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# Signature volatility models: pricing and hedging with Fourier

Eduardo Abi Jaber<EMAIL_ADDRESS>The first author is
grateful for the financial support from the Chaires FiME-FDD, Financial Risks,
Deep Finance & Statistics and Machine Learning and systematic methods in
finance at École Polytechnique. École Polytechnique, CMAP  Louis-Amand Gérard
<EMAIL_ADDRESS>We would like to thank Olivier Guéant
for fruitful discussions, and Yuxing Huang for his precious comments.
Université Paris 1 Panthéon-Sorbonne, CES  Gefip

###### Abstract

We consider a stochastic volatility model where the dynamics of the volatility
are given by a possibly infinite linear combination of the elements of the
time extended signature of a Brownian motion. First, we show that the model is
remarkably universal, as it includes, but is not limited to, the celebrated
Stein-Stein, Bergomi, and Heston models, together with some path-dependent
variants. Second, we derive the joint characteristic functional of the log-
price and integrated variance provided that some infinite-dimensional extended
tensor algebra valued Riccati equation admits a solution. This allows us to
price and (quadratically) hedge certain European and path-dependent options
using Fourier inversion techniques. We highlight the efficiency and accuracy
of these Fourier techniques in a comprehensive numerical study.

MSC 2020:

60L10, 91G20, 91G60

Keywords:

stochastic volatility, path signature, pricing, hedging, calibration, Fourier
methods

##  1 Introduction

An important challenge of stochastic volatility modeling is the construction
of realistic models that remain tractable for option pricing, risk hedging,
trading and calibration purposes. Notable realistic features encompass various
aspects, such as inter-temporal and path dependencies, which are inherent
phenomena in financial markets. These phenomena have been established
empirically at different timescales through extensive research, either in the
form of long/short range dependence since Mandelbrot and Van Ness [50] and as
documented in [9, 22, 40, 41, 23] or in the form of self-excitation of
financial markets [11]. They can also be understood more strategically using
the transitory nature of the decisions of market participants and their impact
on prices [16, 17].

Incorporating path dependencies and memory effects into the modeling framework
gives rise to non-Markovian models, which generally pose computational
challenges. Recent developments have identified interesting mathematical
classes of models that address this issue. One such class comprises stochastic
Volterra models, which provide a flexible framework for capturing certain
inter-temporal dependencies while maintaining computational tractability under
specific affine and quadratic structures [1, 3, 25, 26, 32]. Another avenue
worth exploring lies in the application of signature-based methods. The
signature of a path, initially introduced by Chen [20] in 1957, consists of
the (infinite) sequence of iterated integrals of a path. It plays a crucial
role in the theory of rough paths [37, 46] and is recently gaining
considerable momentum in the fields of Machine Learning [21, 35, 51] and
Mathematical Finance [10, 13, 14, 18, 19, 27, 28, 31, 48] namely due to its
universal linearization property: any functional of the path can be
approximated by a linear combination of the elements of the signature of the
path, provided some regularity. In our present work, we explore the modeling
and numerical aspects of these signature-based approaches for stochastic
volatility modeling. We uncover their potential in addressing the
computational challenges associated with path-dependent models and we
highlight their versatility and universality.

We consider a class of stochastic volatility models for a stock price $S$ in
the form

$dS_{t}=S_{t}\Sigma_{t}\left(\rho
dW_{t}+\sqrt{1-\rho^{2}}dW_{t}^{\perp}\right),$

where the stochastic volatility process $\Sigma$ is of a general path-
dependent form

$f(t,W_{0\leq s\leq t}),\quad t\leq T,$

for a certain class of measurable functions $f$ and $(W,W^{\perp})$ a two-
dimensional Brownian motion. The correlation $\rho\in[-1,1]$ accounts for the
leverage effect. More precisely, we assume that the volatility process
$\Sigma_{t}$ is a (possibly infinite) linear combination of the elements of
the signature process $\widehat{\mathbb{W}}_{t}$ of the time extended Brownian
motion $\widehat{W}_{t}:=(t,W_{t})$ defined by the infinite sequence of
iterated Stratonovich [60] integrals:

$\displaystyle\widehat{\mathbb{W}}_{t}$
$\displaystyle=\left(1,\begin{pmatrix}t\\\
W_{t}\end{pmatrix},\begin{pmatrix}\frac{t^{2}}{2!}&\int_{0}^{t}sdW_{s}\\\
\int_{0}^{t}W_{s}ds&\frac{W_{t}^{2}}{2!}\end{pmatrix},\begin{pmatrix}\frac{t^{3}}{3!}&\int_{0}^{t}\frac{s^{2}}{2!}dW_{s}&\\\
\int_{0}^{t}\int_{0}^{s}udW_{u}ds&\int_{0}^{t}\int_{0}^{s}udW_{u}\circ
dW_{s}&\\\
&\int_{0}^{t}\int_{0}^{s}W_{u}duds&\int_{0}^{t}\int_{0}^{s}W_{u}du\circ
dW_{s}\\\
&\int_{0}^{t}\frac{W_{s}^{2}}{2!}ds&\frac{W_{t}^{3}}{3!}\end{pmatrix},\cdots\right).$
(1.1)

We call such models signature volatility models. They have been introduced by
Arribas et al. [10], with $\Sigma_{t}$ a finite linear combination of elements
of $\widehat{\mathbb{W}}_{t}$, and their theoretical and empirical properties
have been studied further in Cuchiero et al. [27, 28] for pricing and
calibration purposes. They allow to naturally embed inter-temporal and path
dependencies. The construction is very elementary, yet, from the mathematical
perspective, such class of models enjoys a beautiful and powerful universality
feature. So far, the universal approximation property of signatures have been
invoked to argue that “the framework is universal in the sense that classical
models can be approximated arbitrarily well” by signature volatility models,
see for instance [27, proposition 2.14]. In contrast to the existing
literature on signature volatility models, we allow $\Sigma_{t}$ to be an
infinite linear combination of the elements of the signature process
$\widehat{\mathbb{W}}_{t}$. This introduces intricate theoretical challenges,
particularly regarding convergence issues, but provides a more profound
comprehension of the elegant universal structure inherent to these models. Our
approach draws inspiration from Cuchiero et al. [29] where infinite linear
combinations of signature elements are considered in the context of signature
stochastic differential equations.

##### Universality and flexibility of signature volatility models.

In a first step, by considering infinite linear combinations of signature
elements, we go beyond the ‘approximated universality’ and we prove that the
class of signature volatility models is universal in the sense of exact
representations. We show that many classical and popular Markovian models, and
even more advanced not necessarily Markovian models, belong to the class of
signature volatility models, using novel exact representations formulas
derived in the sequel and in the accompanying paper [6]. This includes:

  1. (i)

Affine Markovian models: The models of Stein and Stein [58], Schöbel and Zhu
[55] and Heston [42] which became popular because of the explicit knowledge of
the characteristic function of the log-price, allowing for fast and accurate
pricing and hedging using Fourier inversion techniques.

  2. (ii)

Non-affine Markovian models: The models of Bergomi [15] and Hull and White
[43], which are more flexible than their affine counterparts but less
tractable. In addition, the recently introduced Quintic Ornstein-Uhlenbeck
model of Abi Jaber et al. [4], which is able to jointly capture SPX and VIX
smiles, also belongs to the class of signature volatility models.

  3. (iii)

Non-Markovian models: A large class of models based on stochastic delayed
equations and Volterra processes, including the class of polynomial Gaussian
volatility models [5], Volterra Stein-Stein model [1], the Volterra [2] and
rough [12] Bergomi models.

The above list is far from being exhaustive and we believe that more known and
important volatility models can be exactly re-written as a signature
stochastic volatility model. In addition, when such exact representation
cannot be found for a specific model, building an approximated signature
volatility model is possible thanks to the universal approximation property of
the signature. The representations are derived in Section 3 and illustrated on
numerical examples.

##### Tractability of signature volatility models.

In a second step, we develop a generic framework for pricing and quadratic
hedging certain vanilla and path-dependent options on the log-price and the
integrated variance using Fourier inversion technology. More specifically, we
obtain in Theorem 4.1 that for any signature volatility model, the joint
characteristic functional of the log-price and the integrated variance is
known up to the solution of an infinite dimensional Riccati equation. This
result opens the door to fast and accurate Fourier pricing and hedging going
beyond the standard affine classes (i) including the whole list of Markovian
and non-Markovian models in (ii)-(iii) above, for which pricing and hedging is
an intricate task. Our representation formula for the characteristic function
can be directly related to the ones that have appeared in Cuchiero et al.
[29], and share similarities with the formulas in Friz et al. [38] and Lyons
et al. [49].

Using a comprehensive numerical study we highlight the efficiency and the
accuracy of the Fourier techniques for pricing and hedging in signature
volatility models in Sections 5 and 6. We stress that the numerics are not
straightforward, since they involve several approximations and truncations. We
use ideas in the spirit of ‘control variate’ with Black-Scholes prices and
deltas to stabilize the Fourier inversions and reduce the number of
evaluations of the characteristic functional. We also point out that the
proposed implementation is both generic and scalable, as it can be applied, or
fine-tuned if needed, to any signature volatility model. It only requires as
inputs the coefficients (i.e. the parameters) of the signature volatility
model, which essentially define the model itself whether it is Markovian/non-
Markovian/non-semimartingale, and it generates as outputs option prices and
hedging strategies by Fourier methods.

Outline. Section 2 introduces the framework of signatures with a focus on
infinite linear combinations of signature elements. In Section 3, we define
signature volatility models and study their representation properties. Section
4 derives the characteristic functional of the log-price and integrated
variance. As applications, pricing and hedging of various European and Asian
options under the signature volatility model are implemented in Sections 5 and
6, where we also include calibration examples on simulated and market data.
Finally, Appendix A collects some proofs for the representations found in
Subsection 3.1.

##  2 A primer on signatures

###  2.1 Tensor algebra

In this section, we setup the framework for dealing with signatures of
semimartingales. One can also refer to the first sections in [13, 28, 48].

Let $d\in\mathbb{N}$ and denote by $\otimes$ the tensor product over
$\mathbb{R}^{d}$, e.g. $(x\otimes y\otimes z)_{ijk}=x_{i}y_{j}z_{k}$, for
$i,j,k=1,\dots,d$, for $x,y,z\in\mathbb{R}^{d}$. For $n\geq 1$, we denote by
$(\mathbb{R}^{d})^{\otimes n}$ the space of tensors of order $n$ and by
$(\mathbb{R}^{d})^{\otimes 0}=\mathbb{R}$. In the sequel, we will consider
mathematical objects, path signatures, that live on the extended tensor
algebra space $T((\mathbb{R}^{d}))$ over $\mathbb{R}^{d}$, that is the space
of (infinite) sequences of tensors defined by

$T((\mathbb{R}^{d})):=\left\\{\bm{\ell}=(\bm{\ell}^{n})_{n=0}^{\infty}:\bm{\ell}^{n}\in(\mathbb{R}^{d})^{\otimes
n}\right\\}.$

Similarly, for $M\geq 0$, we define the truncated tensor algebra
$T^{M}(\mathbb{R}^{d})$ as the space of sequences of tensors of order at most
$M$ defined by

$T^{M}(\mathbb{R}^{d}):=\left\\{\bm{\ell}\in
T((\mathbb{R}^{d})):\bm{\ell}^{n}=0,\text{ for all }n>M\right\\},$

and the tensor algebra $T(\mathbb{R}^{d})$ as the space of all finite
sequences of tensors defined by

$T(\mathbb{R}^{d}):=\bigcup_{M\in\mathbb{N}}T^{M}(\mathbb{R}^{d}).$

We clearly have $T(\mathbb{R}^{d})\subset T((\mathbb{R}^{d}))$. For
$\bm{\ell}=(\bm{\ell}^{n})_{n\in\mathbb{N}},\bm{p}=(\bm{p}^{n})_{n\in\mathbb{N}}\in
T((\mathbb{R}^{d}))$ and $\lambda\in\mathbb{R}$, we define the following
operations:

$\displaystyle\bm{\ell}+\bm{p}:$
$\displaystyle=(\bm{\ell}^{n}+\bm{p}^{n})_{n\in\mathbb{N}}$
$\displaystyle\bm{\ell}\otimes\bm{p}:$
$\displaystyle=\left(\sum_{k=0}^{n}\bm{\ell}^{k}\otimes\bm{p}^{n-k}\right)_{n\in\mathbb{N}}$
$\displaystyle\lambda\bm{\ell}:$
$\displaystyle=(\lambda\bm{\ell}^{n})_{n\in\mathbb{N}}.$

These operations induce analogous operations on $T^{M}(\mathbb{R}^{d})$ and
$T(\mathbb{R}^{d})$.

Important notations. Let $\\{e_{1},\dots,e_{d}\\}\subset\mathbb{R}^{d}$ be the
canonical basis of $\mathbb{R}^{d}$ and
${A}_{d}=\\{{\mathcolor{NavyBlue}{\mathbf{1}}},{\mathcolor{NavyBlue}{\mathbf{2}}},\dots,{\mathcolor{NavyBlue}{\mathbf{d}}}\\}$
be the corresponding alphabet. To ease reading, for $i\in\\{1,\dots,d\\}$, we
write $e_{i}$ as the blue letter ${\mathcolor{NavyBlue}{\mathbf{i}}}$ and for
$n\geq 1,i_{1},\dots,i_{n}\in\\{1,\dots,d\\}$, we write
$e_{i_{1}}\otimes\cdots\otimes e_{i_{n}}$ as the concatenation of letters
${\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots i_{n}}}}$, that we call a word of
length $n$. We note that $(e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}})_{(i_{1},\dots,i_{n})\in\\{1,\dots,d\\}^{n}}$ is a basis of
$(\mathbb{R}^{d})^{\otimes n}$ that can be identified with the set of words of
length $n$ defined by

$V_{n}:=\\{{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n}}}}:{\mathcolor{NavyBlue}{\mathbf{i_{k}}}}\in{A}_{d}\text{ for
}k=1,2,\dots,n\\}.$ (2.1)

Moreover, we denote by ø the empty word and by
$V_{0}=\\{{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}\\}$ which
serves as a basis for $(\mathbb{R}^{d})^{\otimes 0}=\mathbb{R}$. It follows
that $V:=\cup_{n\geq 0}V_{n}$ represents the standard basis of
$T((\mathbb{R}^{d}))$. In particular, every $\bm{\ell}\in
T((\mathbb{R}^{d}))$, can be decomposed as

$\bm{\ell}=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}{\mathcolor{NavyBlue}{\mathbf{v}}},$
(2.2)

where $\bm{\ell}^{\mathcolor{NavyBlue}{\mathbf{v}}}$ is the real coefficient
of $\bm{\ell}$ at coordinate ${\mathcolor{NavyBlue}{\mathbf{v}}}$.
Representation (2.2) will be frequently used in the paper. We stress again
that in the sequel, every blue ‘word‘ ${\mathcolor{NavyBlue}{\mathbf{v}}}\in
V$ represents an element of the canonical basis of $T((\mathbb{R}^{d}))$, i.e.
there exists $n\geq 0$ such that ${\mathcolor{NavyBlue}{\mathbf{v}}}$ in the
form
${\mathcolor{NavyBlue}{\mathbf{v}}}={\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n}}}}$, which represents the element $e_{i_{1}}\otimes\cdots\otimes
e_{i_{n}}$. The concatenation $\bm{\ell}{\mathcolor{NavyBlue}{\mathbf{v}}}$ of
elements $\bm{\ell}\in T((\mathbb{R}^{d}))$ and the word
${\mathcolor{NavyBlue}{\mathbf{v}}}={\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n}}}}$ means $\bm{\ell}\otimes e_{i_{1}}\otimes\cdots\otimes e_{i_{n}}$.

In addition to the decomposition (2.2) of elements $\bm{\ell}\in
T((\mathbb{R}^{d}))$, we introduce the projection
$\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{u}}}}\in T((\mathbb{R}^{d}))$ as

$\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{u}}}}:=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{vu}}}}{\mathcolor{NavyBlue}{\mathbf{v}}}$
(2.3)

for all ${\mathcolor{NavyBlue}{\mathbf{u}}}\in V$. The projection plays an
important role in the space of iterated integrals as it is closely linked to
partial differentiation, in contrast with the concatenation that relates to
integration. It will be used throughout the paper.

######  Remark 2.0.1.

The projection allows us to decompose elements of the extended tensor algebra
$\bm{\ell}\in T((\mathbb{R}^{d}))$ as

$\displaystyle\bm{\ell}=\bm{\ell}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\sum_{{\mathcolor{NavyBlue}{\mathbf{i}}}\in{A}_{d}}\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{i}}}}{\mathcolor{NavyBlue}{\mathbf{i}}}.$
(2.4)

This decomposition is quite natural as, when iterated, it gives back the
decomposition in (2.2). $\blacksquare$

######  Example 2.0.1.

Take the alphabet
${A}_{3}=\\{{\mathcolor{NavyBlue}{\mathbf{1}}},{\mathcolor{NavyBlue}{\mathbf{2}}},{\mathcolor{NavyBlue}{\mathbf{3}}}\\}$
and let
$\bm{\ell}=4\cdot{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+3\cdot{\mathcolor{NavyBlue}{\mathbf{1}}}-1\cdot{\mathcolor{NavyBlue}{\mathbf{12}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{2212}}}$,
then (2)
$\bm{\ell}^{{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}}=4\cdot{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$,
$\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}=3\cdot{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$,
$\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}=-1\cdot{\mathcolor{NavyBlue}{\mathbf{1}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{221}}}$,
$\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{3}}}}=0$. $\blacksquare$

We now define the bracket between $\bm{\ell}\in T(\mathbb{R}^{d})$ and
$\bm{p}\in T((\mathbb{R}^{d}))$ by

$\displaystyle\langle\bm{\ell},\bm{p}\rangle:=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}\bm{p}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}.$
(2.5)

Notice that it is actually well defined as $\bm{\ell}$ has finitely many non-
zero terms. For $\bm{\ell}\in T((\mathbb{R}^{d}))$, the series in (2.5)
involves infinitely many terms and requires special care, this is discussed in
Subsection 2.4.

######  Example 2.0.2.

Take the alphabet
${A}_{2}=\\{{\mathcolor{NavyBlue}{\mathbf{1}}},{\mathcolor{NavyBlue}{\mathbf{2}}}\\}$
and let
$\bm{p}:=1\cdot{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{2}}}-3\cdot{\mathcolor{NavyBlue}{\mathbf{12}}}+4\cdot{\mathcolor{NavyBlue}{\mathbf{111}}}\in
T((\mathbb{R}^{d}))$, then (2)
$\langle{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}},\bm{p}\rangle=1$,
$\langle{\mathcolor{NavyBlue}{\mathbf{2}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{12}}},\bm{p}\rangle=2-6=-4$,
$\langle-2\cdot{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+{\mathcolor{NavyBlue}{\mathbf{111}}},\bm{p}\rangle=-2+4=2$,
$\langle\bm{p},\bm{p}\rangle=1+4+9+16=20$. $\blacksquare$

We will also consider another operation on the space of words, the shuffle
product. The shuffle product plays a crucial role for an integration by parts
formula on the space of iterated integrals, see Proposition 2.6 below.

######  Definition 2.1 (Shuffle product).

The shuffle product $\mathrel{\sqcup\mkern-3.2mu\sqcup}:V\times V\to
T(\mathbb{R}^{d})$ is defined inductively for all words
${\mathcolor{NavyBlue}{\mathbf{v}}}$ and ${\mathcolor{NavyBlue}{\mathbf{w}}}$
and all letters ${\mathcolor{NavyBlue}{\mathbf{i}}}$ and
${\mathcolor{NavyBlue}{\mathbf{j}}}$ in ${A}_{d}$ by

$\displaystyle({\mathcolor{NavyBlue}{\mathbf{v}}}{\mathcolor{NavyBlue}{\mathbf{i}}})\mathrel{\sqcup\mkern-3.2mu\sqcup}({\mathcolor{NavyBlue}{\mathbf{w}}}{\mathcolor{NavyBlue}{\mathbf{j}}})$
$\displaystyle=({\mathcolor{NavyBlue}{\mathbf{v}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}({\mathcolor{NavyBlue}{\mathbf{w}}}{\mathcolor{NavyBlue}{\mathbf{j}}})){\mathcolor{NavyBlue}{\mathbf{i}}}+(({\mathcolor{NavyBlue}{\mathbf{v}}}{\mathcolor{NavyBlue}{\mathbf{i}}})\mathrel{\sqcup\mkern-3.2mu\sqcup}{\mathcolor{NavyBlue}{\mathbf{w}}}){\mathcolor{NavyBlue}{\mathbf{j}}},$
$\displaystyle{\mathcolor{NavyBlue}{\mathbf{w}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$
$\displaystyle={\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}{\mathcolor{NavyBlue}{\mathbf{w}}}={\mathcolor{NavyBlue}{\mathbf{w}}}.$

With some abuse of notation, the shuffle product on $T((\mathbb{R}^{d}))$
induced by the shuffle product on $V$ will also be denoted by
$\mathrel{\sqcup\mkern-3.2mu\sqcup}$. The shuffle product is clearly
commutative. See [53] and [39] for more information on the shuffle product.

The shuffle product corresponds to all riffle shuffles of two decks of cards
together, which keeps the order of each single deck, as illustrated in the
following example:

######  Example 2.1.1.

We have

  * •

${\mathcolor{NavyBlue}{\mathbf{12}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}{\mathcolor{NavyBlue}{\mathbf{34}}}={\mathcolor{NavyBlue}{\mathbf{1234}}}+{\mathcolor{NavyBlue}{\mathbf{1324}}}+{\mathcolor{NavyBlue}{\mathbf{3124}}}+{\mathcolor{NavyBlue}{\mathbf{1342}}}+{\mathcolor{NavyBlue}{\mathbf{3142}}}+{\mathcolor{NavyBlue}{\mathbf{3412}}},$

  * •

${\mathcolor{NavyBlue}{\mathbf{1}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}{\mathcolor{NavyBlue}{\mathbf{21}}}={\mathcolor{NavyBlue}{\mathbf{121}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{211}}},$

  * •

$({\mathcolor{NavyBlue}{\mathbf{1}}}+{\mathcolor{NavyBlue}{\mathbf{31}}})\mathrel{\sqcup\mkern-3.2mu\sqcup}{\mathcolor{NavyBlue}{\mathbf{21}}}={\mathcolor{NavyBlue}{\mathbf{121}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{211}}}+{\mathcolor{NavyBlue}{\mathbf{2131}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{2311}}}+{\mathcolor{NavyBlue}{\mathbf{3121}}}+2\cdot{\mathcolor{NavyBlue}{\mathbf{3211}}}.$

$\blacksquare$

###  2.2 Resolvent and linear equation

For $n\in\mathbb{N}$ and $\bm{\ell}\in T((\mathbb{R}^{d}))$, we define the
concatenation power of $\bm{\ell}$ by

$\bm{\ell}^{\otimes
n}:=\overbrace{\bm{\ell}\otimes\bm{\ell}\otimes\cdots\otimes\bm{\ell}}^{\text{$n$
times}},$

with the convention that $\bm{\ell}^{\otimes
0}={\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$. For $\bm{\ell}\in
T((\mathbb{R}^{d}))$ such that
$\bm{\ell}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}=0$, we
define the resolvent of $\bm{\ell}$ by

$\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\bm{\ell}\right)^{-1}:=\sum_{n=0}^{\infty}\bm{\ell}^{\otimes
n}.$ (2.6)

The assumption
$\bm{\ell}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}=0$ assures
that it is well-defined.

The resolvent allows us to solve linear algebraic equations, for instance:

######  Proposition 2.2.

Let $\bm{p},\bm{q}\in T((\mathbb{R}^{d}))$ such that
$\bm{q}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}=0$, then the
unique solution $\bm{\ell}\in T((\mathbb{R}^{d}))$ to the linear algebraic
equation

$\displaystyle\bm{\ell}=\bm{p}+\bm{\ell}\bm{q}$ (2.7)

is given by

$\displaystyle\bm{\ell}=\bm{p}\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\bm{q}\right)^{-1},$
(2.8)

with
$\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\cdot\right)^{-1}$
as defined in (2.6).

###### Proof.

It is easy to verify that
$\bm{\ell}=\bm{q}\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\bm{p}\right)^{-1}$
is a solution of (2.7). On the other hand, $\bm{\ell}=\bm{\ell}\bm{p}$,
together with the fact that
$\bm{p}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}\neq 1$,
implies that $\bm{\ell}=0$, verifying the uniqueness. ∎

Interestingly, whenever $\bm{\ell}$ is a linear combination of single letters,
the resolvent of $\bm{\ell}$ is equal to the shuffle exponential
$e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}$ defined by

$e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}:=\sum_{n=0}^{\infty}\frac{\bm{\ell}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}}{n!},$
(2.9)

where

$\bm{\ell}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}:=\overbrace{\bm{\ell}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}\mathrel{\sqcup\mkern-3.2mu\sqcup}\cdots\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}^{\text{$n$
times}},\quad n\geq
1,\quad\bm{\ell}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}0}={\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}.$

######  Proposition 2.3.

Whenever $\bm{\ell}$ is of the form
$\sum_{i\in{A}_{d}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i}}}}{\mathcolor{NavyBlue}{\mathbf{i}}}$,
with $\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i}}}}\in\mathbb{R}$, we have
that

$\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\bm{\ell}\right)^{-1}=e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}.$

In particular, this implies

$e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}={\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}\bm{\ell}={\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\bm{\ell}e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}}.$

###### Proof.

Using Lemma 2.4 below, it is easy to see that
$\frac{1}{n!}\bm{\ell}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}=\bm{\ell}^{\otimes
n}$ whenever
$\bm{\ell}=\sum_{{\mathcolor{NavyBlue}{\mathbf{i}}}\in{A}_{d}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i}}}}{\mathcolor{NavyBlue}{\mathbf{i}}}$,
hence proving the proposition. ∎

######  Lemma 2.4.

Let $\bm{p}\in T((\mathbb{R}^{d}))$ and
$\bm{\ell}=\sum_{{\mathcolor{NavyBlue}{\mathbf{i}}}\in{A}_{d}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i}}}}{\mathcolor{NavyBlue}{\mathbf{i}}}$
with $\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i}}}}\in\mathbb{R}$, then,

$(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}=n\left[\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n-1}\right]\bm{\ell},\quad
n\in\mathbb{N}.$

###### Proof.

By induction, assume that the property holds for some $n\in\mathbb{N}$, then

$\displaystyle(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n+1}$
$\displaystyle=(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{p}\bm{\ell}$
$\displaystyle=n[\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n-1}]\bm{\ell}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{p}\bm{\ell}$
$\displaystyle=n([\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n-1}]\bm{\ell}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{p})\bm{\ell}+n([\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n-1}]\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{p}\bm{\ell})\bm{\ell}$
$\displaystyle=((\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{p})\bm{\ell}+n(\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n})\bm{\ell}$
$\displaystyle=(n+1)[\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}]\bm{\ell}.$

Finally,
$(\bm{p}\bm{\ell})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}1}=\bm{p}\bm{\ell}=[\bm{p}\mathrel{\sqcup\mkern-3.2mu\sqcup}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}]\bm{\ell}$
concludes the proof. ∎

###  2.3 Signatures

We define the (path) signature of a semimartingale process as the sequence of
iterated stochastic integrals in the sense of Stratonovich. Throughout the
paper, the Itô integral is denoted by $\int_{0}^{\cdot}Y_{t}dX_{t}$ and the
Stratonovich integral by $\int_{0}^{\cdot}Y_{t}\circ dX_{t}.$ If both $X$ and
$Y$ are semimartingales then, we have the relation $\int_{0}^{\cdot}Y_{t}\circ
dX_{t}=\int_{0}^{\cdot}Y_{t}dX_{t}+\frac{1}{2}[X,Y]_{\cdot}$.

######  Definition 2.5 (Signature).

Fix $T>0$. Let $(X_{t})_{t\geq 0}$ be a continuous semimartingale in
$\mathbb{R}^{d}$ on some filtered probability space
$(\Omega,\mathcal{F},(\mathcal{F}_{t})_{t\geq 0},\mathbb{P})$. The signature
of $X$ is defined by

$\displaystyle\mathbb{X}:\Omega\times[0,T]$ $\displaystyle\to
T((\mathbb{R}^{d}))$ $\displaystyle(\omega,t)$
$\displaystyle\mapsto\mathbb{X}_{t}(\omega):=(1,\mathbb{X}_{t}^{1}(\omega),\dots,\mathbb{X}_{t}^{n}(\omega),\dots),$

where

$\mathbb{X}_{t}^{n}:=\int_{0<u_{1}<\cdots<u_{n}<t}\circ
dX_{u_{1}}\otimes\cdots\otimes\circ dX_{u_{n}}$

takes value in $(\mathbb{R}^{d})^{\otimes n}$, $n\geq 0$. Similarly, the
truncated signature of order $M\in\mathbb{N}$ is defined by

$\displaystyle\mathbb{X}^{\leq M}:[0,T]$ $\displaystyle\to
T^{M}(\mathbb{R}^{d})$ (2.10) $\displaystyle(\omega,t)$
$\displaystyle\mapsto\mathbb{X}_{t}^{\leq
M}(\omega):=(1,\mathbb{X}_{t}^{1}(\omega),\dots,\mathbb{X}_{t}^{M}(\omega),0,\dots,0,\dots).$
(2.11)

The signature plays a similar role to polynomials on path-space. Indeed, in
dimension $d=1$, the signature of $X$ is the sequence of monomials
$\left(\frac{1}{n!}(X_{t}-X_{0})^{n}\right)_{n\in\mathbb{N}}$. In particular,
any finite combination of elements of the signature
$\left\langle\bm{\ell},\mathbb{X}_{t}\right\rangle$, defined in (2.5) for
$\bm{\ell}\in T^{M}(\mathbb{R}^{d})$, is a polynomial of degree $M$ in
$X_{t}$.

######  Remark 2.5.1.

Explicitly we can write the term $\mathbb{X}_{t}^{n}$ as
$(\mathbb{X}_{t}^{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n}}}})_{({\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots i_{n}}}})\in V_{n}}$. So
the definition can be written in a iterated form as

$\displaystyle\mathbb{X}_{t}^{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n}}}}=\int_{0}^{t}\mathbb{X}_{s}^{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{n-1}}}}\circ dX_{s}^{{\mathcolor{NavyBlue}{\mathbf{i_{n}}}}}.$ (2.12)

$\blacksquare$

In what will follow, we are exclusively interested in the case $d=1$ and
$X_{t}=\widehat{W}_{t}:=(t,W_{t})$ where $W$ is a 1-dimensional Brownian
motion. Its first few signature orders are given by

$\widehat{\mathbb{W}}_{t}^{0}=1,\quad\widehat{\mathbb{W}}_{t}^{1}=\begin{pmatrix}t\\\
W_{t}\end{pmatrix},\quad\widehat{\mathbb{W}}_{t}^{2}=\begin{pmatrix}\frac{t^{2}}{2!}&\int_{0}^{t}sdW_{s}\\\
\int_{0}^{t}W_{s}ds&\frac{W_{t}^{2}}{2!}\end{pmatrix}$ (2.13)

and

$\widehat{\mathbb{W}}_{t}^{3}=\begin{pmatrix}\frac{t^{3}}{3!}&\int_{0}^{t}\frac{s^{2}}{2!}dW_{s}&\\\
\int_{0}^{t}\int_{0}^{s}udW_{u}ds&\int_{0}^{t}\int_{0}^{s}udW_{u}\circ
dW_{s}&\\\
&\int_{0}^{t}\int_{0}^{s}W_{u}duds&\int_{0}^{t}\int_{0}^{s}W_{u}du\circ
dW_{s}\\\
&\int_{0}^{t}\frac{W_{s}^{2}}{2!}ds&\frac{W_{t}^{3}}{3!}\end{pmatrix},$ (2.14)

recall (1.1).

###  2.4 Infinite linear combinations of signature elements

In this section, we recall some results on infinite linear combinations
$\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle$ for certain
admissible $\bm{\ell}\in T((\mathbb{R}^{d}))$ for which the infinite series
will make sense. Two crucial ingredients for our paper are the shuffle product
(Proposition 2.6) and an Itô’s formula (Lemma 2.7). We follow the presentation
in [6, Section 2], and we refer to [29] for more general results.

We first introduce the space $\mathcal{A}$ of admissible elements $\bm{\ell}$
below, using the associated semi-norm:

$||\bm{\ell}||_{t}^{\mathcal{A}}:=\sum_{n=0}^{\infty}\left|\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}^{\mathcolor{NavyBlue}{\mathbf{v}}}\widehat{\mathbb{W}}_{t}^{\mathcolor{NavyBlue}{\mathbf{v}}}\right|,\quad
t\geq 0,$

recall the definition of $V_{n}$ in (2.1) and the decomposition (2.2).
Whenever, $||\bm{\ell}||_{t}^{\mathcal{A}}<\infty$ a.s., the infinite linear
combination

$\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}$

is well-defined. This leads to the following definition for the admissible set
$\mathcal{A}$:

$\mathcal{A}:=\left\\{\bm{\ell}\in
T((\mathbb{R}^{2})):||\bm{\ell}||_{t}^{\mathcal{A}}<\infty\text{ for all
}t\in[0,T]\text{ a.s.}\right\\}.$

Note that $T(\mathbb{R}^{2})\subset\mathcal{A}$ and that
$\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle$ is an extension
of (2.5), as the two bracket operations $\langle\cdot,\cdot\rangle$ coincide
whenever $\bm{\ell}\in T(\mathbb{R}^{2})$.

The admissible set $\mathcal{A}$ has another very interesting property, as it
allows us to linearize polynomials on infinite linear combination of the
signature, see Proposition 2.6. This is what most of the literature refers to
when putting forth the linearization power of the signature.

######  Proposition 2.6 (Shuffle property).

If $\bm{\ell}_{1},\bm{\ell}_{2}\in\mathcal{A}$, then
$\bm{\ell}_{1}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}_{2}\in\mathcal{A}$
and

$\left\langle\bm{\ell}_{1},\widehat{\mathbb{W}}_{t}\right\rangle\left\langle\bm{\ell}_{2},\widehat{\mathbb{W}}_{t}\right\rangle=\left\langle\bm{\ell}_{1}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}_{2},\widehat{\mathbb{W}}_{t}\right\rangle.$

###### Proof.

This follows from a particular instance of shuffle compatible partitions, see
[29, Lemma 4.4.]. Alternatively, we refer to [6, Proposition 3.2]. ∎

######  Example 2.6.1.

When $d=1$, for $\bm{\ell}_{1}\in T^{M}(\mathbb{R})$ and $\bm{\ell}_{2}\in
T^{N}(\mathbb{R})$,
$\langle\bm{\ell}_{1}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\ell}_{2},\widehat{\mathbb{W}}_{t}\rangle$
can be seen as analogous to the Cauchy product (scaled by $\frac{1}{n!}$) and
is thus a polynomial of degree $M+N$. Actually, the shuffle product is closely
related to the integration by parts in the space of iterated Stratonovich
integrals, see [39]. $\blacksquare$

For elements $\bm{\ell}\in\mathcal{A}$, the process
$(\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\rangle)_{t\leq T}$ is well-
defined. An important question is to know whether it is a semimartingale and
compute its Itô’s decomposition. The answer is positive, thanks to an Itô’s
formula in Lemma 2.7 below, for elements in the set

$\mathcal{I}:=\left\\{\bm{\ell}\in\mathcal{A}:\text{ for all
}t\in[0,T],\>||\bm{\ell}||_{t}^{\mathcal{I}}<\infty\text{ and
}\int_{0}^{T}||\bm{\ell}||_{t}^{\mathcal{I}}dt<\infty\text{ a.s.}\right\\},$
(2.15)

where

$||\bm{\ell}||_{t}^{\mathcal{I}}:=||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}||_{t}^{\mathcal{A}}+||\tfrac{1}{2}\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}||_{t}^{\mathcal{A}}+\left(||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}||_{t}^{\mathcal{A}}\right)^{2}.$

More generally, we state the result for time dependent linear combinations
$(\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\rangle)_{t\leq T}$ with
$\bm{\ell}:[0,T]\to\mathcal{A}$ in the set

$\mathcal{I}^{\prime}:=\left\\{\bm{\ell}:[0,T]\to\mathcal{I}:\begin{matrix}\text{
for all }t\in[0,T],\>\bm{\ell}_{t}^{\mathcolor{NavyBlue}{\mathbf{v}}}\in
C^{1}([0,T])\text{ for all }{\mathcolor{NavyBlue}{\mathbf{v}}}\in V,\\\ \text{
and }||\dot{\bm{\ell}}_{t}||_{t}^{\mathcal{A}}<\infty\text{ and
}\int_{0}^{T}||\dot{\bm{\ell}}_{t}||_{t}^{\mathcal{A}}dt<\infty\text{
a.s.}\hfill\end{matrix}\right\\},$ (2.16)

where $\dot{\bm{\ell}_{t}}:=\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V}\frac{d}{dt}\bm{\ell}_{t}^{\mathcolor{NavyBlue}{\mathbf{v}}}{\mathcolor{NavyBlue}{\mathbf{v}}}$
for all $t\in[0,T]$.

######  Lemma 2.7 (Itô’s decomposition).

Let $\bm{\ell}\in\mathcal{I}$, then

$\displaystyle
d\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle=\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (2.17)

Let $\bm{\ell}\in\mathcal{I}^{\prime}$, then

$\displaystyle
d\left\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\right\rangle=\left\langle\dot{\bm{\ell}}_{t}+\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (2.18)

###### Proof.

The full proof can be found in [6, Section 3], however we will give the reader
a sketch of the proof to illustrate the algebraic computations assuming a
finite number of non-zero terms, i.e. $\bm{\ell}:[0,T]\to T(\mathbb{R}^{2})$.
To obtain the full proof, it suffices then to apply dominated convergence
theorems. First we show in (i) that

$d\left\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\right\rangle=\left\langle\dot{\bm{\ell}}_{t}+\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\circ
dW_{t},$

and then in (ii) that

$\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\circ
dW_{t}=\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+\frac{1}{2}\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt.$

  1. (i)

First recall the definition of the bracket (2.5)

$\left\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\right\rangle=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\bm{\ell}_{t}^{\mathcolor{NavyBlue}{\mathbf{v}}}\widehat{\mathbb{W}}_{t}^{\mathcolor{NavyBlue}{\mathbf{v}}},$

which allows us to write

$\displaystyle
d\left\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\right\rangle$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\left(\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}d\bm{\ell}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}+\bm{\ell}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}d\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}\right)$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n}}\dot{\bm{\ell}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}dt+\sum_{n=1}^{\infty}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{n-1}}\left(\bm{\ell}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v1}}}}\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}dt+\bm{\ell}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v2}}}}\widehat{\mathbb{W}}_{t}^{{\mathcolor{NavyBlue}{\mathbf{v}}}}\circ
dW_{t}\right)$
$\displaystyle=\left\langle\dot{\bm{\ell}}_{t},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\circ
dW_{t}.$

The last equality comes from the definition of the projection (2.3).

  2. (ii)

We then make the transition between Stratonovich and Itô integrals:

$\displaystyle\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\circ
dW_{t}$
$\displaystyle=\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+\frac{1}{2}d\left[\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{\cdot}\right\rangle,W_{\cdot}\right]_{t}$
$\displaystyle
d\left[\left\langle\bm{\ell}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{\cdot}\right\rangle,W_{\cdot}\right]_{t}$
$\displaystyle=d\left[\int_{0}^{\cdot}\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{s}\right\rangle\circ
dW_{s},W_{\cdot}\right]_{t}$
$\displaystyle=\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt.$

∎

We note that we clearly have $T(\mathbb{R}^{2})\in\mathcal{I}$, i.e. showing
that finite linear combinations of the signature are always semimartingales.
The following example highlights that Lemma 2.7 is indeed an extension of the
usual Itô’s formula.

######  Example 2.7.1.

Fix an analytic function $f(y):=\sum_{n\geq 0}a_{n}y^{n}$ with infinite radius
of convergence which we apply to $W$. It is clear that

$f(W_{t})=\sum_{n\geq
0}a_{n}W_{t}^{n}=\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle,$

where

$\bm{\ell}=\sum_{n\geq
0}a_{n}{\mathcolor{NavyBlue}{\mathbf{2}}}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}n}=\sum_{n\geq
0}n!a_{n}{\mathcolor{NavyBlue}{\mathbf{2}}}^{\otimes n}.$

In particular, the projections read:

$\displaystyle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}$
$\displaystyle=0,$
$\displaystyle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}$
$\displaystyle=\sum_{n\geq
0}(n+1)a_{n+1}{\mathcolor{NavyBlue}{\mathbf{2}}}^{\otimes n},$
$\displaystyle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}$
$\displaystyle=\sum_{n\geq
0}(n+2)a_{n+2}{\mathcolor{NavyBlue}{\mathbf{2}}}^{\otimes n}.$

It is easy to verify that
$\bm{\ell},\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}\in\mathcal{A}$
since $f$ has infinite radius of convergence. We can thus further derive that

$\displaystyle
f^{\prime}(W_{t})=\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad
f^{\prime\prime}(W_{t})=\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle.$

On the other hand we can see that
$||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}||_{t}^{\mathcal{A}}=\sum_{n=0}^{\infty}(n+1)|a_{n+1}||W_{t}|^{n}$,
and since $g(x):=\sum_{n\geq 0}(n+1)|a_{n+1}|x^{n}$ is also analytic and
$||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}||_{t}^{\mathcal{A}}$ has
continuous sample path almost surely, then

$\sup_{t\in[0,T]}||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}||_{t}^{\mathcal{A}}<\infty.$

With similar arguments we can show that
$\sup_{t\in[0,T]}||\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}||_{t}^{\mathcal{A}}<\infty.$
This allows us to verify that $\bm{\ell}\in\mathcal{I}$. An application of
Itô’s formula (2.17) yields

$\left\langle\bm{\ell},\widehat{\mathbb{W}}_{t}\right\rangle=\bm{\ell}^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\int^{t}_{0}\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{s}\right\rangle
dW_{s}+\frac{1}{2}\int^{t}_{0}\left\langle\bm{\ell}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{s}\right\rangle
ds,$

which is equivalent to

$f(W_{t})=f(0)+\int_{0}^{t}f^{\prime}(W_{s})dW_{s}+\frac{1}{2}\int_{0}^{t}f^{\prime\prime}(W_{s})ds,$

the standard Itô’s formula. $\blacksquare$

##  3 The signature volatility model

Let $(\Omega,\mathcal{F},\mathbb{Q})$ be a probability space supporting a two-
dimensional Brownian motion $(W,W^{\perp})$. We denote by
$(\mathcal{F}_{t})_{t\geq 0}$ the filtration generated by $(W,W^{\perp})$. We
set

$\displaystyle B=\rho W+\sqrt{1-\rho^{2}}W^{\perp},$ (3.1)

for some $\rho\in[-1,1]$. We define the time-augmented process
$\widehat{W}_{t}=(t,W_{t})$ and we consider that the dynamics of the risky
asset $S$, under the risk neutral probability measure $\mathbb{Q}$, are given
by a stochastic volatility model where the volatility process $\Sigma$ is a
(possibly infinite) linear combination of the signature of $\widehat{W}$:

$\displaystyle\frac{dS_{t}}{S_{t}}$ $\displaystyle=\Sigma_{t}dB_{t},$ (3.2)
$\displaystyle\Sigma_{t}$
$\displaystyle=\left\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t}\right\rangle,$
(3.3)

where $\bm{\sigma}:[0,T]\to\mathcal{A}$ corresponds to the parameters of the
volatility process and is such that

$\displaystyle\int_{0}^{T}\mathbb{E}\left[\Sigma_{t}^{2}\right]dt<\infty.$
(3.4)

We recall the definition of the set $\mathcal{A}$ in (2.4). The condition
(3.4) ensures that the stochastic integral

$\int_{0}^{\cdot}\Sigma_{s}dB_{s}$

is well defined as an Itô integral, so that there exists a unique solution to
(3.2) given by

$S_{t}=S_{0}\exp\left(-\frac{1}{2}\int_{0}^{t}\Sigma_{s}^{2}ds+\int_{0}^{t}\Sigma_{s}dB_{s}\right),\quad
t\geq 0.$

Condition (3.4) can be made more explicit by observing that the instantaneous
variance $\Sigma^{2}_{t}$ is also linear in the signature, i.e.

$\displaystyle\Sigma_{t}^{2}=\left\langle\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\right\rangle,$
(3.5)

thanks to the shuffle product in Proposition 2.6. We note that the time-
independent case $\bm{\sigma}_{t}=\bm{\sigma}$ for all $t\in[0,T]$ and some
$\bm{\sigma}\in\mathcal{A}$, leads to
$\int_{0}^{T}\mathbb{E}\left[\Sigma_{t}^{2}\right]dt=\langle\bm{\sigma}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}{\mathcolor{NavyBlue}{\mathbf{1}}},\mathbb{E}[\widehat{\mathbb{W}}_{T}]\rangle$
where the quantity $\mathbb{E}[\widehat{\mathbb{W}}_{T}]$ can be computed
explicitly using Fawcett’s formula [33] extended to time-augmented Brownian
motions in [47, Proposition 4.10]:

$\displaystyle\mathbb{E}\left[\widehat{\mathbb{W}}_{t}\right]=\sum_{n\geq
0}\frac{t^{n}}{n!}\left({\mathcolor{NavyBlue}{\mathbf{1}}}+\frac{1}{2}{\mathcolor{NavyBlue}{\mathbf{22}}}\right)^{\otimes
n}.$ (3.6)

In practice, we will usually be interested in truncated elements
$\bm{\sigma}:[0,T]\to T^{M}(\mathbb{R}^{2})$, for some $M\in\mathbb{N}$, which
automatically satisfy the condition (3.4) since in this case $\bm{\sigma}$ has
only a finite number of non-zero terms and all the integrated moments of the
Brownian motion are finite.

Notice that for general $\bm{\sigma}:[0,T]\to\mathcal{A}$ the process $\Sigma$
is not necessarily Markovian nor a semimartingale. It is a semimartingale if
in addition $\bm{\sigma}\in\mathcal{I}^{\prime}$ thanks to Itô’s formula in
Lemma 2.7. For truncated elements $\bm{\sigma}:[0,T]\to T^{M}(\mathbb{R}^{2})$
the process $\Sigma$ is a semimartingale. So far, truncated elements have been
considered in the related literature [10, 28].

In the next subsection, we highlight the flexibility introduced by infinite
linear combinations of signature elements in terms of exact representations
and provide numerical implementations of their truncated form, i.e.

$\Sigma_{t}^{\leq M}:=\left\langle\bm{\sigma}^{\leq
M},\widehat{\mathbb{W}}_{t}\right\rangle,\quad M\geq 0,$

where $\bm{\sigma}^{\leq M}:[0,T]\to T^{M}(\mathbb{R}^{2})$ is the truncated
form of $\bm{\sigma}$ at order $M$, i.e. its $M$ first levels coincide with
$\bm{\sigma}$ and everything else is set to $0$.

###  3.1 Examples of exact representations

We first highlight the flexibility of the signature volatility model
(3.2)-(3.3) by showing that it subsumes several known and useful Markovian and
non-Markovian models based on Ornstein-Uhlenbeck processes, mean-reverting
geometric Brownian motions, square-root processes111The theoretical
justification for the representation of the square-root process is a work in
progress, it is validated numerically in Section 3.1.3 below. and processes
with path-dependent dynamics including stochastic Volterra processes.

####  3.1.1 Models based on the Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck (OU) process $X$ given by

$\displaystyle dX_{t}=\kappa(\theta-X_{t})dt+\eta dW_{t},\quad
X_{0}=x\in\mathbb{R},$ (3.7)

for $\kappa,\theta,\eta\in\mathbb{R}$, can be represented as an infinite
linear combination of the signature elements of the time-extended Brownian
motion, either in a time-independent or time-dependent way as shown in the
next Lemma.

######  Lemma 3.1.

The unique solution $X$ to (3.7) is given by

$\displaystyle
X_{t}=\left\langle\bm{\ell}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\bm{\ell}^{\textnormal{OU}}=(x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\kappa\theta{\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}})e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}-\kappa{\mathcolor{NavyBlue}{\mathbf{1}}}},$
(3.8)

such that $\bm{\ell}^{\textnormal{OU}}\in\mathcal{A}$, with
$e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}~{}}$ the exponential shuffle defined in
(2.9). Furthermore, $X$ can also be written in terms of time-dependent
coefficients:

$\displaystyle
X_{t}=\left\langle\tilde{\bm{\ell}}_{t}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\tilde{\bm{\ell}}_{t}^{\textnormal{OU}}=\theta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+e^{-\kappa
t}\left((x-\theta){\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\eta
e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\kappa{\mathcolor{NavyBlue}{\mathbf{1}}}}{\mathcolor{NavyBlue}{\mathbf{2}}}\right),$
(3.9)

###### Proof.

The proof is detailed in Appendix A.1. ∎

######  Example 3.1.1.

To be more explicit, up to order 3, the linear form of the Ornstein-Uhlenbeck
process in (3.8) reads

$\bm{\ell}^{\textnormal{OU}}=\left(x,\begin{pmatrix}-\kappa(x-\theta)\\\
\eta\end{pmatrix},\begin{pmatrix}\kappa^{2}(x-\theta)&0\\\
-\kappa\eta&0\end{pmatrix},\begin{pmatrix}-\kappa^{3}(x-\theta)&0&\\\ 0&0&\\\
&\kappa^{2}\eta&0\\\ &0&0\end{pmatrix},\cdots\right).$

$\blacksquare$

Although the representations (3.8) and (3.9) are equivalent, from a numerical
perspective (3.9) is more advantageous. In the time-independent representation
(3.8), $e^{-\kappa t}$ is approximated by the first terms of its Taylor
expansion $\sum_{n\geq 0}\frac{(-\kappa t)^{n}}{n!}$, which becomes
numerically unstable for lower orders of truncation in the region
$t>1/|\kappa|$, see Figure 1. However, in the time-dependent representation
(3.9), $e^{-\kappa t}$ is exact and even though the approximate solution
deteriorates with time, it stays stable and converges towards $\theta$, see
Figure 2.

(a) $\kappa=1,\theta=0.25,\eta=1.2$

(b) $\kappa=4,\theta=0.25,\eta=2$

Figure 1: Trajectories of an Ornstein-Uhlenbeck process against their
truncated time-independent linear representation (3.8), i.e.
$\left\langle\bm{\ell}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

(a) $\kappa=1,\theta=0.25,\eta=1.2$ (b) $\kappa=4,\theta=0.25,\eta=2$

Figure 2: Trajectories of an Ornstein-Uhlenbeck process against their
truncated time-dependent linear representation (3.9), i.e.
$\left\langle\tilde{\bm{\ell}}_{t}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

If $|\kappa t|<1$, we can see in the left-hand side of Figure 1 and Figure 2
that the truncated linear representations seems to converge quite quickly to
the explicit solution of the Ornstein-Uhlenbeck and a truncation order $M=4$
is sufficient to get a relatively close fit.

Going back to our signature volatility model, the representations of the
Ornstein-Uhlenbeck process in Lemma 3.1 combined with the shuffle property of
Proposition 2.6 show that the model (3.2)-(3.3) nests any stochastic
volatility model based on an Ornstein-Uhlenbeck process of the form

$\frac{dS_{t}}{S_{t}}=f(t,X_{t})dB_{t},\quad f(t,x)=\sum_{k\geq
0}\alpha_{k}(t)x^{k},$

for some coefficients $\alpha_{k}:[0,T]\to\mathbb{R}$ that ensure the absolute
convergence of the infinite sum. More precisely, using the representation
(3.8) or (3.9) and the shuffle property in Lemma 2.6, we can write

$\displaystyle f(t,X_{t})=\sum_{k\geq
0}\alpha_{k}(t)\left\langle\bm{\ell}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle^{k}=\left\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t}\right\rangle\quad\text{with
}\bm{\sigma}_{t}:=\sum_{k\geq
0}\alpha_{k}(t)\left(\bm{\ell}^{\textnormal{OU}}\right)^{\mathrel{\sqcup\mkern-3.2mu\sqcup}k}.$
(3.10)

This clearly includes:

  * •

The Stein-Stein model [58] for $f(t,x)=x$,

  * •

The Bergomi model [30, 15] for

$f(t,x)=\xi_{0}(t)e^{\eta x}=\xi_{0}(t)\sum_{k\geq 0}\frac{(\eta x)^{k}}{k!},$

for some $\eta\in\mathbb{R}$ and some deterministic input curve $\xi_{0}$,

  * •

The Quintic OU model [4] for

$f(t,x)=\xi_{0}(t)(\alpha_{0}+\alpha_{1}x+\alpha_{3}x^{3}+\alpha_{5}x^{5}),$

for some $\alpha_{i}\geq 0$ and some deterministic input curve $\xi_{0}$, and
any other finite polynomial of the Ornstein-Uhlenbeck process.

####  3.1.2 Models based on the mean-reverting geometric Brownian motion

More generally, the mean-reverting geometric Brownian motion (mGBM) $Y$, given
by

$\displaystyle dY_{t}=\kappa(\theta-Y_{t})dt+(\eta+\alpha Y_{t})dW_{t},\quad
Y_{0}=y\in\mathbb{R},$ (3.11)

for $\kappa,\theta,\eta,\alpha\in\mathbb{R}$ can be represented as an infinite
linear combination of the signature of the time-extended Brownian motion,
either in a time-independent or time-dependent way.

######  Lemma 3.2.

The unique solution $Y$ to (3.11) is given by

$\displaystyle
Y_{t}=\left\langle\bm{\ell}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\bm{\ell}^{\textnormal{mGBM}}=\left(y{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\left(\kappa\theta-\frac{\alpha\eta}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\left(-\left(\kappa+\frac{\alpha^{2}}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\alpha{\mathcolor{NavyBlue}{\mathbf{2}}}\right)},$
(3.12)

such that $\bm{\ell}^{\textnormal{mGBM}}\in\mathcal{A}$, with
$e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}~{}}$ as defined in (2.9). Equivalently,
$Y$ can also be written in terms of time-dependent coefficients:

$\displaystyle
Y_{t}=\left\langle\tilde{\bm{\ell}}_{t}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\tilde{\bm{\ell}}_{t}^{\textnormal{mGBM}}=\theta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+e^{-\lambda
t}\left((\bm{\ell}^{\textnormal{mGBM}}-\theta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}})\mathrel{\sqcup\mkern-3.2mu\sqcup}e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\lambda{\mathcolor{NavyBlue}{\mathbf{1}}}}\right),$
(3.13)

for some $\lambda\in\mathbb{R}$.

###### Proof.

The proof is detailed in Appendix A.2. ∎

######  Example 3.2.1.

Up to order 3, the linear form of a mean-reverting geometric Brownian motion
reads

$\bm{\ell}^{\textnormal{mGBM}}=\left(y,\begin{pmatrix}\beta y+\gamma\\\ \alpha
y+\eta\end{pmatrix},\begin{pmatrix}(\beta y+\gamma)\beta&(\beta
y+\gamma)\alpha\\\ (\alpha y+\eta)\beta&(\alpha
y+\eta)\alpha\end{pmatrix},\begin{pmatrix}(\beta y+\gamma)\beta^{2}&(\beta
y+\gamma)\beta\alpha&\\\ (\beta y+\gamma)\beta\alpha&(\beta
y+\gamma)\alpha^{2}&\\\ &(\alpha y+\eta)\beta^{2}&(\alpha
y+\eta)\beta\alpha\\\ &(\alpha y+\eta)\beta\alpha&(\alpha
y+\eta)\alpha^{2}\end{pmatrix},\cdots\right),$

where $\beta=-\left(\kappa+\frac{\alpha^{2}}{2}\right)$ and
$\gamma=\kappa\theta-\frac{\alpha\eta}{2}$. $\blacksquare$

The behaviour and numerical limitations of the linear representations of the
mean-reverting geometric Brownian motion are similar to those of the Ornstein-
Uhlenbeck, as shown in Figure 3 and Figure 4.

(a) $\kappa=1,\theta=0.25,\eta=0.5,\alpha=1.6$

(b) $\kappa=4,\theta=0.25,\eta=0.5,\alpha=2$

Figure 3: Trajectories of a mean-reverting geometric Brownian motion against
their truncated time-independent linear representation (3.12), i.e.
$\left\langle\bm{\ell}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

(a) $\kappa=1,\theta=0.25,\eta=0.5,\alpha=1.6,\lambda=0.3$ (b)
$\kappa=4,\theta=0.25,\eta=0.5,\alpha=2,\lambda=0.5$

Figure 4: Trajectories of a mean-reverting geometric Brownian motion against
their truncated time-dependent linear representation (3.13), i.e.
$\left\langle\tilde{\bm{\ell}}_{t}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

######  Remark 3.2.1.

For Monte Carlo simulations, it is better to have explicit processes. When
$\alpha\neq 0$, the mGBM solution to (3.11) can also be formulated explicitly
with

$\displaystyle
Y_{t}:=\left(y+\frac{\eta}{\alpha}+\kappa\left(\theta+\frac{\eta}{\alpha}\right)\int_{0}^{t}e^{\left(\kappa+\frac{\alpha^{2}}{2}\right)s-\alpha
W_{s}}ds\right)e^{-\left(\kappa+\frac{\alpha^{2}}{2}\right)t+\alpha
W_{t}}-\frac{\eta}{\alpha}.$ (3.14)

When $\alpha=0$, the mGBM is an Ornstein-Uhlenbeck process. $\blacksquare$

Going back to the signature volatility model (3.3). The mGBM representation
encompasses the following volatility processes

  * •

The volatility in the Hull-White model [43], i.e.

$\displaystyle\frac{d\sigma_{t}}{\sigma_{t}}=\left(\mu-\tfrac{1}{2}\xi^{2}\right)dt+\xi
dW_{t},\quad\sigma_{0}\in\mathbb{R}.$ (3.15)

  * •

The volatility in Dupire [30], i.e.

$\displaystyle\frac{d\sigma_{t}}{\sigma_{t}}=\frac{1}{2}\left(\frac{\partial\log
V_{t}(0)}{\partial
t}-\frac{b^{2}}{4}\right)dt+\frac{b}{2}dW_{t},\quad\sigma_{0}\in\mathbb{R}^{+},$
(3.16)

for some deterministic forward variance curve $V_{t}(0)$. Note that the linear
representation of $\sigma$ would be time-dependent.

####  3.1.3 Models based on the square-root process

A square-root or Cox-Ingersoll-Ross [24] (CIR) process $V$, driven by

$\displaystyle dV_{t}=\kappa(\theta-V_{t})dt+\eta\sqrt{V_{t}}dW_{t},\quad
V_{0}=v>0,$ (3.17)

seems to admit a conjectured linear representation

$\displaystyle
V_{t}=\left\langle\bm{\ell}^{\textnormal{CIR}},\widehat{\mathbb{W}}_{t}\right\rangle=\left(\left\langle\bm{\sigma}^{\textnormal{CIR}},\widehat{\mathbb{W}}_{t}\right\rangle\right)^{2},$
(3.18)

where
$\bm{\ell}^{\textnormal{CIR}}:=(\bm{\sigma}^{\textnormal{CIR}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}$
with $\bm{\sigma}^{\textnormal{CIR}}$ satisfying the non-linear algebraic
equation

$(\bm{\sigma}^{\textnormal{CIR}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}=v{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\left(\left(\kappa\theta-\frac{\eta^{2}}{4}\right){\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\kappa(\bm{\sigma}^{\textnormal{CIR}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\eta\bm{\sigma}^{\textnormal{CIR}}{\mathcolor{NavyBlue}{\mathbf{2}}}.$

The theoretical convergence, i.e. proving that
$\bm{\sigma}^{\textnormal{CIR}}\in\mathcal{A}$, seems intricate to obtain but
is ongoing in a separate work. From the numerical perspective, Figure 5 seems
to validate our formula under the Feller condition $2\kappa\theta>\eta$ [34].
Figure 15 below also provides a numerical convergence of prices of our
representation (3.18) in the context of the Heston model.

######  Example 3.2.2.

Up to order 3, the linear representation of the Cox-Ingersoll-Ross process
reads

$\displaystyle\bm{\ell}^{\textnormal{CIR}}$
$\displaystyle=\left(v,\begin{pmatrix}\beta\\\
\alpha\end{pmatrix},\begin{pmatrix}-\kappa\beta&\lambda\beta\\\
-\kappa\alpha&\lambda\alpha\end{pmatrix},\begin{pmatrix}\kappa^{2}\beta&-\left(\kappa+\frac{\beta}{2v}\right)\lambda\beta&\\\
-\kappa\lambda\beta&0&\\\
&\kappa^{2}\alpha&-\left(\kappa+\frac{\beta}{2v}\right)\lambda\alpha\\\
&-\kappa\lambda\alpha&0\end{pmatrix},\cdots\right),$

where $\beta:=-\kappa(v-\theta)-\frac{\eta^{2}}{4},\alpha:=\eta\sqrt{v}$ and
$\lambda:=\frac{\eta}{2\sqrt{v}}$. $\blacksquare$

(a) $\kappa=1,\theta=0.25,\eta=0.7$

(b) $\kappa=4,\theta=0.25,\eta=1.4$

Figure 5: Trajectories of a Cox-Ingersoll-Ross process against their truncated
time-independent linear representation (3.18), i.e.
$\left\langle\bm{\ell}^{\textnormal{CIR}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

Going back to our signature volatility model, the representation (3.18) allows
us to include in our framework the volatility of the Heston model [42].

####  3.1.4 Models based on path-dependent processes

As shown in [6], several path-dependent stochastic Volterra processes:

$Z_{t}=Z_{0}+\int_{0}^{t}K(t-s)(a_{0}+a_{1}Z_{s})ds+\int_{0}^{t}K(t-s)(b_{0}+b_{1}Z_{s})dW_{s},$

with $Z_{0},a_{0},a_{1},b_{0},b_{1}\in\mathbb{R}$, for certain locally square-
integrable kernels $K$ also admit linear representations in the form
$Z_{t}=\left\langle\bm{\ell}_{t},\widehat{\mathbb{W}}_{t}\right\rangle$. This
includes non-semimartingale processes such as the Riemann-Liouville fractional
Brownian motion

$W_{t}^{H}=\int_{0}^{t}(t-s)^{H-1/2}dW_{s},\quad H\in(0,1).$

For instance, the (time-dependent) representation of $W^{H}$ reads

$W_{t}^{H}=\left\langle\bm{\ell}_{t}^{\textnormal{RL}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\bm{\ell}_{t}^{\textnormal{RL}}=t^{H-\frac{1}{2}}\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}-H\right)^{\bar{n}}}{t^{n}}{\mathcolor{NavyBlue}{\mathbf{1}}}^{\otimes
n}{\mathcolor{NavyBlue}{\mathbf{2}}},$

where $(\cdot)^{\bar{n}}$ is the rising factorial. Please refer to [6, Section
4] for more details on such representations. Again any volatility process that
is an analytic function of such processes falls into the framework of
signature volatility models (3.2)-(3.3) thanks to the shuffle property, recall
(3.10). This includes for instance the class of Volterra polynomial models
[5], in particular Volterra [2] and rough Bergomi models [12].

As as final example, the delayed equation (DE) process $U$, given by

$\displaystyle dU_{t}$
$\displaystyle=\left(a+\int_{0}^{t}e^{\alpha(t-s)}U_{s}ds\right)dt+\left(b+\int_{0}^{t}e^{\beta(t-s)}U_{s}ds\right)dW_{t},\quad
U_{0}=u\in\mathbb{R},$ (3.19)

for some $a,b,\alpha,\beta\in\mathbb{R}$, can be represented as a linear
combination of the signature of the time-extended Brownian motion with

$\displaystyle
U_{t}=\left\langle\bm{\ell}^{\textnormal{DE}},\widehat{\mathbb{W}}_{t}\right\rangle,\quad\bm{\ell}^{\textnormal{DE}}=(u{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+a{\mathcolor{NavyBlue}{\mathbf{1}}}+b{\mathcolor{NavyBlue}{\mathbf{2}}})\left({\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-{\mathcolor{NavyBlue}{\mathbf{1}}}\left(e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\alpha{\mathcolor{NavyBlue}{\mathbf{1}}}}{\mathcolor{NavyBlue}{\mathbf{1}}}+e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\beta{\mathcolor{NavyBlue}{\mathbf{1}}}}{\mathcolor{NavyBlue}{\mathbf{2}}}\right)\right)^{-1}.$
(3.20)

The reader can refer to [6, Theorem 4.4] for more details on the linear
delayed equation process.

######  Example 3.2.3.

Up to order 3, the linear form of a delayed equation process reads

$\bm{\ell}^{\textnormal{DE}}=\left(x,\begin{pmatrix}a\\\
b\end{pmatrix},\begin{pmatrix}z&z\\\ 0&0\end{pmatrix},\begin{pmatrix}\alpha
z+a&\beta z+a&\\\ 0&0&\\\ &b&b\\\ &0&0\end{pmatrix},\cdots\right),$

see Figure 6 for a numerical illustration of the signature representation.
$\blacksquare$

(a) $a=-2,b=-0.5,z=0.25,\alpha=1,\beta=1.5$

(b) $a=-2,b=-0.5,z=0.25,\alpha=4,\beta=1.5$

Figure 6: Trajectories of a delayed equation process against their truncated
time-independent linear representation (3.20), i.e.
$\left\langle\bm{\ell}^{\textnormal{DE}},\widehat{\mathbb{W}}_{t}\right\rangle$,
for several truncation orders $M$.

###  3.2 Leverage effect

The leverage effect in the model (3.2)-(3.3) is defined as the instantaneous
correlation between the log-price $S$ and its instantaneous volatility
$|\Sigma|$ defined by

$\displaystyle\frac{d[\log S,|\Sigma|]_{t}}{\sqrt{d[\log
S]_{t}}\sqrt{d[|\Sigma|]_{t}}}.$

In practice, the leverage effect is negative on equity markets and one usually
would like to control its sign via the correlation parameter $\rho$ between
$B$ and $W$ in (3.1). Since $\Sigma$ is not necessarily positive at all time,
the leverage effect can flip sign in the model (3.2)-(3.3), which is not
realistic. The next Lemma provides a necessary and sufficient condition on the
coefficients $\bm{\sigma}$ for the leverage effect to be equal to $\rho$, at
all time.

######  Lemma 3.3.

Assume $\bm{\sigma}\in\mathcal{I}^{\prime}$, then the leverage effect is given
by

$\displaystyle\frac{d[\log S,|\Sigma|]_{t}}{\sqrt{d[\log
S]_{t}}\sqrt{d[|\Sigma|]_{t}}}=\rho\,\textnormal{sign}\left(\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\right),$

where $\textnormal{sign}(x)=1$ if $x\geq 0$ and $-1$ otherwise. In particular,
it is equal to $\rho$ for all $t\geq 0$ if and only if $\bm{\sigma}$ satisfies

$\displaystyle\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\geq
0,\quad t\geq 0.$ (3.21)

###### Proof.

The dynamics of $\log S$ are in the form

$\displaystyle d\log
S_{t}=-\frac{1}{2}\Sigma_{t}^{2}dt+\Sigma_{t}dB_{t}=-\frac{1}{2}\left\langle\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t}\right\rangle dB_{t}.$
(3.22)

The instantaneous volatility of the model is $|\Sigma|$. By an application of
Itô-Tanaka’s formula on $|\Sigma|$ and of Lemma 2.7, we have

$\displaystyle d|\Sigma|_{t}$
$\displaystyle=\textnormal{sign}\left(\Sigma_{t}\right)d\Sigma_{t}+dL^{0}(t)$
$\displaystyle=\textnormal{sign}\left(\Sigma_{t}\right)\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+\textnormal{sign}\left(\Sigma_{t}\right)\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}+\dot{\bm{\sigma}}_{t},\widehat{\mathbb{W}}_{t}\right\rangle
dt+dL^{0}(t),$

where $L^{0}$ is the local time of $\Sigma$ at 0. It follows that the leverage
effect is given by

$\displaystyle\frac{d[\log S,|\Sigma|]_{t}}{\sqrt{d[\log
S]_{t}}\sqrt{d[|\Sigma|]_{t}}}=\rho\frac{\Sigma_{t}\,\textnormal{sign}\left(\Sigma_{t}\right)\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle}{|\Sigma_{t}|\left|\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\right|}=\rho\,\textnormal{sign}\left(\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle\right).$

∎

The condition (3.21) can be made explicit in the Stein-Stein, Quintic and
Bergomi models as shown in the next example.

######  Example 3.3.1.

  * •

For the Stein-Stein model driven by a Brownian motion,
$\bm{\sigma}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}=\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$,
see (3.8), and
$\left\langle\bm{\sigma}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle=\eta\geq
0$,

  * •

For the Quintic model of [4] constructed on a Brownian motion,
$\bm{\sigma}=\alpha_{0}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha_{1}{\mathcolor{NavyBlue}{\mathbf{2}}}+\alpha_{3}{\mathcolor{NavyBlue}{\mathbf{222}}}+\alpha_{5}{\mathcolor{NavyBlue}{\mathbf{22222}}}$
so that
$\bm{\sigma}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}=\alpha_{1}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha_{3}{\mathcolor{NavyBlue}{\mathbf{22}}}+\alpha_{5}{\mathcolor{NavyBlue}{\mathbf{2222}}}$,
which gives

$\left\langle\bm{\sigma}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle=\alpha_{1}+\frac{\alpha_{3}}{2}W^{2}_{t}+\frac{\alpha_{5}}{4!}W^{4}_{t}\geq
0,\quad t\geq 0,$

as long as $\alpha_{1},\alpha_{3},\alpha_{5}$ are non negative,

  * •

For the Bergomi model, see [30, 15],
$\bm{\sigma}_{t}=\xi_{0}(t)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\eta{\mathcolor{NavyBlue}{\mathbf{2}}}}$
so that
$\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}=\eta\xi_{0}(t)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}\eta{\mathcolor{NavyBlue}{\mathbf{2}}}}$,
which gives

$\left\langle\bm{\sigma}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle=\eta\xi_{0}(t)e^{\eta
W_{t}}\geq 0,$

as long as $\xi_{0}(t),\eta$ are non negative, for all $t\leq T$.

$\blacksquare$

###  3.3 Approximated representations

More generally, if exact linear representations are not available for certain
processes, approximate representations can be obtained thanks to the universal
approximation property of path-signatures:

######  Theorem 3.4 (Universal approximation theorem [27]).

For some $t\in[0,T]$, let $K_{t}$ be a compact subset of
$\\{(\widehat{\mathbb{W}}_{s}^{\leq
2})_{s\in[0,t]}(\omega):\omega\in\Omega\\}$ and consider a continuous map
$F:K_{t}\to\mathbb{R}$. Then for every $\epsilon>0$, there exists some
$M^{\epsilon}\in\mathbb{N}$ and some $\bm{f}^{\epsilon}\in
T^{M^{\epsilon}}(\mathbb{R}^{2})$ such that

$\sup_{W\in K_{t}}\left|F\left((\widehat{\mathbb{W}}_{s}^{\leq
2})_{s\in[0,t]}\right)-\left\langle\bm{f}^{\epsilon},\widehat{\mathbb{W}}_{t}\right\rangle\right|<\epsilon,$

almost surely.

In practice, one would perform a linear regression on trajectories of a given
process $X$ against a finite linear combination of the signature elements
using the following steps:

######  Algorithm 3.5 (Regression against truncated signature).

Assume the spot volatility is of the following form

$X_{t}=F(t,(W_{s})_{s\leq t}).$

  1. 1.

Generate $N$ realizations of the Brownian motion $W$, denoted by
$W^{(1)},\ldots,W^{(N)}$,

  2. 2.

For each realization $n=1,\ldots,N$, compute $X^{(n)}$ and the truncated
signature $\widehat{\mathbb{W}}_{~{}}^{(n),\leq M}$ up to order $M$,

  3. 3.

Regress $\left(X^{(1)},\ldots,X^{(N)}\right)$ against
$\left(\widehat{\mathbb{W}}_{~{}}^{(1),\leq
M},\ldots,\widehat{\mathbb{W}}_{~{}}^{(N),\leq M}\right)$ with $L^{1}$ and
$L^{2}$ regularization $\beta_{1}$ and $\beta_{2}$ to learn the coefficients
of $\bm{f}$ that minimize

$\min_{\bm{f}\in
T^{M}(\mathbb{R}^{d})}\frac{1}{N}\frac{1}{J}\sum_{n=1}^{N}\sum_{j=1}^{J}\left|X^{(n)}_{t_{j}}-\left\langle\bm{f},\widehat{\mathbb{W}}_{t_{j}}^{(n),\leq
M}\right\rangle\right|^{2}+\sum_{k=0}^{M}\sum_{{\mathcolor{NavyBlue}{\mathbf{v}}}\in
V_{k}}\left(\beta_{1}\left|\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{k}}}}}\right|+\beta_{2}\left(\bm{\ell}^{{\mathcolor{NavyBlue}{\mathbf{i_{1}\cdots
i_{k}}}}}\right)^{2}\right).$

Such approach appeared in Cuchiero et al. [27], Lyons et al. [48], Fermanian
[36], Arribas et al. [10].

(a) $T=0.25$.

(b) $T=2$.

Figure 7: Trajectories of an inverse CIR process against their linear
regression. $\kappa=4,\theta=0.5,\eta=2$

We recall the dynamics of the inverse CIR process, being the variance in the
3/2 model [52], driven by

$dV_{t}=\kappa V_{t}(\theta-V_{t})dt+\eta V_{t}^{\frac{3}{2}}dW_{t}.$

We see in Figure 7 that a process without known linear representation can
still be approximated quite well through a linear regression.

(a) $H=0.1$.

(b) $H=0.3$.

Figure 8: Trajectories of a fractional Brownian motion against their linear
regression for $M=6$.

However, as shown on Figure 8, the regression approach does not always work
with small truncation levels. For instance, the ability to capture high
roughness in a linear functional is not trivial to achieve whilst keeping the
natural embedding $\widehat{W}_{t}=(t,W_{t})$.

We end this section with a brief comparison of approximate vs exact
representations. As seen in the previous subsections, the convergence of the
linear representation is quick for short horizons. However, when the horizon
gets too large, relatively to the parameters of the represented models, e.g.
the mean reversion rate, the truncated representations drastically
deteriorates, see Figures 1 to 6. Yet, as seen previously, linear regressions
make quite stable representations over their training time and can thus be
made over targeted horizons to control the stability. In Figures 9, 10 and 11,
we can remark that for short horizons (relative to $\kappa$), the linear
regression doesn’t fit as well as the linear representation. However for
longer horizons, where the linear representation lacks in stability, the
linear regression retains hers.

(a) $T=0.25$.

(b) $T=2$.

Figure 9: Trajectories of an Ornstein-Uhlenbeck process against their linear
representation and linear regression. $\kappa=4,\theta=0.25,\eta=2$

(a) $T=0.25$.

(b) $T=2$.

Figure 10: Trajectories of a mean-reverting geometric Brownian motion against
their linear representation and linear regression.
$\kappa=4,\theta=0.25,\eta=1,\alpha=2$

(a) $T=0.25$.

(b) $T=2$.

Figure 11: Trajectories of a square-root Cox-Ingersoll-Ross process against
their linear representation and linear regression.
$\kappa=4,\theta=0.0625,\eta=1$

However, this stability across the training horizon could easily be
interpreted as overfitting. Table 1 below displays the mean squared error
between explicit simulations of an Ornstein-Uhlenbeck process and their linear
representations (exact truncated and from a regression) where
$\kappa=4,\theta=0.25,\eta=2$ and $v=0.25$. The number of simulations and
training horizon have been set to 100 000 and 1 year with 252 time steps
respectively, see also Figure 12.

MSE | Test horizon
---|---
3 months | 6 months | 1 year | 2 years | 4 years
$M=2$ | Exact | 3.065e-05 | 5.370e-03 | 1.969e+00 | 2.536e+01 | 2.625e+02
Regression | 6.149e-03 | 8.564e-03 | 2.405e-02 | 6.239e-01 | 1.189e+01
$M=4$ | Exact | 6.541e-08 | 1.736e-05 | 1.718e+00 | 4.243e+02 | 8.177e+04
Regression | 8.002e-06 | 3.801e-05 | 1.079e-04 | 1.516e+00 | 1.213e+03
$M=6$ | Exact | 5.908e-08 | 1.167e-06 | 3.173e-01 | 1.410e+03 | 4.872e+06
Regression | 5.735e-06 | 8.426e-06 | 1.596e-07 | 1.066e+00 | 3.188e+04
Table 1: Mean-squared-error between simulations of an Ornstein-Uhlenbeck
process against its linear representation and linear regression when trained
over only 1 year. $\kappa=4,\theta=0.25,\eta=2$.

(a) $T=1$.

(b) $T=2$.

Figure 12: Trajectories of an Ornstein-Uhlenbeck process against their linear
representation and linear regression when trained over only 1 year.
$\kappa=4,\theta=0.25,\eta=2$.

##  4 The joint characteristic functional

The following theorem provides the joint conditional characteristic functional
of the log-price $\log S$ and the integrated variance
$\bar{V}:=\int_{0}^{\cdot}\Sigma^{2}_{s}ds$ in the model (3.2)–(3.3) in terms
of a solution to an infinite-dimensional system of time-dependent
$T((\mathbb{R}^{2}))$-valued Riccati differential equations.

######  Theorem 4.1.

Let $f,g:[0,T]\to\mathbb{C}$ be measurable and bounded functions and
$\bm{\sigma}:[0,T]\to\mathcal{A}$. Assume that there exists
$\bm{\psi}\in\mathcal{I}^{\prime}$, solution to the following infinite-
dimensional system of time-dependent Riccati equations

$\displaystyle-\dot{\bm{\psi}}_{t}$
$\displaystyle=\frac{1}{2}(\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}+\rho
f(t)(\bm{\sigma}_{t}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}})+\frac{1}{2}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}+\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\left(\frac{f(t)^{2}-f(t)}{2}+g(t)\right)\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},$
(4.1) $\displaystyle\bm{\psi}_{T}$ $\displaystyle=0.$ (4.2)

Define the processes

$\displaystyle U_{t}$
$\displaystyle=\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle+\int_{0}^{t}f(s)d\log
S_{s}+\int_{0}^{t}g(s)d\bar{V}_{s},$ (4.3) and $\displaystyle M_{t}$
$\displaystyle=e^{U_{t}}.$ (4.4)

Then $M$ is a local martingale. If in addition it is a true martingale, then
the following expression holds for the joint characteristic functional of
$(\log S,\bar{V})$:

$\mathbb{E}\left[\left.\exp\left(\int_{t}^{T}f(s)d\log
S_{s}+\int_{t}^{T}g(s)d\bar{V}_{s}\right)\right|\mathcal{F}_{t}\right]=\exp\left(\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle\right),\quad
t\leq T.$ (4.5)

###### Proof.

To show that $M$ is a local martingale we show that its part in $dt$ vanishes
using Itô’s formula. The dynamics of $M$ read

$\displaystyle
dM_{t}=M_{t}\left(dU_{t}+\frac{1}{2}d\left[U\right]_{t}\right).$ (4.6)

We start by deriving the dynamics of $U$ in (4.3). By using Lemma 2.7, since
$\bm{\psi}\in\mathcal{I}^{\prime}$ by assumption, we have that
$\left(\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle\right)_{t\geq
0}$ is a semimartingale with dynamics

$\displaystyle
d\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle$
$\displaystyle=\left\langle\dot{\bm{\psi}}_{t}+\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$

Combining the previous identity with
$d\bar{V}_{t}=\langle\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\rangle
dt$, recall (3.5), and the dynamics of $\log S$ in (3.22), we obtain that

$\displaystyle dU_{t}$
$\displaystyle=d\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle+f(t)d\log
S_{t}+g(t)d\bar{V}_{t}$
$\displaystyle=\left\langle\dot{\bm{\psi}}_{t}+\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}$
$\displaystyle\qquad+\left(g(t)-\tfrac{1}{2}f(t)\right)\left\langle\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\right\rangle
dt+f(t)\left\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t}\right\rangle
dB_{t}$
$\displaystyle=\left\langle\dot{\bm{\psi}}_{t}+\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}+\left(g(t)-\tfrac{1}{2}f(t)\right)\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\right\rangle
dt$
$\displaystyle\qquad+\left\langle\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+f(t)\left\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t}\right\rangle
dB_{t}.$

Using the shuffle product of Proposition 2.6 and the fact that $B$ and $W$ are
correlated, recall (3.1), we get that the quadratic variation of $U$ is

$\displaystyle d\left[U\right]_{t}$
$\displaystyle=\left\langle(\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}+2\rho
f(t)(\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\sigma}_{t})+f(t)^{2}\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2},\widehat{\mathbb{W}}_{t}\right\rangle
dt.$

This yields that the drift of $dM_{t}/M_{t}$ in (4.6) is given by

$\displaystyle\left\langle\dot{\bm{\psi}}_{t}+\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}+\left(g(t)-\tfrac{1}{2}f(t)\right)\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}\right.$
$\displaystyle\qquad\left.+\tfrac{1}{2}\left[(\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}})^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}+2\rho
f(t)(\bm{\psi}_{t}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}\mathrel{\sqcup\mkern-3.2mu\sqcup}\bm{\sigma}_{t})+f(t)^{2}\bm{\sigma}_{t}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}\right],\widehat{\mathbb{W}}_{t}\right\rangle,$

which is equal to $0$ from the Riccati equations (4.1). This shows that $M$ is
a local martingale. By assumption $M$ is even a true martingale. After
observing that the terminal value of $M$, is given by

$M_{T}=\exp\left(\int_{0}^{T}f(s)d\log
S_{s}+\int_{0}^{T}g(s)d\bar{V}_{s}\right),$

recall that $\bm{\psi}_{T}=0$, we obtain

$\displaystyle\mathbb{E}\left[\exp\left(\int_{0}^{T}f(s)d\log
S_{s}+\int_{0}^{T}g(s)d\bar{V}_{s}\right)\bigg{|}\mathcal{F}_{t}\right]=\mathbb{E}\left[M_{T}\big{|}\mathcal{F}_{t}\right]=M_{t}=\exp\left(U_{t}\right),$

which yields (4.5). ∎

Theorem 4.1 is a verification result to obtain the exponentially affine
representation of the joint characteristic functional (4.5). It disentangles
the algebraic affine structure in infinite dimension. It can be related to
[29, Theorem 4.24] if one considers the signature of the three dimensional
process $((t,W_{t},B_{t}))_{t\geq 0}$ there. It relies on the two crucial
assumptions that a well-behaved $\mathcal{I}$-valued solution $\bm{\psi}$
exists to the Riccati equation (4.1), together with the true martingality of
$M$. These assumptions seem very intricate to prove, even in one dimensional
settings, partial results in these directions for $T((\mathbb{R}))$-valued
Riccati equations can be found in [29, Section 6] and [7]. We note that no
semimartingality assumption for
$\Sigma_{t}=\langle\bm{\sigma}_{t},\widehat{\mathbb{W}}_{t},\rangle$ is
required in Theorem 4.1.

In the following two sections, we will validate the representation (4.5)
numerically and we will highlight the application of Theorem 4.1 to the
pricing and the hedging of several contingent claims by Fourier inversion
techniques. The general functions $f$ and $g$ allow for enough flexibility to
cover a broad set of contingent claims. For instance:

  1. 1.

Certain vanilla options that depend on the values $(S_{T},\bar{V}_{T})$, like
European call and put options and volatility swaps, are recovered using
constant $f$ and $g$.

  2. 2.

Geometric Asian options on the average $\frac{1}{T}\int_{0}^{T}\log S_{s}ds$
can be recovered by setting $f(s):=iu\frac{T-s}{T}$, since

$\displaystyle\int_{t}^{T}f(s)d\log S_{s}$
$\displaystyle=iu\int_{t}^{T}\frac{T-s}{T}d\log S_{s}$ (4.7)
$\displaystyle=iu\frac{1}{T}\int_{t}^{T}\log S_{s}ds-iu\frac{T-t}{T}\log
S_{t}.$ (4.8)

##  5 Pricing by Fourier methods

In this section, we show how Theorem 4.1 can be applied to price European and
Asian call and put options as well as $q$-Volatility swaps using Fourier
inversion techniques in our signature volatility model (3.2)-(3.3). All of our
numerical results validate the exponentially affine representation (4.5). We
also provide a calibration to market volatility surface for the S&P 500.

For the numerical implementation, we consider a truncated version of $S$ from
(3.2)-(3.3), denoted by $S^{\leq M}$ and defined by

$dS_{t}^{\leq M}=S_{t}^{\leq M}\left\langle\bm{\sigma}_{t}^{\leq
M},\widehat{\mathbb{W}}_{t}\right\rangle dB_{t},$

where $\bm{\sigma}_{t}^{\leq M}\in T^{M}(\mathbb{R}^{2})$ has its first $M$
levels coincide with $\bm{\sigma}_{t}$ and is $0$ elsewhere. Finally, in order
to ease notations in the sequel we assume $S_{0}=1$ (recall that the short
rate here is assumed to be $0$).

###  5.1 European options

Let us consider a European call option on $S$ with maturity $T>0$ and strike
$K>0$. Its price at time $t\leq T$ is given by
$C_{t}(T,K)=\mathbb{E}[(S_{T}-K)^{+}|\mathcal{F}_{t}]$. From Lewis [45], one
can price this option using the Fourier inversion formula

$C_{t}(S_{t};T,K)=S_{t}-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{i(u-\frac{i}{2})k_{t}}\phi_{t}\left(u-\tfrac{i}{2}\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)},$
(5.1)

where $\phi$ is the conditional characteristic function
$\phi_{t}(u)=\mathbb{E}\left[e^{iu\log\frac{S_{T}}{S_{t}}}|\mathcal{F}_{t}\right]$
and $k_{t}=\log\frac{S_{t}}{K}$.

Moreover, we add a ‘control variate’ to quicken the convergence, as in
Andersen and Andreasen [8]. Given $\sigma_{\textnormal{BS}}<\infty$, one has

$C_{t}(S_{t};T,K)=C_{t}^{\textnormal{BS}}(S_{t};T,K)-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{i(u-\frac{i}{2})k_{t}}\left(\phi_{t}\left(u-\tfrac{i}{2}\right)-\phi_{t}^{\textnormal{BS}}\left(u-\tfrac{i}{2}\right)\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)},$
(5.2)

where

$\displaystyle C_{t}^{\textnormal{BS}}(S_{t};K,T)$
$\displaystyle=\mathcal{N}(d_{1})S_{t}-\mathcal{N}(d_{2})K,$ (5.3)

with

$\displaystyle
d_{1}=\frac{1}{\sigma_{\textnormal{BS}}\sqrt{T-t}}\left(\log\frac{S_{t}}{K}+\frac{\sigma_{\textnormal{BS}}^{2}}{2}(T-t)\right),\quad
d_{2}=d_{1}-\sigma_{\textnormal{BS}}\sqrt{T-t}$ (5.4)

and

$\displaystyle\phi_{t}^{\textnormal{BS}}(u)$
$\displaystyle=\exp\left[-\frac{\sigma_{\textnormal{BS}}^{2}}{2}\left(u^{2}+iu\right)(T-t)\right].$
(5.5)

This Black-Scholes control variate can also be applied to other products and,
as will be seen in Section 6, to Fourier hedging. The quantity
$\sigma_{\textnormal{BS}}$ can be determined to ensure approximate moment
matching for instance, and one can use the characteristic function $\phi_{t}$
to approximate (by finite differences) the second order cumulant of the
distribution of the log price $\log S$.

Numerically, for the discretization of the Fourier integral, our numerical
experiments show that Gauss-Laguerre quadrature outperform other quadrature
rules in our class of models, which is in line with the empirical findings
that appeared in [54]. In particular, the higher the maturity is, the lower
the degree of the quadrature needs to be for the same set of parameters. In
addition, the use of a control variate makes computations much quicker
yielding fewer calls of the characteristic function to make the quadrature
converge, see Figure 13. Moreover, we can see that the added value of the
control variate, in terms of speed of convergence, grows with the time
horizon. See [54, Section 6] for more details on numerical refinements.

For a given signature volatility model (3.2)-(3.3), i.e. a set of parameters
$\bm{\sigma}:[0,T]\to\mathcal{A}$, an application of Theorem 4.1 with
$f(t)=iu$ and $g=0$ provides the characteristic function $\phi_{t}(u)$ modulo
the solution $\bm{\psi}$ to the $T((\mathbb{R}^{2}))$-valued Riccati equation
(4.1), which allows us to compute the price of call options in the signature
volatility model using (5.2) together with a suitable truncation and
discretization of the Riccati equation (4.1). The truncation rule for
$\bm{\psi}$ is not trivial and requires a little bit of care, because the
shuffle products cannot be exact, as each step of the discretized ODE would
double the truncation order of $\bm{\psi}$. For the numerical implementation,
we decided to fix the order $\tilde{M}$ of $\bm{\psi}$ for each step and hence
only have a shuffle product projected on $T^{\tilde{M}}(\mathbb{R}^{2})$, i.e.
$\widetilde{\mathrel{\sqcup\mkern-3.2mu\sqcup}}:(T^{\tilde{M}}(\mathbb{R}^{2}))^{2}\to
T^{\tilde{M}}(\mathbb{R}^{2})$. Obviously, choosing $\tilde{M}$ lower than
$2M$, where $M$ is the truncation order of $\bm{\sigma}$, also induces an
approximation of shuffle product in
$\bm{\sigma}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}$, which greatly
deteriorated the quality of the convergence, if not prevented it altogether.
We thus fixed $\tilde{M}=2M$ throughout our experiments. Said differently for
a given signature volatility $\bm{\sigma}\in T^{M}(\mathbb{R}^{2})$ the
truncated solution $\bm{\psi}$ to the Riccati equation (4.1) is an element of
$T^{2M}(\mathbb{R}^{2})$. Regarding the numerical discretization and in order
for the Riccati to converge in a realistic amount of time, we use Runge-Kutta
to the 4th order to solve the ODE, which computes 4 times as many points as
the Euler direct algorithm, but converges more than 4 times as fast. We also
compute the characteristic function both JIT and in parallel so that it is
drastically faster.

Figure 13: Gauss-Laguerre quadrature mean-squared-error of put pricing with
and without control variate, under Stein-Stein model.
$\kappa=1,\theta=0.25,\eta=1.2$ and $\rho=-0.5$.

In Figures 14 and 15, we compare our Signature volatility pricing using the
linear representations of the Ornstein-Uhlenbeck (3.8) and Cox-Ingersoll-Ross
(3.18) representations, truncated at order $M=4$, to the explicit Fourier
pricing of the Stein-Stein [58] and Heston [42] models. Lewis’ approach
together with Black-Scholes control variate, see (5.2), was used. This serves
as a numerical validation for Theorem 4.1 as well as for our conjectured
representation for the square-root process (3.18).

Figure 14: European call Fourier pricing of Stein-Stein model vs Signature
Ornstein-Uhlenbeck volatility model (3.8). $\kappa=1,\theta=0.25,\eta=1.2$ and
$\rho=-0.5$. Figure 15: European call Fourier pricing of Heston model vs
Signature square-root Cox-Ingersoll-Ross volatility model (3.18).
$\kappa=4,\theta=0.0625,\eta=0.7$ and $\rho=-0.7$.

In Figure 16 we compare our signature volatility pricing using the linear
mean-reverting geometric Brownian motion representation (3.12) truncated at
order $M=4$ to Monte Carlo simulations, see Remark 3.14. Lewis’ approach
together with Black-Scholes control variate was also used.

Figure 16: European call Monte Carlo pricing of mGBM volatility model vs
Fourier pricing of Signature mean-reverting geometric Brownian motion
volatility model (3.12). $\kappa=1,\theta=0.25,\eta=1.2,\alpha=0.6$ and
$\rho=-0.5711$.

In Figure 17 we work on a linear functional drawn at random under the leverage
effect condition (LER), see (3.21). In this example, the coefficients
$\bm{\sigma}^{\textnormal{LER}}$ are drawn such that

  1. 1.

$\bm{\sigma}^{\textnormal{LER}{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}}=x$,

  2. 2.

the coefficients that must remain positive have been drawn from a
$\mathcal{U}_{[0,0.5]}$,

  3. 3.

the unconstrained coefficients have been drawn from a
$\mathcal{U}_{[-0.5,0.5]}$.

Bellow is the draw used in Figure 17:

$\bm{\sigma}^{\textnormal{LER}}=\left(0.25,\begin{pmatrix}-0.0644715\\\
0.36712438\end{pmatrix},\begin{pmatrix}0.05250405&0\\\
-0.02796828&0\end{pmatrix},\begin{pmatrix}-0.3698671&0&\\\ 0.05678161&0&\\\
&0.3307244&0\\\ &0.2119773&0.2379793\end{pmatrix},\bm{0}\right).$ (5.6)

Then, using $\bm{\sigma}^{\textnormal{LER}}$, we compute Monte Carlo
simulations, i.e. we simulate Brownian motions, augment them with time and
compute their signature. We then compute the linear combination of
$\bm{\sigma}^{\textnormal{LER}}$ against each simulated signature. This
further validates Theorem 4.5 and highlights the efficiency of our numerical
implementation using Gauss-Laguerre quadrature, a Black-Scholes control
variate and the fourth-order Runge-Kutta scheme for the truncated tensor
algebra valued Riccati equation.

Figure 17: European put Monte Carlo pricing of LER linear volatility model vs
Fourier pricing of Signature LER volatility model.
$\bm{\sigma}^{\textnormal{LER}}$ defined in (5.6).

###  5.2 Asian options

Let us consider a geometric Asian call option with price
$\bar{C}_{t}(T,K)=\mathbb{E}[(\bar{S}_{T}-K)^{+}|\mathcal{F}_{t}]$, where
$\bar{S}_{t}=\exp\left(\frac{1}{T}\int_{0}^{t}\log S_{s}ds\right)$. Similarly
to (5.2), we obtain the Fourier representation

$\bar{C}_{t}(\bar{S}_{t};T,K)=\bar{C}_{t}^{\textnormal{BS}}(\bar{S}_{t};T,K)-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{i(u-\frac{i}{2})\bar{k}_{t}}\left(\bar{\phi}_{t}\left(u-\tfrac{i}{2}\right)-\bar{\phi}_{t}^{\textnormal{BS}}\left(u-\tfrac{i}{2}\right)\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)},$
(5.7)

where $\bar{\phi}$ is the characteristic function
$\bar{\phi}_{t}(u)=\mathbb{E}\Big{[}e^{iu\log\frac{\bar{S}_{T}}{\bar{S}_{t}}}\Big{|}\mathcal{F}_{t}\Big{]}$
and $\bar{k}_{t}=\log\frac{\bar{S}_{t}}{K}$ with

$\bar{\phi}_{t}^{\textnormal{BS}}=\exp\left(-\frac{\sigma_{\textnormal{BS}}^{2}}{2}\left(u^{2}\frac{(T-t)^{3}}{3T^{2}}+iu\frac{(T-t)^{2}}{2T}\right)\right),$
(5.8)

and where

$\bar{C}_{t}^{\textnormal{BS}}(\bar{S}_{t};T,K):=\mathcal{N}(\bar{d}_{1})\bar{S}_{t}\exp\left(\frac{T-t}{T}\log
S_{t}-\frac{\sigma^{2}_{\textnormal{BS}}}{2}\left(\frac{(T-t)^{2}}{2T}-\frac{(T-t)^{3}}{3T^{2}}\right)\right)+\mathcal{N}(\bar{d}_{2})K,$
(5.9)

and

$\bar{d}_{2}:=\frac{1}{\sigma_{\textnormal{BS}}\sqrt{\frac{(T-t)^{3}}{3T^{2}}}}\left(\log\frac{\bar{S}_{t}}{K}+\frac{T-t}{T}\log
S_{t}-\frac{\sigma^{2}_{\textnormal{BS}}}{2}\frac{(T-t)^{2}}{2T}\right),\quad\bar{d}_{1}:=\bar{d}_{2}-\sigma_{\textnormal{BS}}\sqrt{\frac{(T-t)^{3}}{3T^{2}}}.$
(5.10)

In Figure 18, we compare our signature volatility Fourier pricing for Asian
options using the truncated, at order $M=4$, representation of the linear
Ornstein-Uhlenbeck (3.8) to Monte Carlo simulations. Lewis’ approach together
with Black-Scholes control variate was also used.

Figure 18: Asian put Monte Carlo pricing of Stein-Stein model vs Signature
Ornstein-Uhlenbeck volatility model. $\kappa=1,\theta=0.25,\eta=1.2$ and
$\rho=-0.9$.

###  5.3 q-Volatility swaps

The payoff of a $q$-volatility swap is $R_{T}^{q}-K^{q}$ where
$R_{T}^{q}=\left(\frac{1}{T}\bar{V}_{T}\right)^{q}$ is the realized
$q$-volatility and $K^{q}$ the $q$-volatility strike. The aim in pricing
$q$-volatility swaps is to find the fair strike price, i.e.
$K^{q}=\mathbb{E}[R_{T}^{q}]$. This is made possible by Laplace inversion:

######  Theorem 5.1 (Schürger [56]).

Let $X\geq 0$ be a random variable, then

$\mathbb{E}[X^{q}]=\frac{q}{\Gamma(1-q)}\int_{0}^{\infty}\frac{1-\mathbb{E}\left[e^{-uX}\right]}{u^{q+1}}du,\quad
q\in(0,1).$

It is straightforward to see that setting $f=0$ and $g(t)=-\frac{u}{T}$ in
(4.3) gives $\tilde{M}_{0}(u)=\mathbb{E}[e^{-\frac{u}{T}\bar{V}_{T}}]$, which
allows us to compute analytically $q$-volatility swaps in the framework of the
signature volatility models with

$\mathbb{E}\left[\left(\frac{1}{T}\bar{V}_{T}\right)^{q}\right]=\frac{q}{\Gamma(1-q)}\int_{0}^{\infty}\frac{1-\tilde{M}_{0}(u)}{u^{q+1}}du.$

Specifically, the fair strike of the volatility swap, i.e. $q=\frac{1}{2}$, is
thus of the form

$\mathbb{E}\left[\sqrt{\frac{1}{T}\bar{V}_{T}}\right]=\frac{1}{2\sqrt{\pi}}\int_{0}^{\infty}\frac{1-\tilde{M}_{0}(u)}{u^{3/2}}du.$

Moreover, the fair strike of the variance swap, i.e. $q=1$, can be written in
closed form thanks to Fawcett’s formula (3.6) for time-independent
representations, i.e.

$K_{T}^{1}=\frac{1}{T}\left\langle\bm{\sigma}^{\mathrel{\sqcup\mkern-3.2mu\sqcup}2}{\mathcolor{NavyBlue}{\mathbf{1}}},\mathbb{E}[\widehat{\mathbb{W}}_{T}]\right\rangle.$

(a) $q=\frac{1}{2}$ (volatility swap).

(b) $q=1$ (variance swap).

Figure 19: Strikes of $q$-volatility swaps as a function of the maturity, for
a Signature Ornstein-Uhlenbeck volatility model.
$\kappa=1,\theta=0.25,\eta=1.2$ and $\rho=-0.7$.

###  5.4 Calibration examples

Another way to learn the dynamics $\bm{\sigma}$ of the signature volatility is
to calibrate it against implied volatility surfaces. In this section, we
consider $\bm{\sigma}\in T^{3}(\mathbb{R}^{2})$. We recall that the number of
non-zero terms in an object of $T^{M}(\mathbb{R}^{d})$, is
$\frac{d^{M+1}-1}{d-1}$. In our case, $d=2$ and we use $M=3$ as it is
sufficient to make relatively good fits, i.e. we calibrate 15 parameters. For
Figures 20, 21 and 22, we minimized the MSE between a given implied volatility
and the one of the signature model, using the differential evolution global
minimizer [59].

Figure 20: Implied volatility of time-independent Signature model calibrated
against Heston implied volatility surface.
$V_{0}=0.25^{2},\kappa=4,\theta=0.0625,\eta=0.7$ and $\rho=-0.6$. Figure 21:
Implied volatility of time-dependent Signature model calibrated against
Quintic implied volatility surface. $\xi_{0}(t)\equiv
0.25^{2},H=-0.1,\epsilon=\frac{1}{52},\alpha=\\{0.01,1,0,0.214,0,0.227\\}$ and
$\rho=-0.65$.

Finally, we show how our signature volatility model is well adapted to produce
a good fit with daily SPX implied volatility surface data purchased from the
CBOE website https://datashop.cboe.com/.

Figure 22: Implied volatility of time-dependent Signature model calibrated
against SPX data 2017-05-19.

##  6 Quadratic hedging by Fourier methods

The signature volatility model (3.2)-(3.3) generates an incomplete market in
general (unless the correlation $\rho$ between the two Brownian motions $B$
and $W$ is $\pm 1$). Therefore contingent claims on the stock $S$ cannot be
perfectly hedged. We will consider quadratic hedging methods instead, we refer
to [57] for a detailed overview of quadratic hedging approaches. We will show
that in our setup, quadratic hedging remains highly tractable in signature
volatility models using Fourier techniques on the conditional characteristic
function (4.5). We do this in two steps:

  1. 1.

We first solve in Section 6.1 the quadratic hedging problem for a generic
contingent claim $\xi$, in a general stochastic volatility model driven by the
two dimensional Brownian motion $(W,W^{\perp})$. The hedging strategy depends
on the quantities that appear in the martingale representation theorem of
$\mathbb{E}[\xi|\mathcal{F}_{t}]$. We provide a concise proof in this setting.

  2. 2.

We then show in Section 6.2 that in the setting of a signature volatility
model and for contingent claims that admit Fourier representation, such as
European and Asian call and put options, the hedging strategy can be recovered
from the conditional characteristic function (4.5).

###  6.1 A generic solution

Let $\xi$ be an $\mathcal{F}_{T}$-measurable non-negative random variable such
that $\mathbb{E}[\xi^{2}]<\infty$ that we are looking to hedge using a self-
financing portfolio. We recall that $(\mathcal{F}_{t})_{t\geq 0}$ is the
filtration generated by $(W,W^{\perp})$. A self-financing hedging portfolio
$X$ consists of an initial wealth $X_{0}\in\mathbb{R}$ and a progressively
measurable strategy $(\vartheta_{u})_{u\leq T}$ of the amount of shares
invested in asset $S$ given in (3.2) at time $u\leq T$. It has the following
dynamics

$\displaystyle
X_{t}^{\alpha}=X_{0}+\int_{0}^{t}\vartheta_{u}dS_{u}=X_{0}+\int_{0}^{t}\alpha_{u}dB_{u},$
(6.1)

with $\alpha_{u}:=\vartheta_{u}S_{u}\Sigma_{u}$. The set of admissible hedging
strategies $\alpha$ is defined by

$\displaystyle\mathcal{H}=\left\\{\alpha\text{ progressively measurable such
that }\int_{0}^{T}\mathbb{E}\left[\alpha_{s}^{2}\right]ds<\infty\right\\}.$
(6.2)

We stress that in this section we do not impose specific dynamics for the
stochastic volatility $\Sigma$, i.e. (3.3), $\Sigma$ is only assumed to be
adapted to the Brownian motion $(W,W^{\perp})$.

A quadratic hedging strategy aims at minimizing the following objective
function

$\displaystyle
J(X_{0},\alpha)=\mathbb{E}\left[\left(X_{T}^{\alpha}-\xi\right)^{2}\right]$
(6.3)

over $X_{0}\in\mathbb{R}$ and $\alpha\in\mathcal{H}$.

The next theorem provides a solution of the quadratic hedging problem using
the martingale representation theorem. Note that
$(\mathbb{E}[\xi|\mathcal{F}_{t}])_{t\leq T}$ is a square integrable
martingale with terminal value $\xi$ at $T$. An application of the martingale
representation theorem [44, Theorem 4.15] yields the existence of two
progressively measurable and square integrable processes $Z$ and $Z^{\perp}$
such that

$\displaystyle\mathbb{E}\left[\xi|\mathcal{F}_{t}\right]=\xi-\int_{t}^{T}Z_{s}dW_{s}-\int_{t}^{T}Z^{\perp}_{s}dW^{\perp}_{s}.$
(6.4)

######  Theorem 6.1.

The value of the quadratic hedging problem is given by

$\displaystyle\inf_{X_{0}\in\mathbb{R},\alpha\in\mathcal{H}}J(X_{0},\alpha)=\mathbb{E}\left[\int_{0}^{T}\left(Z_{t}^{2}+(Z_{t}^{\perp})^{2}\right)dt-\int_{0}^{T}\left(\rho
Z_{t}+\sqrt{1-\rho^{2}}Z_{t}^{\perp}\right)^{2}dt\right],$ (6.5)

where the optimum is attained for $(X_{0}^{*},\alpha^{*})$ given by

$\displaystyle
X_{0}^{*}=\mathbb{E}\left[\xi\right]\quad\text{and}\quad\alpha^{*}_{t}=\rho
Z_{t}+\sqrt{1-\rho^{2}}Z_{t}^{\perp},\quad t\leq T.$ (6.6)

###### Proof.

We start by expanding the square inside the objective function $J$ in (6.3):

$(X_{T}^{\alpha}-\xi)^{2}=(X_{T}^{\alpha})^{2}-2\xi X_{T}^{\alpha}+\xi^{2},$

and we write

$\displaystyle(X_{T}^{\alpha})^{2}$
$\displaystyle=X_{0}^{2}+2X_{0}\int_{0}^{T}\alpha_{u}dB_{u}+\left(\int_{0}^{T}\alpha_{u}dB_{u}\right)^{2}$
$\displaystyle\xi X_{T}^{\alpha}$
$\displaystyle=\mathbb{E}[\xi]X_{0}+\int_{0}^{T}\mathbb{E}[\xi|\mathcal{F}_{s}]dX_{s}^{\alpha}+\int_{0}^{T}X_{s}^{\alpha}d\mathbb{E}[\xi|\mathcal{F}_{s}]+\int_{0}^{T}\alpha_{s}\left(\rho
Z_{s}+\sqrt{1-\rho^{2}}Z_{s}^{\perp}\right)ds$

so that by using the fact that stochastic integrals are centered and by Itô’s
isometry we obtain that

$\displaystyle J(X_{0},\alpha)$
$\displaystyle=\mathbb{E}\left[(X_{0}-\mathbb{E}[\xi])^{2}+\left(\xi^{2}-\mathbb{E}[\xi]^{2}\right)+\int_{0}^{T}\left(\alpha_{s}^{2}-2\alpha_{s}\left(\rho
Z_{s}+\sqrt{1-\rho^{2}}Z_{s}^{\perp}\right)\right)ds\right]$
$\displaystyle=\mathbb{E}\left[(X_{0}-\mathbb{E}[\xi])^{2}+\int_{0}^{T}\left(Z_{s}^{2}+(Z_{s}^{\perp})^{2}\right)ds+\int_{0}^{T}\left(\alpha_{s}-\alpha_{s}^{*}\right)^{2}ds-\int_{0}^{T}\left(\rho
Z_{s}+\sqrt{1-\rho^{2}}Z_{s}^{\perp}\right)^{2}ds\right].$

It immediately follows that the minimum is clearly attained for $X_{0}^{*}$
and $\alpha^{*}$ given by (6.6) and the claimed expression for the value
function (6.5) follows. ∎

###  6.2 Fourier implementation in the signature volatility model

We now illustrate how the optimal hedging strategy $X_{0}^{*}$ and
$\alpha^{*}$ given in (6.6) can be recovered numerically from the knowledge of
the conditional characteristic function (4.5) in the specific case of a
signature volatility model, i.e. when $\Sigma$ is of the form (3.3) and for
contingent claims that admit a Fourier representation.

####  6.2.1 European call option

In this section, we consider a European call options. In order to implement
the quadratic hedging, the idea is to re-express the Fourier inversion formula
(5.2) in terms of the process $M_{t}(u)$ in (4.4) with $f(t)=iu$ and $g=0$ and
apply Itô. Since in this case

$M_{t}(u)=\phi_{t}(u)e^{iu\log S_{t}},$

the representation (5.2) directly leads to

$C_{t}(S_{t};T,K)=C_{t}^{\textnormal{BS}}(S_{t};T,K)-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{i(u-\frac{i}{2})\log
K}\left(M_{t}\left(u-\tfrac{i}{2}\right)-M_{t}^{\textnormal{BS}}\left(u-\tfrac{i}{2}\right)\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)},$
(6.7)

where

$M_{t}^{\textnormal{BS}}(u)=\phi_{t}^{\textnormal{BS}}(u)e^{iu\log S_{t}},$

and $\phi_{t}^{\textnormal{BS}}$ as defined in (5.5). Setting
$w(u):=\frac{K}{\pi}\frac{e^{-i(u-\frac{i}{2})\log K}}{u^{2}+\frac{1}{4}}$ and
$\tilde{u}:=(u-\frac{i}{2})$, an application of Itô’s formula yields that

$\displaystyle dC_{t}$
$\displaystyle=dC_{t}^{\textnormal{BS}}+\int_{0}^{\infty}\,\mathfrak{Re}\left[w(u)\left(dM_{t}(\tilde{u})-dM_{t}^{\textnormal{BS}}(\tilde{u})\right)\right]du$
$\displaystyle=\Delta_{t}^{\textnormal{BS}}dS_{t}+\left(\Theta_{t}^{\textnormal{BS}}+\tfrac{1}{2}\Gamma_{t}^{\textnormal{BS}}(S_{t}\Sigma_{t})^{2}\right)dt$
$\displaystyle\qquad+\int_{0}^{\infty}\,\mathfrak{Re}\Big{[}w(u)\Big{(}M_{t}(\tilde{u})\left(dU_{t}+\tfrac{1}{2}d[U]_{t}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-
M_{t}^{\textnormal{BS}}(\tilde{u})\left(d\log
M_{t}^{\textnormal{BS}}(\tilde{u})+\tfrac{1}{2}d[\log
M^{\textnormal{BS}}(\tilde{u})]_{t}\right)\Big{)}\Big{]}du,$

where $\Delta_{t}^{\textnormal{BS}}=\frac{\partial}{\partial
S_{t}}C_{t}^{\textnormal{BS}}$,
$\Theta_{t}^{\textnormal{BS}}=\frac{\partial}{\partial
t}C_{t}^{\textnormal{BS}}$ and
$\Gamma_{t}^{\textnormal{BS}}=\frac{\partial^{2}}{\partial
S_{t}^{2}}C_{t}^{\textnormal{BS}}$. Furthermore, using equalities between
Black-Scholes Greeks, one can show that

$\displaystyle dC_{t}$
$\displaystyle=\Delta_{t}^{\textnormal{BS}}S_{t}\Sigma_{t}dB_{t}+\frac{\Theta_{t}^{\textnormal{BS}}}{\sigma_{\textnormal{BS}}^{2}}\left(\sigma_{\textnormal{BS}}^{2}-\Sigma_{t}^{2}\right)dt$
$\displaystyle\qquad+\int_{0}^{\infty}\,\mathfrak{Re}\Big{[}w(u)\Big{(}M_{t}(\tilde{u})\left(\left\langle\bm{\psi}_{t}(\tilde{u})|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+i\tilde{u}\Sigma_{t}dB_{t}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-
M_{t}^{\textnormal{BS}}(\tilde{u})\left[\tfrac{1}{2}\left(\tilde{u}^{2}+i\tilde{u}\right)\left(\sigma_{\textnormal{BS}}^{2}-\Sigma_{t}^{2}\right)dt+i\tilde{u}\Sigma_{t}dB_{t}\right]\Big{)}\Big{]}du$
$\displaystyle=\Delta_{t}^{\textnormal{BS}}S_{t}\Sigma_{t}dB_{t}+$
$\displaystyle\qquad+\int_{0}^{\infty}\,\mathfrak{Re}\Big{[}w(u)\Big{(}M_{t}(\tilde{u})\left(\left\langle\bm{\psi}_{t}(\tilde{u})|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}+i\tilde{u}\Sigma_{t}dB_{t}\right)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad-
M_{t}^{\textnormal{BS}}(\tilde{u})i\tilde{u}\Sigma_{t}dB_{t}\Big{)}\Big{]}du.$

The final equality comes from the fact that $\Theta_{t}^{\textnormal{BS}}$ can
also be written as a Fourier integral. This gives option price dynamics of the
following form

$\displaystyle dC_{t}$
$\displaystyle=Z_{t}dW_{t}+Z_{t}^{\perp}dW_{t}^{\perp},$ (6.8)

where $Z$ and $Z^{\perp}$ are defined as follows

$\displaystyle Z_{t}$
$\displaystyle=\Sigma_{t}S_{t}\Delta_{t}^{\textnormal{BS}}\rho+\int_{0}^{\infty}\,\mathfrak{Re}\left[\zeta_{t}(\tilde{u})w(u)\right]du$
$\displaystyle Z_{t}^{\perp}$
$\displaystyle=\Sigma_{t}S_{t}\Delta_{t}^{\textnormal{BS}}\sqrt{1-\rho^{2}}+\int_{0}^{\infty}\,\mathfrak{Re}\left[\zeta_{t}^{\perp}(\tilde{u})w(u)\right]du,$
with $\displaystyle\zeta_{t}(u):$
$\displaystyle=iu\Sigma_{t}\left(M_{t}(u)-M_{t}^{\textnormal{BS}}(u)\right)\rho+M_{t}(u)\left\langle\bm{\psi}_{t}(u)|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle$
$\displaystyle\zeta_{t}^{\perp}(u):$
$\displaystyle=iu\Sigma_{t}\left(M_{t}(u)-M_{t}^{\textnormal{BS}}(u)\right)\sqrt{1-\rho^{2}}.$

This allows us to solve the quadratic hedging problem in (6.6) numerically for
European call options in the framework of signature volatility models.
Moreover, applying the put-call parity allows us to easily extend it to
European put options.

In Figure (23), we simulate price trajectories under the Stein-Stein model
[58] and compare the performance of the explicit hedging strategy to the
Fourier hedging of the signature Ornstein-Uhlenbeck volatility model for a
European put option with multiple strikes and two horizons.

Figure 23: P&L of Stein-Stein model (blue) vs signature Ornstein-Uhlenbeck
volatility model (orange) quadratic hedging strategies.
$\kappa=1,\theta=0.25,\eta=1.2$ and $\rho=-0.6$.

Remark that both strategies mostly coincide. It illustrates that both the
method in Section 6.2 works well within our framework and that the truncated
signature representation of the volatility process is a good approximation
when horizons are short enough, relative to the stiffness of the model’s
parameters.

####  6.2.2 Asian call option

In the same spirit as for the European call option, we will express the
Fourier inversion formula (5.2) in terms of the process $\bar{M}_{t}(u)$ in
(4.5) with $f(s):=iu\frac{T-s}{T}$ and $g=0$ and apply Itô. Using the same
notations as in Section 5.2, for all $t\leq T$

$\displaystyle\exp\left(\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle\right)$
$\displaystyle=\mathbb{E}\left[\left.\exp\left(\int_{t}^{T}f(s)d\log
S_{s}\right)\right|\mathcal{F}_{t}\right]$ (6.9)
$\displaystyle=\mathbb{E}\left[\left.\exp\left(iu\log\frac{\bar{S}_{T}}{\bar{S}_{t}}-iu\frac{T-t}{T}\log
S_{t}\right)\right|\mathcal{F}_{t}\right],$ (6.10)

so that, recall (4.3),

$\displaystyle\bar{\phi}_{t}(u)$
$\displaystyle=\exp\left(\left\langle\bm{\psi}_{t},\widehat{\mathbb{W}}_{t}\right\rangle+iu\frac{T-t}{T}\log
S_{t}\right)$ $\displaystyle=\bar{M}_{t}(u)\exp\left(-\int_{0}^{t}f(s)d\log
S_{s}+iu\frac{T-t}{T}\log S_{t}\right)$
$\displaystyle=\bar{M}_{t}(u)e^{-iu\log\bar{S}_{t}}.$

Thus one can now write $\bar{C}$ as a Fourier integral on $\bar{M}$

$\displaystyle\bar{C}_{t}(\bar{S}_{t};T,K)$
$\displaystyle=\bar{S}_{t}-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{-i(u-\frac{i}{2})\log
K}M_{t}\left(u-\tfrac{i}{2}\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)},$

together with its Black-Scholes control variate version

$\bar{C}_{t}(\bar{S}_{t};T,K)=\bar{C}_{t}^{\textnormal{BS}}(\bar{S}_{t};T,K)-\frac{K}{\pi}\int_{0}^{\infty}\,\mathfrak{Re}\left[e^{i(u-\frac{i}{2})\log
K}\left(\bar{M}_{t}\left(u-\tfrac{i}{2}\right)-\bar{M}_{t}^{\textnormal{BS}}\left(u-\tfrac{i}{2}\right)\right)\right]\frac{du}{\left(u^{2}+\tfrac{1}{4}\right)}$
(6.11)

with

$\bar{M}_{t}^{\textnormal{BS}}(u):=\bar{\phi}_{t}^{\textnormal{BS}}(u)e^{iu\log\bar{S}_{t}},$
(6.12)

where $\bar{\phi}^{\textnormal{BS}}$ is defined in (5.8) and
$\bar{C}^{\textnormal{BS}}$ in (5.9).

Finally, $w(u)$ and $\tilde{u}$ are defined as in Subsection 6.2.1 and with
very similar computations, one can get $\bar{Z}$ and $\bar{Z}^{\perp}$ as
follows

$\displaystyle\bar{Z}_{t}$
$\displaystyle=\Sigma_{t}S_{t}\bar{\Delta}_{t}^{\textnormal{BS}}\rho+\int_{0}^{\infty}\,\mathfrak{Re}\left[\bar{\zeta}_{t}(\tilde{u})w(u)\right]du$
$\displaystyle\bar{Z}_{t}^{\perp}$
$\displaystyle=\Sigma_{t}S_{t}\bar{\Delta}_{t}^{\textnormal{BS}}\sqrt{1-\rho^{2}}+\int_{0}^{\infty}\,\mathfrak{Re}\left[\bar{\zeta}_{t}^{\perp}(\tilde{u})w(u)\right]du,$
with $\displaystyle\bar{\zeta}_{t}(u):$
$\displaystyle=\frac{T-t}{T}iu\Sigma_{t}\left(\bar{M}_{t}(u)-\bar{M}_{t}^{\textnormal{BS}}(u)\right)\rho+\bar{M}_{t}(u)\left\langle\bm{\psi}_{t}(u)|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle$
$\displaystyle\bar{\zeta}_{t}^{\perp}(u):$
$\displaystyle=\frac{T-t}{T}iu\Sigma_{t}\left(\bar{M}_{t}(u)-\bar{M}_{t}^{\textnormal{BS}}(u)\right)\sqrt{1-\rho^{2}},$

where
$\bar{\Delta}_{t}^{\textnormal{BS}}=\frac{\partial}{\partial\bar{S}_{t}}\bar{C}_{t}^{\textnormal{BS}}$.

This allows us to solve the quadratic hedging problem in (6.6) numerically for
Asian call and put options in the framework of sig-volatility models.

In Figure 24, we simulate price trajectories under mean-reverting geometric
Brownian motion volatility model and compare the performance of the Black-
Scholes hedging strategy, simply as a point of reference with
$\sigma_{\textnormal{BS}}=\theta$, to the Fourier hedging of the signature
mean-reverting geometric Brownian motion volatility model for an Asian put
option.

Figure 24: P&L of mGBM volatility model (blue) vs signature mGBM volatility
model (orange) quadratic hedging strategies.
$\kappa=1,\theta=0.25,\eta=1.2,\alpha=0.6$ and $\rho=-0.6$.

Remark that the signature volatility systematically outperforms, in terms of
minimized squared P&L, the naive Black-Scholes quadratic hedging strategy by
10 to 25%. This suggests that our framework might specifically be relevant for
path-dependent options and more complex volatility dynamics where hedging
strategies are not known explicitly or tractable.

##  Appendix A Proofs

###  A.1 Proof of Lemma 3.1

######  Lemma A.1.

$\bm{\sigma}^{\textnormal{OU}}=\left(x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\kappa\theta{\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}-\kappa{\mathcolor{NavyBlue}{\mathbf{1}}}}$
solves the equation

$\bm{\sigma}^{\textnormal{OU}}=(x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\kappa\theta{\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}})-\kappa\bm{\sigma}^{\textnormal{OU}}{\mathcolor{NavyBlue}{\mathbf{1}}}.$
(A.1)

###### Proof.

Straightforward by applying Proposition 2.3. ∎

Using [6, Theorem 4.2], one has

$\bm{\sigma}^{\textnormal{OU}}=\left(x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\kappa\theta{\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}-\kappa{\mathcolor{NavyBlue}{\mathbf{1}}}}\in\mathcal{A}.$

We can therefore define the process

$X_{t}=\left\langle\bm{\sigma}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle.$

Now, showing

$\displaystyle\left(\bm{\sigma}^{\textnormal{OU}}\right)^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$
$\displaystyle=y{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}},$
(A.2)
$\displaystyle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}$
$\displaystyle=\kappa\theta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\kappa\bm{\sigma}^{\textnormal{OU}},$
(A.3)
$\displaystyle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}$
$\displaystyle=\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}},$
(A.4)
$\displaystyle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}$
$\displaystyle=0.$ (A.5)

makes it clear that
$||\bm{\sigma}^{\textnormal{OU}}||_{t}^{\mathcal{I}}<\infty\text{ a.s.}$.
Moreover, remarking that
$\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}{\mathcolor{NavyBlue}{\mathbf{1}}}=\bm{\sigma}^{\textnormal{OU}}-x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\eta{\mathcolor{NavyBlue}{\mathbf{2}}}$
by using (A.1), allows us to write

$\int_{0}^{t}\left\langle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}},\widehat{\mathbb{W}}_{s}\right\rangle
ds=\left\langle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}{\mathcolor{NavyBlue}{\mathbf{1}}},\widehat{\mathbb{W}}_{t}\right\rangle<\infty,$

and thus have
$\int_{0}^{t}||\bm{\sigma}^{\textnormal{OU}}||_{s}^{\mathcal{I}}ds<\infty\text{
a.s.}$ and $\bm{\sigma}^{\textnormal{OU}}\in\mathcal{I}$.

We are now ready to apply Proposition 2.7

$\displaystyle dY_{t}$
$\displaystyle=d\left\langle\bm{\sigma}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle$
(A.6)
$\displaystyle=\left\langle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\sigma}^{\textnormal{OU}}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (A.7)
$\displaystyle=\left\langle\kappa\theta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\kappa\bm{\sigma}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (A.8)
$\displaystyle=\left(\kappa\theta-\kappa\left\langle\bm{\sigma}^{\textnormal{OU}},\widehat{\mathbb{W}}_{t}\right\rangle\right)dt+\eta
dW_{t}$ (A.9) $\displaystyle=\kappa\left(\theta-Y_{t}\right)dt+\eta dW_{t}.$
(A.10)

By uniqueness of the solution of the Ornstein-Uhlenbeck, the representation
(3.8) follows.

###  A.2 Proof of Lemma 3.2

######  Lemma A.2.

$\bm{\sigma}^{\textnormal{mGBM}}=\left(y{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\left(\kappa\theta-\frac{\alpha\eta}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}-\left(\kappa+\frac{\alpha^{2}}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\alpha{\mathcolor{NavyBlue}{\mathbf{2}}}}$
solves the equation

$\displaystyle\bm{\sigma}^{\textnormal{mGBM}}=\left(x{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\left(\kappa\theta-\frac{\alpha\eta}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)+\bm{\sigma}^{\textnormal{mGBM}}\left(-\left(\kappa+\frac{\alpha^{2}}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\alpha{\mathcolor{NavyBlue}{\mathbf{2}}}\right).$
(A.11)

###### Proof.

Straightforward by applying Proposition 2.3. ∎

Using [6, Theorem 4.2], one has

$\bm{\sigma}^{\textnormal{mGBM}}=\left(y{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\left(\kappa\theta-\frac{\alpha\eta}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\eta{\mathcolor{NavyBlue}{\mathbf{2}}}\right)e^{\mathrel{\sqcup\mkern-3.2mu\sqcup}-\left(\kappa+\frac{\alpha^{2}}{2}\right){\mathcolor{NavyBlue}{\mathbf{1}}}+\alpha{\mathcolor{NavyBlue}{\mathbf{2}}}}\in\mathcal{A}.$

We can therefore define the process

$Y_{t}=\left\langle\bm{\sigma}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle.$

Moreover, showing

$\displaystyle\left(\bm{\sigma}^{\textnormal{mGBM}}\right)^{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}$
$\displaystyle=y{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}},$
(A.12)
$\displaystyle\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}$
$\displaystyle=\left(\kappa\theta-\tfrac{\alpha\eta}{2}\right){\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\left(\kappa+\tfrac{\alpha^{2}}{2}\right)\bm{\sigma}^{\textnormal{mGBM}},$
(A.13)
$\displaystyle\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}}$
$\displaystyle=\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha\bm{\sigma}^{\textnormal{mGBM}},$
(A.14)
$\displaystyle\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}}$
$\displaystyle=\alpha\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha^{2}\bm{\sigma}^{\textnormal{mGBM}},$
(A.15)

makes it clear that
$||\bm{\sigma}^{\textnormal{mGBM}}||_{t}^{\mathcal{I}}<\infty$. As it requires
much more care to prove
$\int_{0}^{t}||\bm{\sigma}^{\textnormal{mGBM}}||_{s}^{\mathcal{I}}ds<\infty$,
it is left for the interested reader to refer to [6, Section 7] for a detailed
proof.

We can thus finally use Proposition 2.7

$\displaystyle dY_{t}$
$\displaystyle=d\left\langle\bm{\sigma}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle$
(A.16)
$\displaystyle=\left\langle\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{1}}}}+\tfrac{1}{2}\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{22}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dt+\left\langle\bm{\sigma}^{\textnormal{mGBM}}|_{{\mathcolor{NavyBlue}{\mathbf{2}}}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (A.17)
$\displaystyle=\left\langle\left(\kappa\theta-\tfrac{\alpha\eta}{2}\right){\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}-\left(\kappa+\tfrac{\alpha^{2}}{2}\right)\bm{\sigma}^{\textnormal{mGBM}}+\tfrac{1}{2}\left(\alpha\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha^{2}\bm{\sigma}^{\textnormal{mGBM}}\right),\widehat{\mathbb{W}}_{t}\right\rangle
dt$ (A.18)
$\displaystyle\qquad+\left\langle\eta{\color[rgb]{0.0600000000000001,0.46,1}\textup{{\o{}}}}+\alpha\bm{\sigma}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle
dW_{t}.$ (A.19)
$\displaystyle=\left(\kappa\theta-\kappa\left\langle\bm{\sigma}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle\right)dt+\left(\eta+\alpha\left\langle\bm{\sigma}^{\textnormal{mGBM}},\widehat{\mathbb{W}}_{t}\right\rangle\right)dW_{t}$
(A.20) $\displaystyle=\kappa\left(\theta-Y_{t}\right)dt+\left(\eta+\alpha
Y_{t}\right)dW_{t}$ (A.21)

By uniqueness of the solution of the mean-reverting geometric Brownian motion,
the representation (3.12) follows.

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