∗ Johann Radon Institute for Computational and Applied Mathematics (RICAM),
Austrian Academy of Sciences, AT-4040 Linz, Austria † Institute for Financial
Mathematics and Applied Number Theory, University of Linz, AT-4040 Linz,
Austria

# Pricing of geometric Asian options in the Volterra-Heston model

Florian Aichinger∗ Johann Radon Institute for Computational and Applied
Mathematics (RICAM), Austrian Academy of Sciences, AT-4040 Linz, Austria
<EMAIL_ADDRESS>and  Sascha Desmettre† Institute for
Financial Mathematics and Applied Number Theory, University of Linz, AT-4040
Linz, Austria<EMAIL_ADDRESS>missingmissing

###### Abstract.

Geometric Asian options are a type of options where the payoff depends on the
geometric mean of the underlying asset over a certain period of time. This
paper is concerned with the pricing of such options for the class of Volterra-
Heston models, covering the rough Heston model. We are able to derive semi-
closed formulas for the prices of geometric Asian options with fixed and
floating strikes for this class of stochastic volatility models. These
formulas require the explicit calculation of the conditional joint Fourier
transform of the logarithm of the stock price and the logarithm of the
geometric mean of the stock price over time. Linking our problem to the theory
of affine Volterra processes, we find a representation of this Fourier
transform as a suitably constructed stochastic exponential, which depends on
the solution of a Riccati-Volterra equation. Finally we provide a numerical
study for our results in the rough Heston model.

Key words :

Volterra-Heston model, Rough volatility; Asian options; Fourier inversion
method; affine Volterra processes

MSC classification (2020) :

45D05, 60B15, 60L20, 91G20

##  1\. Introduction

The stochastic volatility model of Heston, cf. [21], is nowadays a standard
model for the pricing of financial derivatives. In contrast to the Black-
Scholes model with a constant volatility, in the Heston model, the volatility
follows itself a stochastic process, in particular one obtains the following
dynamics for the stock price and its volatility under a risk-neutral measure
$\mathbb{Q}$:

$dS_{t}=S_{t}\left(rdt+\sqrt{\nu_{t}}dB_{t}^{S}\right)\,,$ (1.1)
$d\nu_{t}=\kappa(\theta-\nu_{t})dt+\sigma\sqrt{\nu_{t}}dB_{t}^{\nu}\,,\vspace{1mm}$
(1.2)

where the process $\nu$ with $\nu_{0}\geq 0$ is a Cox-Ingersoll-Ross process.
The constants $\kappa,\theta,\sigma$ are assumed to be positive and satisfy
the Feller condition $2\kappa\theta\geq\sigma^{2}$, $(B^{\nu}_{t})$ and
$(B^{S}_{t})$ are $(\mathcal{F}_{t})$-adapted Brownian motions, where
$\mathcal{F}_{t}:=\sigma(B_{u}^{S},B_{u}^{\nu}:u\leq t)$ such that $\langle
B^{S},B^{\nu}\rangle_{t}=\rho t$, and $r$ denotes the risk-free interest rate.
This model fixes on the one hand the problem of a non-constant volatility
smile which the Black-Scholes model cannot produce and on the other hand is
more suitable to reproduce stylized facts of financial data such as the
leverage effect and the mean reversion of volatility. Most importantly, the
Heston model still gives a (semi-)explicit option pricing formula in terms of
the solution of a Riccati ordinary differential equation which is crucial for
a (fast) calibration to market data.

In recent times, since the observation was made that the paths of realized
volatilities are rougher than established volatility models would suggest, cf.
[15], there is a growing research interest in developing new models that
better fit empirical data. Therefore, the popular Heston model was adapted to
the rough volatility framework in [11] by using a fractional process with
Hurst index $H<\frac{1}{2}$ as driver of the volatility process. Thus, the
dynamics of the volatility process (1.2) become

$\nu_{t}=\nu_{0}+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\kappa(\theta-\nu_{s})ds+\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}\sigma\sqrt{\nu_{s}}dB^{\nu}_{s}\,,$
(1.3)

where $\Gamma$ denotes the Gamma-function. The parameter
$\alpha\in(\frac{1}{2},1)$ governs the roughness of the paths and is related
to the Hurst parameter of the fractional Brownian motion. In [11], the
characteristic function of the log-price in rough Heston models is computed
using a link between fractional volatility models and its microstructural
foundation, the so called Hawkes processes. Similar to the classical case, the
characteristic function can be calculated (semi-)explicitly in terms of the
solution of a fractional Riccati equation.
A more general class of volatility models covering this rough Heston model is
obtained by modeling the volatility process as a stochastic Volterra equation
of convolution type, cf. [1, 23, 2]. The Volterra-Heston model under a risk-
neutral measure $\mathbb{Q}$ is then given as

$dS_{t}=rS_{t}dt+\sqrt{\nu_{t}}S_{t}dB_{t}^{S}\,,$ (1.4)
$\displaystyle\nu_{t}=\nu_{0}+\int_{0}^{t}\mathcal{K}(t-s)(\kappa(\theta-\nu_{s}))ds+\int_{0}^{t}\mathcal{K}(t-s)\sigma\sqrt{\nu_{s}}dB_{s}^{\nu}\,,$
(1.5)

with integral kernel $\mathcal{K}\in L^{2}([0,T],\mathbb{R})$. The constants
$\kappa,\theta,\sigma$ are again assumed to be positive, $(B^{\nu}_{t})$ and
$(B^{S}_{t})$ are Brownian motions such that $\langle
B^{S},B^{\nu}\rangle_{t}=\rho t$, and $r$ denotes the risk-free interest rate.
We consider a filtration $\\{\mathcal{F}_{t},t\geq 0\\}$ where
$\mathcal{F}_{t}:=\sigma(B_{u}^{S},B_{u}^{\nu}:u\leq t)$. Existence of a
unique in law $\mathbb{R}^{+}$-valued continuous weak solution of (1.5) for
any initial condition $\nu_{0}\in\mathbb{R}^{+}$ is ensured by [1, Theorem
6.1] under suitable assumptions on the kernel $\mathcal{K}$. In particular
this holds for the fractional integral kernels
$\mathcal{K}(t-s)=(t-s)^{\alpha-1}/\Gamma(\alpha)$, $\alpha\in(\frac{1}{2},1)$
in the rough Heston model (1.3) (cf. [1, Theorem 7.1]).

Concerning option pricing in general Volterra models only few explicit results
exist. The seminal paper [6] shows how the rough fractional stochastic
volatility (RFSV) model as in [15] can be used to price claims on both the
underlying and integrated variance, however no explicit call price formula is
given yet. This is done in [20] for the rough Heston model in the sense of
[11], but not for general Volterra kernels. The next step of complexity are
then exotic options, cf. the well-known monograph [29]. Using forward variance
methods along the lines of [23], [22] develops efficient Monte Carlo methods
and asymptotic approximations for computing option prices and hedge ratios in
models where the log-volatility follows a (modulated) Gaussian Volterra
processes. Using kernel-based approximation methods, the price of American
options in the general Volterra Heston model is given in [8]. [14] deals with
the pricing and hedging of index options in a rough volatility model, using
large deviation methods. To conclude, [26] and [25] deal with the numerical
and analytical pricing of Basket and Barrier options based on Fourier methods
in rough (versions of) Heston models, however once again not in general
Volterra models.
The pricing of Asian options, i.e. options where the payoff depends on the
mean of the underlying asset over a certain period of time, has yet not been
studied for the rough Heston model or the more general Volterra-Heston model.
For the classical Heston model, [24] derive semi-analytical pricing formulas
for Asian options with fixed and floating strike. First they establish
formulas for the option prices in terms of the conditional joint Fourier
transform of the logarithm of the stock price at maturity and the logarithm of
the geometric mean of the stock price over a certain time period. This
approach is based on methods such as numeraire changes and the Fourier
inversion formula, which go back to e.g. [18] and [17]. It is important to
note that the proofs do not depend on the underlying volatility model. The
joint Fourier transform is then represented explicitly as an exponential
affine function of the volatility. Finding this representation in the
classical model involves Itô calculus and is heavily relying on the
Markovianity of the volatility process (1.2).
In this paper we present pricing formulas for fixed- and floating strike Asian
options in the Volterra-Heston model (1.1), (1.5). The pricing formulas in
terms of the joint Fourier transform as derived in [24] for the classical
Heston model, are still valid in the Volterra-Heston model. The challenging
part is thus to explicitly calculate this Fourier transform, when the
volatility of the stock is modeled as a Volterra square-root process given via
(1.5). In contrast to the classical case, the process (1.5) clearly does not
have to be Markovian, and thus different techniques have to be applied in our
case. Linking our problem to the theory of affine Volterra processes developed
in [1], we show that in the Volterra-Heston model, the joint Fourier transform
can be represented as a suitably constructed stochastic exponential in terms
of the so called forward variance process and the solution of a Volterra-
Riccati equation. Combining this explicit representation of the Fourier
transform with a general, model-independent pricing approach based on the
Fourier inversion formula, we are finally able to give semi-analytical pricing
formulas for fixed- and floating-strike geometric Asian put and call options
in the Volterra- Heston model. These formulas are a generalization of the
results derived in [24] for the classical Heston model and we also show that
their results can be recovered as a special case of the general Volterra
approach, even if the respective representations of the Fourier transform
appear to be quite different at first sight. We also provide an extensive
numerical study of Asian option prices in the rough Heston model, which is of
particular practical interest. Implementing the pricing formulas, we
numerically calculate prices for the different types of geometric Asian
options for varying strikes, maturities and roughness levels and compare our
results to those of [24] in the classical case. We observe, that the effect of
the roughness on the price is highly dependent on the maturity of the option.

The paper is structured as follows. Section 1 serves as an introduction to the
class of Volterra-Heston models. In Section 2 we present a different
parametrization of the Volterra-Heston model via a family of Markovian
processes and demonstrate, how this can be used for pricing European options.
In Section 3, we introduce geometric Asian options and a general pricing
approach linked to the conditional joint Fourier transform of the logarithm of
the stock price and the logarithm of the geometric mean of the stock price
over time. In Section 4, we derive an explicit representation of this
conditional joint Fourier transform for the Volterra-Heston model. Section 5
provides semi-closed pricing formulas for fixed and floating strike Asian call
and put options in terms of the Fourier transform calculated in the previous
section. In Section 6, we compare our results to those obtained in the
classical Heston model. Finally, Section 7 features a numerical study of our
results for the rough Heston model.

##  2\. European option pricing in the Volterra Heston model

To start off, we give a short overview on how pricing formulas for European
options can be obtained in the Volterra-Heston model. Based on Fourier
inversion methods, it is possible to obtain semi-closed formulas for European
options in terms of the characteristic function of the stock price $S_{T}$. In
the classical Heston model [21], determining this characteristic function
involves Itô calculus and hence heavily relies on the Markovianity of the
processes $S$ and $\nu$ given via (1.1) and (1.2). The process $\nu$ defined
via (1.5) does not have to be Markovian, in particular, this is clearly the
case for fractional integral kernels in the rough Heston model (1.3). Thus, in
order to apply Itô calculus, one considers an equivalent parametrization of
the Volterra Heston model in terms of the family of forward variance processes

$\displaystyle\xi_{t}(T):=\mathbb{E}^{\mathbb{Q}}(\nu_{T}|\mathcal{F}_{t})\,,$
(2.1)

indexed by $T$ (cf. [16], [23]). It turns out that for fixed $T$, the process
$\xi_{t}(T)$ is again a Markov process. As detailed in [23], one can find the
dynamics of the forward variance process as follows.

######  Lemma 2.1 (Prop. 3.2, [23]).

Let $\xi_{t}(T)=\mathbb{E}^{\mathbb{Q}}(\nu_{T}|\mathcal{F}_{t})$ and
$R_{\kappa}$ be the resolvent of $\kappa\mathcal{K}$, i.e.

$R_{\kappa}\ast(\kappa\mathcal{K})=(\kappa\mathcal{K})\ast
R_{\kappa}=\kappa\mathcal{K}-R_{\kappa}.$

Then the dynamics of $\xi_{t}(T)$ are given by

$d\xi_{t}(T)=\frac{1}{\kappa}R_{\kappa}(T-t)\sigma\sqrt{\nu_{t}}dB_{t}^{\nu}$

with

$\xi_{0}(T)=\nu_{0}(1-\int_{0}^{T}R_{\kappa}(s)ds)+\theta\int_{0}^{T}R_{\kappa}(s)ds.$

The following table gives some examples for kernels and their corresponding
resolvents. Here
$E_{\alpha,\beta}(z)=\sum\limits_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\alpha
n+\beta)}$ denotes the Mittag-Leffler function.

Type | $K(t)$ | $R(t)$
---|---|---
Constant | $c$ | $ce^{-ct}$
Fractional | $c\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ | $ct^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})$
Exponential | $ce^{-\lambda t}$ | $ce^{-\lambda t}e^{-ct}$
Gamma | $ce^{-\lambda t}\frac{t^{\alpha-1}}{\Gamma(\alpha)}$ | $ce^{-\lambda t}t^{\alpha-1}E_{\alpha,\alpha}(-ct^{\alpha})$
Table 1. Different integral kernels with their corresponding resolvents.

The forward variance form of the Volterra Heston model in log-price notation
under a risk-neutral measure $\mathbb{Q}$ is then given by

$d\operatorname{log}S_{t}=(r-\frac{1}{2}\nu_{t})dt+\sqrt{\nu_{t}}dB_{t}^{S},$
(2.2)
$d\xi_{t}(T)=\frac{1}{\kappa}R_{\kappa}(T-t)\sigma\sqrt{\nu_{t}}dW_{t},\quad\xi_{0}(T)=\nu_{0}(1-\int_{0}^{T}R_{\kappa}(s)ds)+\theta\int_{0}^{T}R_{\kappa}(s)ds.\vspace{3mm}$
(2.3)

Note that the variance $\nu_{t}$ can be recovered from the forward variance
process via $\nu_{t}=\xi_{t}(t)$. The next theorem then ensures that the
characteristic function of the log-spot price in the Volterra Heston model can
be determined as a suitably constructed exponential in terms of the forward
variance and the solution of a Riccati-Volterra equation:

######  Theorem 2.2 (Thm. 3.4, [23]).

Let $\phi$ be the solution of the Riccati-Volterra equation

$\phi=\mathcal{K}\ast(Q(iu,\phi)-\kappa\phi),\text{ with}$
$Q(iu,\phi)=-\frac{1}{2}(u^{2}+iu)+\sigma\rho
iu\phi+\frac{\sigma^{2}}{2}\phi^{2}.$

Then the auxiliary process $(M_{\tau})_{0\leq\tau\leq T}$ defined as

$\displaystyle
M_{\tau}:=\operatorname{exp}\left(iu(\operatorname{log}S_{\tau}+r(T-\tau))+\int_{\tau}^{T}\xi_{\tau}(s)Q(iu,\phi(T-s))ds\right)$
(2.4)

is a true martingale and the characteristic function
$\psi_{\operatorname{log}S_{T}|\mathcal{F}_{t}}$ is given by

$\psi_{\operatorname{log}S_{T}|\mathcal{F}_{t}}(u)=\mathbb{E}^{\mathbb{Q}}(\operatorname{exp}(iu\operatorname{log}S_{T})|\mathcal{F}_{t})=\mathbb{E}^{\mathbb{Q}}(M_{T}|\mathcal{F}_{t})=M_{t}.$

Applying the general model independent call price formula that connects the
distribution of the stock price with its Fourier transform (cf. e.g. [5]) and
using the above explicit representation of the characteristic function yields
the following semi-analytic formula for the European call price in the
Volterra Heston model.

######  Theorem 2.3.

The price $C(0)$ of a European call option with strike price $K$ and maturity
$T$ is given by

$C(0)=S_{0}\Pi_{1}-e^{-rT}K\Pi_{2},$ (2.5)

where

$\Pi_{1}=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left[\frac{e^{-iu\operatorname{ln}(K)}\psi_{\operatorname{log}S_{T}}(u-i)}{iu\psi_{\operatorname{log}S_{T}}(-i)}\right]du,$

$\Pi_{2}=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left[\frac{e^{-iu\operatorname{ln}(K)}\psi_{\operatorname{log}S_{T}}(u)}{iu}\right]du.$

and $\psi_{\operatorname{log}S_{T}}$ is the characteristic function of the
logarithmic stock price at time $T$. In the general Volterra Heston model
(1.5), the characteristic function $\psi_{\operatorname{log}S_{T}}$ is given
by

$\psi_{\operatorname{log}S_{T}}(u)=\operatorname{exp}\left(iu(\operatorname{log}S_{0}+rT)+\int_{0}^{T}Q(iu,\phi(T-s))\xi_{0}(s)ds\right),$
(2.6)

with

$\xi_{0}(s)=\nu_{0}(1-\int_{0}^{s}R_{\kappa}(y)dy)+\theta\int_{0}^{s}R_{\kappa}(y)dy.$

_Proof:_ The proof for the model independent pricing formula (2.5) is standard
and relies on a combination of changes of measure and an application of the
Fourier inversion formula. For the sake of completeness, a detailed derivation
is given in the Appendix. The explicit representation (2.6) of the
characteristic function follows immediately from Theorem 2.2 for the case
$t=0$.

∎

The result of Theorem 2.3 is a generalization of the result presented in [20]
for the rough Heston model to general kernels $\mathcal{K}$, in case of a
constant interest rate. In [20], the interest rate can also be a stochastic
process.

##  3\. Geometric Asian Options

Asian options are a type of options where the payoff depends on the mean of
the underlying asset over a certain period of time. We distinguish between two
different types of such options, namely arithmetic and geometric Asian option.
Arithmetic Asian options are options where the payoff depends on the
arithmetic mean

$A_{[0,T]}=\frac{1}{T}\int_{0}^{T}\operatorname{S}_{u}du,$

of the underlying over time. For such options, no explicit closed form
solution is known in the Heston model (cf. [24]), and this is already the case
for the Black-Scholes model. There is a vast literature on the numerical
valuation of such arithmetic options in the Heston model (e.g. [27], [13],
[7]), and therefore we focus on finding explicit formulas for Asian options on
the geometric mean.
Denote by $G_{[0,T]}$ the geometric mean of the the stock price $S_{t}$ over
the time period $[0,T]$, i.e.

$G_{[0,T]}=\operatorname{exp}\left(\frac{1}{T}\int_{0}^{T}\operatorname{log}S_{u}du\right).$
(3.1)

Then there are four types of geometric Asian option contracts: fixed strike
Asian call options, fixed strike Asian put options, floating strike Asian call
options and floating strike Asian put options. The payoffs of fixed strike
geometric Asian call and put options with strike $K$ and maturity $T$ are
given respectively by

$\operatorname{max}\\{G_{[0,T]}-K,0\\},\quad\operatorname{max}\\{K-G_{[0,T]},0\\}.$
(3.2)

The payoffs of floating strike geometric Asian call and put options with
maturity $T$ are given respectively by

$\operatorname{max}\\{G_{[0,T]}-S_{T},0\\},\quad\operatorname{max}\\{S_{T}-G_{[0,T]},0\\}.\vspace{3mm}$
(3.3)

Starting from a risk-neutral pricing approach, it can be shown (cf. [24]) that
prices of Asian options can be expressed in terms of the conditional joint
Fourier transform $\psi$ of the logarithm of the stock price at maturity and
the logarithm of the geometric mean of the stock price over a certain time
period

$\psi_{t}(s,w)=\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\operatorname{log}G_{t,T}+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right],$
(3.4)

with

$G_{t,T}=\operatorname{exp}\left(\frac{1}{T}\int_{t}^{T}\operatorname{log}S_{u}du\right).\vspace{3mm}$
(3.5)

Similarly to the derivation of the formula (2.5) for European options, this
pricing approach is again based on changes of measure and the Fourier
inversion formula, which go back to [18] and [17] and can be applied
independently of the underlying volatility model. The main difficulty is thus
to find an explicit representation of the Fourier transform (3.4) in the
respective market model. In the classical Heston model, [24] derive such an
explicit representation as an exponential in terms of the solution of a
Riccati equation. However, their proof heavily relies on the Markovianity of
the classical Heston volatility process.
We want to derive pricing formulas for geometric Asian options in the
Volterra-Heston model (1.4), (1.5). Thus we have to find an explicit
representation for (3.4) when the volatility is modeled as a (non-Markovian)
Volterra square-root process. This will be done in the next Section. In
Section 5 we then present semi-closed pricing formulas for fixed- and floating
strike geometric Asian call and put options.

##  4\. The joint conditional Fourier transform

In this Section we show that in the Volterra-Heston model (1.4), (1.5), the
conditional joint Fourier transform (3.4) can be represented as a suitably
constructed exponential in terms of the forward variance and the solution of a
Riccati-Volterra equation. We start with the following Lemma, which links our
problem to the theory of affine Volterra processes developed in [1].

######  Lemma 4.1.

Define a stochastic process $(M_{\tau})_{0\leq\tau\leq T}$ as

$M_{\tau}:=\operatorname{exp}(Y_{\tau})$ (4.1)

with

$\displaystyle
Y_{\tau}=\mathbb{E}^{\mathbb{Q}}\left[\frac{s}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}|\mathcal{F}_{\tau}\right]$
(4.2)
$\displaystyle+\frac{1}{2}\int_{\tau}^{T}\left(\phi_{1}^{2}(T-u)+2\rho\sigma\phi_{1}(T-u)\phi_{2}(T-u)+\sigma^{2}\phi^{2}(T-u)\right)\xi_{\tau}(u)du,$

where $\xi_{t}(s)=\mathbb{E}^{\mathbb{Q}}[\nu_{s}|\mathcal{F}_{t}]$ is the
forward variance process, $\phi_{1}\in L^{2}([0,T],\mathbb{C})$ is an affine
linear function given by

$\phi_{1}(t)=s\frac{t}{T}+w,$ (4.3)

and $\phi_{2}\in L^{2}([0,T],\mathbb{C})$ is the solution of the Volterra-
Riccati equation

$\phi_{2}(t)=\int_{0}^{t}\mathcal{K}(t-u)\left[\frac{1}{2}\left(\phi_{1}^{2}(u)-\phi_{1}(u)\right)-\kappa\phi_{2}(u)+\frac{1}{2}\left(\sigma^{2}\phi_{2}^{2}(u)+2\rho\sigma\phi_{1}(u)\phi_{2}(u)\right)\right]du.$
(4.4)

Then for all
$s,w\in\mathcal{D}:=\\{(s,w)\in\mathbb{C}^{2}:\operatorname{Re}(s)\geq
0,\operatorname{Re}(w)\geq
0,0\leq\operatorname{Re}(s)+\operatorname{Re}(w)\leq 1\\}$, the process
$(M_{\tau})_{0\leq\tau\leq T}$ is a true martingale.

_Proof:_ Our aim is to write the process $(Y_{\tau})_{0\leq\tau\leq T}$ in
terms of a two dimensional affine Volterra process in order to apply results
for this class of affine stochastic processes studied in [1]. To this end, we
observe that equations (1.4) and (1.5) of the log-spotprice and its volatility
can be written as

$\displaystyle\begin{pmatrix}\operatorname{log}S\\\ \nu\end{pmatrix}_{t}$
$\displaystyle=\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}_{0}+\int_{0}^{t}\begin{pmatrix}1&0\\\
0&\mathcal{K}(t-u)\end{pmatrix}\left[\begin{pmatrix}0\\\
\kappa\theta\end{pmatrix}+\begin{pmatrix}0&-\frac{1}{2}\\\
0&-\kappa\end{pmatrix}\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}_{u}\right]du$ (4.5)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad+\int_{0}^{t}\begin{pmatrix}1&0\\\
0&\mathcal{K}(t-u)\end{pmatrix}\sqrt{\nu_{u}}\begin{pmatrix}\sqrt{1-\rho^{2}}&\rho\\\
0&\sigma\end{pmatrix}\begin{pmatrix}d\tilde{W}_{u}\\\
dW_{u}^{\nu}\end{pmatrix}$

Defining the matrix valued integral kernel $\bm{\mathcal{K}}$ as
$\bm{\mathcal{K}}:=\begin{pmatrix}1&0\\\ 0&K\end{pmatrix}$, we see that the
two-dimensional process $(\operatorname{log}S,\nu)^{\top}$ solves the Volterra
equation

$(\operatorname{log}S,\nu)_{t}^{\top}=(\operatorname{log}S,\nu)_{0}^{\top}+\int_{0}^{t}\bm{\mathcal{K}}(t-u)b((\operatorname{log}S,\nu)_{u}^{\top})du+\int_{0}^{t}\bm{\mathcal{K}}(t-u)\sigma((\operatorname{log}S,\nu)_{u}^{\top})d(\tilde{W},W^{\nu})_{u}^{\top},$

where

$a((S,\nu)^{\top})=\sigma((S,\nu)^{\top})\sigma^{\top}((S,\nu)^{\top})=A^{0}+\operatorname{log}SA^{1}+\nu
A^{2}\text{ with
}A^{0}=\bm{0},A^{1}=\bm{0},A^{2}=\begin{pmatrix}1&\rho\sigma\\\
\rho\sigma&\sigma^{2}\end{pmatrix},$

and

$b((S,\nu)^{\top})=b^{0}+\operatorname{log}Sb^{1}+\nu b^{2}\text{ with
}b^{0}=\begin{pmatrix}0\\\ \kappa\theta\end{pmatrix},b^{1}=\begin{pmatrix}0\\\
0\end{pmatrix},b^{2}=\begin{pmatrix}-\frac{1}{2}\\\ \kappa\end{pmatrix}.$

Thus $(\operatorname{log}S,\nu)^{\top}$ is an affine process in the sense of
[1]. Equations (4.3) and (4.4) can be written as a two-dimensional Volterra-
Riccati equation

$(\phi_{1},\phi_{2})=(w,0){\bm{\mathcal{K}}}+\left(\left(\frac{s}{T},0\right)+({\phi_{1}},\phi_{2})B+\frac{1}{2}A(({\phi_{1}},\phi_{2}))\right)\ast{\bm{\mathcal{K}}}$
(4.6)

where

$B=(b^{1},b^{2})\text{ and
}A((\phi_{1},\phi_{2}))=((\phi_{1},\phi_{2})A^{1}(\phi_{1},\phi_{2})^{\top},(\phi_{1},\phi_{2})A^{2}(\phi_{1},\phi_{2})^{\top})$

with solution $(\phi_{1},\phi_{2})\in L^{2}([0,T],(\mathbb{C}^{2})^{*})$.
Finally the process $Y_{\tau}$ can be expressed in terms of the two
dimensional affine Volterra process $(\operatorname{log}S,\nu)^{\top}$ and the
solution $(\phi_{1},\phi_{2})$ of the Volterra-Riccati equation (4.6) as

$\displaystyle Y_{\tau}$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[(\frac{s}{T},0)\ast\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}(T)+(w,0)\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}_{T}\Big{|}\mathcal{F}_{\tau}\right]$ (4.7)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\tau}^{T}(\phi_{1},\phi_{2})_{t-u}\left[\mathbb{E}^{\mathbb{Q}}(\nu_{u}|\mathcal{F}_{\tau})\begin{pmatrix}1&\rho\sigma\\\
\rho\sigma&\sigma^{2}\end{pmatrix}\right]\begin{pmatrix}\phi_{1}\\\
\phi_{2}\end{pmatrix}_{t-u}du.$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[(\frac{s}{T},0)\ast\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}(T)+(w,0)\begin{pmatrix}\operatorname{log}S\\\
\nu\end{pmatrix}_{T}\Big{|}\mathcal{F}_{\tau}\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\frac{1}{2}\int_{\tau}^{T}(\phi_{1},\phi_{2})_{t-u}a(\mathbb{E}^{\mathbb{Q}}[(\operatorname{log}S,\nu)^{\top}_{u}|\mathcal{F}_{\tau}])(\phi_{1},\phi_{2})_{t-u}^{\top}du,$

where

$a(\mathbb{E}^{\mathbb{Q}}[(\operatorname{log}S,\nu)^{\top}_{u}|\mathcal{F}_{\tau}])=A^{0}+\mathbb{E}^{\mathbb{Q}}[\operatorname{log}S_{u}|\mathcal{F}_{\tau}]A^{1}+\mathbb{E}^{\mathbb{Q}}[\nu_{u}|\mathcal{F}_{\tau}]A^{2}.$

Thus Theorem 4.3 of [1] for $X=(\operatorname{log}S,\nu)$, $u=(w,0)$ and
$f=(s/T,0)$ can be applied to prove that
$M_{\tau}=\operatorname{exp}(Y_{\tau})$ is a local martingale. It remains to
show that $M$ is indeed a true martingale. We note that for
$s,w\in\mathcal{D}$ it holds that for $\phi_{1}$ defined via (4.3), we have
$\phi_{1}\in[0,1]$. Since $\operatorname{Re}(u_{2})=0$ and
$\operatorname{Re}(f_{2})=0$, Theorem 7.1 of [1] yields that $\phi_{2}$
defined via (4.4) has a unique solution on $[0,T]$ which satisfies
$\operatorname{Re}(\phi_{2})\leq 0$ and $M_{\tau}$ is a true martingale.

∎

The following Lemma ensures the martingale property for a type of processes
that appear in the proof of Theorem 4.3.

######  Lemma 4.2.

Let $(P_{t})_{t\in[0,T]}$ be a stochastic process given by

$P_{t}:=\int_{0}^{t}f(u)\sqrt{\nu_{u}}dW_{u},$

where $f\in C^{\infty}([0,T])$ and $W$ is a Brownian motion. Then $P$ is a
true martingale on $[0,T]$.

_Proof:_ Clearly $P$ is a local martingale, since it is an integral with
reference to a Brownian motion. In order to prove that $P$ is a true
martingale, we show that $\mathbb{E}^{\mathbb{Q}}(\langle
P,P\rangle_{t})<\infty$ for all $t\in[0,T]$. This is the case, since for every
$t\in[0,T]$

$\mathbb{E}^{\mathbb{Q}}[\langle
P,P\rangle_{t}]=\mathbb{E}^{\mathbb{Q}}[\int_{0}^{t}f^{2}(u)\nu_{u}du]\leq
C\int_{0}^{t}\mathbb{E}^{\mathbb{Q}}[\nu_{u}]du<\infty.$

∎

Given these auxiliary results, we can now proceed to prove that (3.4) has an
explicit representation. It will become evident in the next Section, that the
condition $s,w\in\mathcal{D}$ on the arguments of $\psi$ does not constitute a
restriction for deriving the option pricing formulas.

######  Theorem 4.3.

Let $(s,w)\in\mathcal{D}$. Then for $t\in[0,T]$, the conditional joint Fourier
transform $\psi_{t}$ defined as

$\psi_{t}(s,w)=\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\operatorname{log}G_{t,T}+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right]$

is given by

$\displaystyle\psi_{t}(s,w)=\operatorname{exp}\bigg{(}s\left(\frac{T-t}{T}\operatorname{log}S_{t}+r\frac{(T-t)^{2}}{2{T}}\right)+w\left(\operatorname{log}S_{t}+r(T-t)\right)$
(4.8)
$\displaystyle+\int_{t}^{T}\left(\frac{1}{2}\left(\phi_{1}^{2}(T-u)-\phi_{1}(T-u)\right)+\sigma\rho\phi_{1}(T-u)\phi_{2}(T-u)+\frac{1}{2}\sigma^{2}\phi_{2}^{2}(T-u)\right)\xi_{t}(u)du\bigg{)},$

where $\xi_{t}(s)$ is the forward variance process, $\phi_{1}\in
L^{2}([0,T],\mathbb{C})$ is an affine linear function given by (4.3) and
$\phi_{2}\in L^{2}([0,T],\mathbb{C})$ is the solution of the Volterra-Riccati
equation (4.4).

_Proof:_ Starting from the definition of $\psi_{t}$ and $G_{t,T}$ in (3.4) and
(3.5), we have

$\displaystyle\psi_{t}(s,w)$
$\displaystyle:=\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\operatorname{log}G_{t,T}+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right]$
$\displaystyle=\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\frac{1}{T}\int_{t}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right]$
$\displaystyle=\operatorname{exp}(-s\frac{1}{T}\int_{0}^{t}\operatorname{log}S_{u}du)\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\frac{1}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right].$

Since for the process $M$ defined in (4.1) it holds that

$M_{T}=\operatorname{exp}\left(s\frac{1}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}\right)$

and $M$ is a martingale by Lemma 4.1, we have

$\mathbb{E}^{\mathbb{Q}}\left[\operatorname{exp}\left(s\frac{1}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}\right)|\mathcal{F}_{t}\right]=\mathbb{E}^{\mathbb{Q}}[M_{T}|\mathcal{F}_{t}]=M_{t},$

and thus

$\displaystyle\psi_{t}(w,s)$
$\displaystyle=\operatorname{exp}(-s\frac{1}{T}\int_{0}^{t}\operatorname{log}S_{u}du)\operatorname{exp}(\mathbb{E}^{\mathbb{Q}}(w\operatorname{log}S_{T}+\frac{s}{T}\int_{0}^{T}\operatorname{log}S_{u}du|\mathcal{F}_{\tau}))$
(4.9)
$\displaystyle\times\operatorname{exp}(\frac{1}{2}\int_{\tau}^{T}\left(\phi_{1}^{2}(T-u)+2\rho\sigma\phi_{1}(T-u)\phi_{2}(T-u)+\sigma^{2}\phi^{2}(T-u)\right)\xi_{\tau}(u)du).$

We simplify the term
$\mathbb{E}^{\mathbb{Q}}(\frac{s}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}|\mathcal{F}_{\tau})$.
From the dynamics of $\operatorname{log}S$ in (1.4), we can write
$\operatorname{log}S_{T}$ as

$\operatorname{log}S_{T}=\operatorname{log}S_{t}+(T-t)r-\frac{1}{2}\int_{t}^{T}\nu_{u}du+\int_{t}^{T}\sqrt{\nu_{u}}dW_{u}^{S}.$

Thus for the conditional expectation we get

$\displaystyle\mathbb{E}^{\mathbb{Q}}(\operatorname{log}S_{T}|\mathcal{F}_{t})$
$\displaystyle=\operatorname{log}S_{t}+(T-t)r-\frac{1}{2}\mathbb{E}^{\mathbb{Q}}(\int_{t}^{T}\nu_{u}du|\mathcal{F}_{t})+\mathbb{E}(\int_{t}^{T}\sqrt{\nu_{u}}dW_{u}^{S}|\mathcal{F}_{t})$
$\displaystyle=\operatorname{log}S_{t}+(T-t)r-\frac{1}{2}\int_{t}^{T}\xi_{t}(u)du,$

where we have used the fact that the process
$P_{x}^{1}:=\int_{0}^{x}\sqrt{\nu_{u}}dW_{u}^{S}$ is a martingale by Lemma
4.2.

Similarly for the integrated log-spotprice, we have

$\displaystyle\int_{0}^{T}\operatorname{log}S_{z}dz=\int_{0}^{t}\operatorname{log}S_{z}dz+\int_{t}^{T}(\operatorname{log}S_{t}+(z-t)r-\frac{1}{2}\int_{t}^{z}\nu_{u}du+\int_{t}^{z}\sqrt{\nu_{u}}dW_{u}^{S})dz$
$\displaystyle=\int_{0}^{t}\operatorname{log}S_{z}dz+(T-t)\operatorname{log}S_{t}-\frac{1}{2}(T-t)^{2}r-\frac{1}{2}\int_{t}^{T}(T-u)\nu_{u}du+\int_{t}^{T}(T-u)\sqrt{\nu_{u}}dW_{u}^{S},$

where we have used the stochastic Fubini theorem from [28]. Note that the
stochastic Fubini theorem is applicable since

$\int_{t}^{T}(\int_{t}^{z}|\sqrt{\nu_{u}}|^{2}d\langle
W^{S},W^{S}\rangle_{u})^{1/2}dz\leq T(\int_{0}^{T}\nu_{u}du)^{1/2}$

is finite almost surely. Since
$P_{x}^{2}:=\int_{0}^{x}(T-u)\sqrt{\nu_{u}}dW_{u}^{S}$ is again a martingale
by Lemma 4.2, we have

$\displaystyle\mathbb{E}^{\mathbb{Q}}(\int_{0}^{T}\operatorname{log}S_{z}dz|\mathcal{F}_{t})$
$\displaystyle=\int_{0}^{t}\operatorname{log}S_{u}du+(T-t)\operatorname{log}S_{t}+\frac{1}{2}(T-t)^{2}r-\frac{1}{2}\int_{t}^{T}(T-u)\xi_{t}(u)du$

This gives us

$\displaystyle\mathbb{E}^{\mathbb{Q}}(\frac{s}{T}\int_{0}^{T}\operatorname{log}S_{u}du+w\operatorname{log}S_{T}|\mathcal{F}_{\tau})=\frac{s}{T}\int_{0}^{t}\operatorname{log}S_{u}du+s\left(\frac{T-t}{T}\operatorname{log}S_{t}+r\frac{(T-t)^{2}}{2T}\right)$
(4.10)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad+w\left(\operatorname{log}S_{t}+r(T-t)\right)-\frac{1}{2}\int_{t}^{T}(s\frac{T-u}{T}+w)\xi_{t}(u)du.$

Note that $s\frac{T-u}{T}+w=\phi_{1}(T-u)$. Taking the exponential and
inserting (4.10) back into (4.9) thus yields

$\displaystyle\psi_{t}(s,w)=\operatorname{exp}\bigg{(}s\left(\frac{T-t}{T}\operatorname{log}S_{t}+r\frac{(T-t)^{2}}{2T}\right)+w\left(\operatorname{log}S_{t}+r(T-t)\right)$
$\displaystyle+\int_{t}^{T}\left(\frac{1}{2}\left(\phi_{1}^{2}(T-u)-\phi_{1}(T-u)\right)+\sigma\rho\phi_{1}(T-u)\phi_{2}(T-u)+\frac{1}{2}\sigma^{2}\phi_{2}^{2}(T-u)\right)\xi_{t}(u)du\bigg{)},$

which completes the proof

∎

##  5\. Pricing Geometric Asian Options in the Volterra-Heston Model

In this Section we present semi-closed formulas for the prices of geometric
Asian call and put options with fixed or floating strike in the Volterra-
Heston model. Combining the general pricing approach that links the price of
geometric Asian options to the joint Fourier transform (3.4) with the explicit
representation this joint Fourier transform (4.8) derived in the previous
section we obtain the following results.

###  5.1. Fixed-strike Asian options

For a fixed-strike geometric Asian call option, i.e. an option with payoff
$\operatorname{max}\\{G_{[0,T]}-K,0\\}$, the price of the option at time $t$
is given as follows.

######  Theorem 5.1.

The price $C_{[0,T]}(t)$ of a fixed-strike geometric Asian call option with
strike $K$ and maturity $T$ and $t\in[0,T]$ is given by

$\displaystyle C_{[0,T]}(t)$
$\displaystyle=e^{-r(T-t)+\frac{1}{T}\int_{0}^{t}\operatorname{log}S_{u}du}$
(5.1)
$\displaystyle\qquad\times\left[\frac{\psi_{t}(1,0)-K_{t,T}}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{t}(1+iz,0)-K_{t,T}\psi_{t}(iz,0)\big{)}\frac{e^{-iz\operatorname{log}K_{t,T}}}{iz}\right)dz\right],$

where

$K_{t,T}:=Ke^{-\frac{1}{T}\int_{0}^{t}\operatorname{log}S_{u}du}$ (5.2)

and the Fourier transform $\psi_{t}$ is given by (4.8).

_Proof:_ The formula (5.1) which was originally obtained in [24, Thm. 4.1] for
the classical Heston model, holds for every model for which the joint Fourier
transform (3.4) is known. It remains to prove that we can apply Theorem 4.3 to
obtain the exponential representation (4.8) of $\psi_{t}$ in the Volterra-
Heston model. This is the case since all arguments $(s,w)$ of $\psi$ that
appear in (5.1) fulfill

$s,w\in\mathcal{D}=\\{(s,w)\in\mathbb{C}^{2}:\operatorname{Re}(s)\geq
0,\operatorname{Re}(w)\geq
0,0\leq\operatorname{Re}(s)+\operatorname{Re}(w)\leq 1\\}.$

∎

For a fixed-strike geometric Asian put option, i.e. an option with payoff
$\operatorname{max}\\{K-G_{[0,T]},0\\}$, the price of the option at time $t$
is given as follows.

######  Corollary 5.2.

The price $P_{[0,T]}(t)$ of a fixed-strike geometric Asian put option with
strike $K$ and maturity $T$ and $t\in[0,T]$ is given by

$\displaystyle P_{[0,T]}(t)$
$\displaystyle=e^{-r(T-t)+\frac{1}{T}\int_{0}^{t}\operatorname{log}S_{u}du}$
(5.3)
$\displaystyle\qquad\times\left[\frac{K_{t,T}-\psi_{t}(1,0)}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{t}(1+iz,0)-K_{t,T}\psi_{t}(iz,0)\big{)}\frac{e^{-iz\operatorname{log}K_{t,T}}}{iz}\right)dz\right],$

where $K_{t,T}$ is defined in (5.2) and the Fourier transform $\psi_{t}$ is
given by (4.8).

_Proof:_ The formula (5.3) follows from (5.1) and the put-call parity (cf.
[24, Cor. 4.2]). Theorem 4.3 is again applicable since all arguments $(s,w)$
of $\psi$ that appear in (5.3) fulfill $s,w\in\mathcal{D}$ and hence (4.8)
holds. ∎

###  5.2. Floating-strike Asian options

The price of a floating strike geometric Asian call option, i.e. an option
with payoff $\operatorname{max}\\{G_{[0,T]}-S_{T},0\\}$, at time $t$ is given
as follows.

######  Theorem 5.3.

The price $\tilde{C}_{[0,T]}(t)$ of a floating-strike geometric Asian call
option with maturity $T$ and $t\in[0,T]$ is given by

$\displaystyle\tilde{C}_{[0,T]}(t)$
$\displaystyle=e^{-r(T-t)}\bigg{[}\frac{1}{2}\Big{(}e^{r(T-t)}S_{t}-e^{(1/T)\int_{0}^{t}\operatorname{log}S_{u}du}\psi_{t}(1,0)\Big{)}$
(5.4)
$\displaystyle+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}e^{(1/T)\int_{0}^{t}\operatorname{log}S_{u}du}\psi_{t}(1+iz,-iz)-\psi_{t}(iz,1-iz)\big{)}\frac{e^{i(z/T)\int_{0}^{t}\operatorname{log}S_{u}du}}{iz}\right)dz\bigg{]},$

where $K_{t,T}$ is defined in (5.2) and the Fourier transform $\psi_{t}$ is
given by (4.8).

_Proof:_ The formula (5.4) is derived in [24, Cor. 4.4]). Theorem 4.3 is
applicable since all arguments $(s,w)$ of $\psi$ that appear in (5.4) fulfill
$s,w\in\mathcal{D}$ and hence (4.8) holds. ∎

Again by the put–call parity the price of a floating-strike geometric Asian
put option, i.e. an option with payoff
$\operatorname{max}\\{S_{T}-G_{[0,T]},0\\}$, at time $t$ is given as follows.

######  Corollary 5.4.

The price $\tilde{P}_{[0,T]}(t)$ of a floating-strike geometric Asian put
option with maturity $T$ and $t\in[0,T]$ is given by

$\displaystyle\tilde{P}_{[0,T]}(t)$
$\displaystyle=e^{-r(T-t)}\bigg{[}\frac{1}{2}\Big{(}e^{(1/T)\int_{0}^{t}\operatorname{log}S_{u}du}\psi_{t}(1,0)-e^{r(T-t)}S_{t}\Big{)}$
(5.5)
$\displaystyle+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}e^{(1/T)\int_{0}^{t}\operatorname{log}S_{u}du}\psi_{t}(1+iz,-iz)-\psi_{t}(iz,1-iz)\big{)}\frac{e^{i(z/T)\int_{0}^{t}\operatorname{log}S_{u}du}}{iz}\right)dz\bigg{]},$

where $K_{t,T}$ is defined in (5.2) and the Fourier transform $\psi_{t}$ is
given by (4.8).

_Proof:_ The formula (5.5) is derived in [24, Thm. 4.3]). Theorem 4.3 is
applicable since all arguments $(s,w)$ of $\psi$ that appear in (5.5) fulfill
$s,w\in\mathcal{D}$ and hence (4.8) holds. ∎

##  6\. Equality of results

The representation (4.8) of the Fourier transform (3.4) in the Volterra Heston
model which we derive in Section 4 is of a different structure than the
representation derived in the classical Heston model in [24]. In this section,
we explicitly show that in the case of $K=1$, both representations are
identical using calculus of convolutions and resolvents and theory of Riccati
equations. In the following, we will denote the representation (4.8) for the
special case $\mathcal{K}=1$ and $t=0$ as $\psi_{0}^{V}(s,w)$ and the
classical representation of [24] as $\psi_{0}^{C}(s,w)$. We start with a
detailed overview of the two representations.

###  6.1. Volterra approach

For $K=1$, the approach developed in Section 4 leads to

$\displaystyle\psi_{0}^{V}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+\frac{rT}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}\times$
(6.1)
$\displaystyle\operatorname{exp}\bigg{(}\int_{0}^{T}\left(\frac{1}{2}\left(f^{2}(T-u)-f(T-u)\right)+\sigma\rho
f(T-u)\phi(T-u)+\frac{1}{2}\sigma^{2}\phi^{2}(T-u)\right)\xi_{0}(u)du\bigg{)},$

where

$\xi_{0}(u)=\mathbb{E}^{\mathbb{Q}}[\nu_{u}|\mathcal{F}_{0}]=\nu_{0}(1-\int_{0}^{u}R_{\kappa}(y)dy)+\theta\int_{0}^{u}R_{\kappa}(y)dy$

is the forward variance for the classical Heston volatility ($K=1$) at $t=0$,
$f\in L^{2}([0,T],\mathbb{C})$ is an affine linear function given by

$f(u)=s\frac{u}{T}+w,$

and $\phi\in L^{2}([0,T],\mathbb{C})$ is the solution of the Riccati equation

$\phi(u)=\int_{0}^{u}\frac{1}{2}\left(f^{2}(y)-f(y)\right)-\kappa\phi(y)+\frac{1}{2}\left(\sigma^{2}\phi^{2}(y)+2\rho\sigma
f(y)\phi(y)\right)dy.$ (6.2)

###  6.2. Classical approach

The classical approach of [24] leads to

$\psi_{0}^{C}(s,w)=\operatorname{exp}(z_{0})\times\operatorname{exp}\left(\nu_{0}C(T)+\kappa\theta\int_{0}^{T}C(u)du\right),$
(6.3)

where

$z_{0}=s\left(\operatorname{log}S_{0}+r\frac{T}{2}-\frac{\kappa\theta\rho
T}{2\sigma}-\frac{\rho}{\sigma}\nu_{0}\right)+w\left(\operatorname{log}S_{0}+rT-\frac{\kappa\theta\rho
T}{\sigma}-\frac{\rho}{\sigma}\nu_{0}\right)$

$z_{1}=\frac{s^{2}(1-\rho^{2})}{2T^{2}},\quad
z_{2}=\frac{s(2\rho\kappa-\sigma)}{2\sigma T}+\frac{sw(1-\rho^{2})}{T},\quad
z_{3}=\frac{s\rho}{\sigma
T}+\frac{w(2\rho\kappa-\sigma)}{2\sigma}+\frac{w^{2}(1-\rho^{2})}{2},$

and $C\in L^{2}([0,T],\mathbb{C})$ is the solution of the Riccati equation

$C(u)=\frac{\rho}{\sigma}w+\int_{0}^{u}z_{1}y^{2}+z_{2}y+z_{3}-\kappa
C(y)+\frac{\sigma^{2}}{2}C^{2}(y)dy.$ (6.4)

###  6.3. Equality of the two approaches

We now show that both representations are indeed equal, i.e.
$\psi_{0}^{V}(s,w)=\psi_{0}^{C}(s,w)$. As a first step, we start by
simplifying the convolution appearing in (6.1) to show that

$\psi_{0}^{V}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}\times\operatorname{exp}\left(\nu_{0}\phi(T)+\kappa\theta\int_{0}^{T}\phi(u)du\right).$

To this end, we define

$Q(y):=\frac{1}{2}\left(f^{2}(y)-f(y)\right)+\sigma\rho
f(y)\phi(y)+\frac{1}{2}\sigma^{2}\phi^{2}(y)$

Thus we can write

$\psi_{0}^{V}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}\times\operatorname{exp}\left(\int_{0}^{T}Q(T-u)\xi_{0}(u)du\right).$

The result now follows since

$\displaystyle\int_{0}^{T}Q(T-u)\xi_{0}(u)du$
$\displaystyle=\int_{0}^{T}Q(T-u)\left(\nu_{0}-\nu_{0}\int_{0}^{u}R_{\kappa}(z)dz+\theta\int_{0}^{u}R_{\kappa}(z)dz\right)du$
$\displaystyle=\nu_{0}\int_{0}^{T}Q(T-u)du-\nu_{0}Q\ast\bm{1}\ast
R_{\kappa}(T)+\theta Q\ast\bm{1}\ast R_{\kappa}(T)$
$\displaystyle=\nu_{0}\int_{0}^{T}Q(u)du-\nu_{0}\bm{1}\ast R_{\kappa}\ast
Q(T)+\theta\bm{1}\ast R_{\kappa}\ast Q(T)$
$\displaystyle=\nu_{0}\left(\phi(T)+\kappa\int_{0}^{T}\phi(u)du\right)-\nu_{0}\kappa\int_{0}^{T}\phi(u)du+\kappa\theta\int_{0}^{T}\phi(u)du$
$\displaystyle=\nu_{0}\phi(T)+\kappa\theta\int_{0}^{T}\phi(u)du.$

Here we have used the commutativity of convolution for the third and the
identity $\phi=\frac{1}{\kappa}R_{\kappa}\ast Q$ for the fourth equality. A
short proof for this identity is provided in the Appendix. In the next step,
we show that the solutions of the Riccati equations (6.4) and (6.2) are
related to each other. More precisely, we show that if $C(u)$ is the solution
of (6.4), then $\phi(u):=C(u)-\frac{\rho}{\sigma}f(u)$ is the solution of
(6.2). This is true since

$\displaystyle
C(u)-\frac{\rho}{\sigma}f(u)=\int_{0}^{u}\frac{1}{2}(f^{2}(y)-f(y))-\kappa(C(y)-\frac{\rho}{\sigma}f(y))$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\rho\sigma
f(y)(C(y)-\frac{\rho}{\sigma}f(y))+\frac{\sigma^{2}}{2}(C(y)-\frac{\rho}{\sigma}f(y))^{2}dy$
$\displaystyle\iff
C(u)-\frac{\rho}{\sigma}f(u)=\int_{0}^{u}\frac{1}{2}(f^{2}(y)-f(y))+\frac{\rho\kappa}{\sigma}f(y)-\frac{\rho^{2}}{2}f^{2}(y)-\kappa
C(y)+\frac{\sigma^{2}}{2}C^{2}(y)dy$ $\displaystyle\iff
C(u)-\frac{\rho}{\sigma}f(u)=\int_{0}^{u}z_{1}y^{2}+z_{2}y+z_{3}-\frac{\rho
s}{\sigma T}-\kappa C(y)+\frac{\sigma^{2}}{2}C^{2}(y)dy$ $\displaystyle\iff
C(u)-\frac{\rho}{\sigma}(\frac{su}{T}+w)=-\frac{\rho}{\sigma}\frac{su}{T}+\int_{0}^{u}z_{1}y^{2}+z_{2}y+z_{3}-\kappa
C(y)+\frac{\sigma^{2}}{2}C^{2}(y)dy$ $\displaystyle\iff
C(u)=\frac{\rho}{\sigma}w+\int_{0}^{u}z_{1}y^{2}+z_{2}y+z_{3}-\kappa
C(y)+\frac{\sigma^{2}}{2}C^{2}(y)dy.$

Finally we combine the results of the two previous steps to obtain

$\displaystyle\psi_{0}^{V}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}\times\operatorname{exp}\left(\nu_{0}\phi(T)+\kappa\theta\int_{0}^{T}\phi(u)du\right)$
$\displaystyle=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}$
$\displaystyle\qquad\qquad\qquad\times\operatorname{exp}\left(\nu_{0}(C(T)-\frac{\rho}{\sigma}(s+w))+\kappa\theta\int_{0}^{T}C(u)-\frac{\rho}{\sigma}(\frac{s}{T}u+w)du\right)$
$\displaystyle=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}$
$\displaystyle\qquad\qquad\qquad\times\operatorname{exp}\left(-s\frac{\rho}{\sigma}\nu_{0}-w\frac{\rho}{\sigma}\nu_{0}-s\frac{\kappa\theta\rho
T}{2\sigma}-w\frac{\kappa\theta\rho
T}{\sigma}\right)\times\operatorname{exp}\left(\nu_{0}C(T)+\kappa\theta\int_{0}^{T}C(u)du\right)$
$\displaystyle=\operatorname{exp}\left(s\left(\operatorname{log}S_{0}+r\frac{T}{2}-\frac{\kappa\theta\rho
T}{2\sigma}-\frac{\rho}{\sigma}\nu_{0}\right)+w\left(\operatorname{log}S_{0}+rT-\frac{\kappa\theta\rho
T}{\sigma}-\frac{\rho}{\sigma}\nu_{0}\right)\right)$
$\displaystyle\qquad\qquad\qquad\times\operatorname{exp}\left(\nu_{0}C(T)+\kappa\theta\int_{0}^{T}C(u)du\right)$
$\displaystyle=\operatorname{exp}(z_{0})\times\operatorname{exp}\left(\nu_{0}C(T)+\kappa\theta\int_{0}^{T}C(u)du\right)=\psi_{0}^{C}(s,w).$

This is the desired equality. ∎

##  7\. Numerical results

In this section, we present numerical results for fixed- and floating-strike
geometric Asian call and put options for the class of fractional Heston
models, i.e. we consider the model (1.4), (1.5) for the fractional integral
kernel $K(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}$. For $\alpha=1$, we recover
the classical Heston model and hence in this special case, our results
correspond to those derived in [24]. Note that the parameter $\alpha$, which
governs the roughness of the paths is related to the Hurst parameter $H$ of
the fractional Brownian motion via $H=\alpha-\frac{1}{2}$. Consequently, for
$\alpha\in(\frac{1}{2},1)$ paths of the volatility process are rougher, and
for $\alpha\in(1,\frac{3}{2})$ they are smoother than in the classical case.
As already lined out, since the observation was made, that volatility is rough
(cf. [15]), the class of rough Heston models (i.e.
$\alpha\in(\frac{1}{2},1)$), has become of particular theoretical and
practical interest, and thus we will put particular emphasis on the impact of
the roughness parameter on the option price.

###  7.1. Useful representation of the option prices

To start off, we summarize the pricing formulas derived in the previous
Sections for the case $t=0$.

######  Corollary 7.1.

For $t=0$, the price of a fixed-strike geometric Asian call option is given by

$\displaystyle C_{[0,T]}(0)$
$\displaystyle=e^{-rT}\left[\frac{\psi_{0}(1,0)-K}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{0}(1+iz,0)-K\psi_{0}(iz,0)\big{)}\frac{e^{-iz\operatorname{log}K}}{iz}\right)dz\right],$
(7.1)

the price of a fixed-strike geometric Asian put option is given by

$\displaystyle P_{[0,T]}(0)$
$\displaystyle=e^{-rT}\left[\frac{K-\psi_{0}(1,0)}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{0}(1+iz,0)-K\psi_{0}(iz,0)\big{)}\frac{e^{-iz\operatorname{log}K}}{iz}\right)dz\right],$
(7.2)

the price of a floating-strike geometric Asian call option is given by

$\displaystyle\tilde{C}_{[0,T]}(0)$
$\displaystyle=e^{-rT}\bigg{[}\frac{e^{rT}S_{0}-\psi_{0}(1,0)}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{0}(1+iz,-iz)-\psi_{0}(iz,1-iz)\big{)}\frac{1}{iz}\right)dz\bigg{]},$
(7.3)

the price of a floating-strike geometric Asian put option is given by

$\displaystyle\tilde{P}_{[0,T]}(0)$
$\displaystyle=e^{-rT}\bigg{[}\frac{\psi_{0}(1,0)-e^{rT}S_{0}}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left(\big{(}\psi_{0}(1+iz,-iz)-\psi_{0}(iz,1-iz)\big{)}\frac{1}{iz}\right)dz\bigg{]},$
(7.4)

and the conditional joint Fourier transform $\psi_{0}$ can be written as

$\displaystyle\psi_{0}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+\frac{rT}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)+\int_{0}^{T}\nu_{0}Q(u)+(\kappa\theta-\nu_{0})\phi(u)du\bigg{)},$
(7.5)

where $\phi\in L^{2}([0,T],\mathbb{C})$ is the solution of the Volterra
Riccati equation

$\phi(u)=\int_{0}^{u}K(u-y)\left(\frac{1}{2}\left(f^{2}(y)-f(y)\right)-\kappa\phi(y)+\frac{1}{2}\left(\sigma^{2}\phi^{2}(y)+2\rho\sigma
f(y)\phi(y)\right)\right)dy,$ (7.6)

$f\in L^{2}([0,T],\mathbb{C})$ is an affine linear function given by

$f(u)=s\frac{u}{T}+w,$

and $Q$ is defined as

$Q(u)=Q(\phi,f;u):=\frac{1}{2}\left(f^{2}(u)-f(u)\right)+\frac{1}{2}\left(\sigma^{2}\phi^{2}(u)+2\rho\sigma
f(u)\phi(u)\right).\vspace{3mm}$

_Proof:_ The formulas (7.1), (7.2), (7.3) and (7.4) are immediately obtained
by setting $t=0$ in equations (5.1), (5.3) (5.4), and (5.5). For the
representation of the Fourier transform $\psi_{0}$, we first observe that the
formula (4.8) evaluated at $t=0$ yields

$\psi_{0}(s,w)=\operatorname{exp}\bigg{(}s\left(\operatorname{log}S_{0}+r\frac{T}{2}\right)+w\left(\operatorname{log}S_{0}+rT\right)\bigg{)}\times\operatorname{exp}\left(\int_{0}^{T}Q(T-u)\xi_{0}(u)du\right).$

Substituting
$\xi_{0}(u)=\nu_{0}(1-\int_{0}^{u}R_{\kappa}(y)dy)+\theta\int_{0}^{u}R_{\kappa}(y)dy$
and applying calculus of convolutions and resolvents to the last term yields

$\displaystyle\int_{0}^{T}Q(T-u)\xi_{0}(u)du$
$\displaystyle=\int_{0}^{T}Q(T-u)\left(\nu_{0}-\nu_{0}\int_{0}^{u}R_{\kappa}(z)dz+\theta\int_{0}^{u}R_{\kappa}(z)dz\right)du$
$\displaystyle=\nu_{0}\int_{0}^{T}Q(T-u)du-\nu_{0}Q\ast\bm{1}\ast
R_{\kappa}(T)+\theta Q\ast\bm{1}\ast R_{\kappa}(T)$
$\displaystyle=\nu_{0}\int_{0}^{T}Q(u)du-\nu_{0}\bm{1}\ast R_{\kappa}\ast
Q(T)+\theta\bm{1}\ast R_{\kappa}\ast Q(T)$
$\displaystyle=\nu_{0}\int_{0}^{T}Q(u)du-\nu_{0}\kappa\int_{0}^{T}\phi(u)du+\kappa\theta\int_{0}^{T}\phi(u)du.\vspace{3mm}$

Here we have used the commutativity of convolution for the third and the
identity $\phi=\frac{1}{\kappa}R_{\kappa}\ast Q$ (cf. Lemma 9.1 in the
Appendix) for the fourth equality. This now gives us (7.5). ∎

Note that in contrast to (4.8), the resolvent does no longer appear in the
representation (7.5) of $\psi_{0}$ anymore. This fact is particularly useful
in the case of fractional integral kernels appearing in the rough Heston
model, since we do not have to approximate the Mittag-Leffler function
appearing in the resolvent (cf. Table 1). Therefore, this allows us to improve
the accuracy as well as the computational efficiency of the implementation of
our formulas.

###  7.2. Concrete results

We now present concrete numerical results for fixed- and floating-strike
geometric Asian call and put option prices at time $0$. In order to have a
benchmark parameter set, analogous to [24], we use the calibrated market
parameters

$\kappa=1.15\,,\,\theta=0.348\,,\,\sigma=0.39\mbox{ and }\rho=-0.64\,,$

which correspond to the daily averages of the respective implied parameters
reported in Table IV of [4] as well as their choices

$r=0.05\,,\,S_{0}=100\mbox{ and }\nu_{0}=0.09\,.$

For the numerical implementation of the analytic formulas, two approximations
are carried out. In order to calculate the Fourier transform (7.5), one first
has to numerically solve the Volterra Riccati equation (7.6) for
$K(t)=\frac{t^{\alpha-1}}{\Gamma(\alpha)}$, which is done using the fractional
Adams method, developed and studied in [9], [10]. Secondly, the infinite
integral appearing in (7.1), (7.3) must be truncated. Here we take an upper
limit of $10^{2}$. This is in line with the observation made in [24], that for
higher limits the numerical results remain unchanged.

In Table 2, we present prices for fixed-strike geometric Asian call options
for different maturities, strike prices and roughness levels of the
volatility. We observe the following characteristics: The option price
decreases with increasing strike. Moreover, the smaller the maturity, the
higher is the impact of the strike on the option price. Larger maturities lead
to higher option prices across all different strikes. All these observations
are in line with the results of [24] for the classical Heston model. Note in
particular that for $\alpha=1$ in Table 2, we recover their results presented
in [24, Table 5].

In addition to the classical Heston model, we now face a further parameter
which has to be taken into account, namely the roughness parameter $\alpha$.
We observe that the effect of the roughness depends on the maturity of the
option. For smaller maturities the price of the option increases with the
roughness level, whereas for large maturities, there is a smile effect, i.e.
option prices increase if the volatility is either smoother or rougher than in
the classical case.

For the fixed-strike put option prices in Table 3, the option price rises with
increasing strike and maturity. As it is also the case for the corresponding
call options, only for very large maturities prices start to decline slightly.
The effect of roughness again changes depending on the maturity of the option.
Here, the price of the option again increases with the roughness for smaller
maturities. However, for larger maturities, we observe the reverse effect,
i.e. option prices decrease with increasing roughness levels. A possible
explanation for this changed behavior is that rough processes are more
volatile than smooth processes in the short term but less volatile in the long
term. This is in line with the fact that time-dependent effects of the
roughness are well-established in the context of portfolio optimization
theory, where the preference between buying a rough or a smooth stock changes
with the time-horizon (cf. [2], [19], [3]). However, in addition, the prices
listed in the corresponding tables are related to each other via the put-call
parity

$C_{[0,T]}(0)-P_{[0,T]}(0)=e^{-rT}(\psi_{0}(1,0)-K).$

The value of the Fourier transform $\psi_{0}$ also depends on the roughness
parameter $\alpha$ and hence effects of roughness on put and call option
prices cannot be expected to be observed on a one-for-one basis. Please note
that the put prices have at the same time been checked for correctness with
the help of this put-call parity.

The results for floating-strike geometric Asian call and put options given in
Tables 4 and 5 exhibit a similar behaviour. Larger maturities again lead to
higher option prices for both put and call options. As for fixed-strike
options, the effect of the roughness depends on the maturity of the option.
For maturities $T<1$, rougher volatility paths lead to higher option prices,
while the reverse effect can be observed for $T>1$ for both put and call
options. For very large maturities, we observe a reversed smile effect for the
floating strike put option, i.e. option prices decline if the volatility is
smoother or rougher than the classical Heston volatility. Note that in
contrast to the fixed-strike case, the stock price at maturity $S_{T}$, which
appears in the payoff, is dependent on the roughness parameter as well. Here,
the prices listed in the corresponding tables are related to each other via
the put-call parity

$\tilde{C}_{[0,T]}(0)-\tilde{P}_{[0,T]}(0)=S_{0}-e^{-rT}\psi_{0}(1,0)\,,$

and have once more been checked for correctness using this parity.

| Option value
---|---
T (years) | K | $\alpha=1.40$ | $\alpha=1.25$ | $\alpha=1.00$ | $\alpha=0.75$ | $\alpha=0.60$
0.2 | 90 | 10.5515 | 10.5770 | 10.6571 | 10.8151 | 10.9546
| 95 | 6.4252 | 6.4670 | 6.5871 | 6.7931 | 6.9556
| 100 | 3.3060 | 3.3433 | 3.4478 | 3.6182 | 3.7484
| 105 | 1.3902 | 1.4073 | 1.4552 | 1.5338 | 1.5976
| 110 | 0.4672 | 0.4684 | 0.4724 | 0.4841 | 0.5016
0.4 | 90 | 11.4604 | 11.5317 | 11.7112 | 11.9773 | 12.1707
| 95 | 7.7825 | 7.8730 | 8.0894 | 8.3894 | 8.5983
| 100 | 4.8671 | 4.9551 | 5.1616 | 5.4415 | 5.6345
| 105 | 2.7774 | 2.8442 | 3.0018 | 3.2178 | 3.3695
| 110 | 1.4363 | 1.4759 | 1.5715 | 1.7089 | 1.8103
0.5 | 90 | 11.9228 | 12.0151 | 12.2329 | 12.5314 | 12.7380
| 95 | 8.3898 | 8.5021 | 8.7553 | 9.0844 | 9.3050
| 100 | 5.5389 | 5.6502 | 5.8971 | 6.2119 | 6.4211
| 105 | 3.4107 | 3.5027 | 3.7072 | 3.9697 | 4.1460
| 110 | 1.9504 | 2.0143 | 2.1589 | 2.3501 | 2.4823
1 | 90 | 14.0831 | 14.2511 | 14.5779 | 14.9390 | 15.1588
| 95 | 11.0148 | 11.2017 | 11.5551 | 11.9353 | 12.1637
| 100 | 8.3978 | 8.5893 | 8.9457 | 9.3239 | 9.5499
| 105 | 6.2352 | 6.4179 | 6.7559 | 7.1130 | 7.3263
| 110 | 4.5056 | 4.6693 | 4.9723 | 5.2932 | 5.4859
1.5 | 90 | 15.9436 | 16.1478 | 16.5030 | 16.8533 | 17.0544
| 95 | 13.1767 | 13.3931 | 13.7625 | 14.1213 | 14.3261
| 100 | 10.7472 | 10.9670 | 11.3374 | 11.6939 | 11.8969
| 105 | 8.6500 | 8.8650 | 9.2245 | 9.5687 | 9.7647
| 110 | 6.8701 | 7.0735 | 7.4122 | 7.7358 | 7.9205
2 | 90 | 17.5283 | 17.7451 | 18.0914 | 18.4083 | 18.5845
| 95 | 14.9958 | 15.2160 | 15.5640 | 15.8803 | 16.0558
| 100 | 12.7317 | 12.9505 | 13.2933 | 13.6034 | 13.7755
| 105 | 10.7286 | 10.9414 | 11.2728 | 11.5719 | 11.7379
| 110 | 8.9740 | 9.1772 | 9.4921 | 9.7761 | 9.9339
3 | 90 | 20.0259 | 20.2258 | 20.5102 | 20.7529 | 20.8857
| 95 | 17.8492 | 18.0381 | 18.3060 | 18.5346 | 18.6598
| 100 | 15.8616 | 16.0389 | 16.2895 | 16.5034 | 16.6208
| 105 | 14.0550 | 14.2202 | 14.4531 | 14.6523 | 14.7617
| 110 | 12.4200 | 12.5730 | 12.7882 | 12.9725 | 13.0740
8 | 90 | 24.8414 | 24.7887 | 24.7980 | 24.9009 | 24.9967
| 95 | 23.5658 | 23.5012 | 23.4852 | 23.5573 | 23.6319
| 100 | 22.3630 | 22.2874 | 22.2483 | 22.2920 | 22.3471
| 105 | 21.2282 | 21.1429 | 21.0827 | 21.1003 | 21.1374
| 110 | 20.1572 | 20.0633 | 19.9840 | 19.9778 | 19.9983
12 | 90 | 24.2787 | 24.2082 | 24.1840 | 24.3052 | 24.4514
| 95 | 23.3792 | 23.3013 | 23.2584 | 23.3520 | 23.4766
| 100 | 22.5244 | 22.4397 | 22.3795 | 22.4472 | 22.5516
| 105 | 21.7114 | 21.6205 | 21.5442 | 21.5878 | 21.6734
| 110 | 20.9375 | 20.8409 | 20.7498 | 20.7710 | 20.8388

Table 2. Fixed-strike geometric Asian call option prices for different
maturities, strike prices, and roughness levels for model parameters
$S_{0}=100$, $\nu_{0}=0.09$, $t=0$, $r=0.05$, $\kappa=1.15$, $\theta=0.348$,
$\sigma=0.39$.

| Option value
---|---
T (years) | K | $\alpha=1.40$ | $\alpha=1.25$ | $\alpha=1.00$ | $\alpha=0.75$ | $\alpha=0.60$
0.2 | 90 | 0.3194 | 0.3524 | 0.4526 | 0.6424 | 0.8061
| 95 | 1.1433 | 1.1927 | 1.3329 | 1.5705 | 1.7575
| 100 | 2.9743 | 3.0192 | 3.1438 | 3.3459 | 3.5005
| 105 | 6.0087 | 6.0334 | 6.1014 | 6.2118 | 6.3000
| 110 | 10.0360 | 10.0448 | 10.0689 | 10.1124 | 10.1542
0.4 | 90 | 1.0483 | 1.1453 | 1.3799 | 1.7123 | 1.9469
| 95 | 2.2713 | 2.3876 | 2.6591 | 3.0254 | 3.2755
| 100 | 4.2570 | 4.3706 | 4.6323 | 4.9785 | 5.2128
| 105 | 7.0682 | 7.1608 | 7.3735 | 7.6558 | 7.8489
| 110 | 10.6282 | 10.6935 | 10.8442 | 11.0478 | 11.1906
0.5 | 90 | 1.4436 | 1.5722 | 1.8619 | 2.2389 | 2.4909
| 95 | 2.7871 | 2.9357 | 3.2609 | 3.6685 | 3.9345
| 100 | 4.8128 | 4.9604 | 5.2792 | 5.6726 | 5.9273
| 105 | 7.5612 | 7.6895 | 7.9659 | 8.3069 | 8.5288
| 110 | 10.9774 | 11.0776 | 11.2941 | 11.5638 | 11.7417
1 | 90 | 3.4998 | 3.7429 | 4.1801 | 4.6239 | 4.8798
| 95 | 5.1877 | 5.4497 | 5.9135 | 6.3763 | 6.6408
| 100 | 7.3268 | 7.5934 | 8.0602 | 8.5211 | 8.7830
| 105 | 9.9204 | 10.1782 | 10.6266 | 11.0663 | 11.3155
| 110 | 12.9469 | 13.1858 | 13.5991 | 14.0027 | 14.2312
1.5 | 90 | 5.5830 | 5.8470 | 6.2606 | 6.6283 | 6.8267
| 95 | 7.4548 | 7.7310 | 8.1588 | 8.5350 | 8.7371
| 100 | 9.6640 | 9.9437 | 10.3725 | 10.7464 | 10.9465
| 105 | 12.2055 | 12.4804 | 12.8983 | 13.2599 | 13.4529
| 110 | 15.0643 | 15.3277 | 15.7246 | 16.0658 | 16.2473
2 | 90 | 7.6004 | 7.8065 | 8.0965 | 8.3323 | 8.4548
| 95 | 9.5920 | 9.8016 | 10.0933 | 10.3285 | 10.4502
| 100 | 11.8522 | 12.0603 | 12.3467 | 12.5758 | 12.6939
| 105 | 14.3732 | 14.5754 | 14.8504 | 15.0684 | 15.1804
| 110 | 17.1429 | 17.3363 | 17.5939 | 17.7968 | 17.9005
3 | 90 | 11.1708 | 11.1558 | 11.1123 | 11.0617 | 11.0298
| 95 | 13.2977 | 13.2717 | 13.2116 | 13.1469 | 13.1073
| 100 | 15.6137 | 15.5760 | 15.4986 | 15.4193 | 15.3717
| 105 | 18.1106 | 18.0609 | 17.9658 | 17.8716 | 17.8159
| 110 | 20.7792 | 20.7172 | 20.6044 | 20.4954 | 20.4316
8 | 90 | 19.0304 | 18.7038 | 18.0809 | 17.3813 | 16.9256
| 95 | 21.1066 | 20.7679 | 20.1198 | 19.3894 | 18.9125
| 100 | 23.2552 | 22.9057 | 22.2345 | 21.4757 | 20.9792
| 105 | 25.4720 | 25.1127 | 24.4204 | 23.6356 | 23.1211
| 110 | 27.7528 | 27.3848 | 26.6733 | 25.8646 | 25.3336
12 | 90 | 19.7611 | 19.5295 | 18.9998 | 18.2823 | 17.7561
| 95 | 21.6057 | 21.3667 | 20.8183 | 20.0731 | 19.5254
| 100 | 23.4950 | 23.2491 | 22.6834 | 21.9124 | 21.3445
| 105 | 25.4260 | 25.1740 | 24.5922 | 23.7971 | 23.2103
| 110 | 27.3961 | 27.1384 | 26.5418 | 25.7243 | 25.1198

Table 3. Fixed-strike geometric Asian put option prices for different
maturities, strike prices, and roughness levels for model parameters
$S_{0}=100$, $\nu_{0}=0.09$, $t=0$, $r=0.05$, $\kappa=1$, $\theta=0.15$,
$\sigma=0.6$.

| Option value
---|---
T (years) | $\alpha=1.40$ | $\alpha=1.25$ | $\alpha=1.00$ | $\alpha=0.75$ | $\alpha=0.60$
0.2 | 3.6731 | 3.7730 | 3.9925 | 4.2467 | 4.3812
0.4 | 5.8948 | 6.0960 | 6.4473 | 6.7508 | 6.8805
0.5 | 6.9698 | 7.2020 | 7.5727 | 7.8563 | 7.9658
1 | 12.2170 | 12.4048 | 12.5761 | 12.5965 | 12.5621
1.5 | 17.1044 | 17.0459 | 16.8305 | 16.5500 | 16.3745
2 | 21.4506 | 21.1133 | 20.5478 | 20.0250 | 19.7323
3 | 28.4519 | 27.7957 | 26.8612 | 26.0414 | 25.5759
8 | 48.7588 | 48.6256 | 48.0585 | 47.0888 | 46.3269
12 | 59.7564 | 59.6788 | 59.2926 | 58.4512 | 57.6810

Table 4. Floating-strike geometric Asian call option prices for different
maturities, and roughness levels for model parameters $S_{0}=100$,
$\nu_{0}=0.09$, $t=0$, $r=0.05$, $\kappa=1.15$, $\theta=0.348$, $\sigma=0.39$.

| Option value
---|---
T (years) | $\alpha=1.40$ | $\alpha=1.25$ | $\alpha=1.00$ | $\alpha=0.75$ | $\alpha=0.60$
0.2 | 3,0098 | 3.1020 | 3.3015 | 3.5240 | 3.6341
0.4 | 4.5248 | 4.7002 | 4.9965 | 5.2336 | 5.3222
0.5 | 5.2269 | 5.4228 | 5.7216 | 5.9267 | 5.9908
1 | 8.4109 | 8.5236 | 8.5845 | 8.5222 | 8.4516
1.5 | 10.9619 | 10.8435 | 10.5698 | 10.2720 | 10.0988
2 | 12.8138 | 12.4873 | 11.9781 | 11.5364 | 11.2964
3 | 14.7706 | 14.3294 | 13.7228 | 13.1961 | 12.8935
8 | 14.8984 | 15.0392 | 15.1044 | 14.9363 | 14.7224
12 | 13.6670 | 13.7505 | 13.8699 | 13.8658 | 13.7619

Table 5. Floating-strike geometric Asian put option prices for different
maturities, and roughness levels for model parameters $S_{0}=100$,
$\nu_{0}=0.09$, $t=0$, $r=0.05$, $\kappa=1.15$, $\theta=0.348$, $\sigma=0.39$.

##  8\. Conclusion

We provide semi-analytic pricing formulas for fixed- and floating-strike
geometric Asian options in the class of Volterra-Heston models, which include
the rough Heston model. These formulas are obtained by combining a general
pricing approach based on the Fourier inversion theorem with an explicit
representation of the conditional joint Fourier transform of the logarithm of
the stock price at maturity and the logarithm of the geometric mean of the
stock price over a certain time period. In contrast to the classical Heston
model, the volatility process in the Volterra-Heston model is no longer
Markovian and hence we have to apply different techniques to obtain this
representation as a suitably constructed stochastic exponential in terms of
the solution of a Volterra-Riccati equation. This construction requires a link
to the theory of affine Volterra processes developed in [1]. Our work is an
extension of the results of [24] for the classical Heston model to the class
of Volterra-Heston models. We also implement our pricing formulas and provide
a numerical study for the rough Heston model, in which we investigate the
influence of the roughness of the volatility process on the option prices.

## Acknowledgements

F. Aichinger is supported by the Austrian Science Fund (FWF) Project P34808.
For the purpose of open access, the authors have applied a CC BY public
copyright licence to any author accepted manuscript version arising from this
submission.

##  9\. Appendix

###  9.1. Proof of the general pricing approach in Theorem 2.3:

Under a risk-neutral measure ${\mathbb{Q}}$, the price of the European call
option is given as

$\displaystyle C_{0}$
$\displaystyle=e^{-rT}\mathbb{E}^{\mathbb{Q}}(\operatorname{max}(S_{T}-K,0))$
$\displaystyle=e^{-rT}[\mathbb{E}^{{\mathbb{Q}}}(S_{T}\mathbbm{1}_{\\{S_{T}>K\\}})-\mathbb{E}^{{\mathbb{Q}}}(K\mathbbm{1}_{\\{S_{T}>K\\}})].$

For the second expectation we get

$\mathbb{E}^{{\mathbb{Q}}}(K\mathbbm{1}_{\\{S_{T}>K\\}})=K{\mathbb{Q}}(S_{T}>K).$

Define a probability measure ${\mathbb{Q}}_{S}$ via
${\mathbb{Q}}_{S}(A):=\mathbb{E}^{{\mathbb{Q}}}\left(\frac{S_{T}}{\mathbb{E}^{{\mathbb{Q}}}(S_{T})}\mathbbm{1}_{A}\right)$.
Then for the first expectation we obtain

$\displaystyle\mathbb{E}^{{\mathbb{Q}}}(S_{T}\mathbbm{1}_{\\{S_{T}>K\\}})$
$\displaystyle=\mathbb{E}^{{\mathbb{Q}}}\left(\mathbb{E}^{{\mathbb{Q}}}(S_{T})\frac{S_{T}}{\mathbb{E}^{{\mathbb{Q}}}(S_{T})}\mathbbm{1}_{\\{S_{T}>K\\}}\right)$
$\displaystyle=\mathbb{E}^{{\mathbb{Q}}}(S_{T})\mathbb{E}^{{\mathbb{Q}}}\left(\frac{S_{T}}{\mathbb{E}^{{\mathbb{Q}}}(S_{T})}\mathbbm{1}_{\\{S_{T}>K\\}}\right)=e^{rT}S_{0}{\mathbb{Q}}_{S}(S_{T}>K).$

Thus we have

$\displaystyle C_{0}$
$\displaystyle=S_{0}{\mathbb{Q}}_{S}(S_{T}>K)-e^{-rT}K{\mathbb{Q}}(S_{T}>K)$
(9.1)
$\displaystyle=S_{0}{\mathbb{Q}}_{S}(\operatorname{log}S_{T}>\operatorname{log}K)-e^{-rT}K{\mathbb{Q}}(\operatorname{log}S_{T}>\operatorname{log}K)$

Using the identity

$\displaystyle\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}_{S}}(u)$
$\displaystyle=\mathbb{E}^{{\mathbb{Q}}_{S}}(e^{iu\operatorname{log}S_{T}})=\mathbb{E}^{{\mathbb{Q}}}\left(e^{iu\operatorname{log}S_{T}}\frac{d{\mathbb{Q}}_{S}}{d{\mathbb{Q}}}\right)$
$\displaystyle=\mathbb{E}^{{\mathbb{Q}}}\left(e^{iu\operatorname{log}S_{T}}\frac{S_{T}}{\mathbb{E}^{{\mathbb{Q}}}(S_{T})}\right)=\frac{\mathbb{E}^{{\mathbb{Q}}}(e^{iu\operatorname{log}S_{T}}S_{T})}{{\mathbb{E}^{{\mathbb{Q}}}(S_{T})}}=\frac{\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}}(u-i)}{\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}}(-i)},$

an application of the Fourier inversion formula now yields

$\displaystyle{\mathbb{Q}}_{S}(\operatorname{log}S_{T}>\operatorname{log}K)$
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left[\frac{e^{-iu\operatorname{log}K}\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}_{S}}(u)}{iu}\right]du$
(9.2)
$\displaystyle=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left[\frac{e^{-iu\operatorname{log}(K)}\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}}(u-i)}{iu\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}}(-i)}\right]du,$

and

${\mathbb{Q}}(\operatorname{log}S_{T}>\operatorname{log}K)=\frac{1}{2}+\frac{1}{\pi}\int_{0}^{\infty}\operatorname{Re}\left[\frac{e^{-iu\operatorname{log}K}\psi_{\operatorname{log}S_{T}}^{{\mathbb{Q}}}(u)}{iu}\right]du.$
(9.3)

Inserting (9.2) and (9.3) into equation (9.1) yields the desired result

∎

###  9.2. Equivalent form of the Riccati-Volterra equation

We derive an equivalent representation for the Riccati-Volterra equations
appearing in Sections 6 and 7.

######  Lemma 9.1.

The Riccati-Volterra equation

$\phi=K\ast(Q(\phi,f)-\kappa\phi)$

with

$Q(\phi,f)=\frac{1}{2}\left(f^{2}-f\right)+\frac{1}{2}\left(\sigma^{2}\phi^{2}+2\rho\sigma
f\phi\right)\vspace{3mm}$

has equivalent representation

$\phi=\frac{1}{\kappa}\ast Q(\phi,f).$

_Proof:_ We start with the Volterra-Riccati equation

$\phi=K\ast(Q(\phi,f)-\kappa\phi)$

and subtract $R_{\kappa}\ast\phi$ from both sides. This yields

$\displaystyle\phi-R_{\kappa}\ast\phi$
$\displaystyle=K\ast(Q(\phi,f)-\kappa\phi)-R_{\kappa}\ast
K\ast(Q(\phi,f)-\kappa\phi)$ $\displaystyle=(K-R_{\kappa}\ast
K)\ast(Q(\phi,f)-\kappa\phi)$
$\displaystyle=\frac{1}{\kappa}R_{\kappa}\ast(Q(\phi,f)-\kappa\phi)$
$\displaystyle=\frac{1}{\kappa}R_{\kappa}\ast Q(\phi,f)-R_{\kappa}\ast\phi.$

And consequently, we get the equivalent representation

$\phi=\frac{1}{\kappa}R_{\kappa}\ast Q(\phi,f).$

∎

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