# Signature-based models: theory and calibration

Christa Cuchiero Vienna University, Department of Statistics and Operations
Research, Data Science Uni Vienna, Kolingasse 14-16 1, A-1090 Wien, Austria,
<EMAIL_ADDRESS>Guido Gazzani Vienna University, Department of
Statistics and Operations Research, Kolingasse 14-16 1, A-1090 Wien, Austria,
<EMAIL_ADDRESS>Sara Svaluto-Ferro University of Verona, Department
of Economics, Via Cantarane 24, 37129 Verona, Italy,
<EMAIL_ADDRESS>
The authors gratefully acknowledge financial support by the FWF project I 3852
and through grant Y 1235 of the FWF START-program.

###### Abstract

We consider asset price models whose dynamics are described by linear
functions of the (time extended) signature of a primary underlying process,
which can range from a (market-inferred) Brownian motion to a general
multidimensional continuous semimartingale. The framework is universal in the
sense that classical models can be approximated arbitrarily well and that the
model’s parameters can be learned from all sources of available data by simple
methods. We provide conditions guaranteeing absence of arbitrage as well as
tractable option pricing formulas for so-called sig-payoffs, exploiting the
polynomial nature of generic primary processes. One of our main focus lies on
calibration, where we consider both time-series and implied volatility surface
data, generated from classical stochastic volatility models and also from S&P
500 index market data. For both tasks the linearity of the model turns out to
be the crucial tractability feature which allows to get fast and accurate
calibrations results.

Keywords: signature methods, calibration of financial models, Monte Carlo
methods, linear (infinite dimensional) systems, polynomial processes
MSC (2010) Classification: 91B70, 62P05, 65C20.

###### Contents

  1. 1 Introduction
    1. 1.1 Relation to the literature
  2. 2 Signature: definition and properties
    1. 2.1 Basic notions
    2. 2.2 Universal approximation theorem
    3. 2.3 Stochastic Stratonovich Taylor expansion
  3. 3 The model
    1. 3.1 Definition and first properties
    2. 3.2 Absence of arbitrage and universality properties
    3. 3.3 The expected signature of ${S}_{n}(\ell)$
    4. 3.4 Pricing of sig-payoffs
  4. 4 Calibration
    1. 4.1 Calibration to time-series data
      1. 4.1.1 Regression using price data
      2. 4.1.2 Regression using spot volatility data
      3. 4.1.3 Numerical examples
    2. 4.2 Calibration to option prices
      1. 4.2.1 Monte Carlo pricing
      2. 4.2.2 Variance reduction with sig-payoffs
      3. 4.2.3 Model’s performance
      4. 4.2.4 Time-dependent parameters
    3. 4.3 Joint calibration to time-series data and option prices
  5. A Expected signature of correlated Brownian motions
  6. B Calibration using sig-payoffs

##  1 Introduction

In the past few years data-driven models have successfully entered the area of
stochastic modeling and mathematical finance. The paradigm of calibrating a
few well interpretable parameters has changed to learning the model’s
characteristics as a whole, thereby exploiting all available sources of data.
Thus highly parametric and overparametrized models have gained more and more
importance. On the one hand this has opened the door to robust and more data-
driven model selection mechanisms, while on the other hand model classes still
have to be chosen in a way to guarantee first principles from finance like “no
arbitrage”. Relying on different universal approximation theorems leads to
different well-suited classes of dynamic processes that can serve both
purposes.

One class of such financial models are so-called _neural stochastic
differential equations_ (SDEs) which are defined as Itô-diffusions where the
drift and the volatility function are parameterized via neural networks (see
e.g. Gierjatowicz et al. (2020); Cuchiero et al. (2020); Cohen et al. (2021)).
Another class of models, considered in Perez Arribas et al. (2020) and
inspiring the current work, are so-called _Sig-SDEs_. These are again Itô-
diffusions, however in this case the characteristics are linear functions of
the _signature_ (more precisely introduced below) of some (properly extended)
driving Brownian motion.

We consider here a related approach, where the asset price model itself is
parameterized as a linear function of the signature of a primary underlying
process. This primary process can either be a classical driving signal, e.g. a
Brownian motion, but also a more general tractable stochastic model describing
well observable quantities.

Before going into the details of the current model framework, let us first
explain the mathematical significance of _signature_ , a notion which goes
back to Chen (1977, 1957) and plays a particular important role in the context
of rough path theory initiated by Lyons (1998). Indeed, we consider here the
_time extended signature_ of an $\mathbb{R}^{d}$-valued path which serves as
linear regression basis for continuous path functionals, since

  * •

it is _point-separating_ , as its final value uniquely determines the
underlying path;

  * •

linear functions on the signature form an algebra of continuous functions
(with respect to a certain variation distance) that contains 1. More
precisely, every polynomial on the signature may be realized as a linear
function via the so-called shuffle product $\shuffle$.

The Stone-Weierstrass theorem therefore yields a Universal Approximation
Theorem (UAT), telling that continuous path functionals on compact sets can be
uniformly approximated by a linear function of the time extended signature.

Therefore signature-based methods provide a non-parametric way to extract
characteristic features (linearly) from time series data, which is essential
in machine learning tasks in finance. This explains why these techniques
become more and more popular in econometrics and mathematical finance, see
e.g., Buehler et al. (2020); Kalsi et al. (2020); Perez Arribas et al. (2020);
Lyons et al. (2020); Ni et al. (2021); Bayer et al. (2021); Min and Hu (2021);
Akyildirim et al. (2022) and the references therein.

We consider here signature-based models with the goal to provide a data-
driven, universal, tractable and easy to calibrate model for a set of traded
assets $S$. For the sake of exposition we shall assume throughout that $S$ is
one-dimensional. As already mentioned above the main ingredient of our
modeling framework is a $d$-dimensional _primary process_
$X=(X_{t}^{1},\dots,X_{t}^{d})_{t\geq 0}$, in most case augmented with time
$t$, where $X$ is supposed to be a continuous semimartingale. We shall denote
the time augmented process via $(\widehat{X}_{t})_{t\geq
0}=(t,X_{t}^{1},\dots,X_{t}^{d})_{t\geq 0}$ and assume that its signature
denoted by $\widehat{\mathbb{X}}$ serves a linear regression basis for $S$.
Indeed, $S$ is modeled/approximated via a process $S_{n}(\ell)$ defined as

$\displaystyle S_{n}(\ell)_{t}:=\ell(\widehat{\mathbb{X}}_{t}),$ (1.1)

where $\ell$ is a linear map of the signature of $\widehat{X}$ up to degree
$n\in\mathbb{N}$ that has to be inferred from data (see Definition 3.3 for
further details). The attractiveness of this model class arises from several
important features that we summarize in the sequel.

No arbitrage:

As shown in Section 3.2, the model of form (1.1) can also be expressed in
terms of stochastic integrals. From this representation conditions
guaranteeing no-arbitrage can in turn be easily deduced.

Universality:

As we argue in Example 3.15 classical stochastic volatility models driven by
Brownian motions with sufficiently regular coefficients can be arbitrarily
well approximated by models of form (1.1), when choosing the driving Brownian
motions (possibly modified with some drift) as primary process $X$. Locally in
time this can be inferred from the stochastic Taylor expansion that we state
in Proposition 2.14, which gives quantitative estimates. It can also seen as a
consequence of the Stone-Weierstrass theorem in the way formulated e.g. in
Lemma 5.2 in Bayer et al. (2021) (see also Cuchiero and Möller (2022)),
however without a convergence rate.

Tractable option pricing formulas for sig-payoffs:

By relying on the above UAT and in turn on approximations via so-called sig-
payoffs of the form $L(\widehat{\mathbb{S}}_{T}({\ell}))$ (see also Lyons et
al. (2020)), where $\widehat{\mathbb{S}}$ denotes the signature of
$(\widehat{S}_{t})_{t}:=(t,S_{t})_{t}$ and $L$ a linear map, this kind of
approximate options pricing reduces to the computation of the expected
signature of $\widehat{X}$. Indeed, in Section 3.3 we first show how to
express $\widehat{\mathbb{S}}_{T}({\ell})$ in terms of the signature of
$\widehat{X}$ and then translate this to the computation of
$\mathbb{E}_{\mathbb{Q}}[L(\widehat{\mathbb{S}}_{T}({\ell})\rangle]$ under
some pricing measure $\mathbb{Q}$. These formulas can then be applied whenever
$\mathbb{E}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}]$ can be easily computed.
This is the case for highly generic primary processes of the form

$\displaystyle\mathrm{d}\widehat{X}_{t}=\mathbf{b}(\widehat{\mathbb{X}}_{t})\mathrm{d}t+\sqrt{\mathbf{a}(\widehat{\mathbb{X}}_{t})}\mathrm{d}B_{t},$

where $\mathbf{b}$ and $\mathbf{a}$ are linear maps. Indeed, as shown in
Cuchiero et al. (2022b) these processes can be seen as projections of extended
tensor algebra valued polynomial processes (introduced in Cuchiero et al.
(2012); Filipović and Larsson (2016)), which implies that the expected
signature can be computed by solving a linear ODE. This ODE is usually
infinite dimensional, but if $\widehat{X}$ is itself a polynomial process it
becomes finite dimensional. We exemplify this polynomial process point of view
by means of time extended correlated Brownian motions, a particular simple
polynomial process, in Section A.

Note that similarly to polynomial approximations the approximation of vanilla
call and put options via sig-payoffs is not straightforward and can involve
several problems as we point out in Section B. Nevertheless sig-payoffs can be
used for variance reduction techniques (see Section 4.2.2) and are interesting
in their own right as certain path dependent options like Asian forwards fall
into this class.

Calibration to time series data:

The tractability of the model class given by (1.1) becomes particularly clear
in view of calibration tasks. Indeed, when the goal is to calibrate to times
series data of prices $(S_{t_{i}})_{i=1}^{N}$ or spot volatility/spot variance
$(V_{t_{i}})_{i=1}^{N}$, this task reduces to a simple linear regression. We
illustrate this well working procedure on out-of-sample data generated from
Heston, SABR and multivariate Black-Scholes models in Section 4.1.

Calibration to options:

When calibrating to the market’s volatility surface (see Section 4.2), we
exploit again the linearity of the model by precomputing Monte Carlo samples
of $\widehat{\mathbb{X}}$ and then performing a standard optimization to find
the parameters of the linear map $\ell$. By initializing the parameters of
$\ell$ appropriately this optimization task actually becomes a convex problem,
which makes it particularly tractable. On simulated and real market data
(S$\&$P 500 index) we show that a full calibration to the volatility surface,
in particular when using time dependent parameters, is highly accurate and
very fast. Note that _sampling from the calibrated model_ is also particularly
easy since it just means computing a scalar product between the parameters and
the trajectory of the signature.111The codes which we used to generate the
results in Section 4 are available at https://github.com/GuidoGazzani-
ai/sigsde$\\_$calibration.

Let us remark that in (1.1) the signature process of the time extended primary
process $\widehat{X}$ could be replaced by other processes representing the
features of $S$. Indeed, instead of $\widehat{\mathbb{X}}$ one could for
instance use randomized signature as introduced in Cuchiero et al. (2021a, b)
and work with a linear model of this randomized signature. Not all of the
above properties, in particular the pricing formulas of the sig-payoffs, could
then be preserved, but the tractability properties for calibration would
remain valid and could potentially be speeded up even further, if the analog
of $\widehat{\mathbb{X}}$ can be precomputed faster. We shall exploit this in
a follow-up paper on VIX-calibration.

###  1.1 Relation to the literature

Let us here briefly summarize some relevant and recent literature on
calibration of (data-driven) asset price models. Closely related to the
current work is the inspiring paper by Perez Arribas et al. (2020) (see also
Perez Arribas (2020)). Indeed, the class of Sig-SDEs considered there can be
embedded in our framework by choosing a properly extended one-dimensional
Brownian motion (namely a lead-lag transformation) as primary process. This is
due to the fact that (1.1) can also be expressed in terms of stochastic
integrals as stated in Proposition 3.9. While in Perez Arribas et al. (2020)
calibration was considered for certain options within the Black-Scholes model,
we here provide calibration results to both, time-series data and volatility
surfaces generated from classical stochastic volatility models and also from
real market data. A crucial difference is that the model in Perez Arribas et
al. (2020) is calibrated to sig-payoffs that approximate vanilla call and
puts. As we encountered some problems with this procedure (as outlined in
Appendix B), we rely on Monte Carlo pricing which is fast due to the
possibility to precompute all samples of $\widehat{\mathbb{X}}$ in advance.

Other promising calibration results, also in view of joint calibration of SPX
and VIX options, have for instance been achieved in the following recent
papers Gierjatowicz et al. (2020); Guo et al. (2022); Guyon (2021, 2022);
Jacquier et al. (2021); Rosenbaum and Zhang (2021); Rømer (2022); Vadori
(2022), see also the references therein.

The remainder of the article is structured as follows: in Section 2 we recall
important concepts concerning the signature of continuous semimartingales,
while Section 3 builds on these results to introduce our model framework.
Section 4 is then dedicated to present our calibration results.

##  2 Signature: definition and properties

We start by introducing basic notions related to the definition of the
signature of an $\mathbb{R}^{d}$-valued continuous semimartingale. For similar
introductions to the concept of signature we refer e.g. to Section 2.2. in
Bayer et al. (2021).

###  2.1 Basic notions

For each $n\in\mathbb{N}_{0}$ consider the $n$-fold tensor product of
$\mathbb{R}^{d}$ given by

$(\mathbb{R}^{d})^{\otimes 0}:=\mathbb{R},\qquad(\mathbb{R}^{d})^{\otimes
n}:=\underbrace{\mathbb{R}^{d}\otimes\cdots\otimes\mathbb{R}^{d}}_{n}.$

For $d\in{\mathbb{N}}$, we define the extended tensor algebra on
$\mathbb{R}^{d}$ as

$T((\mathbb{R}^{d})):=\\{\textbf{a}:=(a_{0},\dots,a_{n},\dots):a_{n}\in(\mathbb{R}^{d})^{\otimes
n}\\}.$

Similarly we introduce the truncated tensor algebra of order $N\in\mathbb{N}$

$T^{(N)}(\mathbb{R}^{d}):=\\{\textbf{a}\in T((\mathbb{R}^{d})):a_{n}=0,\forall
n>N\\},$

and the tensor algebra
$T(\mathbb{R}^{d}):=\bigcup_{N\in{\mathbb{N}}}T^{(N)}(\mathbb{R}^{d}).$ Note
that $T^{(N)}({\mathbb{R}}^{d})$ has dimension
$\sum_{i=0}^{N}d^{i}=(d^{N+1}-1)/(d-1)$.

For each $\textbf{a},\textbf{b}\in T((\mathbb{R}^{d}))$ and
$\lambda\in{\mathbb{R}}$ we set

$\displaystyle\textbf{a}+\textbf{b}$
$\displaystyle:=(a_{0}+b_{0},\dots,a_{n}+b_{n},\dots),$
$\displaystyle\lambda\textbf{a}$ $\displaystyle:=(\lambda a_{0},\dots,\lambda
a_{n},\dots),$ $\displaystyle\textbf{a}\otimes\textbf{b}$
$\displaystyle:=(c_{0},\dots,c_{n},\dots),$

where $c_{n}:=\sum_{k=0}^{n}a_{k}\otimes b_{n-k}$. Observe that
$(T((\mathbb{R}^{d})),+,\cdot,\otimes)$ is a real non-commutative algebra with
neutral element 1$=(1,0,\dots,0,\dots)$.

For a multi-index $I:=(i_{1},\ldots,i_{n})$ we set $|I|:=n$. We also consider
the empty index $I:=\emptyset$ and set $|I|:=0$. If $n\geq 1$ or $n\geq 2$ we
set $I^{\prime}:=(i_{1},\ldots,i_{n-1})$, and
$I^{\prime\prime}:=(i_{1},\ldots,i_{n-2})$, respectively. We also use the
notation

$\\{I\colon|I|=n\\}:=\\{1,\ldots,d\\}^{n},$

omitting the parameter $d$ whenever this does not introduce ambiguity. Observe
that multi-indices can be identified with words, as it is done for instance in
Lyons et al. (2020).

Next, for each $|I|\geq 1$ we set

$e_{I}:=e_{i_{1}}\otimes\cdots\otimes e_{i_{n}},$

where $e_{1},\ldots,e_{d}$ denote the canonical basis vectors of
$\mathbb{R}^{d}$. Observe that the set $\\{e_{I}\colon|I|=N\\}$ is an
orthonormal basis of $(\mathbb{R}^{d})^{\otimes N}$. Denoting by
$e_{\emptyset}$ the basis element corresponding to
$({\mathbb{R}}^{d})^{\otimes 0}$, each element of $\textbf{a}\in
T(({\mathbb{R}}^{d}))$ can thus be written as

$\textbf{a}=\sum_{|I|\geq 0}a_{I}e_{I},$

for some $a_{I}\in{\mathbb{R}}$. Finally, for each $\textbf{a}\in
T({\mathbb{R}}^{d})$ and each $\textbf{b}\in T(({\mathbb{R}}^{d}))$ we set

$\langle\textbf{a},\textbf{b}\rangle:=\sum_{|I|\geq 0}\langle
a_{I},b_{I}\rangle.$

Observe in particular that $b_{I}=\langle e_{I},\textbf{b}\rangle$.

Throughout the paper we consider a filtered probability space
$(\Omega,{\mathcal{F}},({\mathcal{F}}_{t})_{t\geq 0},{\mathbb{P}})$. Whenever
not specified, stochastic processes are always supposed to be defined there.
We are now ready to define the signature of an $\mathbb{R}^{d}$-valued
continuous semimartingale.

######  Definition 2.1.

Let $(X_{t})_{t\in[0,T]}$ be a continuous $\mathbb{R}^{d}$-valued
semimartingale. The _signature of $X$_ is the $T(({\mathbb{R}}^{d}))$-valued
process $(s,t)\mapsto{\mathbb{X}}_{s,t}$ whose components are recursively
defined as

$\langle e_{\emptyset},\mathbb{X}_{s,t}\rangle:=\textup{1},\qquad\langle
e_{I},\mathbb{X}_{s,t}\rangle:=\int_{s}^{t}\langle
e_{I^{\prime}},\mathbb{X}_{s,r}\rangle\circ\mathrm{d}X_{r}^{i_{n}},$

for each $I=(i_{1},\ldots,i_{n})$ , $I^{\prime}=(i_{1},\ldots,i_{n-1})$ and
$0\leq s\leq t\leq T$, where $\circ$ denotes the Stratonovich integral. Its
projection ${\mathbb{X}}^{N}$ on $T^{(N)}(\mathbb{R}^{d})$ is given by

$\mathbb{X}_{s,t}^{N}=\sum_{|I|\leq N}\langle e_{I},\mathbb{X}_{s,t}\rangle
e_{I}$

and is called _signature of $X$ truncated at level $N$_. If $s=0$, we use the
notation ${\mathbb{X}}_{t}$ and ${\mathbb{X}}_{t}^{N}$, respectively.

Observe that the signature of $X$ and the signature of $X-c$ coincides for
each $c\in{\mathbb{R}}$. Moreover, with an equivalent notation we can write

$\displaystyle{\mathbb{X}}_{t}$
$\displaystyle=\bigg{(}1,\int_{0}^{t}1\circ\mathrm{d}X_{s}^{1},\ldots,\int_{0}^{t}1\circ\mathrm{d}X_{s}^{d},\int_{0}^{t}\bigg{(}\int_{0}^{s}1\circ\mathrm{d}X_{r}^{1}\bigg{)}\circ\mathrm{d}X_{s}^{1},$
$\displaystyle\qquad\int_{0}^{t}\bigg{(}\int_{0}^{s}1\circ\mathrm{d}X_{r}^{1}\bigg{)}\circ\mathrm{d}X_{s}^{2},\ldots,\int_{0}^{t}\bigg{(}\int_{0}^{s}1\circ\mathrm{d}X_{r}^{d}\bigg{)}\circ\mathrm{d}X_{s}^{d},\ldots\bigg{)}.$

Using Itô integrals this can thus be rewritten as

$\displaystyle{\mathbb{X}}_{t}$
$\displaystyle=\bigg{(}1,X_{t}^{1}-X^{1}_{0},\ldots,X_{t}^{d}-X_{0}^{d},\int_{0}^{t}(X_{s}^{1}-X^{1}_{0})\mathrm{d}X_{s}^{1}+\frac{1}{2}[X^{1}]_{t},$
$\displaystyle\qquad\int_{0}^{t}(X_{s}^{1}-X^{1}_{0})\mathrm{d}X_{s}^{2}+\frac{1}{2}[X^{1},X^{2}]_{t},\ldots,\int_{0}^{t}(X_{s}^{d}-X^{d}_{0})\mathrm{d}X_{s}^{d}+\frac{1}{2}[X^{d}]_{t},\ldots\bigg{)},$

where the square brackets denote the quadratic variation and covariation
processes. In particular, by the definition of the Stratonovich integral and
Itô’s formula we can write

$\langle
e_{(i,i)},{\mathbb{X}}_{t}\rangle=\int_{0}^{t}\bigg{(}\int_{0}^{s}1\circ\mathrm{d}X^{i}_{r}\bigg{)}\circ\mathrm{d}X^{i}_{s}=\frac{1}{2}(X_{t}^{i}-X_{0}^{i})^{2}=\langle
e_{i},{\mathbb{X}}_{t}\rangle^{2},$

showing that the quadratic expression on the right hand side has a linear
representation. This property generalizes to every polynomial function. For
the precise statement we first need to introduce the shuffle product.

######  Definition 2.2.

For every two multi-indices $I:=(i_{1},\ldots,i_{n})$ and
$J:=(j_{1},\ldots,j_{m})$ the _shuffle product_ is defined recursively as

$\displaystyle e_{I}\shuffle e_{J}:=(e_{I^{\prime}}\shuffle e_{J})\otimes
e_{i_{n}}+(e_{I}\shuffle e_{J^{\prime}})\otimes e_{j_{m}},$

with $e_{I}\shuffle e_{\emptyset}:=e_{\emptyset}\shuffle e_{I}=e_{I}$. It
extends to $\textbf{a},\textbf{b}\in T({\mathbb{R}}^{d})$ as

$\textbf{a}\shuffle\textbf{b}=\sum_{|I|,|J|\geq 0}a_{I}b_{J}(e_{I}\shuffle
e_{J}).$

Observe that $(T({\mathbb{R}}^{d}),+,\shuffle)$ is a commutative algebra,
which in particular means that the shuffle product is associative and
commutative.

The proof of the rough paths version of the next result for can be found for
instance in Ree (1958) or Lyons et al. (2007).

######  Proposition 2.3 (Shuffle property).

Let $(X_{t})_{t\in[0,T]}$ be a continuous $\mathbb{R}^{d}$-valued
semimartingale and $I,J$ be two multi-indices. Then

$\langle e_{I},\mathbb{X}\rangle\langle e_{J},\mathbb{X}\rangle=\langle
e_{I}\shuffle e_{J},\mathbb{X}\rangle.$ (2.1)

###### Proof.

The result follows by induction using the chain rule for Stratonovich
integrals. ∎

######  Example 2.4.

Let $(X_{t})_{t\in[0,T]}$ be a continuous $\mathbb{R}^{d}$-valued
semimartingale with $X_{0}=0$. Then the (Stratonovich) integration by parts
formula yields, for any $i,j\in\\{1,\dots,d\\}$

$\displaystyle\langle e_{i},\mathbb{X}_{T}\rangle\langle
e_{j},\mathbb{X}_{T}\rangle=X_{T}^{i}X_{T}^{j}$
$\displaystyle=\int_{0}^{T}X_{t}^{i}\circ{\mathrm{d}}X_{t}^{j}+\int_{0}^{T}X_{t}^{j}\circ{\mathrm{d}}X_{t}^{i},$
$\displaystyle=\langle e_{i}\otimes e_{j},\mathbb{X}_{T}\rangle+\langle
e_{j}\otimes e_{i},\mathbb{X}_{T}\rangle$ $\displaystyle=\langle e_{i}\shuffle
e_{j},\mathbb{X}_{T}\rangle.$

######  Example 2.5.

Let $(X_{t})_{t\in[0,T]}$ be a continuous $\mathbb{R}$-valued semimartingale
with $X_{0}=0$. Then $\langle e_{1},{\mathbb{X}}_{T}\rangle=X_{T}-X_{0}$.
Since

$\underbrace{e_{1}\shuffle\ldots\shuffle
e_{1}}_{k\text{-times}}=k!\underbrace{e_{1}\otimes\ldots\otimes
e_{1}}_{k\text{-times}},$

Proposition 2.3 yields

$\displaystyle{\mathbb{X}}_{T}=(1,X_{T}-X_{0},\frac{1}{2}(X_{T}-X_{0}),\frac{1}{3!}(X_{T}-X_{0})^{3},\ldots,\frac{1}{k!}(X_{T}-X_{0})^{k},\ldots).$

We recall now an important property of the signature. The result is known in
the rough paths literature (see for instance Boedihardjo et al. (2016)), but
can also be proved directly in the simpler situation of a continuous
semimartingale that contains time as strictly monotone component.

######  Lemma 2.6 (Uniqueness of the signature).

Let $(X_{t})_{t\in[0,T]}$ and $(Y_{t})_{t\in[0,T]}$ be two continuous
${\mathbb{R}}^{d}$-valued semimartingales with $X_{0}=Y_{0}=0$. Set
$\widehat{X}_{t}:=(t,X_{t})$, $\widehat{Y}_{t}:=(t,Y_{t})$ and let
$\widehat{\mathbb{X}}$ and $\widehat{\mathbb{Y}}$ be the corresponding
signature processes. Then $\widehat{\mathbb{X}}_{T}=\widehat{\mathbb{Y}}_{T}$
if and only if $X_{t}=Y_{t}$ for each $t\in[0,T]$.

###### Proof.

Denote by $0,\ldots,d$ the indices of $\widehat{X}$ and $\widehat{Y}$, where 0
corresponds to the time component. The claim follows by noticing that
$\langle(e_{i}\shuffle(e_{0}^{\otimes k}))\otimes
e_{0},\widehat{\mathbb{X}}_{T}\rangle=\int_{0}^{T}X_{t}^{i}\frac{t^{k}}{k!}\mathrm{d}t$,
which implies that $\widehat{\mathbb{X}}_{T}$ uniquely determines
$\widehat{X}_{t}$ for every $t\in[0,T]$. ∎

######  Remark 2.7.

Observe that $(t^{k}/k!)_{k\in{\mathbb{N}}_{0}}$ is a basis of
$L^{2}([0,T],\mathrm{d}t)$ and

$\langle(e_{i}\shuffle(e_{0}^{\otimes k}))\otimes
e_{0},\widehat{\mathbb{X}}_{T}\rangle=\int_{0}^{T}X_{t}^{i}\frac{t^{k}}{k!}\mathrm{d}t$

is the corresponding projection of $X^{i}$ on $t^{k}/k!$. This property can be
used to explicitly construct a sequence of polynomials with random
coefficients on $[0,T]$ converging almost surely to $X^{i}$ on
$L^{2}([0,T],\mathrm{d}t)$. It also establishes a link between signature-based
methods and quantization methods (see for instance Luschgy and Pagès (2002);
Pagès and Printems (2005) and Bonesini et al. (2021); Tissot-Daguette (2021)
for recent advances).

Let us now introduce the definition of the half-shuffle product (see also
Ebrahimi-Fard and Patras (2015), Diehl et al. (2020), where it is usually
denoted with $\prec$). Given an ${\mathbb{R}}^{d}$-valued continuous
semimartingale, this operation permits to write the signature of
${\mathbb{X}}^{N}$ as a linear map of ${\mathbb{X}}$. This result will be
useful later when we will need to compute the signature of a model given by a
linear combination of terms of $\widehat{\mathbb{X}}^{N}$ (see Section 3.4).

######  Definition 2.8.

Set $I=(i_{1},\ldots,i_{n})$ and $J=(j_{1},\ldots,j_{m})$ for some
$n\in{\mathbb{N}}$ and $m>0$. We define the half-shuffle
${\widetilde{\shuffle}}$ as $e_{I}{\widetilde{\shuffle}}e_{\emptyset}:=0$ and

$e_{I}{\widetilde{\shuffle}}e_{J}:=(e_{I}\shuffle e_{J^{\prime}})\otimes
e_{j_{m}}.$

######  Lemma 2.9.

For each multi-indices $I=(i_{1},\ldots,i_{n})$ and $J=(j_{1},\ldots,j_{m})$
it holds

$\int_{0}^{t}\langle e_{I},{\mathbb{X}}_{s}\rangle\circ\mathrm{d}\langle
e_{J},{\mathbb{X}}_{s}\rangle=\langle
e_{I}{\widetilde{\shuffle}}e_{J},{\mathbb{X}}_{t}\rangle,$ (2.2)

for each $t\geq 0$.

###### Proof.

By definition of the Stratonovich integral, Proposition 2.3, and by the
definition of the signature we know that

$\int_{0}^{t}\langle e_{I},{\mathbb{X}}_{s}\rangle\circ\mathrm{d}\langle
e_{J},{\mathbb{X}}_{s}\rangle=\int_{0}^{t}\langle
e_{I},{\mathbb{X}}_{s}\rangle\langle
e_{J^{\prime}},{\mathbb{X}}_{s}\rangle\circ\mathrm{d}\langle
e_{j_{m}},{X}_{s}\rangle=\langle(e_{I}\shuffle e_{J^{\prime}})\otimes
e_{j_{m}},{\mathbb{X}}_{t}\rangle.$

∎

In order to combine the value of the signature on different time intervals
Chen’s identity going back to Chen (1957, 1977) turns out to be fundamental.
For the reader convenience, we propose here a direct proof using the
definition of Stratonovich integrals.

######  Lemma 2.10 (Chen’s identity).

Let $(X_{t})_{t\in[0,T]}$ be an ${\mathbb{R}}^{d}$-valued semimartingale. Then

${\mathbb{X}}_{s,t}={\mathbb{X}}_{s,u}\otimes{\mathbb{X}}_{u,t}$

for each $s\leq u\leq t\leq T$. This can equivalently be written as

$\langle e_{I},{\mathbb{X}}_{s,t}\rangle=\sum_{e_{I_{1}}\otimes
e_{I_{2}}=e_{I}}\langle e_{I_{1}},{\mathbb{X}}_{s,u}\rangle\langle
e_{I_{2}},{\mathbb{X}}_{u,t}\rangle,$

for each multi-index $I$.

###### Proof.

We proceed by induction on the length of the multi-index $I$. For $|I|=0$ we
have $I=\emptyset$ and the statement is clear. Suppose now that the claim
holds for each $|J|<n$ and set $I=(i_{1},\ldots,i_{n})$. Then, applying Chen’s
identity to $\langle e_{I^{\prime}},\mathbb{X}_{s,r}\rangle$ yields

$\displaystyle\langle e_{I},\mathbb{X}_{s,t}\rangle$
$\displaystyle=\int_{s}^{t}\langle
e_{I^{\prime}},\mathbb{X}_{s,r}\rangle\circ\mathrm{d}{X}_{r}^{i_{n}}$
$\displaystyle=\int_{s}^{u}\langle
e_{I^{\prime}},\mathbb{X}_{s,r}\rangle\circ\mathrm{d}{X}_{r}^{i_{n}}+\int_{u}^{t}\sum_{e_{I_{1}^{\prime}}\otimes
e_{I_{2}^{\prime}}=e_{I^{\prime}}}\langle
e_{I_{1}^{\prime}},\mathbb{X}_{s,u}\rangle\langle
e_{I_{2}^{\prime}},\mathbb{X}_{u,r}\rangle\circ\mathrm{d}{X}_{r}^{i_{n}}$
$\displaystyle=\langle
e_{I},\mathbb{X}_{s,u}\rangle+\sum_{e_{I_{1}^{\prime}}\otimes
e_{I_{2}^{\prime}}=e_{I^{\prime}}}\langle
e_{I_{1}^{\prime}},\mathbb{X}_{s,u}\rangle\langle e_{I_{2}^{\prime}}\otimes
e_{i_{n}},\mathbb{X}_{u,t}\rangle$ $\displaystyle=\sum_{e_{I_{1}}\otimes
e_{I_{2}}=e_{I}}\langle e_{I_{1}},{\mathbb{X}}_{s,u}\rangle\langle
e_{I_{2}},{\mathbb{X}}_{u,t}\rangle,$

whence the claim follows. ∎

###  2.2 Universal approximation theorem

This section is devoted to the universal approximation result mentioned in the
introduction. Loosely speaking, it states that every quantity of the form

$f\Big{(}(\widehat{\mathbb{X}}_{t}^{2})_{t\in[0,T]}\Big{)}$

for some continuous map $f$ and some $T>0$ can be approximated arbitrarily
well on compact sets by linear functions of the signature of the form
$\langle\ell,\widehat{\mathbb{X}}_{T}\rangle$ where $\ell\in
T(\mathbb{R}^{d})$. Observe that the latter just involves the _final value_
$\widehat{\mathbb{X}}_{T}$ of $\widehat{\mathbb{X}}$, instead of its whole
trajectory. Different versions of this result, e.g. for finite variation paths
or for continuous functions depending on the whole signature (instead of level
2), are available in the literature (see for instance Theorem 3.1 in Levin et
al. (2013), Theorem 1 in Király and Oberhauser (2019), Proposition 4.5 in
Lyons et al. (2020) and Section 3 in Cuchiero et al. (2022a)). For
completeness and to keep the paper self-contained we prove it here in the
current context of continuous semimartingales and continuous functions of the
respective paths lifted up to order 2.

Fix $T>0$, consider a continuous ${\mathbb{R}}^{d}$-valued semimartingale
$(X_{t})_{t\in[0,T]}$, and let $\widehat{X}_{t}:=(t,X_{t})$. Denote by
$0,\ldots,d$ the indices of $\widehat{X}$, where 0 corresponds to the time
component. For each $N\in\mathbb{N}$ define the set

${\mathcal{S}}^{(N)}:=\\{(\widehat{\mathbb{X}}_{t}^{N})_{t\in[0,T]}(\omega)\colon\omega\in\Omega\\},$

which, without loss of generality (passing to a subset of $\Omega$ of measure
$1$), corresponds to a set of signature paths of $\widehat{X}$ up to time $T$.

The next lemma states that the signature of a continuous semimartingale
coincides with the so-called Lyons lift (see for instance Theorem 2.2.1 in
Lyons (1998)). This in particular implies that higher order terms of the
signature can be defined pathwise starting from the trajectories of
$\widehat{\mathbb{X}}^{2}$.

######  Lemma 2.11.

For each $N\in{\mathbb{N}}$ there exists a map
$S^{(N)}:{\mathcal{S}}^{(2)}\to{\mathcal{S}}^{(N)}$ such that

$\widehat{\mathbb{X}}^{N}=S^{(N)}(\widehat{\mathbb{X}}^{2})$ (2.3)

almost surely.

###### Proof.

The claim follows from Exercise 17.2 and Theorem 9.5 in Friz and Victoir
(2010). ∎

Without loss of generality (passing again to a subset of $\Omega$ of measure
$1$) assume that conditions (2.1), (2.2), (2.3) and

$\langle e_{I}\otimes
e_{0},\widehat{\mathbb{X}}_{t}(\omega)\rangle=\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}(\omega)\rangle\mathrm{d}s$

holds for each $\omega\in\Omega$ and each multi-index $I$. Consider a generic
distance $d_{{\mathcal{S}}^{(2)}}$ on the set of trajectories given by
${\mathcal{S}}^{(2)}$, with respect to which the map from
${\mathcal{S}}^{(2)}$ to ${\mathbb{R}}$ given by

$\hat{\textbf{x}}^{2}\mapsto\langle
e_{I},S^{(|I|)}(\hat{\textbf{x}}^{2})_{t}\rangle$

is continuous for each multi-index $I$ and every $t\in[0,T]$.

######  Theorem 2.12 (Universal approximation theorem).

Let $K$ be a compact subset of ${\mathcal{S}}^{(2)}$ and consider a continuous
map $f:K\to{\mathbb{R}}$.222Compactness and continuity are defined with
respect to $d_{{\mathcal{S}}^{(2)}}$. Then for every $\varepsilon>0$ there
exists some $\ell\in T({\mathbb{R}}^{d})$ such that

$\sup_{(\widehat{\mathbb{X}}_{t}^{2})_{t\in[0,T]}\in K}\lvert
f((\widehat{\mathbb{X}}^{2}_{t})_{t\in[0,T]})-\langle\ell,\widehat{\mathbb{X}}_{T}\rangle\lvert<\varepsilon,$

almost surely.

###### Proof.

The result follows by an application of the Stone-Weierstrass theorem. To
check the needed properties, observe that by Proposition 2.3 the set

$A:=\textup{span}\\{\hat{\textbf{x}}^{2}\mapsto\langle
e_{I},S^{(|I|)}(\hat{\textbf{x}}^{2})_{T}\rangle\colon
I\in\\{0,\ldots,d\\}^{|I|}\\}$

is an algebra of continuous maps from $(K,d_{{\mathcal{S}}^{(2)}})$ to
${\mathbb{R}}$. Choosing $I=\emptyset$ we get that $A$ is vanishing nowhere
and we just need to check that it is point separating in
${\mathcal{S}}^{(2)}$. Lemma 2.6 yields the result for $\langle
e_{1},\hat{\textbf{x}}^{2}\rangle,\ldots,\langle
e_{d},\hat{\textbf{x}}^{2}\rangle$. Due to Lemma 2.9, this then applies also
to the remaining components of $\hat{\textbf{x}}^{2}$. ∎

######  Remark 2.13.

  1. (i)

Observe that the assumptions on $d_{{\mathcal{S}}^{(2)}}$ guarantee that every
map of the form

$f\Big{(}(\widehat{\mathbb{X}}_{t}^{2})_{t\in[0,T]}\Big{)}:=\tilde{f}(\langle
e_{I_{1}},\widehat{\mathbb{X}}_{t_{1}}\rangle,\ldots,\langle
e_{I_{n}},\widehat{\mathbb{X}}_{t_{n}}\rangle),$

for some $\tilde{f}\in C({\mathbb{R}}^{n})$, is continuous with respect to
$d_{{\mathcal{S}}^{(2)}}$.

  2. (ii)

The rough paths literature provides several metrics $d_{{\mathcal{S}}^{(N)}}$
with respect to which the maps
$S^{(N)}:({\mathcal{S}}^{(2)},d_{{\mathcal{S}}^{(2)}})\to({\mathcal{S}}^{(N)},d_{{\mathcal{S}}^{(N)}})$
are almost surely continuous on bounded sets. Among them one can for instance
consider

$d_{{\mathcal{S}}^{(N)}}(\hat{\textbf{x}},\hat{\textbf{y}}):=\max_{k\in\\{1,\ldots,N\\}}\sup_{\mathcal{D}}\bigg{(}\sum_{t_{i}\in{\mathcal{D}}}|\pi_{k}(\hat{\textbf{x}}_{t_{i-1},t_{i}}-\hat{\textbf{y}}_{t_{i-1},t_{i}})|^{p/k}\bigg{)}^{k/p}.$

where $p\in(2,3)$, $\pi_{k}(\hat{\textbf{x}}):=\sum_{|I|=k}\langle
e_{I},\hat{\textbf{x}}\rangle e_{I}$, and ${\mathcal{D}}$ denotes the set of
all partitions of $[0,T]$, see Corollary 9.11, Definition 8.6 and Theorem 8.10
in Friz and Victoir (2010) for the result or Cuchiero et al. (2022a) for a
more detailed explanation. Note that

$\pi_{1}(\hat{\textbf{x}}_{t_{i-1},t_{i}})=\sum_{i=1}^{d}\langle
e_{i},\hat{\textbf{x}}_{t_{i}}\rangle-\langle
e_{i},\hat{\textbf{x}}_{t_{i-1}}\rangle$

and

$\pi_{2}(\hat{\textbf{x}}_{t_{i-1},t_{i}})=\sum_{i,j=1}^{d}\langle
e_{ij},\hat{\textbf{x}}_{t_{i}}\rangle-\langle
e_{ij},\hat{\textbf{x}}_{t_{i-1}}\rangle-\langle
e_{i},\hat{\textbf{x}}_{t_{i-1}}\rangle(\langle
e_{j},\hat{\textbf{x}}_{t_{i}}\rangle-\langle
e_{j},\hat{\textbf{x}}_{t_{i-1}}\rangle).$

  3. (iii)

The universal approximation theorem can also be stated without constructing
the involved spaces based on a fixed semimartingale. Since this would require
a more advanced knowledge of rough paths theory, we address the interested
reader to Lyons (1998) or Chevyrev and Friz (2019) for uniqueness and
continuity of
$S^{(N)}:({\mathcal{S}}^{(2)},d_{{\mathcal{S}}^{(2)}})\to({\mathcal{S}}^{(N)},d_{{\mathcal{S}}^{(N)}})$
and to Cuchiero et al. (2022a) for a complete formulation and proof of the
resulting statement in the case of càdlàg semimartingales.

  4. (iv)

In other versions of the universal approximation theorem the map $f$ is
defined on a subset of $T(({\mathbb{R}}^{d}))$-valued paths, which includes
the realisations of ${\mathbb{X}}$, see for instance Theorem 3.1 in Levin et
al. (2013) and Proposition 4.5 in Lyons et al. (2020). Here, the critical
issues are to define the involved metric explicitly and to identify compact
subsets of $T(({\mathbb{R}}^{d}))$-valued paths.

###  2.3 Stochastic Stratonovich Taylor expansion

In addition to the (global) universal approximation theorem formulated above,
we now also state a well-known quantitative approximation result for solutions
of SDEs based on the (stochastic) Stratonovich Taylor expansion, see Kloeden
and Platen (1992). We also refer to Litterer and Oberhauser (2014) for
approximations of non-anticipative path functionals of SDEs via Chen-Fliess
series of iterated stochastic integrals. In contrast to Kloeden and Platen
(1992) and Litterer and Oberhauser (2014), where the $L^{2}$-error is
considered (see Remark 2.15 below), we here directly provide a ‘convergence
rate in probability’ without requiring second moment estimates on the
coefficients of the stochastic Taylor expansion. To this end we rely on the
deterministic estimates provided for rough differential equation as in Section
10 in Friz and Victoir (2010).

######  Proposition 2.14.

Let $\widehat{X}$ be a time extended $d$-dimensional Brownian motion and
$(Y_{t})_{t\in[0,T]}$ be an $D$-dimensional strong solution of the SDE

$\mathrm{d}Y_{t}=\sum_{j=0}^{d}\Psi_{j}(Y_{t})^{\top}\circ\mathrm{d}\widehat{X}_{t}^{j},$

for some smooth $\Psi_{j}:{\mathbb{R}}^{D}\to{\mathbb{R}}^{D}$. For each $I$
let $\Psi_{I}:{\mathbb{R}}^{D}\to{\mathbb{R}}^{D}$ be the map whose $k$-th
component is given by

$\Psi_{I}^{k}(y)=\Psi_{i_{1}}^{\top}\nabla(\Psi_{i_{2}}^{\top}\nabla\cdots(\Psi_{i_{n}}^{\top}e_{k}))(y).$

Then, for each $m\geq 2$ and ${\varepsilon}>0$ there is a constant
$C(m,{\varepsilon})$ such that

${\mathbb{P}}\bigg{(}\bigg{|}Y_{t}-Y_{0}-\sum_{0<|I|\leq
m}\Psi_{I}(Y_{0})\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle\bigg{|}>C(m,{\varepsilon})t^{(m+1)/2}\bigg{)}<{\varepsilon}.$

###### Proof.

For each $K>0$ consider the stopping times $\tau_{K}:=\inf\\{t\geq
0\colon|Y_{t}|\geq K\\}.$ For each $j$ and $K$ fix smooth
$\Psi_{j}^{K}:{\mathbb{R}}^{D}\to{\mathbb{R}}^{D}$ such that
$\Psi_{j}^{K}(y)=\Psi_{j}(y)$ for each $|y|\leq K$ and
$\operatorname{supp}(\Psi_{j}^{K})\subseteq\\{y\in{\mathbb{R}}^{D}\colon|y|\leq
K+1\\}$. Choose $\gamma:=m+1$ and $2<p\leq\gamma$. Exercise 17.2, Theorem 17.3
and Corollary 10.15 in Friz and Victoir (2010)333Observe that in Friz and
Victoir (2010) a vector field $\Psi_{j}$ is identified with the first order
operator $f\mapsto\Psi_{j}\cdot\nabla f$ (see for instance the discussion on
page 125). The Euler scheme $\mathcal{E}_{\Psi}$ is defined in Definition
10.1. applied to

$\mathrm{d}Y_{t}=\sum_{j=0}^{d}\Psi_{j}^{K}(Y_{t})\circ\mathrm{d}\widehat{X}_{t\land{\tau_{k}}}^{j}$

yields the existence of a constant $C$ such that

$\bigg{|}Y_{t}-Y_{0}-\sum_{0<|I|\leq m}\Psi_{I}(y_{0})\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle\bigg{|}1_{\\{\tau_{K}\geq
T\\}}\leq|\Psi|_{\text{Lip}^{\gamma-1}}^{\gamma}C\|\widehat{\mathbb{X}}^{2}\|^{\gamma}_{1/p-\text{Höl};[0,T]}t^{\gamma/p},$

almost surely, where $|\Psi|_{\text{Lip}^{\gamma-1}}^{\gamma}$ and
$\|\cdot\|^{\gamma}_{1/p-\text{Höl};[0,T]}$ are defined in Definition 10.2 and
Equation (8.2) of Friz and Victoir (2010), respectively. Observe that
$\gamma/p\leq(m+1)/2$ and

$|\Psi|_{\text{Lip}^{\gamma-1}}^{\gamma}\leq\sup_{k\leq m+1,|y|\leq
K+1}|\Psi^{(k)}(y)|^{\gamma}\leq\bigg{(}1+\sup_{k\leq m+1,|y|\leq
K+1}|\Psi^{(k)}(y)|\bigg{)}^{m+1}.$

By the discussion at the beginning of Section 3 in Friz and Victoir (2010) we
also get that $\|\widehat{\mathbb{X}}^{2}\|^{\gamma}_{1/p-\text{Höl};[0,T]}$
is finite almost surely and the claim follows. ∎

######  Remark 2.15.

Alternatively, an $L^{2}$-error for the stochastic Taylor expansion is given
in Proposition 5.10.1 in Kloeden and Platen (1992). It provides (under some
technical conditions, in particular second moments of $Y_{t}$) the following
estimate

${\mathbb{E}}\bigg{[}\Big{|}Y_{t}-Y_{0}-\sum_{0<|I|\leq
m}\Psi_{I}(Y_{0})\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle\Big{|}^{2}\bigg{]}\leq Ct^{m+1},$

for some $C\geq 0$. An application of the Markov inequality then yields the
previous estimate

${\mathbb{P}}\bigg{(}\Big{|}Y_{t}-Y_{0}-\sum_{0<|I|\leq
m}\Psi_{I}(Y_{0})\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle\Big{|}>\widetilde{C}^{1/2}t^{(m+1)/2}\bigg{)}\leq{\varepsilon},$

where $\widetilde{C}:=C/{\varepsilon}$, for each ${\varepsilon}>0$.

##  3 The model

We now introduce a framework for signature based asset price models. To this
end we fix a time horizon $T>0$, consider a $d$-dimensional continuous
semimartingale ${X}:=({X}^{1},\ldots,{X}^{d})$ and its matrix-valued quadratic
covariation $[{X}]$. We suppose that ${X}$ encodes all the information to
represent the market’s assets ${S}$. For notational convenience we only
consider a single asset, i.e. assume that ${S}$ is one-dimensional.

We shall furthermore assume that ${X}$ has some tractability properties which
are made precise below and which are for instance satisfied by a
$d$-dimensional Brownian motion (see Section A and Example 4.1).
Alternatively, one can choose ${X}$ to be a collection of liquid financial
products, whose time series are easily accessible and can be interpreted as
realizations of continuous semimartingales.

###  3.1 Definition and first properties

As the continuous semimartingale ${X}$ shall serve as main modeling building
block, we call it _primary process_ and extend it appropriately as made
precise in the next definition.

######  Definition 3.1.

We refer to ${X}$ as _primary process_. Depending on the context, we then
consider one of the following two extensions of ${X}$.

  1. (i)

$\widehat{X}_{t}:=(t,{X}_{t},[{X}]_{t})$, where $[{X}]$ denotes the
$d^{2}$-dimensional process given by the quadratic covariation of ${X}$.

  2. (ii)

$\widehat{X}_{t}:=(t,{X}_{t})$. This extension is always be paired with the
assumption that ${X}$ is an Itô-semimartingale with absolutely continuous
characteristics such that each element of its diffusion matrix can be written
as linear combination of elements of its time extended signature.

######  Remark 3.2.

Note that (i) of Definition 3.1 is more general than (ii) in two respects:
first the components of the diffusion matrix do not need to be linear
functions of the time extended signature, but could for instance be more
general path-dependent functionals; second $t\mapsto[X]_{t}$ does not need to
be absolutely continuous with respect to the Lebesgue measure, hence $X$ does
not need to be an Itô-semimartingale.

Throughout the paper we will always denote with $\widehat{{\mathbb{X}}}_{t}$
the signature of the previous extensions of $X_{t}$. Our goal consists in
describing/approximating the dynamics of ${S}$ with a _signature model_.

######  Definition 3.3.

A _signature model_ is a stochastic process of the form

${S}_{n}(\ell)_{t}:=\ell_{\emptyset}+\sum_{0<|I|\leq n}\ell_{I}\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle,$ (3.1)

where $n\in{\mathbb{N}}$ and $\ell:=\\{\ell_{\emptyset},\ell_{I}\colon
0<|I|\leq n\\}$.

######  Remark 3.4.

By Proposition 3.9 below the class of Sig-SDEs models considered in Perez
Arribas et al. (2020) can be embedded in our framework by choosing a properly
extended one-dimensional Brownian motion as primary process.

In the following we list several important properties which make signature
models a tractable framework for stochastic finance.

  * •

For each $t\in[0,T]$, ${S}_{n}(\ell)_{t}$ is linear in
$\widehat{\mathbb{X}}_{t}$. This in particular implies that having pre-
computed the signature of $\widehat{X}$ an update of the parameters $\ell$
boils down to computing (3.1), which is nothing else than a scalar product.

  * •

The quadratic variation of processes of form (3.1) is again of the form (3.1).

  * •

Form (3.1) remains invariant under polynomial transformations.

  * •

Itô-integrals of processes of form (3.1) with respect to processes of form
(3.1) are again processes of form (3.1). This includes in particular the
signature $\widehat{\mathbb{S}}_{n}(\ell)$ of
$\widehat{S}_{n}(\ell)_{t}:=(t,{S}_{n}(\ell)_{t})$ or expressions of the form

$\int_{0}^{\cdot}S_{n}(\ell)_{s}\mathrm{d}{X}_{s}^{i}.$

  * •

The latter implies that the expected signature of $\widehat{S}_{n}(\ell)$ is
given by

${\mathbb{E}}[\langle
e_{J},\widehat{\mathbb{S}}_{n}(\ell)_{t}\rangle]=P_{J}(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{t}]),$

for some $P_{J}$ such that
$P_{J}({\,\cdot\,},{\mathbb{E}}[\widehat{\mathbb{X}}_{t}])$ is a polynomial of
degree $|J|$ and $P_{J}(\ell,{\,\cdot\,})$ is a linear map for each $\ell$
(see Theorem 3.16 and Remark 3.17 for more details).

  * •

Due to Theorem 2.12 this provides approximations

${\mathbb{E}}\Big{[}f\Big{(}(\widehat{\mathbb{S}}^{2}_{n}(\ell)_{t})_{t\in[0,T]}\Big{)}\Big{]}\approx
P_{f}(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{T}])$

for each map $f$, which is continuous with respect to
$d_{{\mathcal{S}}^{(2)}}$, where $P_{f}$ is given by a finite linear
combination of maps $P_{J}$ as above. This includes representations for

${\mathbb{E}}\Big{[}\tilde{f}(\widehat{S}_{n}(\ell)_{T})\Big{]}\text{\qquad
and
\qquad}{\mathbb{E}}\bigg{[}\tilde{f}\bigg{(}\int_{0}^{T}\widehat{S}_{n}(\ell)_{t}\mathrm{d}t\bigg{)}\bigg{]}$

for maps $\tilde{f}$ being payoff functions and where the expectation is taken
with respect to a pricing measure.

We now prove several representation results for the signature model of form
(3.1). Throughout we shall apply the following notation.

######  Notation 3.5.

We use $e_{0}$ for the component of $\widehat{X}$ corresponding to time,
$e_{k}$ for its component corresponding to ${X}^{k}$, and (if needed)
${\varepsilon}_{ij}$ for the component of $\widehat{X}$ corresponding to
$[{X}^{i},{X}^{j}]$.

Consider the following assumption, which includes in particular the case where
${X}$ is a $d$-dimensional Brownian motion.

######  Assumption 3.6.

For all $i,j\in\\{1,\ldots,d\\}$ it holds

$\mathrm{d}[{X}^{i},{X}^{j}]_{t}=\sum_{|I|\leq m}a_{ij}^{I}\langle
e_{I},\widehat{\mathbb{X}}_{t}\rangle\mathrm{d}t$

for some $m\in{\mathbb{N}}$ where $\widehat{X}_{t}=(t,X_{t})$.

######  Remark 3.7.

Note that for $\widehat{X}$ as of Definition 3.1(ii), Assumption 3.6 is
satisfied by definition.

######  Proposition 3.8.

Suppose that ${S}_{n}(\ell)$ has a representation of form (3.1) with
$\widehat{X}$ as in Definition 3.1(i) and that Assumption 3.6 holds. Then
${S}_{n}(\ell)$ has a representation of the same form but with $\widehat{X}$
as in Definition 3.1(ii).

###### Proof.

Observe that by Assumption 3.6 it holds

$[{X}^{i},{X}^{j}]_{t}=\int_{0}^{t}\sum_{|I|\leq m}a_{ij}^{I}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\circ\mathrm{d}s=\sum_{|I|\leq
m}a_{ij}^{I}\langle e_{I}\otimes e_{0},\widehat{\mathbb{X}}_{t}\rangle.$

The claim follows by Lemma 2.9. ∎

######  Proposition 3.9.

Fix $n\in{\mathbb{N}}$ and suppose that $\widehat{X}$ is given by Definition
3.1(i). Then there is a one to one correspondence between representations of
the form (3.1) and representations of the form

$\displaystyle{S}_{n}(\ell)_{t}=\ell_{\emptyset}+\int_{0}^{t}\Big{(}\ell^{0}_{\emptyset}+\sum_{0<|I|\leq
n-1}\ell^{0}_{I}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\Big{)}\mathrm{d}s+\sum_{k=1}^{d}\int_{0}^{t}\Big{(}\ell^{k}_{\emptyset}+\sum_{0<|I|\leq
n-1}\ell^{k}_{I}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\Big{)}\mathrm{d}{X}_{s}^{k}$
$\displaystyle\qquad+\sum_{k_{1},k_{2}=1}^{d}\int_{0}^{t}\Big{(}\ell^{k_{1},k_{2}}_{\emptyset}+\sum_{0<|I|\leq
n-1}\ell^{k_{1},k_{2}}_{I}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\Big{)}\mathrm{d}[{X}^{k_{1}},{X}^{k_{2}}]_{s},$

for
$\ell:=\\{\ell_{\emptyset},\ell^{0}_{I},\ell_{I}^{k_{1}},\ell_{I}^{k_{1},k_{2}}\colon|I|\leq
n-1\text{ and }k_{1},k_{2}\in\\{1,\ldots,d\\}\\}.$

If $\widehat{X}$ satisfies Assumption 3.6, then the same is true with
$\ell_{I}^{k_{1},k_{2}}=0$ for each $|I|\leq n-1\text{ and
}k_{1},k_{2}\in\\{1,\ldots,d\\}$.

Before presenting the proof of this proposition let us formulate the following
technical lemma.

######  Lemma 3.10.

Suppose that $\widehat{X}$ is given by Definition 3.1(i). Then

$\displaystyle\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}s$ $\displaystyle=\langle
e_{I}\otimes e_{0},\widehat{\mathbb{X}}_{t}\rangle,\qquad\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}{X}_{s}^{k}=\langle\tilde{e}_{I}^{k},\widehat{\mathbb{X}}_{s}\rangle,\text{\quad
and\quad}$ $\displaystyle\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[{X}^{k_{1}},{X}^{k_{2}}]_{s}$
$\displaystyle=\langle
e_{I}\otimes{\varepsilon}_{k_{1}k_{2}},\widehat{\mathbb{X}}_{t}\rangle,$

where $\tilde{e}_{\emptyset}^{k}=e_{k}$ and $\tilde{e}_{I}^{k}=e_{I}\otimes
e_{k}-\frac{1}{2}e_{I^{\prime}}\otimes{\varepsilon}_{i_{|I|}k}$ for each
$|I|>0$. If Assumption 3.6 is in force, then the same holds true with

$\tilde{e}_{I}^{k}=e_{I}\otimes e_{k}-\sum_{|J|\leq
m}\frac{a_{i_{|I|}k}^{J}}{2}(e_{I^{\prime}}\shuffle e_{J})\otimes e_{0}$

for each $|I|>0$.

###### Proof.

The representations of $\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}s$ and $\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[{X}^{k_{1}},{X}^{k_{2}}]_{s}$
follow by definition of the signature. We proceed with the proof of the
representation of $\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}{X}_{s}^{k}$. For $I=\emptyset$
the claim is clear. By definition of the Stratonovich integral we have that

$\displaystyle\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}{X}^{k}_{s}$
$\displaystyle=\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\circ\mathrm{d}X^{k}_{s}-\frac{1}{2}[\langle
e_{I},\widehat{\mathbb{X}}\rangle,X^{k}]_{t}$ $\displaystyle=\langle
e_{I}\otimes
e_{k},\widehat{\mathbb{X}}_{t}\rangle-\frac{1}{2}\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[X^{i_{|I|}},X^{k}]_{s}$
$\displaystyle=\langle e_{I}\otimes
e_{k},\widehat{\mathbb{X}}_{t}\rangle-\frac{1}{2}\langle
e_{I^{\prime}}\otimes{\varepsilon}_{i_{|I|}k},\widehat{\mathbb{X}}_{t}\rangle$
$\displaystyle=\langle\tilde{e}_{I}^{k},\widehat{\mathbb{X}}_{t}\rangle.$

Suppose now that Assumption 3.6 is in force. Then Proposition 2.3 and the
definition of the signature yield

$\displaystyle\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[{X}^{k_{1}},{X}^{k_{2}}]_{s}$
$\displaystyle=\int_{0}^{t}\sum_{|J|\leq m}{a_{k_{1}k_{2}}^{J}}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\langle
e_{J},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}s$ $\displaystyle=\sum_{|J|\leq
m}{a_{k_{1}k_{2}}^{J}}\langle(e_{I}\shuffle e_{J})\otimes
e_{0},\widehat{\mathbb{X}}_{t}\rangle,$

and the claim follows. ∎

We are now ready to provide the proof of Proposition 3.9.

###### Proof of Proposition 3.9.

Let ${S}_{n}(\ell)$ be as in the statement of the proposition. By Lemma 3.10
it holds

$\displaystyle{S}_{n}(\ell)_{t}=\ell_{\emptyset}+\Big{(}\ell^{0}_{\emptyset}\langle
e_{0},\widehat{\mathbb{X}}_{t}\rangle+\sum_{0<|I|\leq n-1}\ell^{0}_{I}\langle
e_{I}\otimes
e_{0},\widehat{\mathbb{X}}_{t}\rangle\Big{)}+\sum_{k=1}^{d}\Big{(}\ell^{k}_{\emptyset}\langle\tilde{e}^{k}_{\emptyset},\widehat{\mathbb{X}}_{t}\rangle+\sum_{0<|I|\leq
n-1}\ell^{k}_{I}\langle\tilde{e}^{k}_{I},\widehat{\mathbb{X}}_{t}\rangle\Big{)}$
$\displaystyle\qquad+\sum_{k_{1},k_{2}=1}^{d}\Big{(}\ell^{k_{1},k_{2}}_{\emptyset}\langle{\varepsilon}_{k_{1}k_{2}},\widehat{\mathbb{X}}_{t}\rangle+\sum_{0<|I|\leq
n-1}\ell^{k_{1},k_{2}}_{I}\langle
e_{I}\otimes{\varepsilon}_{k_{1}k_{2}},\widehat{\mathbb{X}}_{s}\rangle\Big{)},$

which is of the form given by (3.1). Conversely, by Lemma 3.10 we also have

$\displaystyle\langle e_{I}\otimes e_{0},\widehat{\mathbb{X}}_{t}\rangle$
$\displaystyle=\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}s,\qquad\langle
e_{I}\otimes{\varepsilon}_{k_{1}k_{2}},\widehat{\mathbb{X}}_{t}\rangle=\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[{X}^{k_{1}},{X}^{k_{2}}]_{s},\text{\quad
and\quad}$ $\displaystyle\langle e_{I}^{k}\otimes
e_{k},\widehat{\mathbb{X}}_{s}\rangle$ $\displaystyle=\int_{0}^{t}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}{X}_{s}^{k}+\frac{1}{2}\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[{X}^{i_{|I|}},{X}^{k}]_{s},$

and the claim follows. ∎

In the case of Brownian motion the formulas of Lemma 3.10 simplify as follows.

######  Example 3.11.

Suppose that ${X}$ is a vector of correlated Brownian motions with correlation
matrix $\rho$ and $\widehat{X}$ is given by Definition 3.1(ii). Then it holds

$\tilde{e}^{k}_{\emptyset}=e_{k}\qquad\text{and}\qquad\tilde{e}^{k}_{I}=e_{I}\otimes
e_{k}-\frac{\rho_{i_{|I|},k}}{2}1_{\\{i_{|I|}\neq 0\\}}e_{I^{\prime}}\otimes
e_{0}.$

###  3.2 Absence of arbitrage and universality properties

The above representation results will allow us to establish conditions that
guarantee absence of arbitrage. We shall therefore assume that _no free lunch
with vanishing risk_ (Delbaen and Schachermayer (1994)) holds, which is – due
to the continuity of sample paths of ${S}_{n}(\ell)$ – equivalent to the
existence of an equivalent local martingale measure $\mathbb{Q}$, i.e. a
measure $\mathbb{Q}\sim\mathbb{P}$ under which the (discounted) asset price
model $S_{n}(\ell)$ is a local martingale. We shall always assume here that
interest rates are zero and that the asset is already discounted. To formulate
a precise no-arbitrage condition, the following corollary which is a direct
consequence of Proposition 3.9 and Lemma 3.10 is essential.

######  Corollary 3.12.

Suppose that ${X}$ is a local martingale. Then ${S}_{n}(\ell)$ is a local
martingale if and only if it admits a representation of the form

${S}_{n}(\ell)_{t}=\ell_{\emptyset}+\sum_{k=1}^{D}\bigg{(}\ell_{\emptyset}^{k}\langle
e_{k},\widehat{\mathbb{X}}_{t}\rangle+\sum_{0<|I|\leq
n-1}\ell^{k}_{I}\langle\tilde{e}^{k}_{I},\widehat{\mathbb{X}}_{t}\rangle\bigg{)},$
(3.2)

for some $D\in\\{1,\ldots,d\\}$ and
$\ell:=\\{\ell_{\emptyset},\ell_{I}^{k}\colon 0\leq|I|\leq n-1\text{ and
}k\in\\{1,\ldots,d\\}\\}.$

###### Proof.

Due to the local martingale property of $X$, $S_{n}(\ell)$ in the
representation of Proposition 3.9 is a local martingale if and only if all
integrals with respect to time and the quadratic variation process vanish.
This means that $S_{n}(\ell)$ is of form

$\displaystyle
S_{n}(\ell)=\ell_{\emptyset}+\sum_{k=1}^{d}\int_{0}^{t}\Big{(}\ell^{k}_{\emptyset}+\sum_{0<|I|\leq
n-1}\ell^{k}_{I}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\Big{)}\mathrm{d}{X}_{s}^{k}$ (3.3)

and Lemma 3.10 yields the assertion. ∎

We are now ready to formulate sufficient no-arbitrage conditions.

######  Corollary 3.13.

Suppose that there is an equivalent measure $\mathbb{Q}\sim\mathbb{P}$ such
that $X$ is a local $\mathbb{Q}$-martingale. Then the following holds.

  1. (i)

The model $S_{n}(\ell)$ is free of arbitrage if it admits a representation as
of (3.2).

  2. (ii)

If $\mathbb{Q}$ is an equivalent local martingale measure for $S_{n}(\ell)$,
then $S_{n}(\ell)$ is necessarily of form (3.2).

###### Proof.

The first assertion is a direct consequence of Corollary 3.12, as form (3.2)
implies that $S_{n}(\ell)$ is a local $\mathbb{Q}$-martingale and thus
$\mathbb{Q}$ is an equivalent local martingale measure. The assumption of the
second assertion implies that $S_{n}(\ell)$ is a local martingale under
$\mathbb{Q}$, whence by Corollary 3.12 it has to be of form (3.2). ∎

######  Remark 3.14.

  1. (i)

Note that $S_{n}(\ell)$ could be free of arbitrage without being of form
(3.2). This is the case if the primary process $X$ is not a local martingale
under any of the equivalent local martingale measures.

  2. (ii)

Observe also that if $S_{n}(\ell)$ is form (3.2) and a local martingale under
$\mathbb{Q}$, then $X$ does not necessarily need to be a local
$\mathbb{Q}$-martingale. This happens if drift terms in (3.3) cancel out. Note
however that in dimension $d=1$ this cannot occur and $X$ is thus necessarily
a local $\mathbb{Q}$-martingale. In particular, in this case $\mathbb{Q}$ is
unique and the model thus complete.

In the following example we show how classical stochastic volatilty models can
be approximated by arbitrage-free signature models, thus making the announced
universality properties precise.

######  Example 3.15.

We consider a generic stochastic volatility model driven by two correlated
Brownian motions $(B_{t})_{t\in[0,1]}$ and $(W_{t})_{t\in[0,1]}$ with
correlation coefficient $\rho\in[-1,1]$. More precisely, we assume that under
some equivalent local martingale measure ${\mathbb{Q}}$ the dynamics of $S$
are given by

$\displaystyle{\mathrm{d}}S_{t}$
$\displaystyle=g(S_{t},V_{t}){\mathrm{d}}B_{t},$
$\displaystyle{\mathrm{d}}V_{t}$
$\displaystyle=h(S_{t},V_{t}){\mathrm{d}}t+\sigma(S_{t},V_{t}){\mathrm{d}}W_{t},$

for some functions $g,h,\sigma:\mathbb{R}^{2}\to\mathbb{R}$. As a possible
choice for the parameters we can for instance consider

  * •

$g(s,v)=s\sqrt{v}$, $h(s,v)=\kappa(\theta-v)$ and
$\sigma(s,v)=\sigma_{0}\sqrt{v}$, for some parameters $\kappa,\theta\geq 0$
and $\sigma_{0}\in\mathbb{R}$ to retrieve the Heston model (see Heston
(1993));

  * •

$g(s,v)=s^{\beta}v$, $h(s,v)=0$ and $\sigma(s,v)=\alpha v$ for $0<\beta\leq 1$
and $\alpha\in\mathbb{R}$ to retrieve the SABR model (see Hagan et al.
(2002)).

Set now $X_{t}:=(B_{t},W_{t})$ and let $\widehat{X}_{t}$ be given by the
representation of Definition 3.1(ii), i.e.,
$\widehat{X}_{t}:=(t,B_{t},W_{t})$. For some fixed $n\in\mathbb{N}$, consider
then the following model

${S}_{n}(\ell)_{t}:=\ell_{\emptyset}+\int_{0}^{t}\Big{(}\ell_{\emptyset}^{B}+\sum_{0<\lvert
I\lvert\leq n-1}\ell_{I}^{B}\langle
e_{I}^{B},\widehat{\mathbb{X}}_{s}\rangle\Big{)}{\mathrm{d}}B_{s},$

which is a local martingale under ${\mathbb{Q}}$ and thus absence of arbitrage
is guaranteed. Moreover, by Proposition 3.9, we know that this model has a
representation of the form

$\displaystyle{S}_{n}(\ell)_{t}=\ell_{\emptyset}+\ell_{\emptyset}^{B}B_{t}+\sum_{0<|I|\leq
n-1}\ell_{I}^{B}\langle\tilde{e}_{I}^{B},\widehat{\mathbb{X}}_{t}\rangle,$

for each $t\in[0,1]$, where by Lemma 3.10

$\displaystyle\tilde{e}_{I}^{B}$ $\displaystyle:=e_{I}\otimes
e_{1}-\frac{1}{2}(1_{\\{i_{|I|}=1\\}}+\rho
1_{\\{i_{|I|}=2\\}})e_{I^{\prime}}\otimes e_{0},$ (3.4)

for each $I$ with $0<|I|\leq n-1$, implying that it is a signature model as of
Definition 3.3.

Concerning universality note that if the coefficients $\ell_{I}^{B}$ are
chosen such that (3.4) matches the signature approximation of $S$ up to order
$n-1$ as specified in Proposition 2.14 (assuming appropriate regularity
conditions on the coefficients $g,h$, $\sigma$), then at least locally in $t$,
$S_{n}(\ell)$ is close to $S$ in probability.

Let us exemplify this by means of the SABR model for $\beta=1$ and, for
simplicity, $\rho=0$. Set $\ell_{\emptyset}:=S_{0}$,
$\ell_{\emptyset}^{B}:=y_{1}y_{2}$,
$\ell_{(0)}^{B}:=-\frac{1}{2}y_{1}y_{2}^{3}-\frac{1}{2}\alpha^{2}y_{1}y_{2}$,
$\ell_{(1)}^{B}:=y_{1}y_{2}^{2}$, and $\ell_{(2)}^{B}:=\alpha y_{1}y_{2}$. By
an application of Proposition 2.14 for $X_{t}=(t,W_{t},B_{t})$, $Y^{1}=S$,
$Y^{2}=V$, $m=2$,

$\Psi_{0}(y)=(1,-\frac{1}{2}y_{1}y_{2}^{2},-\frac{1}{2}\alpha^{2}y_{2})^{\top},$

$\Psi_{1}(y)=(0,y_{1}y_{2},0)^{\top}$, and $\Psi_{2}(y)=(0,0,\alpha
y_{2})^{\top}$ we can conclude that for each ${\varepsilon}>0$ there is a
constant $C({\varepsilon})$ such that

${\mathbb{Q}}\Big{(}|S_{t}-S_{2}(\ell)_{t}|>C({\varepsilon})t^{3/2}\Big{)}<{\varepsilon},$

where $S_{2}(\ell)$ is given by Definition 3.3 for
$\widehat{X}_{t}=(t,B_{t},W_{t})$. The same procedure for general $n$ leads to
a similar result for an arbitrary speed of convergence.

Another possibility to establish universality is to use the fact that the
solution map of a stochastic differential equation (with sufficiently regular
coefficients) is a continuous map of the signature of the driving signal (see
e.g. Corollary 10.40 Friz and Victoir (2010)). The universal approximation
theorem (Theorem 2.12, see also the formulation of Lemma 5.2 in Bayer et al.
(2021) or Cuchiero and Möller (2022)) then yields the result, however without
quantitative estimates.

###  3.3 The expected signature of ${S}_{n}(\ell)$

For pricing purposes of so-called _sig-payoffs_ treated in Section 3.4 it will
be important to be able to compute the expected signature of
$\widehat{S}_{n}(\ell)_{t}=(t,S_{n}(\ell)_{t})$. In this section we thus
provide formulas which trace this computation back to the calculation of the
expected signature of $\widehat{X}_{t}$, which in many cases is well-known
(see e.g. Fawcett (2003) for Brownian motion) and can often be computed by
techniques of polynomial processes (see Cuchiero et al. (2022b)) or by solving
an infinite dimensional system of linear PDEs corresponding to the Kolmogorov
forward equation of the signature process (see Ni (2012)). For a unified
treatment of signature cumulants, i.e. the logarithm of expected signature, we
refer to Friz et al. (2021).

Here, we first suppose that $\widehat{X}$ is given by Definition 3.1(ii).

######  Theorem 3.16.

Fix $n\in{\mathbb{N}}$, a multi-index $J$, and $D\in\\{1,\ldots,d\\}$ and
denote by $\widehat{\mathbb{S}}_{n}(\ell)_{t}$ the signature of
$\widehat{S}_{n}(\ell)_{t}$. Let $e_{0}$ be the component of
$\widehat{S}_{n}(\ell)$ corresponding to time and $e_{1}$ its component
corresponding to $S_{n}(\ell)$. Define
$e(\emptyset,\ell):=\tilde{e}(\emptyset,\ell):=e_{\emptyset}$ and

$\displaystyle e(J,\ell)$
$\displaystyle={\widetilde{\shuffle}}_{i=1}^{|J|}\Big{(}e_{0}1_{\\{j_{i}=0\\}}+\Big{(}\sum_{0<\lvert
I\lvert\leq n}\ell_{I}e_{I}\Big{)}1_{\\{j_{i}=1\\}}\Big{)},$
$\displaystyle\tilde{e}(J,\ell)$
$\displaystyle={\widetilde{\shuffle}}_{i=1}^{|J|}\Big{(}e_{0}1_{\\{j_{i}=0\\}}+\Big{(}\sum_{k=1}^{D}\Big{(}\ell_{\emptyset}^{k}e_{k}+\sum_{0<\lvert
I\lvert\leq
n-1}\ell_{I}^{k}\tilde{e}^{k}_{I}\Big{)}\Big{)}1_{\\{j_{i}=1\\}}\Big{)},$

for $|J|>0$ with ${\widetilde{\shuffle}}$ the half-shuffle being introduced in
Definition 2.8. Then the following representations hold.

  * •

$\langle e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle=\langle
e(J,\ell),\widehat{{\mathbb{X}}}_{t}\rangle$, if $S_{n}(\ell)$ is given by
(3.1), and

  * •

$\langle
e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle=\langle\tilde{e}(J,\ell),\widehat{{\mathbb{X}}}_{t}\rangle$,
if ${S}_{n}(\ell)$ is given as in Corollary 3.12.

###### Proof.

In order to prove the claim we proceed by induction. Fix $S_{n}(\ell)$ as in
(3.1). For $J=\emptyset$ the claim follows by the definition of signature.
Suppose the claim holds true for each $J$ such that $|J|=m-1$, and fix $J$
with $|J|\leq m$. Then

$\displaystyle\langle e_{J},\widehat{\mathbb{S}}_{n}(\ell)_{t}\rangle$
$\displaystyle=\int_{0}^{t}\langle
e_{J^{\prime}},\widehat{\mathbb{S}}_{n}(\ell)_{s}\rangle\circ{\mathrm{d}}\langle
e_{j_{m}},\widehat{S}_{n}(\ell)_{s}\rangle$ $\displaystyle=\int_{0}^{t}\langle
e_{J^{\prime}},\widehat{\mathbb{S}}_{n}(\ell)_{s}\rangle\circ{\mathrm{d}}\langle
e_{0}1_{\\{j_{m}=0\\}}+\Big{(}\sum_{0<\lvert I\lvert\leq
n}\ell_{I}e_{I}\Big{)}1_{\\{j_{m}=1\\}},\widehat{\mathbb{X}}_{s}\rangle$
$\displaystyle=1_{\\{j_{m}=0\\}}\int_{0}^{t}\langle
e_{J^{\prime}},\widehat{\mathbb{S}}_{n}(\ell)_{s}\rangle\circ{\mathrm{d}}\langle
e_{0},\widehat{\mathbb{X}}_{s}\rangle$
$\displaystyle\qquad+1_{\\{j_{m}=1\\}}\sum_{0<\lvert I\lvert\leq
n}\ell_{I}\int_{0}^{t}\langle
e_{J^{\prime}},\widehat{\mathbb{S}}_{n}(\ell)_{s}\rangle\circ{\mathrm{d}}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle.$

The induction hypothesis and Lemma 2.9 yield the first claim and the second
one is analogous. ∎

######  Remark 3.17.

  1. (i)

Let now ${S}_{n}(\ell)$ be as in (3.1) and observe that

$e(J,\ell)={\widetilde{\shuffle}}_{i=1}^{|J|}\sum_{0<|I|\leq
n}\Big{(}e_{I}1_{\\{j_{i}=0\\}}1_{\\{I=(0)\\}}+\ell_{I}e_{I}1_{\\{j_{i}=1\\}}\Big{)}.$

Setting $c(j,I,\ell):=1_{\\{j=0\\}}1_{\\{I=(0)\\}}+\ell_{I}1_{\\{j=1\\}}$ we
thus obtain that

$\displaystyle{\mathbb{E}}[\langle
e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle]$
$\displaystyle={\mathbb{E}}[\langle
e(J,\ell),\widehat{{\mathbb{X}}}_{t}\rangle]=\sum_{|I_{1}|,\ldots,|I_{|J|}|=1}^{n}{\mathbb{E}}[\langle{\widetilde{\shuffle}}_{i=1}^{{|J|}}e_{I_{i}},\widehat{{\mathbb{X}}}_{t}\rangle]\prod_{i=1}^{|J|}c(j_{i},I_{i},\ell).$

Even if at a first sight this representation appears involved, it is in fact
very handy. Indeed the expectations
${\mathbb{E}}[\langle{\widetilde{\shuffle}}_{i=1}^{{|J|}}e_{I_{i}},\widehat{{\mathbb{X}}}_{t}\rangle]$
can be computed just once in advance. Since $c(j,I,{\,\cdot\,})$ is affine, we
also immediately obtain that the map

$(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}])\mapsto
P_{J}(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{t}]):={\mathbb{E}}[\langle
e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle]$

is polynomial of degree $|J|$ in its first argument and linear in the second
one.

  2. (ii)

Similarly, let ${S}_{n}(\ell)$ be as in Corollary 3.12, set
$\tilde{e}_{\emptyset}^{k}:=e_{k}$, and observe that

$\tilde{e}(J,\ell)={\widetilde{\shuffle}}_{i=1}^{|J|}\sum_{k=1}^{D}\sum_{0\leq|I|\leq
n-1}\Big{(}\tilde{e}_{I}^{k}1_{\\{j_{i}=0\\}}1_{\\{I=(0)\\}}1_{\\{k=1\\}}+\ell_{I}^{k}\tilde{e}^{k}_{I}1_{\\{j_{i}=1\\}}\Big{)}.$

Setting
$\tilde{c}(j,I,k,\ell):=1_{\\{j=0\\}}1_{\\{I=(0)\\}}1_{\\{k=1\\}}+\ell_{I}^{k}1_{\\{j=1\\}}$
yields

$\displaystyle{\mathbb{E}}[\langle
e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle]=\sum_{k_{1},\ldots,k_{|J|}=1}^{D}\sum_{|I_{1}|,\ldots,|I_{|J|}|=0}^{n-1}{\mathbb{E}}[\langle{\widetilde{\shuffle}}_{i=1}^{{|J|}}\tilde{e}_{I_{i}}^{k_{i}},\widehat{{\mathbb{X}}}_{t}\rangle]\prod_{i=1}^{|J|}\tilde{c}(j_{i},I_{i},k_{i},\ell).$

We again obtain that the map

$(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}])\mapsto\widetilde{P}_{J}(\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{t}]):={\mathbb{E}}[\langle
e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle]$

is polynomial of degree $|J|$ in its first argument and linear in the second
one.

The expressions above clearly also provide a formula of the variance of
${S}_{n}(\ell)$.

######  Corollary 3.18.

Let ${S}_{n}(\ell)$ be as in Corollary 3.12 and assume that it is a true
martingale. Then

${\mathbb{V}}\textup{ar}(S_{n}(\ell)_{t})=2\sum_{k_{1},k_{2}=1}^{D}\sum_{|I_{1}|,|I_{2}|=1}^{n-1}\mathbb{E}[\langle\tilde{e}_{I_{1}}^{k_{1}}\shuffle\tilde{e}_{I_{2}}^{k_{2}},\widehat{\mathbb{X}}_{t}\rangle]\ell_{I_{1}}\ell_{I_{2}}.$
(3.5)

###### Proof.

The martingale property guarantees that
${\mathbb{E}}[S_{n}(\ell)_{t}]=S_{n}(\ell)_{0}$ and hence, by Proposition 2.3
(see also Example 2.5),
${\mathbb{V}}\textup{ar}(S_{n}(\ell)_{t})=2\mathbb{E}[\langle e_{1}\otimes
e_{1},\widehat{{\mathbb{S}}}_{n}(\ell)_{t}\rangle]$. By Remark 3.17 we can
conclude that

${\mathbb{V}}\textup{ar}(S_{n}(\ell)_{t})=2\sum_{k_{1},k_{2}=1}^{D}\sum_{|I_{1}|,|I_{2}|=1}^{n-1}\mathbb{E}[\langle\tilde{e}_{I_{1}}^{k_{1}}\shuffle\tilde{e}_{I_{2}}^{k_{2}},\widehat{\mathbb{X}}_{t}\rangle]\tilde{c}(1,I_{1},k_{1},\ell)\tilde{c}(1,I_{2},k_{2},\ell)$

and the claim follows. ∎

Let us now consider the case where $X$ is additionally extended by its
quadratic variation. In that case we need the following representation.

######  Lemma 3.19.

Suppose that $\widehat{X}$ is given by Definition 3.1(i) and fix two multi-
indices $I$ and $J$ such that $|I|>0$ and $|J|>0$. Using the notation of
Remark 3.5 set

$e_{I}[\shuffle]e_{J}:=\sum_{k_{1},k_{2}=1}^{d}1_{\\{i_{|I|}=k_{1},j_{|J|}=k_{2}\\}}(e_{I^{\prime}}\shuffle
e_{J^{\prime}})\otimes{\varepsilon}_{{k_{1}k_{2}}}.$

Then $[\langle e_{\emptyset},\widehat{\mathbb{X}}\rangle,\langle
e_{J},\widehat{\mathbb{X}}\rangle]_{t}=[\langle
e_{\emptyset},\widehat{\mathbb{X}}\rangle]_{t}=0$, and $[\langle
e_{I},\widehat{\mathbb{X}}\rangle,\langle
e_{J},\widehat{\mathbb{X}}\rangle]_{t}=\langle
e_{I}[\shuffle]e_{J},\widehat{\mathbb{X}}_{t}\rangle$. If Assumption 3.6 is in
force the same is true with

$e_{I}[\shuffle]e_{J}:=\sum_{k_{1},k_{2}=1}^{d}\sum_{|H|\leq
m}{a_{k_{1}k_{2}}^{H}}1_{\\{i_{|I|}=k_{1},j_{|J|}=k_{2}\\}}(e_{I^{\prime}}\shuffle
e_{J^{\prime}}\shuffle e_{H})\otimes e_{0}.$

###### Proof.

By definition of the signature and Proposition 2.3 we can compute

$\displaystyle[\langle e_{I},\widehat{\mathbb{X}}\rangle,\langle
e_{J},\widehat{\mathbb{X}}\rangle]_{t}$ $\displaystyle=\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\langle
e_{J^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}[\langle
e_{i_{|I|}},\widehat{\mathbb{X}}\rangle,\langle
e_{j_{|J|}},\widehat{\mathbb{X}}\rangle]_{s}$
$\displaystyle=\sum_{k_{1},k_{2}=1}^{d}1_{\\{i_{|I|}=k_{1},j_{|J|}=k_{2}\\}}\langle
e_{I^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\langle
e_{J^{\prime}},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}\langle{\varepsilon}_{k_{1}k_{2}},\widehat{\mathbb{X}}_{s}\rangle$
$\displaystyle=\langle e_{I}[\shuffle]e_{J},\widehat{\mathbb{X}}_{t}\rangle.$

Since under Assumption 3.6 it holds $\mathrm{d}[\langle
e_{i_{|I|}},\widehat{\mathbb{X}}\rangle,\langle
e_{j_{|J|}},\widehat{\mathbb{X}}\rangle]_{t}=\sum_{|H|\leq
m}{a_{k_{1}k_{2}}^{H}}\langle
e_{H},\widehat{\mathbb{X}}_{t}\rangle\mathrm{d}\langle
e_{0},\widehat{\mathbb{X}}_{t}\rangle,$ the second claim follows again by
Proposition 2.3. ∎

Using the above lemma we can now express the signature components of
$\widehat{S}_{n}(\ell)_{t}$ here defined as
$\widehat{S}_{n}(\ell)_{t}:=(t,S_{n}(\ell)_{t},[S_{n}(\ell)]_{t})$ also via
$\widehat{{\mathbb{X}}}_{t}$.

######  Theorem 3.20.

Fix $n\in{\mathbb{N}}$, a multi-index $J$, and suppose that $\widehat{X}$ is
given by Definition 3.1(i). Set
$\widehat{S}_{n}(\ell)_{t}:=(t,S_{n}(\ell)_{t},[S_{n}(\ell)]_{t})$ and let
$\widehat{\mathbb{S}}_{n}(\ell)_{t}$ denote the corresponding signature. Let
$e_{0}$ be the component of $\widehat{S}_{n}(\ell)_{t}$ corresponding to time,
$e_{1}$ its component corresponding to $S_{n}(\ell)_{t}$, and $e_{2}$ its
component corresponding to $[S_{n}(\ell)]_{t}$. Define
$e(J,\ell):=e_{\emptyset}$ for $J=\emptyset$ and

$e(J,\ell)={\widetilde{\shuffle}}_{k=1}^{|J|}\Big{(}e_{0}1_{\\{j_{k}=0\\}}+\Big{(}\sum_{0<\lvert
I\lvert\leq n}\ell_{I}e_{I}\Big{)}1_{\\{j_{k}=1\\}}+\Big{(}\sum_{0<|I|,|H|\leq
n}\ell_{I}\ell_{H}e_{I}[\shuffle]e_{H}\Big{)}1_{\\{j_{k}=2\\}}\Big{)}.$

Then $\langle e_{J},{\widehat{\mathbb{S}}_{n}(\ell)}_{t}\rangle=\langle
e(J,\ell),\widehat{{\mathbb{X}}}_{t}\rangle.$

###### Proof.

First observe that by Lemma 3.19 it holds

$[{S}_{n}(\ell)]=\Big{[}\Big{(}\ell_{\emptyset}+\sum_{0<\lvert I\lvert\leq
n}\ell_{I}\langle
e_{I},\widehat{\mathbb{X}}\rangle\Big{)}\Big{]}=\sum_{0<|I|,|H|\leq
n}\ell_{I}\ell_{H}\langle e_{I}[\shuffle]e_{H},\widehat{\mathbb{X}}\rangle.$

Next, note that

$\displaystyle\langle e_{j},{\widehat{S}_{n}(\ell)}_{t}\rangle$
$\displaystyle=t1_{\\{j=0\\}}+({{{S}_{n}(\ell)}}_{t}-{{{S}_{n}(\ell)}}_{0})1_{\\{j=1\\}}+[{S}_{n}(\ell)]_{t}1_{\\{j=2\\}}$
$\displaystyle=\langle e_{0}1_{\\{j=0\\}}+\Big{(}\sum_{0<\lvert I\lvert\leq
n}\ell_{I}e_{I}\Big{)}1_{\\{j=1\\}}+\Big{(}\sum_{0<|I|,|H|\leq
n}\ell_{I}\ell_{H}e_{I}[\shuffle]e_{H}\Big{)}1_{\\{j=2\\}},\widehat{{\mathbb{X}}}_{t}\rangle.$

The proof now follows the proof of Theorem 3.16. ∎

###  3.4 Pricing of sig-payoffs

We recall here the notion of _sig-payoffs_ as introduced in Lyons et al.
(2020).

######  Definition 3.21.

Suppose that the price process $S$ is given by a continuous semimartingale. A
payoff $F:\Omega\to\mathbb{R}$ is said to be a sig-payoff if there exists
$m\in{\mathbb{N}}$, and $f:=\\{f_{\emptyset},f_{J}\colon 0<|J|\leq m\\},$ such
that

$\displaystyle F:=f_{\emptyset}+\sum_{0<|J|\leq m}f_{J}\langle
e_{J},\widehat{\mathbb{S}}_{T}\rangle,$

where $\widehat{\mathbb{S}}$ denotes the signature of
$\widehat{S}_{t}=(t,S_{t})$.

######  Example 3.22.

Let $K>0$ be a strike price and $T>0$ a maturity time. Then, Asian forwards
written on a stock $S$ are payoffs of the form

$\frac{1}{T}\int_{0}^{T}S_{t}\mathrm{d}t-K=\frac{1}{T}\int_{0}^{T}(S_{t}-S_{0})\mathrm{d}t-K+S_{0}=\frac{1}{T}\langle
e_{1}\otimes e_{0},\widehat{\mathbb{S}}_{T}\rangle+(K-S_{0})\langle
e_{\emptyset},\widehat{\mathbb{S}}_{T}\rangle,$

and are thus sig-payoffs.

Even though the standard vanilla derivatives such as call and put options are
_not_ sig-payoffs, approximate sig-payoffs can be used as efficient control
variates in Monte Carlo pricing, which we outline in Section 4.2.2.

Note that in this section we shall use $\widehat{S}_{t}$ always for
$\widehat{S}_{t}=(t,S_{t})$ and consider pricing of sig-payoffs when $S$ is
given by a signature model, as made precise in the following corollary.

######  Corollary 3.23.

Let the dynamics of $S_{n}(\ell)$ under a local martingale measure
${\mathbb{Q}}$ be specified as in Corollary 3.12 for $\widehat{X}$ given by
Definition 3.1(ii). Consider a sig-payoff

$F=f_{\emptyset}+\sum_{0<|J|\leq m}f_{J}\langle
e_{J},\widehat{\mathbb{S}}_{n}(\ell)_{T}\rangle.$

Then, using the notation of Remark 3.17 we can write the corresponding price
as

$\displaystyle{\mathbb{E}}_{\mathbb{Q}}[F]$
$\displaystyle=f_{\emptyset}+\sum_{0<|J|\leq
m}f_{J}\widetilde{P}_{J}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}])$
$\displaystyle=f_{\emptyset}+\sum_{|J|=1}^{m}\sum_{k_{1},\ldots,k_{|J|}=1}^{D}\sum_{|I_{1}|,\ldots,|I_{|J|}|=0}^{n}{\mathbb{E}}[\langle{\widetilde{\shuffle}}_{i=1}^{{|J|}}\tilde{e}_{I_{i}}^{k_{i}},\widehat{{\mathbb{X}}}_{T}\rangle]f_{J}\prod_{i=1}^{|J|}\tilde{c}(j_{i},I_{i},k_{i},\ell),$
(3.6)

for
$\tilde{c}(j,I,k,\ell):=1_{\\{j=0\\}}1_{\\{I=(0)\\}}1_{\\{k=1\\}}+\ell_{I}^{k}1_{\\{j=1\\}}$.

###### Proof.

This is a direct consequence of Theorem 3.16 and Remark 3.17. ∎

######  Remark 3.24.

  1. (i)

Expression (3.6) admits also a second representation that turns out to be
useful for coding:

$\displaystyle{\mathbb{E}}_{\mathbb{Q}}[F]$
$\displaystyle=f_{\emptyset}+\sum_{|J|=1}^{m}\sum_{k_{1},\ldots,k_{|J|}=1}^{D}\sum_{I_{1},\ldots,I_{|J|}\in{\mathcal{I}}}{\mathbb{E}}[\langle{\widetilde{\shuffle}}_{i=1}^{{|J|}}\tilde{e}_{I_{i}}^{k_{i}},\widehat{{\mathbb{X}}}_{T}\rangle]$
$\displaystyle\qquad\times\,f_{(1_{\\{I_{1}\neq I^{t},k_{1}\neq
0\\}},\ldots,1_{\\{I_{{|J|}}\neq I^{t},k_{{|J|}}\neq
0\\}})}\prod_{i=1}^{|J|}\Big{(}1_{\\{I_{i}=I^{t},k_{i}=0\\}}+\ell_{I_{i}}^{k}1_{\\{I_{i}\neq
I^{t},k_{i}\neq 0\\}}\Big{)}.$

  2. (ii)

With a similar procedure it is also possible to provide a representation of
${\mathbb{V}}\textup{ar}_{\mathbb{Q}}(F)$. By Proposition 2.3 we know that

$(F-f_{\emptyset})^{2}=\sum_{|J_{1}|,|J_{2}|=1}^{m}f_{J_{1}}f_{J_{2}}\langle
e_{J_{1}}\shuffle e_{J_{2}},\widehat{\mathbb{S}}_{n}(\ell)_{T}\rangle.$

Setting $\widetilde{P}_{I_{1}\shuffle
I_{2}}:=\sum_{i=1}^{K}\widetilde{P}_{J_{i}}$ for $J_{i}$ and $K$ satisfying
$e_{I_{1}}\shuffle e_{I_{2}}=\sum_{i=1}^{K}e_{J_{i}}$ we thus obtain

$\displaystyle{\mathbb{V}}\textup{ar}_{\mathbb{Q}}(F)$
$\displaystyle={\mathbb{E}}_{\mathbb{Q}}[(F-f_{\emptyset})^{2}]-{\mathbb{E}}_{\mathbb{Q}}[F-f_{\emptyset}]^{2}$
$\displaystyle=\sum_{|J_{1}|,|J_{2}|=1}^{m}f_{J_{1}}f_{J_{2}}\Big{(}\widetilde{P}_{J_{1}\shuffle
J_{2}}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}])-\widetilde{P}_{J_{1}}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}])\widetilde{P}_{J_{2}}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}])\Big{)}.$

  3. (iii)

From this representations we can see that the maps
$P_{F}(f,\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{T}]):={\mathbb{E}}[F]$ and
$P_{{\mathbb{V}}\textup{ar}(F)}(f,\ell,{\mathbb{E}}[\widehat{\mathbb{X}}_{T}]):={\mathbb{V}}\textup{ar}(F)$
inherit good properties from $\widetilde{P}_{J}$. In particular, $P_{F}$ is
linear in $f$ and polynomial of degree $m$ in $\ell$, and
$P_{{\mathbb{V}}\textup{ar}(F)}$ is quadratic in $f$ and polynomial of degree
$2m$ in $\ell$. Both maps are linear in
${\mathbb{E}}[\widehat{\mathbb{X}}_{T}]$.

##  4 Calibration

This section is dedicated to illustrate how the signature model can be
calibrated to market data. We here consider three different tasks: first,
calibration to time series data, second calibration to option prices, and
third a combination of these two tasks.

Throughout the section we fix $n\geq 1$ and $d\geq 1$. Consider a model given
as in Corollary 3.12 with $\widehat{X}$ as in Definition 3.1(ii). In order to
simplify the notation we set $\ell_{\emptyset}:=S_{0}$ and we drop the index
$1$ from $\ell^{1}_{I}$ and $\tilde{e}_{I}^{1}$.

###  4.1 Calibration to time-series data

For the time-series calibration we investigate two different methods. In both
cases we suppose that time series data for $\widehat{X}$ and ${S}$ are
available on a time grid $t_{1},\ldots,t_{N}$, with $N>1$.

####  4.1.1 Regression using price data

Our first method consists in performing a _linear_ regression where we
directly regress – according to the signature model as in Corollary 3.12 for
$D=1$ – the trajectories of $S$ on the signature of $\widehat{X}$. For the
computation of $(\widehat{\mathbb{X}}_{t_{i}})_{i=1}^{N}$ based on the
observations $(\widehat{X}_{t_{i}})_{i=1}^{N}$ we can resort to several
available packages, e.g. the package `iisignature` developed by Reizenstein
and Graham (2018) or `signatory` by Kidger and Lyons (2020) in Python.

Let $d^{\ast}:=\frac{(d+1)^{n}-1}{d}$ denote the dimension of the signature
vector of $\widehat{X}$ truncated at level $n-1$. Then the current calibration
problem consists in finding $\ell^{\ast}\in\mathbb{R}^{d^{\ast}}$ such that

$\ell^{\ast}\in\displaystyle{\operatorname{argmin}_{\ell}}\
L_{\text{price},\alpha}(\ell),$

where the loss function $L_{\text{price},\alpha}(\ell)$ is given by

$\displaystyle L_{\text{price},\alpha}(\ell)$
$\displaystyle:=\sum_{i=1}^{N}\Big{(}{S}_{n}(\ell)_{t_{i}}-S_{t_{i}}\Big{)}^{2}+\alpha(\ell)$
(4.1)
$\displaystyle=\sum_{i=1}^{N}\Big{(}S_{0}+\ell_{\emptyset}\langle{e}_{1},\widehat{{\mathbb{X}}}_{t_{i}}\rangle+\sum_{0<|I|\leq
n-1}\ell_{I}\langle\tilde{e}_{I},\widehat{{\mathbb{X}}}_{t_{i}}\rangle-
S_{t_{i}}\Big{)}^{2}+\alpha(\ell),$

where $\alpha$ denotes a fixed penalization function. An example for $\alpha$
is given by a convex combination of $L^{1}$ and $L^{2}$ penalizations, i.e.,
$\alpha(\ell)=\lambda\lVert\ell\lVert_{1}+(1-\lambda)\lVert\ell\lVert_{2}$,
where $\lambda\in[0,1]$. Note here that the number of parameters corresponds
to $d^{\ast}$, i.e. the dimension of the signature truncated at level $n-1$.
This is due to the representation of Corollary 3.12, as also seen from the
second equality in (4.1).

######  Example 4.1.

Depending on the choice of the primary process, the time series for $X$ may
not be directly readable from the market. This is for instance the case if we
assume $X$ to be a vector of correlated Brownian motions, whose trajectories
have to be extracted from the observations of $S$. We discuss such a procedure
in the following within the class of classical stochastic volatility models.

Suppose that the dynamics of the asset price process are described by a
stochastic volatility model under the physical measure ${\mathbb{P}}$, e.g.,

$\displaystyle{\mathrm{d}}S_{t}$
$\displaystyle=\mu(S_{t},V_{t}){\mathrm{d}}t+g(S_{t},V_{t}){\mathrm{d}}B_{t}^{{\mathbb{P}}}$
$\displaystyle{\mathrm{d}}V_{t}$
$\displaystyle=h(S_{t},V_{t}){\mathrm{d}}t+\sigma(S_{t},V_{t}){\mathrm{d}}W_{t}^{{\mathbb{P}}},$

with ${\mathrm{d}}[B^{{\mathbb{P}}},W^{{\mathbb{P}}}]_{t}=\rho{\mathrm{d}}t$
where $\rho\in[-1,1]$ and $\mu,h,g,\sigma$ functions from $\mathbb{R}^{2}$ to
$\mathbb{R}$ such that $g$ and $\sigma$ are strictly positive. Choosing $d=2$,
we aim to estimate the trajectories of a primary process
$X_{t}:=(B_{t}^{\mathbb{Q}},W_{t}^{\mathbb{Q}})$ specified by

$B_{t}^{{\mathbb{Q}}}=\int_{0}^{t}\frac{\mu(S_{s},V_{s})}{g(S_{s},V_{s})}{\mathrm{d}}s+B_{t}^{{\mathbb{P}}},\qquad\qquad
W_{t}^{{\mathbb{Q}}}=\int_{0}^{t}\frac{h(S_{s},V_{s})}{\sigma(S_{s},V_{s})}{\mathrm{d}}s+W_{t}^{{\mathbb{P}}}.$
(4.2)

Note that under sufficient integrability conditions on the market prices of
risk $\mathbb{Q}$ is an equivalent local martingale measure, under which
$(B^{{\mathbb{Q}}},W^{{\mathbb{Q}}})$ are correlated ${\mathbb{Q}}$-Brownian
motions and under which the dynamics of $(S,V)$ are of the following form

$\displaystyle{\mathrm{d}}S_{t}$
$\displaystyle=g(S_{t},V_{t}){\mathrm{d}}B_{t}^{{\mathbb{Q}}},$
$\displaystyle{\mathrm{d}}V_{t}$
$\displaystyle=\sigma(S_{t},V_{t}){\mathrm{d}}W_{t}^{{\mathbb{Q}}}.$

Note that other choices of equivalent local martingale measure
$\widetilde{\mathbb{Q}}$ are of course possible. This then amounts to specify
the second Brownian motion $W_{t}^{\widetilde{{\mathbb{Q}}}}$ (under the
assumption that we stay in a Markovian framework) as

$W_{t}^{\widetilde{{\mathbb{Q}}}}=\int_{0}^{t}\frac{h(S_{s},V_{s})-h^{\widetilde{{\mathbb{Q}}}}(S_{s},V_{s})}{\sigma(S_{s},V_{s})}{\mathrm{d}}s+W_{t}^{{\mathbb{P}}}.$

Under this measure the dynamics of $V$, then read as

${\mathrm{d}}V_{t}=h^{\widetilde{{\mathbb{Q}}}}(S_{t},V_{t})\mathrm{d}t+\sigma(S_{t},V_{t}){\mathrm{d}}W_{t}^{\widetilde{{\mathbb{Q}}}}.$

A simple choice is of course to choose $h^{\widetilde{{\mathbb{Q}}}}=h$ so
that $W^{\widetilde{{\mathbb{Q}}}}=W^{{\mathbb{P}}}$. We consider in the
applications below always the case $h^{\widetilde{{\mathbb{Q}}}}=0$, i.e.
$W^{\mathbb{Q}}$ as defined in (4.2).

In order to extrapolate the trajectories of $(B^{\mathbb{Q}},W^{\mathbb{Q}})$,
we estimate the spot quadratic variation of the price and of the volatility
process (see e.g. Jacod and Protter (2011) for different types of pathwise
spot covariance estimators in general semimartingale contexts), to get an
estimator of $\Sigma_{11}:=g(S_{t},V_{t})^{2}$ and
$\Sigma_{22}:=\sigma(S_{t},V_{t})^{2}$, which in turn allows to compute

${\mathrm{d}}B_{t}^{{\mathbb{Q}}}=\frac{{\mathrm{d}}S_{t}}{\sqrt{\Sigma_{11,t}}},\qquad{\mathrm{d}}W_{t}^{{\mathbb{Q}}}=\frac{{\mathrm{d}}V_{t}}{\sqrt{\Sigma_{22,t}}}.$
(4.3)

Observe that assuming $g$ and $\sigma$ to be strictly positive we get that
$\sqrt{\Sigma_{11,t}}=g(S_{t},V_{t})$ and
$\sqrt{\Sigma_{22,t}}=h(S_{t},V_{t})$. This is important to guarantee that
$\mathrm{d}S_{t}=g(S_{t},V_{t})\mathrm{d}B_{t}^{\mathbb{Q}}$ (as requested in
Corollary 3.12) and
$\mathrm{d}V_{t}=h(S_{t},V_{t})\mathrm{d}W_{t}^{\mathbb{Q}}$.

Having extracted the trajectories of $B^{\mathbb{Q}}$ and $W^{\mathbb{Q}}$ we
can readily compute the trajectories of their time extended signature. If one
wants to work with a different equivalent local martingale measure
$\widetilde{{\mathbb{Q}}}$ parameterized via
$h^{\widetilde{{\mathbb{Q}}}}(S_{s},V_{s})$ as above, then
$W_{t}^{\widetilde{{\mathbb{Q}}}}$ can obtained via

$\mathrm{d}W_{t}^{\widetilde{{\mathbb{Q}}}}=\frac{{\mathrm{d}}V_{t}}{\sqrt{\Sigma_{22,t}}}-\frac{h^{\widetilde{{\mathbb{Q}}}}(S_{t},V_{t})}{\sqrt{\Sigma_{22,t}}}{\mathrm{d}}t.$

####  4.1.2 Regression using spot volatility data

Let us now discuss the second method, which works when the primary process $X$
is given by a vector of correlated Brownian motions similarly as in Example
4.1. Indeed, we then can also calibrate to the time-series data of the spot
volatility of the trajectories of $S$. The same task was tackled in Perez
Arribas et al. (2020) for the one-dimensional case with one driving Brownian
motion as primary process and with a lead-lag extension instead of a time
extension. In our framework the calibration to the spot volatility consists in
finding $\ell^{\ast}\in\mathbb{R}^{d^{\ast}}$ such that

$\ell^{\ast}\in\displaystyle{\operatorname{argmin}_{\ell}}\
L_{\text{vol},\alpha}(\ell),$

where the loss function $L_{\text{vol},\alpha}(\ell)$ is now given by

$L_{\text{vol},\alpha}(\ell):=\sum_{i=1}^{N}\Big{(}\ell_{\emptyset}+\sum_{0<|I|\leq
n}\ell_{I}\langle
e_{I},\widehat{{\mathbb{X}}}_{t_{i}}\rangle-\sqrt{\frac{{\mathrm{d}}}{{\mathrm{d}}t}[S]_{t_{i}}}\Big{)}^{2}+\alpha(\ell),$
(4.4)

where $\frac{{\mathrm{d}}}{{\mathrm{d}}t}[S]$ denotes the spot quadratic
variation process and $\alpha$ a penalty function as in (4.1). Note that we
still work with the signature model as of Corollary 3.12 (for $D=1$)
exploiting its representation in form of (3.3).

######  Remark 4.2.

  1. (i)

As the spot volatility is not directly available but has to be estimated, the
penalty function plays an important role in order to avoid overfitting to
noisy data. Note also that not all coefficients of the signature might be
statistically significant. This includes in particular the case where $d$ is
large and the components of ${X}$ are strongly correlated, and the case where
the number of parameters to be calibrated is much higher than the number of
points we aim to fit.

  2. (ii)

Let us stress that in the context of Example 4.1 the calibration to the price
path and to the volatility process $g(S,V)$ are equivalent in the following
sense. Once we have found $\ell^{\ast}$ solving (4.4), we obtain

$[S]_{t}\approx\int_{0}^{t}\bigg{(}\ell_{\emptyset}^{*}+\sum_{0<|I|\leq
n}\ell_{I}^{\ast}\langle
e_{I},\mathbb{\widehat{X}}_{s}\rangle\bigg{)}^{2}\mathrm{d}s.$

Since by definition
$S_{t}=\int_{0}^{t}g(S_{s},V_{s})\mathrm{d}B_{s}^{\mathbb{Q}},$
$g(S_{t},V_{t})>0$, and $[B^{\mathbb{Q}}]_{t}=t$, this yields

$S_{t}\approx S_{0}+\int_{0}^{t}\bigg{(}\ell_{\emptyset}^{*}+\sum_{0<|I|\leq
n}\ell_{I}^{\ast}\langle
e_{I},\mathbb{\widehat{X}}_{s}\rangle\bigg{)}\mathrm{d}B^{\mathbb{Q}}_{s},$

which by Lemma 3.10 can be rewritten as

$S_{t}\approx S_{0}+\ell_{\emptyset}^{*}B_{t}^{\mathbb{Q}}+\sum_{0<|I|\leq
n}\ell_{I}^{\ast}\langle\tilde{e}_{I},\mathbb{\widehat{X}}_{t}\rangle.$

As we have the same expression for the parameters $\ell^{*}$ obtained via
(4.1), the claim follows.

####  4.1.3 Numerical examples

We now study the above regression tasks by means of several examples based on
classical models. We start with a SABR-type and a Heston model.

######  Example 4.3.

Consider two classical stochastic volatility models, namely a SABR-type model

$\displaystyle{\mathrm{d}}S_{t}$ $\displaystyle=\mu
S_{t}{\mathrm{d}}t+S_{t}V_{t}{\mathrm{d}}B_{t}^{{\mathbb{P}}}$
$\displaystyle{\mathrm{d}}V_{t}$ $\displaystyle=\kappa(\theta-
V_{t}){\mathrm{d}}t+\sigma V_{t}{\mathrm{d}}W_{t}^{{\mathbb{P}}},$

and a Heston model

$\displaystyle{\mathrm{d}}S_{t}$ $\displaystyle=\mu
S_{t}{\mathrm{d}}t+S_{t}\sqrt{V_{t}}{\mathrm{d}}B_{t}^{{\mathbb{P}}}$ (4.5)
$\displaystyle{\mathrm{d}}V_{t}$ $\displaystyle=\kappa(\theta-
V_{t}){\mathrm{d}}t+\sigma\sqrt{V_{t}}{\mathrm{d}}W_{t}^{{\mathbb{P}}},$

where in both cases
${\mathrm{d}}[B^{{\mathbb{P}}},W^{{\mathbb{P}}}]_{t}=\rho{\mathrm{d}}t$ where
$\rho\in[-1,1]$.

We now aim to approximate these models with signature models whose primary
processes are ${\mathbb{Q}}$-Brownian motions extracted from the time series
data of $S$ and $V$ as in Example 4.1. The calibration is performed applying
both procedures described above, namely the calibration to the price’s
trajectory and the calibration to the spot volatility trajectory. The
corresponding loss functions are given by (4.1) and (4.4), respectively. In
both cases the truncation parameter is chosen to be $n=2$ and a Lasso’s
penalty function $\alpha(\ell)=10^{-5}\|\ell\|_{1}$ has been employed (a Ridge
regression has led to similar results). For the experiment, we select the
models parameters

$\\{S_{0},V_{0},\mu,\kappa,\theta,\sigma,\rho\\}:=\\{1,0.08,0.001,0.5,0.15,0.25,-0.5\\}.$

Observe that in the two models the process $V$ has a different meaning.
Indeed, in the SABR it corresponds to the spot volatility while in the Heston
it is the spot variance. As we choose the same parameters, this means that the
volatility $V$ in the SABR model is smaller that the volatility $\sqrt{V}$ in
the Heston model.

For both fit we extract 3 ticks of Brownian motions per day from trajectories
that are assumed to be observable with a frequency of 4 seconds for 8 hours
during 1 calendar year.

The calibrations have thus been performed over 1 calendar year of observations
with 3 ticks per day ($T=1$ and $N=1095$). The obtained parameters $\ell^{*}$
are then tested simulating 0.5 years of new trajectories of the stochastic
volatility models, extracting the corresponding ${\mathbb{Q}}$-Brownian
motions, and computing the trajectories of $S_{3}(\ell^{*})$ as in Corollary
3.12. Results are illustrated in Figure 1.

(a) Regression on the price of a Heston model as described in (4.1).

(b) Regression on the volatility of a SABR-type model as described in (4.4).

Figure 1: Out-of-sample of half calendar year with training sample of one
calendar year.

To assess the accuracy of the above calibration to time-series data we
consider the mean squared error (MSE) between the predicted path and the
observed path, i.e.

$\frac{1}{K}\sum_{i=1}^{K}(S_{n}(\ell^{*})_{t_{i}}-S_{t_{i}})^{2}.$

In the following tables we state the in-sample MSE denoted by
$\text{MSE}_{\text{ex}}^{\text{in}}$ as well as the out-of-sample MSE denoted
by $\text{MSE}_{\text{ex}}^{\text{out}}$. To justify the robustness of the
estimated parameters the out-of-sample MSEs are calculated as an average of
the mean squared errors on 1000 different realizations of the simulated model.

Regression on the price as in (4.1) under a Heston model:

$\text{MSE}_{\text{ex}}^{\text{in}}$ | $\text{MSE}_{\text{ex}}^{\text{out}}$
---|---
$2.574\cdot 10^{-6}$ | $2.841\cdot 10^{-4}$

Regression on the volatility as in (4.4) under the SABR-type model:

$\text{MSE}_{\text{ex}}^{\text{in}}$ | $\text{MSE}_{\text{ex}}^{\text{out}}$
---|---
$3.204\cdot 10^{-6}$ | $1.060\cdot 10^{-6}$

It is worth mentioning that learning the volatility as in (4.4) requires an
estimation of the paths of
$(\sqrt{\frac{{\mathrm{d}}}{{\mathrm{d}}t}[S]_{t}})_{t\in[0,T]}$ from the
price’s trajectories. If such estimation is too noisy the procedure can yield
unsatisfactory results. One reason why this is here not visible in the out-of-
sample MSEs is that in the SABR model the volatility enters linearly in the
price dynamics, and hence is easier to be learned via (4.4), which can be seen
from the above tables. Note also that the smaller volatility in the SABR
models plays role in the order of magnitude of
$\text{MSE}_{\text{ex}}^{\text{out}}$.

In the next example we consider a multivariate asset price process whose
dynamics are given by a multi-dimensional Black-Scholes model.

######  Example 4.4 (The multi-dimensional case).

Consider a $d$-dimensional Black-Scholes model under the physical probability
measure ${\mathbb{P}}$, i.e.,

$\mathrm{d}S_{t}=\mu^{\top}\operatorname{diag}(S_{t})\mathrm{d}t+\operatorname{diag}(S_{t})\sqrt{\Sigma}\mathrm{d}B_{t}^{\mathbb{P}},$

where $B^{\mathbb{P}}$ is a $d$-dimensional standard Brownian motion (with
independent components), $\mu\in\mathbb{R}^{d}$, $\Sigma\in\mathbb{S}_{+}^{d}$
and $\sqrt{\cdot}$ denotes the matrix square root. Similarly as in Example 4.1
we extract independent ${\mathbb{Q}}$-Brownian motions (i.e. $\mathbb{P}$ with
the corresponding market price of risk) which serve as primary process $X$ in
the signature model, now however of dimension $d>1$.

In order to calibrate to the simulated price path $S$, we perform a Ridge
regression using the prices’ trajectories with penalty function
$\alpha(\ell)=5\cdot 10^{-5}\lVert\ell\lVert_{2}$. The truncation parameter is
chosen to be $n=5$, corresponding to 46655 parameters to be calibrated. We
considered the case of five positively correlated components, i.e.
$\Sigma_{ij}=\sigma_{i}\sigma_{j}\rho_{i,j}$ with $\sigma_{i}\in(0.15,0.34)$
and $\rho_{i,j}\in(0.4,0.9)$ for $i,j=1,\dots,5$, with drifts
$\mu_{i}\in(-0.003,0.001)$ for $i=1,\dots,5$. The calibration has been
performed over 1 calendar year of observations with 3 ticks per day ($T=1$ and
$N=1095$). The obtained parameters $\ell^{*}$ are then tested simulating 4
months of new trajectories of the Black-Scholes model, extracting the
corresponding ${\mathbb{Q}}$-Brownian motions, and computing the trajectories
of $S_{6}(\ell^{*})$ as in Corollary 3.12. Results are illustrated in Figure
2. The $\text{MSE}_{\text{ex}}^{\text{out}}$ on the single trajectories are of
order $10^{-5}$.

Figure 2: Regression on the price of a 5-dimensional Black-Scholes model as
described in (4.1). Out-of-sample of 4 months with training sample of 1
calendar year.

###  4.2 Calibration to option prices

We now pass to the more common way of calibrating models in mathematical
finance, namely to calibration to call (and put) option prices, with the goal
to match their implied volatilities as well as possible.

Throughout we here assume that there exists a pricing measure ${\mathbb{Q}}$
and that the primary process ${X}$ is a vector of correlated
${\mathbb{Q}}$-Brownian motions. The goal of this approach is to choose the
parameters $\ell$ in order to fit the prices of options on ${S}$ available on
the market. In the context of Sig-SDEs, a similar approach has been considered
in Perez Arribas et al. (2020). There the authors however only consider option
prices generated from a Black-Scholes model and maturities longer than
approximately 5 months. To our knowledge, in the context of signature based
models for asset prices, we are the first ones dealing with market data, thus
including shorter maturities and realistic volatility smiles. Suppose we are
given prices of $N$ call options
$C^{\ast}(T_{1},K_{1}),\dots,C^{\ast}(T_{N},K_{N})$ with maturities $T_{i}$
and strikes $K_{i}$. Then in spirit of Cont and Ben Hamida (2005) (see also
Cuchiero et al. (2020)) we translate the current calibration task to finding
$\ell$ such that

$\ell\in\operatorname{argmin}_{\ell}\ L_{\text{option}}(\ell),$

where

$L_{\text{options}}(\ell)=\sum_{i=1}^{N}\gamma_{i}\Big{(}C^{\ast}(T_{i},K_{i})-C^{\text{model}}(T_{i},K_{i},\ell)\Big{)}^{2},$
(4.6)

with $\gamma_{i}$ Vega weights and $C^{\text{model}}(T_{i},K_{i},\ell)$
denoting the price of the option under the signature model with parameters
$\ell$, maturity $T_{i}$ and strike $K_{i}$.

######  Remark 4.5.

The loss function described in (4.6) can be adapted depending on the available
data. A common approach is for instance to weigh the options by their bid-ask
spreads, see for instance Cont and Ben Hamida (2005). This reflects the
relative importance of reproducing different option prices precisely. In the
present work we shall employ Vega weights since bid-ask spreads are not always
provided in the data and since they are well-suited to match the implied
volatility surface.

The same calibration procedure would apply to any other type of option, e.g.
path-dependent exotic ones: as we shall see below, the pricing of the latter
with a signature model does not require any additional effort. For liquidity
reasons we however decided to stick to vanilla options and calibrate to the
corresponding implied volatility surface.

To fix notation we briefly recall the definition of implied volatility, which
is our goodness of fit criterium.

######  Definition 4.6.

Let $C(T,K)$ be the price of a call option with maturity $T$ and strike price
$K$ written on the asset $S$. The implied volatility of $C(T,K)$ is defined as
the volatility $\sigma(T,K)$ that solves the equation

$C^{BS}(T,K,\sigma(T,K))=C(T,K),$ (4.7)

where $C^{BS}$ Black-Scholes option price. Then $(\sigma(T,K))_{T,K}$ is
called (implied) volatility surface, and $(\sigma(T,K))_{K}$ is called
(implied) volatility smile for each fixed maturity $T$.

Once the optimal parameters $\ell^{\ast}$ are found via (4.6) we then need to
solve (4.7) numerically for $C(T,K)=C^{\text{model}}(T,K,\ell^{\ast})$ to
assess the goodness of fit. Indeed we measure it in terms of the absolute
error (in basis points) between the implied volatilities from the market and
the model.

####  4.2.1 Monte Carlo pricing

We now address the problem of computing for a fixed strike $K\in\mathbb{R}$
and maturity $T>0$, the price $C^{\text{model}}(T,K,\ell)$ of the
corresponding option under the signature model.

Before presenting our method based on Monte Carlo, recall that Section 3.4
provides a closed-form formula for the computation of sig-payoffs without the
need to resort to Monte Carlo techniques. By the universal approximation
theorem (Theorem 2.12), we also know that call options payoffs can be
approximated arbitrarily well by sig-payoffs, or in this case even just by
polynomials, on compacts. The combination of these two properties leads to the
approach exploited by Perez Arribas et al. (2020) of splitting the computation
of $C^{\text{model}}(T,K,\ell)$ in two parts: an approximation of the call
(put) payoffs using sig-payoffs and pricing the latter with the closed formula
proved in Section 3.4. However, in order to achieve an acceptable error
between the original call payoff and the sig-payoff one needs to choose a sig-
payoff involving signature terms of quite high order. Indeed, such an
approximation needs to be valid for every set of coefficients $\ell$ involved
in the optimization procedure and thus on a very large compact set $K$. From a
computational point of view we observed that this sig-payoff or polynomial
approximation is not tenable and we thus opted for abandoning this approach.
Note however that sig-payoffs can still be used as control variates in Monte
Carlo pricing techniques as we shall outline in Section 4.2.2. For more
details on the issues that can occur in view of sig-payoff or polynomial
approximations we refer to Section B.

For these reasons we therefore adopted a Monte Carlo approach, which is due to
the linearity of the model particularly tractable, making the whole
calibration computationally feasible within a reasonable amount of time while
providing highly accurate results. For the Monte Carlo price we thus fix a
number of samples $N_{MC}>0$ and approximate $C^{\text{model}}(T,K,\ell)$ via

$C^{\text{model}}(T,K,\ell)\approx\frac{1}{N_{MC}}\sum_{i=1}^{N_{MC}}(S_{n}(\ell)_{T}(\omega_{i})-K)^{+}.$

We stress again that this can be computed fast. Indeed, by the linearity of
the model simulating $(S_{n}(\ell)_{T}(\omega_{i}))_{i=1}^{N_{MC}}$ boils down
to the following steps:

  * •

simulate $(X_{t}(\omega_{i}))_{t\in[0,T]}$, which in the current setting are
just trajectories for correlated Brownian motions, for each
$i\in\\{1,\ldots,N_{MC}\\}$;

  * •

compute $\langle e_{I},\widehat{\mathbb{X}}_{T}(\omega_{i})\rangle$ for all
$i=1,\dots,N_{MC}$ and for all multi-indices $I$ such that $\lvert I\lvert\leq
n$;

  * •

take linear combinations to compute
$\langle\tilde{e}_{I},\widehat{\mathbb{X}}_{T}(\omega_{i})\rangle$ for all
$i=1,\dots,N_{MC}$ and for all multi-indices $|I|\leq n-1$ as described in
Lemma 3.10;

  * •

retrieve $(S_{n}(\ell)_{T}(\omega_{i}))_{i=1}^{N_{MC}}$ via (3.2).

Observe that the parameters $\ell$ only enter in the very last step which
allows to precompute and store all other quantities. This is in contrast to
other models where the parameters that need to be calibrated enter at each
time point in the simulation steps of e.g. an Euler scheme or more complicated
schemes used for instance for rough volatility models.

######  Remark 4.7.

Since the calibration problem is usually ill-posed as inverse problem, there
might exist more than one local minima of the loss function which is able to
reproduce the prices on the market, and their implied volatility. We refer to
Cont and Tankov (2004) where the sensitivity, e.g. in terms of initial
starting points in gradient descent optimization algorithms, of the nonlinear
least squares calibration problem in the case of exponential Lévy models is
discussed. In our setting it is worth mentioning that since $S_{n}(\ell)_{T}$
is linear and the call payoffs are convex we have a convex optimization
problem whenever for all maturities and strikes

$C^{\ast}(T,K)\leq\frac{1}{N_{MC}}\sum_{i=1}^{N_{MC}}(S_{n}(\ell)_{T}(\omega_{i})-K)^{+}.$

Therefore the initial random parameter $\ell$ for the optimization can for
instance be initialized according to this condition.

####  4.2.2 Variance reduction with sig-payoffs

Even though Monte Carlo pricing is fast since all essential quantities can be
precomputed as explained above, we here discuss variance reduction techniques
(see e.g. Glasserman (2004)) that can speed up the procedure even further. The
idea is to introduce a control variate, i.e., a random variable $\Phi^{cv}$
such that:

$\mathbb{E}_{{\mathbb{Q}}}[\Phi^{cv}]=0,\qquad\qquad{\mathbb{V}}\textup{ar}\big{(}(S_{n}(\ell)_{T}-K)^{+}-\Phi^{cv}\big{)}<{\mathbb{V}}\textup{ar}\big{(}(S_{n}(\ell)_{T}-K)^{+}\big{)}.$

An example of control variates used for pricing and calibrating neural SDE
models can be found in Cuchiero et al. (2020); Gierjatowicz et al. (2020),
where $\Phi^{cv}$ is constructed from hedging strategies. A possible other
choice of control variates for signature models are sig-payoffs. Indeed, one
can use the pricing formula derived in Section 3.4 to define:

$\Phi^{cv}(T,K,\ell):=f_{\emptyset}+\sum_{0<\lvert J\lvert\leq m}f_{J}\langle
e_{J},\widehat{\mathbb{S}}_{n}(\ell)_{T}\rangle-\widetilde{P}_{f}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}]),$

for some fixed $m>0$ where

$(S_{n}(\ell)_{T}-K)^{+}\approx f_{\emptyset}+\sum_{0<\lvert J\lvert\leq
m}f_{J}\langle e_{J},\widehat{\mathbb{S}}_{n}(\ell)_{T}\rangle,$

for a wide range of $\ell$ and with high probability. This can be done by
performing a linear regression to obtain the coefficients $f$. Alternatively,
a polynomial approximation of the payoff’s function can also be employed.

The properties of $\Phi^{cv}$ then guarantee the accuracy of the approximation

$C^{\text{model}}(T,K,\ell)\approx\frac{1}{N_{MC}}\Big{(}\sum_{i=1}^{N_{MC}}(S_{n}(\ell)_{T}(\omega_{i})-K)^{+}-\Phi^{cv}(T,K,\ell)(\omega_{i})\Big{)},$

already for smaller values of $N_{MC}$.

####  4.2.3 Model’s performance

In the following we discuss the problem of minimizing the functional (4.6)
using the Monte Carlo method as described in Section 4.2.1 to compute the
model prices. We consider the model described in Corollary 3.12 for
$\widehat{X}_{t}=(t,B_{t},W_{t})$ as in Definition 3.1(ii) for two correlated
Brownian motions $B$ and $W$ with correlation coefficient $\rho=-0,5$, and
$D=1$.

As a first example we consider synthetic data, where the implied volatility
surface we aim to fit is generated by a Heston model given by (4.5). We
consider 7 maturities $(T_{k})_{k=1}^{7}$ ranging from 30 days to 2 years and
13 strikes $(K_{j})_{j=1}^{13}$ ranging from 80$\%$ to 120$\%$ of the spot
price. The truncation parameter is fixed to $n=3$ and the number of Monte
Carlo samples to $N_{MC}=10^{6}$. The results for the following two sets of
parameters under the risk neutral measure

$\kappa$ | $\theta$ | $\sigma$ | $\rho$ | $V_{0}$
---|---|---|---|---
0.1 | 0.1 | 0.4 | -0,5 | 0.08
0.2 | 0.3 | 0.5 | -0,5 | 0.08

are displayed in the first and in the second row of Figure 3, respectively.

Figure 3: On the left: blue stars correspond to the implied volatilities of
the Heston models, red dots denote the calibrated implied volatilities of
$S_{n}(\ell)$ with $n=3$ (13 estimated parameters). On the right: absolute
errors between the two surfaces expressed in basis points (Bps).

We stress that these calibrations to Heston generated implied volatility
surfaces can take between 4 to 15 minutes on a standard machine.

Let us now turn to market data. We consider the trading day 17/03/2021 for
call options written on the S$\&$P 500 index. Our dataset provided by
Bloomberg consists of 7 maturities $(T_{k})_{k=1}^{7}$, ranging from 30 days
to 2 years, and 9 strike prices $(K_{j})_{j=1}^{9}$ for each maturity which
vary between 80$\%$ and 120$\%$ of the spot price. Again, the truncation
parameter is fixed to $n=3$ and the Monte Carlo’s parameter to
$N_{MC}=10^{6}$. The results are displayed in Figure 4.

Figure 4: On the left: the upper surface represents the implied volatility of
the S$\&$P 500 index as of 17-03-2021, the lower one is the calibrated implied
volatility of $S_{n}(\ell^{*})$ with $n=3$ (13 parameters). On the right:
absolute error between the two surfaces (Bps).

From Figure 4 we notice that volatility smiles for short maturities have not
been captured. In particular, as for many continuous models, the most
problematic part consists in fitting the shortest maturities (see Chapter 3
and Chapter 7 of Gatheral (2011)).

To investigate the ability of the model to reproduce short maturity smiles we
calibrate it using different loss functions penalizing outliers more severely.
Precisely, we first calibrate the model using the procedure described above
and we denote by $w_{i}$ the absolute error between the target implied
volatility and the approximated one for maturity $T_{i}$ and strike $K_{i}$.
Then, in spirit of generative-adversarial distances as for instance considered
in Cuchiero et al. (2020), we define a new loss function

$L_{p,\alpha}(\ell)=\sum_{i=1}^{N}(\gamma_{i}+\alpha
w_{i})|C^{\ast}(T_{i},K_{i})-C^{\text{model}}(T_{i},K_{i},\ell)|^{p},$ (4.8)

depending on parameters $p$ and $\alpha$ that need to be chosen. By taking
high values for $p$ and $\alpha$ we can approximate the sup-distance between
the two price surfaces, i.e.,

$L_{\infty}(\ell)=\sup_{i=1,\ldots,N}|C^{\ast}(T_{i},K_{i})-C^{\text{model}}(T_{i},K_{i},\ell)|,$

without compromising differentiability with respect to $\ell$. The result for
different choices of the parameters $\alpha$ and $p$ but also the truncation
level $n$ is displayed in Figure 5. As can be guessed from the figure,
although the maximal absolute error for maturities larger than 60 days is
(almost) acceptable (96 Bps for $n=3$, $p=1000$, and $\alpha=500$ and 34 Bps
for $n=4$ and $p=1000$, and $\alpha=500$), the absolute error for the shortest
maturity and the far in and out of the money strikes are still above $270$ Bps
and $395$ Bps respectively. Observe that the performance for $n=4$ is the best
for every maturity larger than 60 days as well as for the at-the-money region
of the shortest maturity.

Figure 5: Comparison between the calibrated implied volatility smiles for
different parameters and the S$\&$P 500 index as of 17-03-2021 smile (in blue)
at maturity $T_{1}=30$ days (on the left) and for maturities ranging from 60
days to 2 years (on the right). Calibration has been performed using (4.8).

Finally, as a further experiment, we calibrate the model to the shortest
maturity alone and to every other maturity together. The first calibration is
thus performed to $T_{1}=30$ days and the second one to 6 maturities ranging
from 60 days to 2 years. In both cases 9 strikes ranging from 80$\%$ to
120$\%$ of the spot price are considered. The parameters are fixed to $n=2$,
$p=2$, and $\alpha=0$ for the calibration to the first maturity and $n=4$,
$p=300$, $\alpha=500$ for the remaining maturities. The result is displayed in
Figure 6. We can see that the absolute error is below 12 Bps for the first
maturity and 45 for every other maturity, thus yielding a high accuracy. The
computational time needed for the calibration to the first smile is the range
of a few minutes, the calibration to the remaining surface can take longer,
which is due to the higher value of $n$ and the joint calibration to all
maturities.

Figure 6: Comparison between the calibrated implied volatility smiles and the
SPX-500 Index as of 17-03-2021 smile (in blue) at maturity $T_{1}=30$ days (on
the left) and for maturities ranging from 60 days to 2 years (on the right).
Two different calibrations have been performed using (4.8).

These results suggest that introducing maturity-dependent parameters and
performing a slice-wise calibration to the individual smiles as for instance
in Cuchiero et al. (2020) or Gierjatowicz et al. (2020) can be of interest to
obtain both, an excellent accuracy and a low computational time. We
investigate this further in Section 4.2.4.

######  Remark 4.8.

Note that a signature model with a two dimensional primary process and $n=4$
as in the Figure 6 (right) has 121 parameters. This model is calibrated to 54
options prices. To investigate possible over-fitting, we computed values of
the implied volatility surface for out-of-sample strikes and maturities. More
precisely, we added in total 76 new points. The result is displayed in Figure
7 where the out-of-sample points are displayed in red and the surface
corresponds to the one of Figure 6 (right) for maturities from 60 days to 2
years. This shows that the model exhibits highly promising generalization
features and is thus well suited for fast arbitrage-free smile inter- and
extrapolation.

Figure 7: Implied volatility surface generated by $S_{4}(\ell^{*})$ with
$\ell^{*}$ as in Figure 6 (right). Red dots denote out of sample points.

####  4.2.4 Time-dependent parameters

Motivated by the above results, we propose in the following a variant of the
initial model where the parameters $\ell$ are allowed to depend on time such
that we can perform a slice-wise calibration. Set $\widehat{X}$ as in
Definition 3.1(ii) and let $\\{T_{1},\dots,T_{N}\\}$ with $N>1$ be a set of
fixed times, usually corresponding to the available maturities, and set

$\overline{S}_{n}(\ell,\ell^{\text{corr}})_{t}:=S_{n}(\ell)_{t}+R_{n}(\ell^{\text{corr}})_{t},$
(4.9)

where $S_{n}(\ell)$ is given as in Corollary 3.12 for $D=1$ and

$R_{n}(\ell^{\text{corr}})_{t}=\sum_{j=1}^{N-1}\sum_{|I|\leq
n-1}\ell_{I}^{\text{corr},j}1_{\\{t\geq
T_{j}\\}}\langle\tilde{e}_{I},\widehat{\mathbb{X}}_{t}-\widehat{\mathbb{X}}_{T_{j}}\rangle.$

Observe that $R_{n}(\ell^{\text{corr}})_{t}=0$ for each $t\leq T_{1}$. This
choice is motivated by the observation that $S_{n}(\ell)$ performs very well
if calibrated on the first maturity only (see Figure 6). On every other
$T_{k}$ we have

$R_{n}(\ell^{\text{corr}})_{T_{k}}=\sum_{j=1}^{k-1}\sum_{|I|\leq
n-1}\ell_{I}^{\text{corr},j}\langle\tilde{e}_{I},\widehat{\mathbb{X}}_{T_{k}}-\widehat{\mathbb{X}}_{T_{j}}\rangle.$

Moreover, for any $t\geq 0$ Lemma 3.10 yields

$\displaystyle\overline{S}_{n}(\ell,\ell^{\text{corr}})_{t}$
$\displaystyle=S_{0}+\int_{0}^{t}\sum_{|I|\leq
n-1}\bigg{(}\ell_{I}+\sum_{j=1}^{N-1}\ell_{I}^{\text{corr},j}1_{\\{s\geq
T_{j}\\}}\bigg{)}\langle
e_{I},\widehat{\mathbb{X}}_{s}\rangle\mathrm{d}X_{s}^{1},$

showing that $\overline{S}_{n}(\ell,\ell^{\text{corr}})$ is still continuous.
We can also see that if $X^{1}$ is a local martingale the same is true for
$\overline{S}_{n}(\ell,\ell^{\text{corr}})$.

######  Remark 4.9.

The added correction term can be interpreted to take a similar role as the so-
called leverage function in local stochastic volatility models going back to
Lipton (2002); Ren et al. (2007). Indeed, the leverage function is an
adjustment to stochastic volatility models that depends on time (and state)
and has the purpose to perfectly fit the observed smiles, which by the
underlying stochastic volatility model can usually not be achieved at the
desired accuracy.

Let us now describe the calibration procedure for the adjusted model with
time-dependent parameters. Suppose (for simplicity) that for each maturity
$T_{1},\ldots,T_{N}$ the price of call options with strikes
$K_{1},\ldots,K_{M}$ is known. The calibration is then performed recursively,
adapting the loss function to the already calibrated parameters. Precisely,
the loss function used to calibrate $\ell$ is given by

$L_{\text{options}}(\ell)=\sum_{j=1}^{M}\gamma_{1,j}\Big{(}C^{\ast}(T_{1},K_{j})-C^{\text{model}}(T_{1},K_{j},\ell)\Big{)}^{2},$
(4.10)

which is a version of (4.6). Next, fix $k\in\\{1,\ldots,N-1\\}$ and let
$\ell^{<k}:=(\ell,\ell^{\text{corr},1},\ldots,\ell^{\text{corr},k-1})$ be the
already calibrated coefficients. The loss function used to calibrate
$\ell^{\text{corr},k}$ is given by

$\displaystyle
L_{\text{options},k}(\ell^{\text{corr},k})=\sum_{j=1}^{M}\gamma_{k,j}\Big{(}C^{\ast}(T_{k},K_{j})-C^{\text{model}}(T_{k},K_{j},\ell^{<k},\ell^{\text{corr},k})\Big{)}^{2}.$
(4.11)

In both cases $\gamma_{k,j}$ denotes the Vega weight corresponding to maturity
$T_{k}$ and strike $K_{j}$ and $C^{\text{model}}$ is computed using Monte
Carlo as outlined in Section 4.2.1, namely

$C^{\text{model}}(T_{k},K_{j},\ell^{<k},\ell^{\text{corr},k})\approx\frac{1}{N_{MC}}\sum_{i=1}^{N_{MC}}(\overline{S}_{n}(\ell^{<k},\ell^{\text{corr},k})_{T_{k}}(\omega_{i})-K_{j})^{+}.$

To test this alternative approach we consider again the trading day 17/03/2021
for call options written on the S$\&$P 500 index, 7 maturities ranging from 30
days to 2 years, and 9 strike prices for each maturity which vary between
80$\%$ and 120$\%$ of the spot price. The underlying process is chosen to be
$X=(B,W)$ for two Brownian motions $B$ and $W$ with correlation $\rho=-0.5$.
The truncation parameter is fixed to $n=2$ and the number of Monte Carlo
samples to $N_{MC}=10^{6}$.

In Figure 8 we report the results after minimizing (4.10) and (4.11) for all
$k=1,\dots,6$. This shows that the signature model with time-dependent
parameters perfectly fits each volatility smile. As illustrated in Figure 9
the model also manages to replicate the observed term structure of the at-the-
money (ATM) volatility skew, defined via
$\psi(T):=\big{\lvert}\frac{\partial\sigma(T,K)}{\partial
K}\big{\lvert}_{K=S_{0}},$ for any maturity $T>0$, very accurately. This is a
feature which can also be reproduced by rough volatility models, but classical
stochastic volatility models rather generate a term structure of the ATM skew
that is constant for small maturities (see Gatheral et al. (2018)).

Finally, we stress the fact that the calibration procedure is also very fast,
as for each smile it only took about approximately 1 minute and 30 seconds.
These results suggest that the signature model with time-dependent
coefficients qualifies for very fast and highly accurate calibration fits.

In Figure 10 we plot the values of the 91 calibrated parameters
$\ell,\ell^{\text{corr},1},\ldots,\ell^{\text{corr},6}$. One can see that the
corrections parameters get smaller and smaller. Note that we could have worked
with a smaller number of corrections leading to a similar performance in terms
of accuracy. As shown at the beginning of the section one set of parameters is
indeed enough to capture long maturities (e.g. maturities longer than one
year).

Figure 8:  On the left: blue stars denotes the implied volatilities on the
market as of 17-03-21 on S$\&$P 500 index and the red dots denote the
calibrated implied volatilities with a signature model as in (4.9) with $n=2$.
On the right: absolute error between the two surfaces (Bps).

Figure 9: Comparison between market term structure of the at-the-money (red
crosses) implied volatility skew $\psi$ and the calibrated one from the SigSDE
(azure circles) minimizing (4.10) and (4.11).

Figure 10: Optimal parameters
$\ell,\ell^{\text{corr},1},\ldots,\ell^{\text{corr},6}$ (consisting of 13
parameters each) found calibrating to the volatility surface of the market as
of 17-03-21 on S$\&$P 500 index.

###  4.3 Joint calibration to time-series data and option prices

In this section we combine the promising results obtained in Section 4.1 and
4.2 to solve the following joint calibration problem. The main focus is to
find a signature model that fits at the same time both, a path realization
coming from time series data as well as a volatility smile implied from option
prices. Let $\widehat{X}$ be as in Definition 3.1(ii), let
$\\{T_{1},\dots,T_{N}\\}$ with $N>1$ be fixed times corresponding to the
options’ maturities and consider a signature model $S_{n}(\ell)$ as in
Corollary 3.12. Suppose that for each maturity $T_{1},\ldots,T_{N}$ the prices
of call options are available for several strikes, and that we have time
series data for the primary process and for the target model on a given time
grid $\Pi$ at our disposal. Consider now the loss function used to calibrate
$\ell$ given by

$L_{\text{joint}}(\ell)=\lambda
L_{\text{options}}(\ell)+(1-\lambda)L_{\text{price},0}(\ell)$

where $L_{\text{price},0}$ is given by (4.1) with $\alpha=0$ and times
$t_{i}\in\Pi$, $L_{\text{options}}$ is given by (4.10), and $\lambda\in[0,1]$.

######  Remark 4.10.

When performing this joint calibration we implicitly assume to have a common
primary process $X$ for both tasks. We here suppose that $X$ is given by
(4.3), i.e. we work with Brownian motions under the local martingale measure
${\mathbb{Q}}$ specified in Example 4.1. Note that under this measure
${\mathbb{Q}}$ the volatility/variance process $V$ is necessarily a local
martingale. Below we shall use a Heston model to generate both the price
trajectory (under ${\mathbb{P}}$) and the option prices (under
${\mathbb{Q}}$). For the latter we specify the parameters to guarantee that
$V$ is a (local) martingale under ${\mathbb{Q}}$. One can of course also work
with different local martingale measures as outlined in Example 4.1.

We consider an example where the traded asset $S$ follows Heston dynamics as
described in (4.5), with the following parameters.

$\mu$ | $\kappa$ | $\theta$ | $\sigma$ | $\rho$ | $V_{0}$ | $S_{0}$
---|---|---|---|---|---|---
0.001 | 0.8 | 0.1 | 0.55 | -0.5 | 0.12 | 1

under the physical measure $\mathbb{P}$. We fix $N=2$ and generate option
prices at maturity $T_{1}=$3 months and $T_{2}=$1 year, under the set of
parameters

$\mu$ | $\kappa$ | $\theta$ | $\sigma$ | $\rho$ | $V_{0}$ | $S_{0}$
---|---|---|---|---|---|---
0 | 0 | 0 | 0.55 | -0.5 | 0.12 | 1

which ensures $S$ and $V$ to be local martingales under $\mathbb{Q}$. We
consider additionally equi-spaced strike prices $\\{K_{1},\dots,K_{9}\\}$ with
moneyness ranging between $70-130\%$ of the spot price. In Figure 11 we report
the fit of the implied volatility smiles for $T_{1},T_{2}$ respectively, with
less than 25 bps of absolute error. For the path calibration we extract daily
Brownian motions from a Heston trajectory which is assumed to be observable
with a frequency of 3 seconds for 8 hours during 9 months. The training sample
consists of 3 months, namely 91 points and the out-of-sample performance in
Figure 12, is tested on 6 months. As hyper-parameters we choose $n=2$,
$\lambda=0.9$, and $N_{MC}=10^{6}$ trajectories for the Monte Carlo pricing.
These plots indicate that the model is also able to jointly fit time-series
and option price data.

Figure 11: Comparison of implied volatilities for $T_{1}=$3 months and
$T_{2}=$1 year. Figure 12: Out-of-sample path calibration performance, with
MSE of $1.497\cdot 10^{-5}$.

##  Appendix A Expected signature of correlated Brownian motions

A possible choice for the primary process ${X}$ is given by a $d$-dimensional
correlated Brownian motion $W$ with correlation matrix $\rho$. Since
Assumption 3.6 is satisfied, we follow Definition 3.1(ii) and set
$\widehat{W}_{t}:=(t,W_{t})$. The signature of $\widehat{W}$ up to time $t$ is
denoted by $\widehat{\mathbb{W}}_{t}$.

We explain now how to compute the expected signature of the extended Brownian
motion $\widehat{W}$, i.e. how to compute ${\mathbb{E}}[\langle
e_{I},\widehat{\mathbb{W}}_{t}\rangle]$ for each multi-index $I$. Several
representations of related quantities including the signature of (independent)
Brownian motions can be found in the literature, see for instance Fawcett
(2003), Lyons and Victoir (2004), Lyons and Ni (2015), Boedihardjo et al.
(2021). Observe that Theorem A.1 below can be seen as a particular case of the
expected signature of a $d+1$-dimensional Lévy process without jumps (but with
correlated Brownian motions), see Friz and Shekhar (2017) or of a
multidimensional Gaussian process, see Rossi Ferrucci and Cass (2022). Our
approach is based on polynomial processes as introduced in Cuchiero et al.
(2012) and Filipović and Larsson (2016). We also refer to Section 4.5 in
Cuchiero and Svaluto-Ferro (2021) for related computations for the signature
of independent Brownian motions.

######  Theorem A.1.

Consider a multi-index $I$ admitting the representation

$e_{I}=e_{0}^{\otimes k_{0}}\otimes e_{J_{1}}\otimes e_{0}^{\otimes
k_{1}}\otimes e_{J_{2}}\otimes\cdots\otimes e_{0}^{\otimes k_{m}},$ (A.1)

for some $J_{i}\in\\{1,\dots,d\\}^{2h_{i}}$ and
$h_{i},k_{i}\in{\mathbb{N}}_{0}$. Then

$\mathbb{E}[\langle
e_{I},\widehat{\mathbb{W}}_{t}\rangle]=\frac{t^{\sum_{i=0}^{m}k_{i}+\sum_{i=1}^{m}h_{i}}}{\left(\sum_{i=0}^{m}k_{i}+\sum_{i=1}^{m}h_{i}\right)!}\bigg{(}\frac{1}{2}\bigg{)}^{\sum_{i=1}^{m}h_{i}}\prod_{i=1}^{m}\rho(J_{i}),$

where $\rho(J):=\prod_{k=1}^{|J|/2}\rho_{j_{2k-1},j_{2k}}$. Moreover, if $I$
does not admit representation (A.1) then ${\mathbb{E}}[\langle
e_{I},\widehat{\mathbb{W}}_{t}\rangle]=0$.

###### Proof.

We first prove that the truncated signature $\widehat{\mathbb{W}}_{t}^{N}$ is
a polynomial process in the sense of Definition 2.1 in Filipović and Larsson
(2016). By Lemma 2.2 in Filipović and Larsson (2016), it suffices to show that
for each $|I|\leq N$ it holds

$\mathrm{d}\langle
e_{I},\widehat{\mathbb{W}}_{t}^{N}\rangle=b(\widehat{\mathbb{W}}_{t}^{N})\mathrm{d}t+\sigma(\widehat{\mathbb{W}}_{t}^{N})\mathrm{d}W_{t}$
(A.2)

for some linear maps $b,\sigma$. Observe that

$\displaystyle\langle e_{I},\widehat{\mathbb{W}}^{N}_{t}\rangle$
$\displaystyle=\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{W}}^{N}_{s}\rangle\circ{\mathrm{d}}\langle
e_{i_{|I|}},\widehat{W}_{s}\rangle$ $\displaystyle=\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{W}}^{N}_{s}\rangle{\mathrm{d}}\langle
e_{i_{|I|}},\widehat{W}_{s}\rangle+\frac{\rho_{i_{{|I|}},i_{{|I|}-1}}}{2}1_{\\{i_{{|I|}},i_{{|I|}-1}\neq
0\\}}\int_{0}^{t}\langle
e_{I^{\prime\prime}},\widehat{\mathbb{W}}^{N}_{s}\rangle{\mathrm{d}}s$
$\displaystyle=\int_{0}^{t}\langle
1_{\\{i_{{|I|}}=0\\}}e_{I^{\prime}}+\frac{\rho_{i_{{|I|}},i_{{|I|}-1}}}{2}1_{\\{i_{{|I|}},i_{{|I|}-1}\neq
0\\}}e_{I^{\prime\prime}},\widehat{\mathbb{W}}^{N}_{s}\rangle\mathrm{d}s$
$\displaystyle\qquad+\int_{0}^{t}\langle
e_{I^{\prime}},\widehat{\mathbb{W}}^{N}_{s}\rangle 1_{\\{i_{{|I|}}\neq
0\\}}{\mathrm{d}}W_{s}^{i_{|I|}}.$

Since $|I^{\prime}|,|I^{\prime\prime}|\leq N$ we can conclude that
$\widehat{\mathbb{W}}^{N}$ satisfies (A.2) and is thus a polynomial process.

Our next step consists in applying Theorem 3.1 in Filipović and Larsson
(2016). Observe that by Itô’s formula we have that the generator
${\mathcal{A}}$ of $\widehat{\mathbb{W}}^{N}$ satisfies

${\mathcal{A}}\langle e_{I},{\,\cdot\,}\rangle:=\langle
1_{\\{i_{|I|}=0\\}}e_{I^{\prime}}+\frac{\rho_{i_{|I|},i_{|I|-1}}}{2}1_{\\{i_{|I|},i_{|I|-1}\neq
0\\}}e_{I^{\prime\prime}},{\,\cdot\,}\rangle=\sum_{e_{I_{1}}\otimes
e_{I_{2}}=e_{I}}\langle e_{I_{1}},Q\rangle\langle
e_{I_{2}},{\,\cdot\,}\rangle,$

for $Q:=e_{0}+\frac{1}{2}\sum_{i,j=1}^{d}\rho_{i,j}e_{i}\otimes e_{j}$.
Shortly, this can be written as ${\mathcal{A}}\langle
e_{I},\cdot\rangle=\langle e_{I},Q\otimes(\cdot)\rangle.$ Theorem 3.1 in
Filipović and Larsson (2016) yields then that

$\displaystyle{\mathbb{E}}[\langle e_{I},\widehat{\mathbb{W}}_{t}\rangle]$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{k!}(t{\mathcal{A}})^{k}\langle
e_{I},{\,\cdot\,}\rangle(\widehat{\mathbb{W}}_{0})=\sum_{k=0}^{\infty}\frac{t^{k}}{k!}\langle
e_{I},Q^{\otimes
k}\otimes\widehat{\mathbb{W}}_{0}\rangle=\sum_{k=0}^{|I|}\frac{t^{k}}{k!}\langle
e_{I},Q^{\otimes k}\rangle.$

Observe that $\langle e_{I},Q^{\otimes
k}\rangle=\sum_{e_{I_{1}}\otimes\cdots\otimes
e_{I_{k}}=e_{I}}\prod_{i=1}^{k}\langle e_{I_{i}},Q\rangle$ is nonzero if and
only if $I$ can be decomposed in multi-indices of length 1 whose elements are
all 0, and multi-indices of length 2, whose elements are all strictly larger
than 0, i.e. if and only if $I$ satisfies (A.1). If this is the case, by
definition of $Q$ it holds

${\mathbb{E}}[\langle
e_{I},\widehat{\mathbb{W}}_{t}\rangle]=\frac{t^{\sum_{i=0}^{m}k_{i}+\sum_{i=1}^{m}h_{i}}}{\left(\sum_{i=0}^{m}k_{i}+\sum_{i=1}^{m}h_{i}\right)!}\langle
e_{0},Q\rangle^{\sum_{i=0}^{m}k_{i}}\prod_{i=1}^{m}\langle
e_{J_{i}},Q^{\otimes h_{i}}\rangle,$

and since $\langle e_{0},Q\rangle=1$ and $\langle e_{J_{i}},Q^{\otimes
h_{i}}\rangle=(\frac{1}{2})^{h_{1}}\prod_{k=1}^{h_{i}}\rho_{j_{2k-1},j_{2k}}$
the claim follows. ∎

A combination of the result of Theorem A.1 with Chen’s identity (Lemma 2.10)
yields the following expression for the conditional expected signature.

######  Lemma A.2.

For each $s\leq t$ it holds

$\mathbb{E}[\langle
e_{I},\widehat{\mathbb{W}}_{s+t}\rangle|{\mathcal{F}}_{s}]=\sum_{e_{I_{1}}\otimes
e_{I_{2}}=e_{I}}\langle
e_{I_{1}},\widehat{\mathbb{W}}_{s}\rangle\mathbb{E}[\langle
e_{I_{2}},\widehat{\mathbb{W}}_{t}\rangle].$

###### Proof.

Observe that applying Chen’s identity and the definition of the tensor product
we can compute

$\displaystyle\mathbb{E}[\langle
e_{I},\widehat{\mathbb{W}}_{s+t}\rangle|{\mathcal{F}}_{s}]$
$\displaystyle=\mathbb{E}[\langle
e_{I},\widehat{\mathbb{W}}_{s}\otimes\widehat{\mathbb{W}}_{s,s+t}\rangle|{\mathcal{F}}_{s}]$
$\displaystyle=\sum_{e_{I_{1}}\otimes e_{I_{2}}=e_{I}}\mathbb{E}[\langle
e_{I_{1}},\widehat{\mathbb{W}}_{s}\rangle\langle
e_{I_{2}},\widehat{\mathbb{W}}_{s,s+t}\rangle|{\mathcal{F}}_{s}]$
$\displaystyle=\sum_{e_{I_{1}}\otimes e_{I_{2}}=e_{I}}\langle
e_{I_{1}},\widehat{\mathbb{W}}_{s}\rangle\mathbb{E}[\langle
e_{I_{2}},\widehat{\mathbb{W}}_{s,s+t}\rangle|{\mathcal{F}}_{s}].$

Note that for each suitable integrand $H$ and each $u\in[0,t]$ it holds

$\int_{s}^{s+u}H_{r}\mathrm{d}\widehat{W}_{r}^{i}=\int_{0}^{u}H_{s+r}\mathrm{d}\widehat{B}_{r}^{i},$

where $\widehat{B}$ is the time extended Brownian motion given by
$\widehat{B}_{r}:=\widehat{W}_{s+r}-\widehat{W}_{s}$. Since $B$ is independent
of ${\mathcal{F}}_{s}$ we can conclude that $\mathbb{E}[\langle
e_{I_{2}},\widehat{\mathbb{W}}_{s,s+t}\rangle|{\mathcal{F}}_{s}]=\mathbb{E}[\langle
e_{I_{2}},\widehat{\mathbb{W}}_{t}\rangle]$, and the claim follows. ∎

##  Appendix B Calibration using sig-payoffs

We now discuss the procedure proposed in Perez Arribas et al. (2020) which
consists in calibrating the model to sig-payoffs that are intended to
approximate the true payoffs. For general payoff functions $F$,

$F:(\widehat{S}_{t})_{t\in[0,T]}\mapsto
F\Big{(}(\widehat{S}_{t})_{t\in[0,T]}\Big{)},$

the first step consists in approximating $F$ with a sig-payoff such that

$\displaystyle F\Big{(}(\widehat{S}_{t})_{t\in[0,T]}\Big{)}\approx
f_{\emptyset}+\sum_{0<|J|\leq m}f_{J}\langle
e_{J},\widehat{\mathbb{S}}_{T}\rangle,$ (B.1)

for some $m\geq 0$ and $f_{\emptyset},f_{J}\in{\mathbb{R}}$. Here,
$\widehat{\mathbb{S}}_{T}$ denotes the signature of $\widehat{S}=(t,S_{t})$ at
time $T$. For our purposes, the approximation of the call payoffs has to be
accurate at least with high probability on the set of models

$\\{\widehat{S}_{n}(\ell)\colon\ell\text{ enters in the optimisation}\\}.$

Under the hypothesis that this can be achieved and proceeding as in Section
3.4, the model call prices $C^{\text{model}}(T,K,\ell)$ (appearing in (4.6))
can then be (approximately) computed via

$C^{\text{model}}(T,K,\ell)={\mathbb{E}}_{\mathbb{Q}}[(S_{n}(\ell)_{T}-K)^{+}]\approx
f_{\emptyset}+\sum_{0<|J|\leq
m}f_{J}\widetilde{P}_{J}(\ell,{\mathbb{E}}_{\mathbb{Q}}[\widehat{\mathbb{X}}_{T}]).$

The benefits from a computational perspective are immediate.

To achieve the approximation of the payoff functions, in Lyons et al. (2020)
and Perez Arribas (2020) it is suggested to use an auxiliary stochastic
process $(Y_{t})_{t\geq 0}$ playing the role of a market generator under
${\mathbb{Q}}$. A standard example for $Y$ is given by the Black-Scholes model

$\mathrm{d}Y_{t}=\sigma Y_{t}\mathrm{d}W_{t},\qquad Y_{0}=S_{0},$

where $W$ denotes a Brownian motion and $\sigma$ is sampled uniformly in the
range of the implied volatility surface of the market, or as in Perez Arribas
(2020) between 5$\%$ and 40$\%$ of the spot price. We set again
$\widehat{Y}_{t}:=(t,Y_{t})$ and denote by $\widehat{\mathbb{Y}}$ the
signature of $\widehat{Y}$. The procedure to find the coefficients $f$ of
(B.1) consists in a linear regression, fitting $N_{MC}>0$ realizations of the
simulated sig-payoffs to the corresponding realizations of the simulated
payoff. The corresponding loss function is given by

$L_{\text{payoff}}(f):=\sum_{j=1}^{N_{MC}}\bigg{(}(Y_{T}(\omega_{j})-K)^{+}-\Big{(}f_{\emptyset}+\sum_{0<|J|\leq
m}f_{J}\langle
e_{J},\widehat{\mathbb{Y}}_{T}(\omega_{j})\rangle\Big{)}\bigg{)}^{2}.$

In Perez Arribas (2020) the author suggests to choose $N_{MC}=10^{4}$, $m=5$
and to sample $\sigma\sim\mathcal{U}([0.05,0.4])$.

We applied this procedure and then compared the call prices obtained under the
Black Scholes model with $\sigma=0.25$ with the the approximation via sig-
payoffs denoted by $F(T,K)$. For $T=0.5$, $Y_{0}=100$, and $K=120$, we obtain
the following results:

$\mathbb{E}_{\mathbb{Q}}[(S_{T}-K)^{+}]$ | $\frac{1}{N_{MC}}\sum_{i=1}^{N_{MC}}(S_{T}(\omega_{i})-K)^{+}$ | $\mathbb{E}_{\mathbb{Q}}[F(T,K)]$ | $\frac{1}{N_{MC}}\sum_{i=1}^{N_{MC}}F(T,K)(\omega_{i})$
---|---|---|---
1.5155 | 1.5145 | 1.6567 | 1.6550

Here, $\mathbb{E}_{{\mathbb{Q}}}[(S_{T}-K)^{+}]$ was computed via the Black-
Scholes formula and $\mathbb{E}_{{\mathbb{Q}}}[F(T,K)]$ using the analytic
expression of the expected signature of the time extended Black-Scholes model.