian
process and

$B(t)=\left(\begin{matrix}B^{S}(t)\\\
B^{Y}(t)\end{matrix}\right)=\operatorname{\mathcal{L}}W(t),$

where by $B^{S}=(B_{1}^{S},...,B_{d}^{S})^{T}$ and
$B^{Y}=(B_{1}^{Y},...,B^{Y}_{d})^{T}$ we denote the first and the last $d$
components of $B$ correspondingly and
$\operatorname{\mathcal{L}}=(\ell_{i,j})_{i,j=1}^{2d}$ is a lower-triangular
matrix such that
$\operatorname{\mathcal{L}}\operatorname{\mathcal{L}}^{T}=\Sigma=(\sigma_{i,j})_{i,j=1}^{2d}$.
Now, define $\mathcal{U}=(u_{i,j})_{i,j=1}^{2d}$ an upper $2d\times
2d$-triangular matrix with positive values on the diagonal such that

$\widetilde{W}:=\mathcal{U}^{-1}B=\mathcal{U}^{-1}\operatorname{\mathcal{L}}W$

is a $2d$-dimensional Brownian motion. Let now
$\widetilde{D}=(\widetilde{D}^{1},...,\widetilde{D}^{2d})$ be the Malliavin
derivative with respect to the new Brownian motion $\widetilde{W}$ and
$\widetilde{\mathbb{D}}^{1,2}$ be its domain. Clearly, the matrix
$\mathcal{U}^{-1}\operatorname{\mathcal{L}}$ is non-degenerate and hence $W$
and $\widetilde{W}$ generate the same filtration $\mathbb{F}$ and, moreover,
it is easy to see that $\widetilde{\mathbb{D}}^{1,2}$. Finally, by
construction, $\widetilde{W}_{1}$ is independent of the processes $B^{S}_{2}$,
…, $B^{S}_{d}$, $B^{Y}_{1}$, …, $B^{Y}_{d}$ coincides with
${\mathbb{D}}^{1,2}$ and therefore the following result holds.

######  Theorem 3.6.

Let Assumptions A and B hold. Then, for any $t\in(0,T]$,

$\widetilde{D}^{1}_{v}X_{i}(t)=\begin{cases}u_{1,1}Y_{1}(v)\mathbbm{1}_{[0,t]}(v),&\quad\text{if
}i=1,\\\ 0,&\quad\text{if }i\neq 1,\end{cases}$

and

$\widetilde{D}^{1}_{v}S_{i}(t)=\begin{cases}u_{1,1}S_{1}(t)Y_{1}(v)\mathbbm{1}_{[0,t]}(v),&\quad\text{if
}i=1,\\\ 0,&\quad\text{if }i\neq 1,\end{cases}$

where $u_{1,1}$ is the upper left element of $\mathcal{U}$. In particular,

$\lVert\widetilde{D}_{\cdot}X_{1}(t)\rVert_{L^{2}([0,T];\mathbb{R}^{2d})}>0\quad
a.s.$

###### Proof.

The expression for $\widetilde{D}^{1}X_{i}(t)$ is obtained straightforwardly
by differentiating the right-hand side of (3.8) and taking into account that

$B^{S}_{1}(t)=\sum_{k=1}^{2d}u_{1,k}\widetilde{W}_{k}(t)$

and $\widetilde{D}^{1}Y_{i}(t)=0$ due to independence of $\widetilde{W}_{1}$
and $B^{Y}_{i}$. The expression for $\widetilde{D}^{1}S_{i}(t)$ can be easily
obtained using the chain rule from Proposition 3.4. ∎

######  Corollary 3.7.

For each $i=1,...,d$ and $t\in(0,T]$, the law of $X_{i}(t)$ (and therefore of
$S_{i}(t)=e^{X_{i}(t)}$) has continuous and bounded density.

###### Proof.

By Theorem 3.6,

$\lVert\widetilde{D}_{\cdot}X_{1}(t)\rVert_{L^{2}([0,T];\mathbb{R}^{2d})}>0\quad
a.s.$

and, moreover, $\widetilde{D}X_{1}(t)$ is adapted to the filtration generated
by $\widetilde{B}^{i}$. Thus the result for $X_{1}(t)$ follows from e.g. [41,
Proposition 2.1.1]. The existence of the density for $X_{i}(t)$, $i=2,...,d$,
can be obtained in the same manner by interchanging the order of indices. ∎

As we can see, each of $X_{i}(t)$, $i=1,...,d$, has a density with respect to
the Lebesgue measure. However, in what follows we will require a stronger
result, namely the existence of the joint density of
$X(t)=(X_{1}(t),...,X_{d}(t))$.

######  Theorem 3.8.

For all $t\in(0,T]$, the law of the random vector $X(t)$ is absolutely
continuous with respect to the Lebesgue measure on $\mathbb{R}^{d}$.

###### Proof.

We will follow the approach developed in [1, Theorem 5.3] for the standard
Brownian Heston model. Namely, we will check that for any $f\in
C^{\infty}(\mathbb{R}^{d};\mathbb{R})$ such that $f$ and all its partial
derivatives are bounded the following relation holds:

$\left|\mathbb{E}\left[\frac{\partial f}{\partial
x_{i}}(X(t))\right]\right|\leq c_{i}\sup_{x\in\mathbb{R}^{d}}|f(x)|,\quad
i=1,...,d.$

Then the result will follow from [41, Lemma 2.1.1].

Note that, by applying sequentially the chain rule for the Malliavin
derivative and the duality relationship between Malliavin derivative and the
Skorokhod integral, we can obtain

$\displaystyle\left|\mathbb{E}\left[\frac{\partial f}{\partial
x_{1}}(X(t))\right]\right|$
$\displaystyle=\frac{1}{t}\left|\operatorname{\mathbb{E}}\left[\int_{0}^{t}\frac{\partial
f}{\partial
x_{1}}(X(t))\widetilde{D}_{u}^{1}X_{1}(t)\frac{1}{\widetilde{D}_{u}^{1}X_{1}(t)}du\right]\right|$
$\displaystyle=\frac{1}{t}\left|\operatorname{\mathbb{E}}\left[\int_{0}^{t}\widetilde{D}_{u}^{1}\left[f(X(t))\right]\frac{1}{\widetilde{D}_{u}^{1}X_{1}(t)}du\right]\right|$
$\displaystyle=\frac{1}{t}\left|\operatorname{\mathbb{E}}\left[f(X(t))\int_{0}^{t}\frac{1}{u_{1,1}Y_{1}(u)}d\widetilde{W}_{1}(u)\right]\right|$
$\displaystyle\leq\frac{1}{tu_{1,1}}\operatorname{\mathbb{E}}\left[\left|\int_{0}^{t}\frac{1}{Y_{i}(u)}d\widetilde{W}_{1}(u)\right|\right]\sup_{x\in\mathbb{R}^{d}}|f(x)|$
$\displaystyle=:c_{1}\sup_{x\in\mathbb{R}^{d}}|f(x)|,$

i.e. the required property holds for $\frac{\partial f}{\partial
x_{1}}(X(t))$. As for $\frac{\partial f}{\partial x_{i}}(X(t))$ with
$i=2,...,d$, one can reorder the elements of $B$ so that by swapping $B_{i}$
is the first one and then apply the same arguments. ∎

##  4 Smoothing the payoff functional by Malliavin integration by parts

In the remaining part of the paper, we will apply the results of Section 3 to
numerical computation of prices of options with discontinuous payoff. We will
utilize the quadrature technique via Malliavin integration by parts proposed
initially in [1] and adapted to fBm-driven stochastic volatility in [10, 40].
The idea of the method is as follows: using the Malliavin calculus techniques,
one can replace the discontinuous functional $f$ under the expectation with a
locally Lipschitz one and the latter expression is far more convenient for
numerical algorithms and guarantees a better convergence rate. In this
Section, we explain this smoothing technique whereas the numerical algorithm
and the corresponding convergence results will be provided in Section 5. Note
that in the papers [1, 10, 40] mentioned above
$\mathbb{E}f(S_{1}(T),...,S_{d}(T))$ is computed w.r.t. the physical measure
and not the martingale one, i.e. the numerical algorithms described there
concern the expected payoff rather then the option price. However, it is
possible to modify the method to account for a measure change and we explain
this modification in Subsection 4.3.

###  4.1 One-dimensional case

We start from a simpler case when $d=1$, i.e. the VSV-model has the form

$\begin{gathered}S(t)=S(0)+\int_{0}^{t}\mu(s)S(s)ds+\int_{0}^{t}Y(s)S(s)dW_{1}(s),\\\
Y(t)=Y(0)+\int_{0}^{t}b(s,Y(s))ds+\int_{0}^{t}\mathcal{K}(t,s)\left({\ell_{2,1}}dW_{1}(s)+{\ell_{2,2}}dW_{2}(s)\right),\end{gathered}$
(4.1)

where $\mathcal{K}=\mathcal{K}_{1}$ satisfies Assumption A with $H_{i}=H$,
$b=b_{1}$ satisfies Assumption B, $\mu=\mu_{1}$ is an arbitrarty $H$-Hölder
continuous function and $\operatorname{\mathcal{L}}=(\ell_{i,j})_{i,j=1}^{2}$
is such that
$\operatorname{\mathcal{L}}\operatorname{\mathcal{L}}^{T}=\Sigma=(\sigma_{i,j})_{i,j=1}^{2}$.
Using the transformation from Subsection 3.2, one can rewrite (4.1) as

$\begin{gathered}S(t)=S(0)+\int_{0}^{t}\mu(s)S(s)ds+\int_{0}^{t}Y(s)S(s)\left(u_{1,1}d\widetilde{W}_{1}(s)+u_{1,2}d\widetilde{W}_{2}(s)\right),\\\
Y(t)=Y(0)+\int_{0}^{t}b(s,Y(s))ds+\int_{0}^{t}\mathcal{K}(t,s)d\widetilde{W}_{2}(s),\end{gathered}$
(4.2)

where $\widetilde{W}=(\widetilde{W}_{1},\widetilde{W}_{2})$ is a 2-dimensional
Brownian motion such that

$\left(\begin{matrix}u_{1,1}&u_{1,2}\\\
0&1\end{matrix}\right)\left(\begin{matrix}\widetilde{W}_{1}\\\
\widetilde{W}_{2}\end{matrix}\right)=\left(\begin{matrix}B^{S}_{1}\\\
B^{Y}_{1}\end{matrix}\right)=\left(\begin{matrix}1&0\\\
\ell_{2,1}&\ell_{2,2}\end{matrix}\right)\left(\begin{matrix}W_{1}\\\
W_{2}\end{matrix}\right).$

Let now $f$ be a measurable payoff function that satisfies the following
assumption.

######  Assumption C.

Function $f$: $\mathbb{R}\to\mathbb{R}$ is locally Riemann integrable function
(not necessarily continuous) of polynomial growth, i.e. there exist constants
$q>0$ and $c_{f}>0$ such that

$|f(s)|\leq c_{f}(1+|s|^{q}),\quad s>0.$

Denote now $F(s):=\int_{0}^{s}f(u)du$, $g(x):=f(e^{x})$,
$G(x):=F(1)+\int_{0}^{x}g(v)dv$, i.e. $f(S(T))=g(X(T))$, where $X(t):=\log
S(t)$, $t\in[0,T]$. By Theorem 3.6, $X(t)\in\widetilde{\mathbb{D}}_{1,2}$ with

$\widetilde{D}^{1}_{v}X(t)=u_{1,1}Y(v)\mathbbm{1}_{[0,t]}(v)$

and, by Theorem 3.8, $X(t)$ has density. Thus one can use Proposition 1.2.4
from [41] and remark after it to claim that

$\widetilde{D}^{1}_{v}G(X(t))=g(X(t))\widetilde{D}^{1}_{v}X(t),\quad
t\in(0,T],$

if $G$ turns out to be globally Lipschitz continuous. However, we will require
the Malliavin chain rule for $G(X(T))$ in the setting provided by Assumption
C, i.e. when $G$ is only locally Lipschitz continuous. To do this, we will
extend [41, Proposition 1.2.4] in a manner similar to [10, Remark 9].

######  Proposition 4.1.

For $G$ defined above and any $t\in(0,T]$,
$G(X(t))\in\widetilde{\mathbb{D}}_{1,2}$ and

$\widetilde{D}G(X(t))=g(X(t))\widetilde{D}X(t),\quad t\in(0,T].$ (4.3)

###### Proof.

For any $n\in\mathbb{N}$, denote

$g_{n}(x):=g(x)\mathbbm{1}_{[-n,n]}(x)$

and

$\displaystyle G_{n}(x)$ $\displaystyle:=F(1)+\int_{0}^{x}g_{n}(v)dv$
$\displaystyle=G(x)\mathbbm{1}_{[-n,n]}(x)+G(n)\mathbbm{1}_{[n,\infty)}(x)+G(-n)\mathbbm{1}_{(-\infty,-n]}(x).$

It is clear that $G_{n}$ is globally Lipschitz and thus
$G_{n}(X(t))\in\widetilde{\mathbb{D}}_{1,2}$ and

$\widetilde{D}G_{n}(X(t))=g_{n}(X(t))\widetilde{D}X(t).$

Thus, taking into account the closedness of the operator $\widetilde{D}$, it
is sufficient to check that

$g_{n}(X(t))\xrightarrow{L^{2}(\Omega)}g(X(t)),\quad n\to\infty,$ (4.4)

and

$g_{n}(X(t))\widetilde{D}X(t)\xrightarrow{L^{2}\left(\Omega;L^{2}([0,T];\mathbb{R}^{2})\right)}g(X(t))\widetilde{D}X(t),\quad
n\to\infty.$ (4.5)

Observe that $g_{n}(X(t))\to g(X(t))$ a.s., $n\to\infty$. By Assumption C, $f$
is of polynomial growth and thus, by Theorem 2.6,
$\mathbb{E}f^{2}(S(t))=\mathbb{E}g^{2}(X(t))<\infty$ and (4.4) follows from
the dominated convergence theorem.

Finally, using polynomial growth of $f$, explicit shape of $\widetilde{D}X(t)$
given by Theorem 3.6 as well as estimates similar to the ones from Step 1 of
the proof of Theorem 3.3, one can deduce that

$\mathbb{E}\left[g^{2}(X(t))\int_{0}^{T}(\widetilde{D}^{i}_{u}X(t))^{2}du\right]<\infty,$

and therefore (4.5) also holds due to dominated convergence. ∎

Now, having Proposition 4.1, we are ready to proceed to the main result of
this Subsection in the spirit of [1, Theorem 4.2] or [10, Lemma 11].

######  Theorem 4.2.

Let $S(T)$ be given by (4.2), $f$: $\mathbb{R}\to\mathbb{R}$ satisfy
Assumption C, $X(t):=\log S(t)$, $F(s):=\int_{0}^{s}f(u)du$, $g(x):=f(e^{x})$,
$G(x):=F(1)+\int_{0}^{x}g(v)dv$. Then

$\mathbb{E}f(S(T))=\frac{1}{Tu_{1,1}}\mathbb{E}\left[G(X(T))\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)\right].$
(4.6)

Alternatively,

$\mathbb{E}f(S(T))=\mathbb{E}\left[\frac{F(S(T))}{S(T)}\left(1+\frac{1}{Tu_{1,1}}\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)\right)\right].$
(4.7)

###### Proof.

In order to obtain representation (4.6), observe that

$\displaystyle\mathbb{E}f(S(T))$
$\displaystyle=\mathbb{E}g(X(T))=\frac{1}{T}\mathbb{E}\left[\int_{0}^{T}g(X(T))\widetilde{D}^{1}_{u}X(T)\frac{1}{\widetilde{D}^{1}_{u}X(T)}du\right]$
$\displaystyle=\frac{1}{Tu_{1,1}}\mathbb{E}\left[\int_{0}^{T}\widetilde{D}^{1}_{u}G(X(T))\frac{1}{Y(u)}du\right]$
$\displaystyle=\frac{1}{Tu_{1,1}}\mathbb{E}\left[G(X(T))\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)\right].$

Here we used the chain rule from Proposition 4.1 and duality relation between
the Malliavin derivative and Skorokhod integral (which coincides in our case
with the Itô integral).

As for the representation (4.7), note that

$G(x)=F(1)+\int_{0}^{x}f(e^{v})e^{v}\frac{1}{e^{v}}dv=\frac{F(e^{x})}{e^{x}}+\int_{0}^{x}\frac{F(e^{v})}{e^{v}}dv.$

Moreover, using the same argument as in Proposition 4.1, it is easy to prove
that

$\widetilde{D}^{1}\int_{0}^{X(T)}\frac{F(e^{v})}{e^{v}}dv=\frac{F(S(T))}{S(T)}\widetilde{D}^{1}X(T),$

therefore, again by the Malliavin duality, we obtain that

$\displaystyle\mathbb{E}\left[\int_{0}^{X(T)}\frac{F(e^{v})}{e^{v}}dv\frac{1}{Tu_{1,1}}\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)\right]$
$\displaystyle=\mathbb{E}\left[\widetilde{D}^{1}_{u}\int_{0}^{X(T)}\frac{F(e^{v})}{e^{v}}dv\frac{1}{Tu_{1,1}}\frac{1}{Y(u)}du\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\frac{F(S(T))}{S(T)}\widetilde{D}^{1}_{u}X(T)\frac{1}{\widetilde{D}^{1}_{u}X(T)}du\right]$
$\displaystyle=\mathbb{E}\left[\frac{F(S(T))}{S(T)}\right],$

which implies (4.7). ∎

###  4.2 Multidimensional case

The approach described in Subsection 4.1 can also be utilized for basked
options in $d$-dimensional VSV models. Namely, consider a $d$-dimensional
volatility process $Y=(Y_{1},...,Y_{d})$, where each $Y_{i}$ is defined by
(2.6), a $d$-dimensional price process $S=(S_{1},...,S_{d})$ given by (2.8)
and the corresponding log-price process $X=(X_{1},...,X_{d})$, $X_{i}=\log
S_{i}$.

Consider now a basket option with a payoff
$f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)$ with $f$:
$\mathbb{R}\to\mathbb{R}$ satisfying Assumption C and $\alpha_{i}>0$,
$i=1,...,d$. As before, define $F(s):=\int_{0}^{s}f(u)du$ and observe that the
function

$G_{1}(x_{1},...,x_{d}):=\frac{F\left(\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}\right)}{\alpha_{1}e^{x_{1}}},\quad(x_{1},...,x_{d})\in\mathbb{R}^{d},$

is locally Lipschitz continuous: if
$\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}>\sum_{i=1}^{d}\alpha_{i}e^{x_{i}^{\prime}}$,
then

$\displaystyle|G_{1}(x_{1},...,x_{d})-G_{1}(x_{1}^{\prime},...,x_{d}^{\prime})|$
$\displaystyle\leq\frac{e^{-x_{1}}}{\alpha_{1}}\int_{\sum_{i=1}^{d}\alpha_{i}e^{x_{i}^{\prime}}}^{\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}}|f(s)|ds$
$\displaystyle\quad+\frac{1}{\alpha_{1}}\int_{0}^{\sum_{i=1}^{d}\alpha_{i}e^{x_{i}^{\prime}}}|f(s)|ds|e^{-x_{1}}-e^{-x_{1}^{\prime}}|$
$\displaystyle\leq
c_{f}\frac{e^{-x_{1}}}{\alpha_{1}}\left(1+\left|\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}\right|^{q}\right)\sum_{j=1}^{d}\alpha_{j}e^{x_{j}\vee
x_{j}^{\prime}}|{x_{j}}-{x_{j}^{\prime}}|$
$\displaystyle\quad+c_{f}\frac{1}{\alpha_{1}}\left(\sum_{i=1}^{d}\alpha_{i}e^{x_{i}^{\prime}}+\frac{1}{q+1}\left|\sum_{i=1}^{d}\alpha_{i}e^{x_{i}^{\prime}}\right|^{q+1}\right)e^{-(x_{1}\wedge
x_{1}^{\prime})}|x_{1}-x_{1}^{\prime}|.$

Note that polynomial growth of $f$ (and hence $F$) together with Theorem 2.6
imply that random variables $F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)$
and $G_{1}(X_{1}(T),...,X_{d}(T))$ have moments of all orders. Moreover,
Theorem 3.8 implies that the law of $X$ (and hence of $S$) is absolutely
continuous w.r.t. the Lebesgue measure on $\mathbb{R}^{d}$. Therefore, by
approximating $G_{1}(X(t))$ with Lipschitz functions just like in Proposition
4.1, it is straightforward to get the following chain rule for $G_{1}(X(t))$.

######  Proposition 4.3.

For $G_{1}$ defined above and any $t\in(0,T]$,
$G_{1}(X(t))\in\widetilde{\mathbb{D}}_{1,2}$ and

$\widetilde{D}G_{1}(X(t))=\sum_{i=1}^{d}\frac{\partial G_{1}(X(t))}{\partial
x_{i}}\widetilde{D}X_{i}(t),\quad t\in(0,T].$ (4.8)

We are now ready to formulate an analogue of Theorem 4.2 for the basket
option.

######  Theorem 4.4.

Let $S=(S_{1},...,S_{d})$ be given by (2.8), $X=(X_{1},...,X_{d})$ be the
corresponding log-price process, $X_{i}=\log S_{i}$, $f$:
$\mathbb{R}\to\mathbb{R}$ satisfy Assumption C, $F(s):=\int_{0}^{s}f(u)du$ and
$\alpha_{i}>0$, $i=1,...,d$. Then

$\mathbb{E}f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)=\mathbb{E}\left[\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{1}{Tu_{1,1}}\int_{0}^{T}\frac{1}{Y_{1}(u)}d\widetilde{W}_{1}(u)\right)\right].$
(4.9)

###### Proof.

Note that

$\frac{\partial G_{1}(x_{1},...,x_{d})}{\partial
x_{1}}=f\left(\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}\right)-\frac{1}{\alpha_{1}e^{x_{1}}}F\left(\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}\right)$

hence, taking into account Proposition 4.3 and Theorem 3.6,

$\displaystyle\mathbb{E}\left[f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\left(f\left(\sum_{i=1}^{d}\alpha_{i}e^{X_{i}(T)}\right)-\frac{1}{\alpha_{1}e^{X_{1}(T)}}F\left(\sum_{i=1}^{d}\alpha_{i}e^{X_{i}(T)}\right)\right)dv\right]$
$\displaystyle\quad+\mathbb{E}\left[\frac{1}{\alpha_{1}e^{X_{1}(T)}}F\left(\sum_{i=1}^{d}\alpha_{i}e^{X_{i}(T)}\right)\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\frac{\partial
G_{1}(X(T))}{\partial x_{1}}dv\right]+\mathbb{E}\left[G_{1}(X(T))\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\frac{\partial
G_{1}(X(T))}{\partial
x_{1}}\widetilde{D}^{1}_{v}X_{1}(T)\frac{1}{\widetilde{D}^{1}_{v}X_{1}(T)}dv\right]+\mathbb{E}\left[G_{1}(X(T))\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\widetilde{D}_{v}^{1}G_{1}(X(T))\frac{1}{u_{1,1}Y_{1}(v)}dv\right]+\mathbb{E}\left[G_{1}(X(T))\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{Tu_{1,1}}G_{1}(X(T))\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)\right]+\mathbb{E}\left[G_{1}(X(T))\right]$
$\displaystyle=\mathbb{E}\left[\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{1}{Tu_{1,1}}\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)\right)\right],$

as required. ∎

###  4.3 Quadrature via Malliavin integration under a change of measure

Theorems 4.2 and 4.4 concern the expectation w.r.t. the physical measure, i.e.
the obtained representation of the expected payoff is not the option price. In
this Section, we will modify the quadrature technique to account for a measure
change.

Consider a martingale measure with the density

$\mathcal{E}_{T}:=\mathcal{E}_{T}\left\\{-\sum_{j=1}^{d}\int_{0}^{\cdot}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right\\},$
(4.10)

where $\ell^{(-1)}_{j,k}$ and $\widetilde{\mu}_{k}(s)$ are given in Subsection
2.2.

######  Remark 4.5.

In the literature, the measure with the density (4.10) is commonly referred to
as the minimal martingale measure, see e.g. [44].

Recall that

$W=\operatorname{\mathcal{L}}^{-1}\mathcal{U}\widetilde{W},$

so denote $\rho_{j,i}$ the values such that

$W_{j}(t)=\sum_{i=1}^{2d}\rho_{j,i}\widetilde{W}_{i}(t),$

i.e.

$\displaystyle\mathcal{E}_{T}$
$\displaystyle=\mathcal{E}_{T}\left\\{-\sum_{j=1}^{d}\sum_{i=1}^{2d}\int_{0}^{\cdot}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)d\widetilde{W}_{i}(s)\right\\}$
$\displaystyle=\mathcal{E}_{T}\left\\{-\sum_{i=1}^{2d}\int_{0}^{\cdot}\left(\sum_{j=1}^{d}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)d\widetilde{W}_{i}(s)\right\\}$
$\displaystyle=\exp\left\\{-\sum_{i=1}^{2d}\int_{0}^{T}\left(\sum_{j=1}^{d}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)d\widetilde{W}_{i}(s)\right\\}\times$
$\displaystyle\qquad\times\exp\left\\{-\frac{1}{2}\sum_{i=1}^{2d}\int_{0}^{T}\left(\sum_{j=1}^{d}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)^{2}ds\right\\}.$

Note that each $Y_{i}$ is bounded away from zero by
$\varphi_{i}^{*}:=\min_{t\in[0,T]}\varphi_{i}(t)$ and hence
$\frac{1}{Y_{i}(s)}$ can be represented as follows:

$\frac{1}{Y_{i}(s)}=h_{i}(Y_{i}(s)),\quad
h_{i}(y):=\begin{cases}\frac{1}{y},&\quad y>\varphi_{i}^{*},\\\
-\frac{1}{(\varphi_{i}^{*})^{2}}y+\frac{2}{\varphi_{i}^{*}},&\quad
y\leq\varphi_{i}^{*}.\end{cases}$

The function $h_{i}$ is bounded, continuously differentiable and has bounded
derivative, hence by the classical chain rule
$\frac{1}{Y_{i}(s)}\in\widetilde{\mathbb{D}}_{1,2}$ and

$\widetilde{D}^{1}\left[\frac{1}{Y_{i}(s)}\right]=0.$

In particular, this implies that

$\widetilde{D}^{1}_{v}\left[-\frac{1}{2}\sum_{i=1}^{2d}\int_{0}^{T}\left(\sum_{j=1}^{d}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)^{2}ds\right]=0$

and

$\displaystyle\widetilde{D}_{v}^{1}$
$\displaystyle\left[-\sum_{i=1}^{2d}\int_{0}^{T}\left(\sum_{j=1}^{d}\rho_{j,i}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)d\widetilde{W}_{i}(s)\right]$
$\displaystyle=-\sum_{j=1}^{d}\sum_{k=1}^{d}\rho_{j,1}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)}$
$\displaystyle=\sum_{k=1}^{d}\left(-\sum_{j=1}^{d}\rho_{j,1}\ell^{(-1)}_{j,k}\right)\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)}$
$\displaystyle=:\sum_{k=1}^{d}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)},$

where $\beta_{k}:=\left(-\sum_{j=1}^{d}\rho_{j,1}\ell^{(-1)}_{j,k}\right)$.
Furthermore, using the boundedness of $Y$ and the chain rule from Proposition
3.4, it is easy to establisth that
$\mathcal{E}_{T}\in\widetilde{\mathbb{D}}_{1,2}$ and

$\widetilde{D}^{1}_{v}\mathcal{E}_{T}=\mathcal{E}_{T}\sum_{k=1}^{d}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)}.$

Let now $f$: $\mathbb{R}\to\mathbb{R}$ satisfy Assumption C, $\alpha_{i}>0$,
$i=1,...,d$. Then the following modification of Theorem 4.4 is true.

######  Theorem 4.6.

Let $S=(S_{1},...,S_{d})$ be given by (2.8), $X=(X_{1},...,X_{d})$ be the
corresponding log-price process, $X_{i}=\log S_{i}$, $f$:
$\mathbb{R}\to\mathbb{R}$ satisfy Assumption C, $F(s):=\int_{0}^{s}f(u)du$ and
$\alpha_{i}>0$, $i=1,...,d$. Then

$\displaystyle\mathbb{E}$
$\displaystyle\left[\mathcal{E}_{T}f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$
(4.11)
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{1}{Tu_{1,1}}\left(\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)Y_{1}(v)}dv\right)\right)\right].$

###### Proof.

Denote

$G_{1}(x_{1},...,x_{d}):=\frac{F\left(\sum_{i=1}^{d}\alpha_{i}e^{x_{i}}\right)}{\alpha_{1}e^{x_{1}}},\quad(x_{1},...,x_{d})\in\mathbb{R}^{d}.$

Using polynomial growth of $f$, boundedness of $Y_{i}$, existence of all
moments of $S_{i}$ and Proposition 3.4, it is easy to prove that

$\mathcal{E}_{T}G_{1}(X(T))\in\widetilde{\mathbb{D}}_{1,2}$

and

$\displaystyle\widetilde{D}_{v}^{1}\left[\mathcal{E}_{T}G_{1}(X(T))\right]$
$\displaystyle=\mathcal{E}_{T}\widetilde{D}_{v}^{1}G_{1}(X(T))+G_{1}(X(T))\widetilde{D}_{v}^{1}\mathcal{E}_{T}$
$\displaystyle=\mathcal{E}_{T}\widetilde{D}_{v}^{1}G_{1}(X(T))+\mathcal{E}_{T}G_{1}(X(T))\sum_{k=1}^{d}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)}.$

Next, observe that

$\displaystyle\mathbb{E}$
$\displaystyle\left[\mathcal{E}_{T}f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]=\mathbb{E}\left[\mathcal{E}_{T}\left(f\left(\sum_{i=1}^{d}\alpha_{i}e^{X_{i}(T)}\right)-G_{1}(X(T))\right)\right]+\mathbb{E}\left[\mathcal{E}_{T}G_{1}(X(T))\right]$

and

$\displaystyle\mathbb{E}$
$\displaystyle\left[\mathcal{E}_{T}\left(f\left(\sum_{i=1}^{d}\alpha_{i}e^{X_{i}(T)}\right)-G_{1}(X(T))\right)\right]=\mathbb{E}\left[\mathcal{E}_{T}\frac{\partial
G_{1}(X(T))}{\partial x_{1}}\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{T}\int_{0}^{T}\mathcal{E}_{T}\frac{\partial
G_{1}(X(T))}{\partial
x_{1}}\widetilde{D}^{1}_{v}X_{1}(T)\frac{1}{\widetilde{D}^{1}_{v}X_{1}(T)}dv\right]$
$\displaystyle=\mathbb{E}\left[\frac{1}{Tu_{1,1}}\int_{0}^{T}\mathcal{E}_{T}\widetilde{D}^{1}_{v}G_{1}(X(T))\frac{1}{Y_{1}(v)}dv\right]$
$\displaystyle=\frac{1}{Tu_{1,1}}\left(\mathbb{E}\left[\int_{0}^{T}\widetilde{D}^{1}_{v}\left[\mathcal{E}_{T}G_{1}(X(T))\right]\frac{1}{Y_{1}(v)}dv\right]-\mathbb{E}\left[\int_{0}^{T}G_{1}(X(T))\widetilde{D}^{1}_{v}\mathcal{E}_{T}\frac{1}{Y_{1}(v)}dv\right]\right)$
$\displaystyle=\frac{1}{Tu_{1,1}}\left(\mathbb{E}\left[\mathcal{E}_{T}G_{1}(X(T))\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)\right]-\mathbb{E}\left[G_{1}(X(T))\mathcal{E}_{T}\int_{0}^{T}\sum_{k=1}^{d}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)Y_{1}(v)}dv\right]\right)$
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\frac{1}{Tu_{1,1}}\left(\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)Y_{1}(v)}dv\right)\right].$

Taking into account all of the above, we can finally write:

$\displaystyle\mathbb{E}$
$\displaystyle\left[\mathcal{E}_{T}f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$
$\displaystyle=\mathbb{E}\left[\mathcal{E}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{1}{Tu_{1,1}}\left(\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)Y_{1}(v)}dv\right)\right)\right]$

which ends the proof. ∎

######  Remark 4.7.

Note that the martingale measure should not necessarily be the minimal one: it
is possible to obtain a representation of the type (4.11) for other martingale
measures provided that the corresponding densities are Malliavin
differentiable and one can use the chain rules as in the proof of Theorem 4.6.

##  5 Approximation of the expected value of an option with discontinuous
payoff

In this Section, we provide a numerical algorithm for computation of
$\mathbb{E}\left[f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$. We
will exploit the representations obtained in Theorems 4.2, 4.4 and 4.6:
absence of discontinuities under the expectation sign allows to avoid
deterioration of the convergence speed.

The Section is structured as follows. In Subsection 5.1, we describe the
numerical scheme used to simulate $Y$, $X$ and $S$ (the corresponding
numerical approximations will be denoted by $\widehat{Y}$, $\widehat{X}$ and
$\widehat{S}$). In Subsection 5.2, we consider a straightforward way of
computing
$\mathbb{E}\left[f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$ by
plugging in $\widehat{Y}$ and $\widehat{S}$ into (4.6)–(4.7), (4.9) or (4.11).
An estimate of the approximation error is provided. In Subsection 5.3, we
discuss another algorithm for computation of the expectation for a one-
dimensional case: the vector
$\left(X(T),\int_{0}^{T}\frac{1}{Y(v)}dv\right)^{T}$ turns out to be
conditionally Gaussian w.r.t. an appropriate $\sigma$-field and one can use
this fact to approximate (4.6) without simulating $\widehat{X}(T)$ at all.

###  5.1 Numerical schemes

#### Numerical scheme for the volatility process.

For the stochastic volatility processes $Y_{i}$, $i=1,...,d$, we will use the
drift-implicit Euler scheme described in [22]. For reader’s convenience, we
briefly describe it below as well as give the necessary convergence results.

Let $\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ be a uniform partition of $[0,T]$,
$t_{k}:=\frac{Tk}{N}$, $k=0,1,...,N$, with the mesh $\Delta_{N}:=\frac{T}{N}$
such that

$\Delta_{N}\max_{i=1,...,d}\sup_{(t,y)\in\mathcal{D}^{i}_{0,0}}\frac{\partial
b_{i}}{\partial y}(t,y)<1.$ (5.1)

######  Remark 5.1.

Note that such $\Delta_{N}$ exists due to Assumption B(iv).

Define $\widehat{Y}_{i}(t)$ as follows:

$\displaystyle\widehat{Y}_{i}(0)$ $\displaystyle=Y_{i}(0),$ (5.2)
$\displaystyle\widehat{Y}_{i}(t_{k+1})$
$\displaystyle=\widehat{Y}_{i}(t_{k})+b_{i}(t_{k+1},\widehat{Y}_{i}(t_{k+1}))\Delta_{N}+(Z_{i}(t_{k+1})-Z_{i}(t_{k})),$
$\displaystyle\widehat{Y}_{i}(t)$ $\displaystyle=\widehat{Y}_{i}(t_{k}),\quad
t\in[t_{k},t_{k+1}),$

where $Z_{i}(t):=\int_{0}^{t}\mathcal{K}_{i}(t,s)dB^{Y}_{i}(s)$. Note that the
second expression in (5.2) is considered as an equation with respect to
$\widehat{Y}_{i}(t_{k+1})$ and, under Assumption B, it has a unique solution
such that
$\widehat{Y}_{i}(t_{k+1})\in(\varphi_{i}(t_{k+1}),\psi_{i}(t_{k+1}))$, see
[22, Section 2] for more details. The convergence results from [22] for this
scheme are summarized in the following theorem.

######  Theorem 5.2.

Let Assumptions A and B hold and the mesh of the partition $\Delta_{N}$
satisfy (5.1). Fix $i=1,...,d$, take an arbitrary $\lambda_{i}\in(0,H_{i})$,
where $H_{i}$ is from Assumption A, and fix $r\geq 1$. Then there exists a
constant $C>0$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{t\in[0,T]}|Y_{i}(t)-\widehat{Y}_{i}(t)|^{r}\right]\leq
C\Delta_{N}^{\lambda_{i}r}$

and

$\mathbb{E}\left[\sup_{t\in[0,T]}\left|\frac{1}{Y_{i}(t)}-\frac{1}{\widehat{Y}_{i}(t)}\right|^{r}\right]\leq
C\Delta_{N}^{\lambda_{i}r}.$

#### Numerical scheme for the price process.

Recall that the log-price $X_{i}$ has the form

$X_{i}(t)=X_{i}(0)+\int_{0}^{t}\left(\mu_{i}(s)-\frac{Y_{i}^{2}(s)}{2}\right)ds+\int_{0}^{t}Y_{i}(s)dB^{S}_{i}(s),$

where $\mu_{i}$ is $H_{i}$-Hölder continuous with $H_{i}$ being from
Assumption A(ii). For a given uniform partition
$\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ satisfying (5.1), consider the random
process $\widehat{X}_{i}=\\{\widehat{X}_{i}(t),~{}t\in[0,T]\\}$ such that for
any point $t_{n}$ of the partition

$\widehat{X}_{i}(t_{n})=X_{i}(0)+\frac{T}{N}\sum_{k=0}^{n-1}\left(\mu_{i}(t_{k})-\frac{\widehat{Y}_{i}^{2}(t_{k})}{2}\right)+\sum_{k=0}^{n-1}\widehat{Y}_{i}(t_{k})\left(B^{S}_{i}(t_{k+1})-B^{S}_{i}(t_{k})\right),$

where $\widehat{Y}_{i}$ is defined by (5.2), and for any $t\in(t_{n},t_{n+1})$

$\widehat{X}_{i}(t)=\widehat{X}_{i}(t_{n}).$

######  Theorem 5.3.

Let Assumptions A and B hold and the mesh of the partition $\Delta_{N}$
satisfy (5.1). Fix $i=1,...,d$, take an arbitrary $\lambda_{i}\in(0,H_{i})$,
where $H_{i}$ is from Assumption A, and fix $r\geq 1$. Then there exists a
constant $C>0$ that does not depend on the partition such that

$\mathbb{E}\left[\sup_{n=0,1,...,N}|X_{i}(t_{n})-\widehat{X}_{i}(t_{n})|^{r}\right]\leq
C\Delta^{r\lambda_{i}}_{N}.$

In particular,

$\mathbb{E}\left[|X_{i}(T)-\widehat{X}_{i}(T)|^{r}\right]\leq
C\Delta^{r\lambda_{i}}_{N}.$

###### Proof.

Observe that for any $n=1,...,N$

$\displaystyle|X(t_{n})-\widehat{X}(t_{n})|^{r}$ $\displaystyle\leq
C\left(\sum_{k=0}^{n-1}\int_{t_{k}}^{t_{k+1}}|\mu_{i}(s)-\mu_{i}(t_{k})|ds\right)^{r}+C\left(\int_{0}^{t_{n}}|Y_{i}^{2}(s)-\widehat{Y}_{i}^{2}(s)|ds\right)^{r}$
$\displaystyle\quad+C\left|\int_{0}^{t_{n}}(Y_{i}(s)-\widehat{Y}_{i}(s))dB^{S}_{i}(s)\right|^{r}$
$\displaystyle\leq
C\left(\sum_{k=0}^{N-1}\int_{t_{k}}^{t_{k+1}}|s-t_{k}|^{\lambda_{i}}ds\right)^{r}+C\left(\int_{0}^{T}|Y_{i}(s)+\widehat{Y}_{i}(s)||Y_{i}(s)-\widehat{Y}_{i}(s)|ds\right)^{r}$
$\displaystyle\quad+C\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}(Y_{i}(s)-\widehat{Y}_{i}(s))dB^{S}_{i}(s)\right|\right)^{r}$
$\displaystyle\leq
C\Delta^{r\lambda_{i}}_{N}+C\sup_{t\in[0,T]}|Y_{i}(t)-\widehat{Y}_{i}(t)|^{r}+C\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}(Y_{i}(s)-\widehat{Y}_{i}(s))dB^{S}_{i}(s)\right|\right)^{r}.$

Hence, by the Burkholder-Davis-Gundy inequality and Theorem 5.2,

$\displaystyle\mathbb{E}\left[\sup_{n}|X(t_{n})-\widehat{X}(t_{n})|^{r}\right]$
$\displaystyle\leq
C\Delta^{r\lambda_{i}}_{N}+C\mathbb{E}\left[\sup_{t\in[0,T]}|Y_{i}(t)-\widehat{Y}_{i}(t)|^{r}\right]$
$\displaystyle\quad+C\mathbb{E}\left[\left(\sup_{t\in[0,T]}\left|\int_{0}^{t}(Y_{i}(s)-\widehat{Y}_{i}(s))dB^{S}_{i}(s)\right|\right)^{r}\right]$
$\displaystyle\leq
C\Delta^{r\lambda_{i}}_{N}+C\mathbb{E}\left[\left(\int_{0}^{T}(Y_{i}(s)-\widehat{Y}_{i}(s))^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq C\Delta^{r\lambda_{i}}_{N}.$

∎

Next, denote

$\widehat{S}_{i}(t):=e^{\widehat{X}_{i}(t)}.$

######  Proposition 5.4.

Let Assumptions A and B hold and the mesh of the partition $\Delta_{N}$
satisfy (5.1). Then, for any $r\in\mathbb{R}$

$\sup_{N\geq
1}\sup_{n=0,1,...,N}\mathbb{E}\left[\widehat{S}_{i}^{r}(t_{n})\right]<\infty.$

###### Proof.

Since both $\mu_{i}$ and $\widehat{Y}_{i}$ are bounded and the bound does not
depend on the partition, it is sufficient to prove that

$\sup_{N\geq
1}\sup_{n=0,1,...,N}\mathbb{E}\left[\exp\left\\{r\int_{0}^{t_{n}}\widehat{Y}_{i}(s)dB_{i}^{S}(s)\right\\}\right]<\infty.$

Consider a stochastic process

$\exp\left\\{r\int_{0}^{t}\widehat{Y}_{i}(s)dB_{i}^{S}(s)\right\\},\quad
t\in[0,T].$

Then, since $\widehat{Y}_{i}$ is bounded, Novikov’s condition implies that

$\mathbb{E}\left[\exp\left\\{\int_{0}^{t}r\widehat{Y}_{i}(s)dB_{i}^{S}(s)-\frac{r^{2}}{2}\int_{0}^{t}\widehat{Y}_{i}^{2}(s)ds\right\\}\right]=1$

and hence there exist a constant $C$ that does not depend on the partition
such that

$\displaystyle\mathbb{E}\left[\exp\left\\{r\int_{0}^{t}\widehat{Y}_{i}(s)dB_{i}^{S}(s)\right\\}\right]$
$\displaystyle=\mathbb{E}\left[\exp\left\\{r\int_{0}^{t}\widehat{Y}_{i}(s)dB_{i}^{S}(s)-\frac{r^{2}}{2}\int_{0}^{t}\widehat{Y}_{i}^{2}(s)ds\right\\}\exp\left\\{\frac{r^{2}}{2}\int_{0}^{t}\widehat{Y}_{i}^{2}(s)ds\right\\}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\exp\left\\{\int_{0}^{t}r\widehat{Y}_{i}(s)dB_{i}^{S}(s)-\frac{r^{2}}{2}\int_{0}^{t}\widehat{Y}_{i}^{2}(s)ds\right\\}\right]=C<\infty,$

which implies the required result. ∎

######  Theorem 5.5.

Let Assumptions A and B hold and the mesh of the partition $\Delta_{N}$
satisfy (5.1). Fix $i=1,...,d$, take an arbitrary $\lambda_{i}\in(0,H_{i})$,
where $H_{i}$ is from Assumption A, and fix $r\geq 1$. Then there exists a
constant $C>0$ that does not depend on the partition such that

$\sup_{N\geq
1}\sup_{n=0,1,...,N}\mathbb{E}\left[|S_{i}(t_{n})-\widehat{S}_{i}(t_{n})|^{r}\right]\leq
C\Delta^{r\lambda_{i}}_{N}.$

In particular,

$\mathbb{E}\left[|S_{i}(T)-\widehat{S}_{i}(T)|^{r}\right]\leq
C\Delta^{r\lambda_{i}}_{N}.$

###### Proof.

For any $n=0,1,...,N$, we use Theorem 2.6, Proposition 5.4 and Theorem 5.3 to
write:

$\displaystyle\mathbb{E}\left[|S_{i}(t_{n})-\widehat{S}_{i}(t_{n})|^{r}\right]$
$\displaystyle=\mathbb{E}\left[|e^{X_{i}(t_{n})}-e^{\widehat{X}_{i}(t_{n})}|^{r}\right]$
$\displaystyle\leq\mathbb{E}\left[\left(e^{X_{i}(t_{n})}+e^{\widehat{X}_{i}(t_{n})}\right)^{r}|X_{i}(t_{n})-\widehat{X}_{i}(t_{n})|^{r}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[\left(e^{X_{i}(t_{n})}+e^{\widehat{X}_{i}(t_{n})}\right)^{2r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[|X_{i}(t_{n})-\widehat{X}_{i}(t_{n})|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}[e^{2rX_{i}(t_{n})}]+\mathbb{E}[e^{2r\widehat{X}_{i}(t_{n})}]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[|X_{i}(t_{n})-\widehat{X}_{i}(t_{n})|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq C\Delta^{r\lambda_{i}}_{N},$

where $C$ does not depend on the partition, which implies the required result.
∎

#### Approximation of the martingale density.

In order to get an approximation of option price (i.e. under the change of
measure), we need a result concerning approximation of the corresponding
density. Let $\mathcal{E}_{T}$ be the minimal martingale measure defined by
(4.10). Fix a uniform partition $\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$,
$t_{n}=\frac{Tn}{N}$, with the mesh satisfying (5.1) and put

$\widehat{\mathcal{E}}_{T}:=\exp\left\\{-\sum_{j=1}^{d}\sum_{n=0}^{N-1}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(t_{n})}\right)(W_{j}(t_{n+1})-W_{j}(t_{n}))-\frac{1}{2}\sum_{j=1}^{d}\frac{T}{N}\sum_{n=0}^{N-1}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(t_{n})}\right)^{2}\right\\}.$
(5.3)

######  Theorem 5.6.

Let Assumptions A and B hold and the mesh of the partition $\Delta_{N}$
satisfy (5.1). Let also $\lambda<\min_{i=1,...,d}H_{i}$ with $H_{i}$ being
from Assumption A. Then, for any $r\geq 1$, there exists a constant $C>0$ that
does not depend on the partition such that

$\mathbb{E}\left[|\mathcal{E}_{T}-\widehat{\mathcal{E}}_{T}|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$

###### Proof.

The proof is straightforward, so we will provide an outline omitting all the
details.

First, observe that for every $j=1,...,d$, using boundedness of each
$\frac{1}{\widehat{Y}_{k}}$, $\lambda$-Hölder continuity of each
$\widetilde{\mu}_{k}$ as well as Theorem 5.2, we can write

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left|\sum_{n=0}^{N-1}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(t_{n})}\right)(W_{j}(t_{n+1})-W_{j}(t_{n}))-\int_{0}^{T}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)dW_{j}(s)\right|^{r}\right]$
$\displaystyle=\mathbb{E}\left[\left|\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\left(\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(s)}-\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)dW_{j}(s)\right|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\left(\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(s)}-\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)\right)^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\sum_{k=1}^{d}\left(\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(s)}-\frac{\widetilde{\mu}_{k}(s)}{Y_{k}(s)}\right)^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\sum_{k=1}^{d}\frac{1}{{\widehat{Y}}^{2}_{k}(s)}\left(\widetilde{\mu}_{k}(t_{n})-\widetilde{\mu}_{k}(s)\right)^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\quad+C\mathbb{E}\left[\left(\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\sum_{k=1}^{d}\left(\frac{1}{{\widehat{Y}}_{k}(s)}-\frac{1}{Y_{k}(s)}\right)^{2}ds\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{k=1}^{d}\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\left(\widetilde{\mu}_{k}(t_{n})-\widetilde{\mu}_{k}(s)\right)^{2}ds\right)^{\frac{r}{2}}\right]+C\mathbb{E}\left[\left(\sum_{k=1}^{d}\sup_{s\in[0,T]}\left(\frac{1}{{\widehat{Y}}_{k}(s)}-\frac{1}{Y_{k}(s)}\right)^{2}\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\sum_{k=1}^{d}\sum_{n=0}^{N-1}\int_{t_{n}}^{t_{n+1}}\left(t_{n}-s\right)^{2\lambda}ds\right)^{\frac{r}{2}}\right]+C\Delta_{N}^{r\lambda}$
$\displaystyle\leq C\Delta_{N}^{r\lambda},$

where $C$ is, as always, a constant that does not depend on the partition and
which may vary from line to line. Similarly, for each $j=1,...,d$,

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left|\sum_{j=1}^{d}\frac{T}{N}\sum_{n=0}^{N-1}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(t_{n})}{{\widehat{Y}}_{k}(t_{n})}\right)^{2}-\sum_{j=1}^{d}\int_{0}^{T}\left(\sum_{k=1}^{d}\ell^{(-1)}_{j,k}\frac{\widetilde{\mu}_{k}(s)}{{\widehat{Y}}_{k}(s)}\right)^{2}\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda},$

therefore we conclude that

$\mathbb{E}\left[\left|\log\mathcal{E}_{T}-\log\widehat{\mathcal{E}}_{T}\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$

Using the arguments similar to Proposition 5.4, one can see that for all
$r\geq 1$ there exists a constant $C$ that does not depend on the partition
such that

$\mathbb{E}\left[\mathcal{E}_{T}^{r}\right]+\max_{N\geq
1}\mathbb{E}\left[\widehat{\mathcal{E}}_{T}^{r}\right]<C$ (5.4)

and hence

$\displaystyle\mathbb{E}\left[|\mathcal{E}_{T}-\widehat{\mathcal{E}}_{T}|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\mathcal{E}_{T}^{r}+\widehat{\mathcal{E}}_{T}^{r}\right)\left|\log\mathcal{E}_{T}-\log\widehat{\mathcal{E}}_{T}\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$

∎

###  5.2 Approximation by discretizing both $Y$ and $S$

After introducing the numerical schemes for $Y$, $X$ and $S$, we are ready to
proceed to main results of this Section. We begin with a theorem that is in
the spirit of [10, Theorem 15].

######  Theorem 5.7.

Let $S=(S_{1},...,S_{d})$ be given by (2.8), $X=(X_{1},...,X_{d})$ be the
corresponding log-price process, $X_{i}=\log S_{i}$, $f$:
$\mathbb{R}\to\mathbb{R}$ satisfy Assumption C, $F(s):=\int_{0}^{s}f(u)du$ and
$\alpha_{i}>0$, $i=1,...,d$. Fix a uniform partition
$\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$, $t_{n}=\frac{Tn}{N}$, with the mesh
satisfying (5.1) and take $\widehat{Y}=(\widehat{Y}_{1},...,\widehat{Y}_{d})$,
$\widehat{X}=(\widehat{X}_{1},...,\widehat{X}_{d})$ and
$\widehat{S}=(\widehat{S}_{1},...,\widehat{S}_{d})$ as described in Subsection
5.1. Finally, let $\lambda<\min_{i=1,...,d}H_{i}$ with $H_{i}$ being from
Assumption A. Then

  * 1)

for any $r\geq 1$, there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}$
$\displaystyle\Bigg{[}\Bigg{|}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{Y_{1}(u)}d\widetilde{W}_{1}(u)}{Tu_{1,1}}\right)$
$\displaystyle\qquad-\frac{F\left(\sum_{i=1}^{d}\alpha_{i}\widehat{S}_{i}(T)\right)}{\alpha_{1}\widehat{S}_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{\widehat{Y}_{1}(u)}d\widetilde{W}_{1}(u)}{Tu_{1,1}}\right)\Bigg{|}^{r}\Bigg{]}\leq
C\Delta_{N}^{r\lambda},$

  * 2)

there exists a constant $C>0$ such that

$\left|\mathbb{E}\left[f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]-\mathbb{E}\left[\frac{F\left(\sum_{i=1}^{d}\alpha_{i}\widehat{S}_{i}(T)\right)}{\alpha_{1}\widehat{S}_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{\widehat{Y}_{1}(u)}d\widetilde{W}_{1}(u)}{Tu_{1,1}}\right)\right]\right|\leq
C\Delta_{N}^{\lambda}.$

###### Proof.

By Theorem 4.4, it is enough to prove item 1. We provide the proof only for
$d=1$ (in this case we can put $\alpha_{1}=1$ without any loss in generality).
For the general case of $d>1$, the arguments are the same but require a more
notation-heavy presentation. For reader’s convenience, we split the proof into
three steps.

Step 1. We begin with noting that, by the Burkholder-Davis-Gundy inequality
and Theorem 5.2,

$\displaystyle\mathbb{E}\left[\left|\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)d\widetilde{W}_{1}(u)\right|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left(\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)^{2}du\right)^{\frac{r}{2}}\right]$
$\displaystyle\leq C\Delta_{N}^{\lambda r}.$

Step 2. Next, let us show that for any $r\geq 1$

$\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$ (5.5)

It is clear that

$\displaystyle\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}-\frac{F(S(T))}{\widehat{S}(T)}\right|^{r}\right]$
(5.6)
$\displaystyle\quad+C\mathbb{E}\left[\left|\frac{F(S(T))}{\widehat{S}(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right|^{r}\right].$

Now we estimate both terms in the right-hand side separately. Observe that $F$
has polynomial growth and thus, by Theorem 2.6,
$\mathbb{E}[|F(S(T))|^{2r}]<\infty$. Therefore

$\displaystyle\mathbb{E}\left[|F(S(T))|^{r}\left|\frac{1}{S(T)}-\frac{1}{\widehat{S}(T)}\right|^{r}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[|F(S(T))|^{2r}\right]\mathbb{E}\left[\left|\frac{1}{S(T)}-\frac{1}{\widehat{S}(T)}\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[\left|\frac{1}{S(T)}-\frac{1}{\widehat{S}(T)}\right|^{2r}\right]\right)^{\frac{1}{2}}.$

Moreover, observe that, by Theorem 2.6 and Proposition 5.4, there exists
constant $C>0$ that does not depend on the partition such that

$\mathbb{E}\left[e^{-4r(X(T)+\widehat{X}(T))}(e^{X(T)}+e^{\widehat{X}(T)})^{4r}\right]<C,$

thus, using Theorem 5.3 the inequality $|e^{x}-e^{y}|\leq(e^{x}+e^{y})|x-y|$,
$x$, $y\in\mathbb{R}$, we can write

$\displaystyle\mathbb{E}$
$\displaystyle\left[\left|\frac{1}{S(T)}-\frac{1}{\widehat{S}(T)}\right|^{2r}\right]=\mathbb{E}\left[e^{-2r(X(T)+\widehat{X}(T))}\left|e^{X(T)}-e^{\widehat{X}(T)}\right|^{2r}\right]$
$\displaystyle\leq\mathbb{E}\left[e^{-2r(X(T)+\widehat{X}(T))}(e^{X(T)}+e^{\widehat{X}(T)})^{2r}\left|X(T)-\widehat{X}(T)\right|^{2r}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[e^{-4r(X(T)+\widehat{X}(T))}(e^{X(T)}+e^{\widehat{X}(T)})^{4r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|X(T)-\widehat{X}(T)\right|^{4r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq C\Delta_{N}^{2r\lambda},$

where $C>0$ is again a constant that does not depend on the partition.
Therefore

$\mathbb{E}\left[|F(S(T))|^{r}\left|\frac{1}{S(T)}-\frac{1}{\widehat{S}(T)}\right|^{r}\right]\leq
C\Delta^{r\lambda}.$ (5.7)

Next, observe that for any $s_{1},s_{2}>0$

$\displaystyle|F(s_{1})-F(s_{2})|$ $\displaystyle=\left|\int_{s_{1}\wedge
s_{2}}^{s_{1}\vee s_{2}}f(u)du\right|\leq
C(1+s_{1}^{q}+s_{2}^{q})|s_{1}-s_{2}|,$

so

$\displaystyle\mathbb{E}$
$\displaystyle\left[\frac{1}{\widehat{S}^{r}(T)}\left|F(S(T))-F(\widehat{S}(T))\right|^{r}\right]$
$\displaystyle\leq\left(\mathbb{E}\left[\frac{1}{\widehat{S}^{2r}(T)}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|F(S(T))-F(\widehat{S}(T))\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[\left|F(S(T))-F(\widehat{S}(T))\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[\left|F(S(T))-F(\widehat{S}(T))\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[|(1+S^{q}(T)+\widehat{S}^{q}(T))|^{2r}\left|S(T)-\widehat{S}(T)\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[|(1+S^{q}(T)+\widehat{S}^{q}(T))|^{4r}\right]\right)^{\frac{1}{4}}\left(\mathbb{E}\left[\left|S(T)-\widehat{S}(T)\right|^{4r}\right]\right)^{\frac{1}{4}}$
$\displaystyle\leq
C\left(\mathbb{E}\left[\left|S(T)-\widehat{S}(T)\right|^{4r}\right]\right)^{\frac{1}{4}},$

where we used Theorem 2.6 together with Proposition 5.4 to estimate
$\mathbb{E}\left[\widehat{S}^{-2r}(T)\right]$ together with
$\mathbb{E}\left[|(1+S^{q}(T)+\widehat{S}^{q}(T))|^{4r}\right]$.

Finally, using the same argument as in the proof of (5.7), we can obtain that

$\left(\mathbb{E}\left[\left|S(T)-\widehat{S}(T)\right|^{4r}\right]\right)^{\frac{1}{4}}\leq
C\Delta_{N}^{r\lambda},$

and thus

$\displaystyle\mathbb{E}$
$\displaystyle\left[\frac{1}{\widehat{S}^{r}(T)}\left|F(S(T))-F(\widehat{S}(T))\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$ (5.8)

Now (5.5) follows from (5.6), (5.7) and (5.8).

Step 3. Now we proceed to the claim of the theorem. Since

$\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}\right|^{2r}\right]<\infty$

due to the polynomial growth of the mapping $x\mapsto\frac{F(x)}{x}$, we can
deduce by Step 1, Step 2 and Theorem 4.2 that

$\displaystyle\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)}{Tu_{1,1}}\right)-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)}{Tu_{1,1}}\right)\right|^{r}\right]$
$\displaystyle\quad\leq
C\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right|^{r}\right]$
$\displaystyle\qquad+C\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)d\widetilde{W}_{1}(u)\right|^{r}\right]$
$\displaystyle\qquad+C\mathbb{E}\left[\left|\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\left(\frac{F(S(T))}{S(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right)\right|^{r}\right]$
$\displaystyle\quad\leq
C\Delta^{r\lambda}_{N}+C\left(\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}\right|^{2r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)d\widetilde{W}_{1}(u)\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\qquad+C\left(\mathbb{E}\left[\left|\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right|^{2r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|\frac{F(S(T))}{S(T)}-\frac{F(\widehat{S}(T))}{\widehat{S}(T)}\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\quad\leq C\Delta_{N}^{r\lambda},$

which ends the proof. ∎

Theorem 5.7 concerns representations (4.7) and (4.9). However, for the one-
dimensional case, we also have the representation (4.6) which can also be used
to approximate the expected payoff. We give the corresponding result below.

######  Theorem 5.8.

Let $S$ be given by (4.2), $X$ be the corresponding log-price process, $X=\log
S$, $f$: $\mathbb{R}\to\mathbb{R}$ satisfy Assumption C,
$F(s):=\int_{0}^{s}f(u)du$, $g(x):=f(e^{x})$, $G(x):=F(1)+\int_{0}^{x}g(v)dv$.
Fix a uniform partition $\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$,
$t_{n}=\frac{Tn}{N}$, with the mesh satisfying (5.1) and take $\widehat{Y}$,
$\widehat{X}$ and $\widehat{S}$ as described in Subsection 5.1. Finally, let
$\lambda<H$ with $H:=H_{1}$ being from Assumption A. Then

  * 1)

for any $r\geq 1$, there exists a constant $C>0$ such that

$\mathbb{E}\left[\left|\frac{1}{Tu_{1,1}}G(X(T))\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)-\frac{1}{Tu_{1,1}}G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda},$

  * 2)

there exists a constant $C>0$ such that

$\left|\mathbb{E}\left[f(S(T))\right]-\mathbb{E}\left[\frac{1}{Tu_{1,1}}G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right]\right|\leq
C\Delta_{N}^{\lambda}.$

###### Proof.

Once again, item 1 implies item 2 by Theorem 4.2 and we sketch the proof of
the former omitting the details. First, observe that for any $x_{1}$,
$x_{2}\in\mathbb{R}$

$\displaystyle|G(x_{1})-G(x_{2})|$ $\displaystyle=\left|\int_{x_{1}\wedge
x_{2}}^{x_{1}\vee x_{2}}g(v)dv\right|\leq\sup_{v\in[x_{1}\wedge
x_{2},x_{1}\vee x_{2}]}|g(v)||x_{1}-x_{2}|$
$\displaystyle=\sup_{v\in[x_{1}\wedge x_{2},x_{1}\vee
x_{2}]}|f(e^{v})||x_{1}-x_{2}|\leq c_{f}(1+e^{q(x_{1}\vee
x_{2})})|x_{1}-x_{2}|$ $\displaystyle\leq
c_{f}(1+e^{qx_{1}}+e^{qx_{2}})|x_{1}-x_{2}|$

and thus, using the same machinery as in the proof of Theorem 5.7, we can
write

$\displaystyle\mathbb{E}\left[\left|G(X(T))-G(\widehat{X}(T))\right|^{r}\right]$
$\displaystyle\leq
C\mathbb{E}\left[\left|(1+S^{q}(T)+\widehat{S}^{q}(T))|X(T)-\widehat{X}(T)|\right|^{r}\right]$
$\displaystyle\leq C\Delta_{N}^{r\lambda}.$

Due to existence of exponential moments of $X(T)$ as well as polynomial growth
of $f$, it is also clear that for any $r\geq 1$

$\mathbb{E}\left[\left|G(X(T))\right|^{r}\right]<\infty.$

Finally, as noted in Step 1 of the proof of Theorem 5.7,

$\mathbb{E}\left[\left|\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)d\widetilde{W}_{1}(u)\right|^{r}\right]\leq
C\Delta_{N}^{r\lambda}.$

Therefore

$\displaystyle
E\left[\left|G(X(T))\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)-G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right|^{r}\right]$
$\displaystyle\quad\leq
CE\left[\left|G(X(T))\left(\int_{0}^{T}\frac{1}{Y(u)}d\widetilde{W}_{1}(u)-\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right)\right|^{r}\right]$
$\displaystyle\qquad+CE\left[\left|\left(G(X(T))-G(\widehat{X}(T))\right)\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right|^{r}\right]$
$\displaystyle\quad\leq
C\left(\mathbb{E}\left[\left|G(X(T))\right|^{2r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|\int_{0}^{T}\left(\frac{1}{Y(u)}-\frac{1}{\widehat{Y}(u)}\right)d\widetilde{W}_{1}(u)\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\qquad+C\left(E\left[\left|\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right|^{2r}\right]\right)^{\frac{1}{2}}\left(\mathbb{E}\left[\left|G(X(T))-G(\widehat{X}(T))\right|^{2r}\right]\right)^{\frac{1}{2}}$
$\displaystyle\quad\leq C\Delta_{N}^{r\lambda}.$

∎

We conclude the Subsection with an approximation result in line with Theorem
5.7 involving a change of measure. We omit the proof since it is
straightforward and utilizes the same machinery as the one of Theorem 5.7 (one
just has to additionally use Theorem 5.6 in order to estimate expectations of
the type
$\mathbb{E}\left[|\mathcal{E}_{T}-\widehat{\mathcal{E}}_{T}|^{r}\right]$).

######  Theorem 5.9.

Let $S=(S_{1},...,S_{d})$ be given by (2.8), $X=(X_{1},...,X_{d})$ be the
corresponding log-price process, $X_{i}=\log S_{i}$, $f$:
$\mathbb{R}\to\mathbb{R}$ satisfy Assumption C, $F(s):=\int_{0}^{s}f(u)du$ and
$\alpha_{i}>0$, $i=1,...,d$, and $\mathcal{E}_{T}$ be the minimal martingale
measure defined by (4.10). Fix a uniform partition
$\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$, $t_{n}=\frac{Tn}{N}$, with the mesh
satisfying (5.1) and take $\widehat{Y}=(\widehat{Y}_{1},...,\widehat{Y}_{d})$,
$\widehat{X}=(\widehat{X}_{1},...,\widehat{X}_{d})$,
$\widehat{S}=(\widehat{S}_{1},...,\widehat{S}_{d})$ and
$\widehat{\mathcal{E}}_{T}$ as described in Subsection 5.1. Finally, let
$\lambda<\min_{i=1,...,d}H_{i}$ with $H_{i}$ being from Assumption A. Then

  * 1)

for any $r\geq 1$, there exists a constant $C>0$ such that

$\displaystyle\mathbb{E}$
$\displaystyle\Bigg{[}\Bigg{|}\mathcal{E}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)}{\alpha_{1}S_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{Y_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{Y_{k}(v)Y_{1}(v)}dv}{Tu_{1,1}}\right)$
$\displaystyle\quad-\widehat{\mathcal{E}}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}\widehat{S}_{i}(T)\right)}{\alpha_{1}\widehat{S}_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{\widehat{Y}_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{\widehat{Y}_{k}(v)\widehat{Y}_{1}(v)}dv}{Tu_{1,1}}\right)\Bigg{|}^{r}\Bigg{]}\leq
C\Delta_{N}^{r\lambda},$

  * 2)

there exists a constant $C>0$ such that

$\displaystyle\Bigg{|}\mathbb{E}$
$\displaystyle\left[\mathcal{E}_{T}f\left(\sum_{i=1}^{d}\alpha_{i}S_{i}(T)\right)\right]$
$\displaystyle\quad-\mathbb{E}\left[\widehat{\mathcal{E}}_{T}\frac{F\left(\sum_{i=1}^{d}\alpha_{i}\widehat{S}_{i}(T)\right)}{\alpha_{1}\widehat{S}_{1}(T)}\left(1+\frac{\int_{0}^{T}\frac{1}{\widehat{Y}_{1}(v)}d\widetilde{W}_{1}(v)-\sum_{k=1}^{d}\int_{0}^{T}\beta_{k}\frac{\widetilde{\mu}_{k}(v)}{\widehat{Y}_{k}(v)\widehat{Y}_{1}(v)}dv}{Tu_{1,1}}\right)\right]\Bigg{|}\leq
C\Delta_{N}^{\lambda}.$

###  5.3 Approximation via conditional Gaussianity for one-dimensional model

Assume now that $d=1$ and the price $S$ is given by (4.2). The core idea of
Subsection 5.2 was to plug in the simulated paths $\widehat{Y}$, $\widehat{S}$
and $\widehat{X}$ into representations (4.6) or (4.7). However, this requires
simulation of both volatility and the price which inevitably leads to higher
variance in Monte Carlo estimations. However, it is possible to avoid
simulation of the price and in this Subsection we describe the corresponding
approach.

First note that the random vector
$\left(\widehat{X}(T),\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right)^{T}$
is conditionally Gaussian with respect to the $\sigma$-field
$\widetilde{\mathcal{F}}_{T}^{2}:=\sigma\\{\widetilde{W}_{2}(t),~{}t\in[0,T]\\}$.
The next lemma presents the parameters of the joint conditional distribution
of the random vector
$\left(\widehat{X}(T),\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right)$.

######  Lemma 5.10.

The random vector
$\left(\widehat{X}(T),\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right)^{T}$
is conditionally Gaussian with respect to the $\sigma$-field
$\widetilde{\mathcal{F}}_{T}^{2}:=\sigma\\{\widetilde{W}_{2}(t),~{}t\in[0,T]\\}$
with conditional mean

$\begin{gathered}\left(\begin{matrix}X(0)+\frac{T}{N}\sum_{k=0}^{N-1}\left(\mu(t_{k})-\frac{1}{2}\widehat{Y}^{2}(t_{k})\right)+u_{1,2}\sum_{k=0}^{N-1}\widehat{Y}(t_{k})(\widetilde{W}_{2}(t_{k+1})-\widetilde{W}_{2}(t_{k}))\\\
0\end{matrix}\right)\\\ \quad=:\left(\begin{matrix}m(\widehat{Y})\\\
0\end{matrix}\right)\end{gathered}$ (5.9)

and conditional covariance matrix

$\displaystyle\widehat{\mathcal{C}}$
$\displaystyle=\left(\begin{matrix}u_{1,1}^{2}\frac{T}{N}\sum_{k=0}^{N-1}\widehat{Y}^{2}(t_{k})&u_{1,1}T\\\
u_{1,1}T&\frac{T}{N}\sum_{k=0}^{N-1}\widehat{Y}^{-2}(t_{k})\end{matrix}\right)$
$\displaystyle=:\left(\begin{matrix}u_{1,1}^{2}V^{2}_{1}&u_{1,1}T\\\
u_{1,1}T&V^{2}_{2}\end{matrix}\right),$

where $V^{2}_{1}:=\frac{T}{N}\sum_{k=0}^{N-1}\widehat{Y}^{2}(t_{k})$ and
$V^{2}_{2}:=\frac{T}{N}\sum_{k=0}^{N-1}\widehat{Y}^{-2}(t_{k})$. Moreover, if
additionally

$\det\widehat{\mathcal{C}}=u_{1,1}^{2}\left(V^{2}_{1}V^{2}_{2}-T^{2}\right)>0$

with probability 1, then the random vector
$\left(\widehat{X}(T),\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right)^{T}$
has a conditional density of the form

$\displaystyle\phi(x,y)=\frac{1}{2\pi\sqrt{\det\widehat{\mathcal{C}}}}\exp\left\\{-\frac{V^{2}_{2}(x-m(\widehat{Y}))^{2}+u_{1,1}^{2}V^{2}_{1}y^{2}-2Tu_{1,1}(x-m(\widehat{Y}))y}{2\det\widehat{\mathcal{C}}}\right\\}.$

###### Proof.

Recall that the random variable $\widehat{X}(T)$ can be represented in the
form

$\displaystyle\widehat{X}(T)$
$\displaystyle=X(0)+\frac{T}{N}\sum_{k=0}^{N-1}\left(\mu(t_{k})-\frac{\widehat{Y}^{2}(t_{k})}{2}\right)+u_{1,1}\sum_{k=0}^{N-1}\widehat{Y}(t_{k})\left(\widetilde{W}_{1}(t_{k+1})-\widetilde{W}_{1}(t_{k})\right)$
$\displaystyle\quad+u_{1,2}\sum_{k=0}^{N-1}\widehat{Y}(t_{k})\left(\widetilde{W}_{2}(t_{k+1})-\widetilde{W}_{2}(t_{k})\right).$

Now the explicit form of conditional mean, conditional covariance matrix and
conditional density (provided that the conditional covariance matrix is non-
degenerate) can be computed directly. ∎

######  Theorem 5.11.

Let $\det\widehat{\mathcal{C}}>0$ with probability 1, where
$\widehat{\mathcal{C}}$ is the conditional covariance matrix from Lemma 5.10.
Then

$\displaystyle\frac{1}{Tu_{1,1}}\mathbb{E}\left[G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right]$
(5.10)
$\displaystyle\quad=\frac{1}{\sqrt{2\pi}u_{1,1}^{2}}\int_{\mathbb{R}}G(x)\mathbb{E}\left[\frac{x-m(\widehat{Y})}{V_{1}^{3}}\exp\left\\{-\frac{(x-m(\widehat{Y}))^{2}}{2u_{1,1}^{2}V_{1}^{2}}\right\\}\right]dx.$

###### Proof.

Using Lemma 5.10, we can write

$\displaystyle\mathbb{E}\left[G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[G(\widehat{X}(T))\int_{0}^{T}\frac{1}{\widehat{Y}(u)}d\widetilde{W}_{1}(u)~{}\bigg{|}~{}\widetilde{\mathcal{F}}^{2}_{T}\right]\right]$
$\displaystyle=\mathbb{E}\left[\int_{\mathbb{R}^{2}}G(x)y\phi(x,y)dxdy\right]$
$\displaystyle=\int_{\mathbb{R}}G(x)\mathbb{E}\left[\int_{\mathbb{R}}y\phi(x,y)dy\right]dx.$

The integral $\int_{\mathbb{R}}y\phi(x,y)dy$ can be simplified. Namely, denote
for simplicity $\widetilde{x}:=x-m(\widehat{Y})$. Then

$\displaystyle\int_{\mathbb{R}}y\phi(x,y)dy$
$\displaystyle=\frac{1}{2\pi\sqrt{\det\widehat{\mathcal{C}}}}\int_{\mathbb{R}}y\exp\left\\{-\frac{V^{2}_{2}\widetilde{x}^{2}+u_{1,1}^{2}V^{2}_{1}y^{2}-2u_{1,1}T\widetilde{x}y}{2\det\widehat{\mathcal{C}}}\right\\}dy$
$\displaystyle=\frac{1}{2\pi\sqrt{\det\widehat{\mathcal{C}}}}\int_{\mathbb{R}}y\exp\left\\{-\frac{1}{2\det\widehat{\mathcal{C}}}\left(\left(u_{1,1}V_{1}y-\frac{T\widetilde{x}}{V_{1}}\right)^{2}+V^{2}_{2}\widetilde{x}^{2}-\frac{T^{2}\widetilde{x}^{2}}{V^{2}_{1}}\right)\right\\}dy$
$\displaystyle=\frac{1}{2\pi\sqrt{\det\widehat{\mathcal{C}}}}\exp\left\\{-\frac{(V_{1}^{2}V_{2}^{2}-T^{2})\widetilde{x}^{2}}{2V_{1}^{2}\det{\widehat{\mathcal{C}}}}\right\\}\int_{\mathbb{R}}y\exp\left\\{-\frac{\left(y-\frac{T\widetilde{x}}{u_{1,1}V_{1}^{2}}\right)^{2}}{2\frac{\det\widehat{C}}{u_{1,1}^{2}V_{1}^{2}}}\right\\}dy$
$\displaystyle=\frac{1}{2\pi\sqrt{\det\widehat{\mathcal{C}}}}\exp\left\\{-\frac{\widetilde{x}^{2}}{2u_{1,1}^{2}V_{1}^{2}}\right\\}\sqrt{2\pi}\frac{T\widetilde{x}\sqrt{\det\widehat{\mathcal{C}}}}{u_{1,1}^{2}V_{1}^{3}}$
$\displaystyle=\frac{T\widetilde{x}}{\sqrt{2\pi}u_{1,1}^{2}V_{1}^{3}}\exp\left\\{-\frac{\widetilde{x}^{2}}{2u_{1,1}^{2}V_{1}^{2}}\right\\}.$

This yields the result. ∎

Equation (5.10) from Theorem 5.11 essentially relies on the condition
$\det\widehat{\mathcal{C}}>0$ with probability 1. A simple sufficient
condition is provided by the following proposition.

######  Proposition 5.12.

Let $Z(t)=\int_{0}^{t}\mathcal{K}(t,s)d\widetilde{W}_{2}(s)$, $t\in[0,T]$, and
$\\{0=t_{0}<t_{1}<...<t_{N}=T\\}$ be a uniform partition of $[0,T]$ with the
mesh $\Delta_{N}:=\frac{T}{N}$ having the following property: $c\Delta_{N}<1$,
where $c$ is an upper bound for $\frac{\partial b}{\partial y}$ from
Assumption B(iv).

If the random vector

$(Z(t_{1})-Z(t_{0}),Z(t_{2})-Z(t_{1}),...,Z(t_{N})-Z(t_{N-1}))^{T}$

has density, then $\det\widehat{\mathcal{C}}>0$ with probability 1 and (5.10)
holds.

###### Proof.

Note that $\det\widehat{\mathcal{C}}\geq 0$ a.s., because
$\widehat{\mathcal{C}}$ is a conditional covariance matrix, i.e. we need to
check that $\det\widehat{\mathcal{C}}\neq 0$ with probability 1. Taking into
account that

$\det\widehat{\mathcal{C}}\neq
0\quad\Longleftrightarrow\quad\frac{1}{N^{2}}\sum_{k=0}^{N-1}\sum_{l=0}^{N-1}\widehat{Y}^{2}(t_{k})\widehat{Y}^{-2}(t_{l})\neq
1,$

it is sufficient to check that the random vector
$(\widehat{Y}(t_{1}),...,\widehat{Y}(t_{N}))^{T}$ has density with respect to
the Lebesgue measure. Denote

$b_{n}(y):=y-b(t_{n},y)\Delta_{N},\quad n=1,...,N,$

and observe that for any $n=1,...,N$ is a 1-to-1 mapping of the interval
$(\varphi(t_{n}),\psi(t_{n}))$ onto $\mathbb{R}$ with $b_{n}^{\prime}>0$. By
(5.2), for any $n=1,...,N$

$\widehat{Y}(t_{n})=b_{n}^{-1}\left(\widehat{Y}(t_{n-1})+Z(t_{n})-Z(t_{n-1})\right)$

and thus

$\displaystyle\left(\begin{matrix}\widehat{Y}(t_{1})\\\ \widehat{Y}(t_{2})\\\
\vdots\\\ \widehat{Y}(t_{N})\end{matrix}\right)$
$\displaystyle=\left(\begin{matrix}b_{1}^{-1}\left(Y(0)+Z(t_{1})-Z(t_{0})\right)\\\
b_{2}^{-1}\left(b_{1}^{-1}\left(Y(0)+Z(t_{1})-Z(t_{0})\right)+Z(t_{2})-Z(t_{1})\right)\\\
\vdots\\\
b_{N}^{-1}\left(b_{N-1}^{-1}\left(\cdots)\right)+Z(t_{N})-Z(t_{N-1})\right)\end{matrix}\right)$
(5.11)
$\displaystyle=:I(Z(t_{1})-Z(t_{0}),Z(t_{2})-Z(t_{1}),...,Z(t_{N})-Z(t_{N-1})).$

It is straightforward to check that function $I$:
$\mathbb{R}^{N}\to\prod_{n=1}^{N}\left(\varphi(t_{n}),\psi(t_{n})\right)$ from
(5.11) is a bijection and its Jacobian $\det J_{I}(\vec{z})$ is strictly
positive for any $\vec{z}\in\mathbb{R}^{N}$. This implies that the random
vector $(\widehat{Y}(t_{1}),...,\widehat{Y}(t_{N}))^{T}$ has density, i.e. is
absolutely continuous w.r.t. the Lebesgue measure, which ends the proof. ∎

##  6 Simulations

Let us illustrate the algorithms described in Section 5 with numerical
simulations. For simplicity, we assume that

  * •

$T=1$, $d=1$;

  * •

an explicit shape of the VSV model is

$\displaystyle S(t)$
$\displaystyle=1+\frac{3}{2}\int_{0}^{t}S(s)ds+\int_{0}^{t}Y(s)S(s)dW_{1}(s),$
$\displaystyle Y(t)$
$\displaystyle=1+\int_{0}^{t}\left(\frac{1}{(Y(s)-0.01)^{4}}-\frac{1}{(5-Y(s))^{4}}\right)ds+Z(t);$

  * •

$Z$ is a fractional Brownian motion with Hurst index $H\in(0,1)$ being
different in different simulations, i.e.

$\displaystyle
Z(t)=\int_{0}^{t}C_{H}\left(\left(\frac{t}{s}\right)^{H-\frac{1}{2}}(t-s)^{H-\frac{1}{2}}-\left(H-\frac{1}{2}\right)s^{\frac{1}{2}-H}\int_{s}^{t}u^{H-\frac{3}{2}}(u-s)^{H-\frac{1}{2}}du\right)dW_{2}(s),$
$\displaystyle
C_{H}=\left(\frac{2H\Gamma(3/2-H)}{\Gamma(H+1/2)\Gamma(2-2H)}\right)^{1/2};$

  * •

$W_{1}$ and $W_{2}$ (and hence $W_{1}$ and $Z$) are independent, which
corresponds to

$\operatorname{\mathcal{L}}=\left(\begin{matrix}1&0\\\
0&1\end{matrix}\right),$

$u_{1,1}=1$, $\beta_{1}=-1$ and $\widetilde{W}_{1}=W_{1}$ in the
representations from Section 4;

  * •

the instantaneous interest rate $\nu(t)\equiv\frac{1}{2}$;

  * •

the option payoff function is given by

$f(s)=\mathbbm{1}_{(1,\infty)}(s)+\mathbbm{1}_{(2,\infty)}(s)+\mathbbm{1}_{(4,\infty)}(s).$

First, we exploit the double discretization method and compute
$\mathbb{E}[f(S(T))]$ (under the physical measure) by simulating 10000
independent realisations of

$\frac{F\left(\widehat{S}(T)\right)}{\widehat{S}(T)}\left(1+\int_{0}^{T}\frac{1}{\widehat{Y}_{1}(u)}dW_{1}(u)\right)$
(6.1)

and then averaging them.

Afterwards, the same expectation is computed using the conditional Gaussianity
method described in Subsection 5.3 i.e. by averaging over 10000 independent
realisations of

$\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}G(x)\left(\frac{x-m(\widehat{Y})}{V_{1}^{3}}\right)\exp\left\\{-\frac{(x-m(\widehat{Y}))^{2}}{2V_{1}^{2}}\right\\}dx.$
(6.2)

For both methods, we considered uniform partitions of sizes $N=10$, $100$,
$1000$ and $10000$ and two choices of Hurst indices: $H=0.7$ (long memory
case) and $H=0.3$ (rough case). Table 1 contains information on means and
standard deviations of (6.1) and (6.2) for different partition sizes and Hurst
indices.

Table 1: Computation of $\mathbb{E}[f(S(T))]$, double discretization vs conditional Gaussianity Part. size, $N$ |  | $H=0.7$ |  | $H=0.3$
---|---|---|---|---
| Double discr. | Cond. Gauss. |  | Double discr. | Cond. Gauss.
| Mean | Sd | Mean | Sd |  | Mean | Sd | Mean | Sd
10 |  | 1.4154 | 2.0449 | 1.4424 | 0.3262 |  | 1.2941 | 1.9904 | 1.2998 | 0.3130
100 |  | 1.3849 | 2.0283 | 1.4117 | 0.3346 |  | 1.2030 | 1.9313 | 1.1991 | 0.2894
1000 |  | 1.4002 | 2.005 | 1.4028 | 0.3391 |  | 1.1444 | 1.8783 | 1.1821 | 0.2829
10000 |  | 1.4081 | 2.0339 | 1.4026 | 0.3333 |  | 1.1648 | 1.8883 | 1.1748 | 0.2802

As we can see, the method based on conditional Gaussianity produces
substantially smaller standard deviations and hence is superior to the
approach where both price and volatility are discritized. This happens because
there is no additional error from simulating $\widehat{S}(T)$ which results in
a far more stable behaviour of (6.2) (c.f. [10]).

In the second series of simulations, we estimate
$\mathbb{E}[\mathcal{E}_{T}f(S(T))]$, where

$\mathcal{E}_{T}:=\exp\left\\{-\int_{0}^{T}\frac{1}{Y(s)}dW_{1}(s)-\frac{1}{2}\int_{0}^{T}\frac{1}{Y^{2}(s)}ds\right\\}$

is the minimal martingale measure. We do that by averaging over 10000
realizations of

$\widehat{\mathcal{E}}_{T}\frac{F\left(\widehat{S}(T)\right)}{\widehat{S}(T)}\left(1+\int_{0}^{T}\frac{1}{\widehat{Y}(v)}dW_{1}(v)+\int_{0}^{T}\frac{1}{\widehat{Y}^{2}(v)}dv\right)$

and, just like in calculations under the physical measure, we perform
simulations for partition sizes $N=10$, $100$, $1000$ and $10000$ and Hurst
indices $H=0.7$ and $H=0.3$. The results are summarized in Table 2.

Table 2: Computation of $\mathbb{E}[\mathcal{E}_{T}f(S(T))]$ Part. size, $N$ |  | $H=0.7$ |  | $H=0.3$
---|---|---|---|---
| Mean | Sd |  | Mean | Sd
10 |  | 0.6732 | 0.6456 |  | 0.6308 | 0.6968
100 |  | 0.6716 | 0.6490 |  | 0.5752 | 0.7059
1000 |  | 0.6693 | 0.6511 |  | 0.5759 | 0.7101
10000 |  | 0.6687 | 0.6506124 |  | 0.5875 | 0.7239

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