# Mathematical modeling of the Chilean riots of 2019,
an epidemiological non–local approach

Carlos Cartes<EMAIL_ADDRESS>Complex Systems Group, Facultad de
Ingeniería y Ciencias Aplicadas.
Universidad de los Andes, Santiago, 7620001, Chile.

###### Abstract

During the second half of October 2019, Chile, and especially its capital
city, Santiago, suffered from widespread violence with public and private
infrastructure destruction. This work aims to expand an epidemiological
non–local model that successfully described the French riots of 2005 to
incorporate the topology of Santiago’s subway network and explain the reported
distribution of rioting activity in the city. Although the model reproduced
the disorders’ aggregated temporal evolution, it could not deliver results
resembling the observed spatial distribution of activity on Santiago. The main
reason for this failure can be attributed to the fact that the model lacks a
population displacement mechanism, which seems vital to explain Santiago’s
unrest episodes.

††preprint: AIP/123-QED

> A wave of generalized civil unrest started in October 2019 in Santiago,
> Chile. The disorders rapidly propagated to other major cities of the
> country, resulting in a massive destruction of public and private premises.
> Epidemiological models have proven helpful in describing the aggregated time
> evolution of a riot and reproducing the spatial distribution of events in a
> geographical region. This work extends one of such models, successfully used
> to describe the French riots of 2005, to incorporate the subway network’s
> influence on disorder distribution, because it has proven that such network
> played a crucial role in the observed patterns of riot spatial density in
> Santiago.

##  I Introduction

On Friday, October 18th of 2019, a generalized civil disorder started in
Santiago, Chile, and, due to social media and news reports, rapidly propagated
to the whole country. The incidents were triggered by an increase of $4$US
cents on the transport fare in Santiago. Nevertheless, this was not the reason
for the generalized riots. Instead, they were attributed to the poor quality
of health, social security, and education systems, among other petitions for
improving living standards. The severity of the large–scale disorder, which
lasted for several days, and intermittently in some areas for months, forced
the government to declare a state of emergency and impose a curfew on the
country’s main cities. The cost of damage to public and private property only
during the first month of the activity was estimated at $4.6$billion US
hribernik47countries  equivalent to $1.1\%$ of the Chilean Gross Domestic
Product.

From a historical point of view, riots were called ‘madness of the crowds’
le2017crowd  and ‘mass enthusiasm’ lorenz2005aggression . They were explained
as a return of the collective to more primitive manners. Later works found
that generalized disorder has a multifactorial origin, where poverty and a low
standard of living are the principal factors explaining the severity and
duration of the activity spilerman1970causes . Qualitative approaches on the
subject were introduced by the innovative work of Burbeck burbeck1978dynamics
, where an epidemiological model successfully explained the aggregated
behavior of the activity registered during the racial riots in the 1960s USA
and the assumption that conscious and rational decisions were involved in
disorder participation granovetter1978threshold . Epstein’s formulation for
social conflict epstein2002modeling  using Agent–Based Models (ABM) well
represented more detailed classical patterns in public disorder, like the long
calm periods interrupted by intermittent bursts of collective violence.

Different approaches to this problem were made by using partial differential
equations, capturing essential features like the contagious nature of rioting
behavior petrovskii2020modelling  or the observed patterns from the French
riots of 2005 berestycki2015model ; bonnasse2018epidemiological . With the
same approach, Davies et al. Davies2013  successfully captured the
quantitative dynamics of the London riots in 2011. This later model was based
on the evidence that rational cost/benefit evaluation is involved at the
moment of choice, to either join or not a riot baudains2013target , in the
same manner as regular criminal activity maximizes the potential reward by
selecting different alternatives bernasco2004residential ; clare2009formal ;
davies2015event . However, some evidence points out that criminal opportunism
does not entirely explain rioters’ target selection tiratelli2018reclaiming ;
drury2020social . Besides, it was found that generalized disorder does not
change baseline criminal activity mechoulan2020civil  and depends on the
available means to manifest disagreement with authority boyer2020origins .

More recently, and as a consequence of the outbreaks of violence in Chile and
other Latin American countries TheConversation2019 ; TheGuardian2019 , Caroca
et al. Caroca2020  proved that Burbeck’s model adequately described the
aggregated rioting activity in Santiago, Chile, during October 2019. To
explain the same events, Cartes and Davies CARTESDavies  developed a numerical
model that reproduces the primary spatial distribution of the riots by
implementing a simple Latin American city layout. As was found by Cartes,
Asahi, and Fernández Cartestransport , those disorder activities were mainly
concentrated around the subway network premises and had a clear association
with the residents’ income. Following the same research line on the Chilean
generalized unrest, we can point out an analysis based on diffusion–reaction
equations, applied to country–wide economic and social variables, able to
predict civil unrest due to large–scale dissatisfaction curilef2021analyzing .

The proposition for this work is to use Bonnasse–Gahot’s
bonnasse2018epidemiological  model, which successfully reproduced the temporal
evolution and spatial distribution of activity for the French riots of 2005,
to explain the spatial distribution of the riots from Santiago during the
first days of widespread violence. Because it was found that the subway
network played a crucial role in promoting the formation of high activity
clusters Cartestransport . This increase of activity was already noted during
the London riots of 2011 baudains2016london , but it is also present for
ordinary crime by creating awareness of new opportunities
brantingham1991public ; swartz2000spatial . Therefore, public transport
networks can change the way that the general population perceives distances. A
similar approach was implemented by extending Davies’ model for the London
riots from 2011 Davies2013 , on a simplistic pattern of a Latin American city
and successfully captured some basic features like the clusterization of
activity in the areas with high connectivity CARTESDavies . Bonnasse-Gahot’s
model was chosen over Davies’ model just because of its relatively simplicity
of implementation, the low number of free parameters to adjust, and the
possibility of trying a new model fed with data from a major Latin American
city.

This article is organized as follows, Section II will present Bonnasse–Gahot’s
model, which briefly consists of space–coupled and non–local version of
Burbeck’s epidemiological model. Section III will show some empirical
observations from the first days of Santiago’s riots, followed by some details
on the numerical simulations, the extension of the model to include the
effects of the subway network, and the main results. Finally, in Section IV a
discussion and analysis of the main results are presented, with comparisons to
other models and perspectives for future work.

##  II  The Model

Burbeck’s model successfully described the quantitative evolution of riots
that happened in several cities in the United States during the 60’s
burbeck1978dynamics . It is based on an epidemiological approach, where the
number of riot events $\lambda(t)$, proportional to the number of rioters, and
the “supply” of riot events $\sigma(t)$, proportional to the number of
available individuals Caroca2020 . Therefore the evolution equations for
$\lambda$ and $\sigma$ are

$\displaystyle\frac{d\lambda}{dt}$ $\displaystyle=$
$\displaystyle-\omega\lambda+\beta\sigma\lambda$ (1)
$\displaystyle\frac{d\sigma}{dt}$ $\displaystyle=$
$\displaystyle-\beta\sigma\lambda$

where $\omega$ represents the retreat from the riot and $\beta$ the
transmission rate from the supply to the disorder event. Complementing those
parameters we need the initial conditions $\lambda(t_{0})=\lambda_{0}$ and
$\sigma(t_{0})=\sigma_{0}$.

A non–local extension of this model was introduced to explain the French riots
of 2005bonnasse2018epidemiological . There Burbeck’s model is generalized, to
consider multiple sites therefore we have now a spatial distribution of the
activity, this distribution can be performed to different spatial scales only
depending on the granularity of the data.

For this work we well consider the number of riot events at the site $k$,
$\lambda_{k}$ and the supply at the same location $\sigma_{k}$. The dynamics
of those quantities can be expressed as

$\displaystyle\frac{d\lambda_{k}}{dt}$ $\displaystyle=$
$\displaystyle-\omega_{k}\lambda_{k}+\sigma_{k}\Phi\left[\Lambda_{k}\right]$
(2) $\displaystyle\frac{d\sigma_{k}}{dt}$ $\displaystyle=$
$\displaystyle-\sigma_{k}\Phi\left[\Lambda_{k}\right]$

where $\Lambda_{k}$ is the non–local coupling which is defined by the
summation

$\Lambda_{k}=\sum_{j}W_{kj}\lambda_{j}$ (3)

here the non–local kernel $W_{kj}$ has to take into account long distance
interactions. For simplicity the non–local coupling function is chosen to be
$\Phi\left[\Lambda_{k}\right]=\beta\Lambda_{k}$, where the $\beta$ parameter
is the same for each $k$ in the domain. Following Bonasse–Gahot indication
bonnasse2018epidemiological  our kernel will have either one of the following
shapes

$W_{kj}\sim\left(1+\frac{|\vec{r}_{k}-\vec{r}_{j}|}{d_{0}}\right)^{-\delta}$

or

$W_{kj}\sim\epsilon+\left(1-\epsilon\right)e^{\frac{|\vec{r}_{k}-\vec{r}_{j}|}{d_{0}}}\,.$

In order to make comparisons between our simulations and the model from eqns.
(2), we integrated them with a fourth-order Runge-Kutta method. A wide range
of parameters was explored to fit with our simulation results.

##  III  Results

###  III.1 Empirical Observations

This section will show the main results from the data analysis belonging to
the first four days of the Santiago riots in October 2019. Cartes, Asahi and
Fernández recently performed a complete analysis of the dataset
Cartestransport . All the data was provided by SOSAFE, an open platform where
users can report any event they found relevant. The SOSAFE platform registers
the user’s time, geographical coordinates, and a short description of the
event. They range from relatively soft, like the temporal traffic interruption
or street sign destruction, to very extreme, like the intentional burning of
public and private buildings. The raw data in question is publicly available
datasosafe . The surface taken into account is delimited by Santiago’s public
transport system. More specifically, the geographic region is bounded by the
parallels $-33.32^{\circ}$ and $-33.67^{\circ}$ latitude South and the
meridians $-70.49^{\circ}$ and $-70.87^{\circ}$ longitude West.

Figure 1:  Heatmap of the registered activity distribution in Santiago, from
October 2019, alongside the subway network.

Data analysis reported that the temporal evolution of the disorder activity
followed approximately Burbeck’s model. Thus it showed an exponential growth
and, after peaking, a fast decay. The activity peaked at around 22 hours, and
the lowest level was at 8 hours. The spatial distribution of the riots
manifested an evident clustering around the subway stations, this spatial
distribution is shown in FIG. 1, there the riot level is normalized to a scale
between $0$ and $1$, being $0$ no reported activity and $1$ the maximum of all
covered surface. It is important to mention that Santiago’s subway network is
composed by $7$ lines, $136$ stations and $140$km, serving around $12.4\%$ of
the total public transport system covering surface. Nevertheless half of the
total disorders were registered at 1 km or less from the subway network and
$80\%$ at 3 km or less. This correlation was found significant at the lower
and higher-income neighborhoods, therefore the authors hypothesize that at
least a fraction of the rioters traveled from the lower-income to the high-
income regions to manifest themselves. Primarily motivated by the historical
and symbolic importance of the city’s most affected areas.

###  III.2 Numerical Simulations

A fourth–order Runge–Kutta method was implemented to solve the model. The
space discretization was performed over a grid with dimensions $71\times 78$.
Thus each element has the dimensions $500$m$\times 500$m. Thus covering the
same geographic region as the aforementioned analysis Cartestransport  and
keeping the grid element size at a typical walking distance tyler2002 .
Because the model needs some preliminary activity to trigger the disorder, the
$20$ grid elements with the highest number of denounces were taken as the
initial condition for the $\lambda$ field. This initial condition was
determined with the data provided by SOSAFE. Because it is crucial to keep the
relative importance between the different grid elements belonging to the
initial condition, it was multiplied by a constant $\alpha<1$.

Different kernels for the long–distance coupling and values for the numerical
constants were tried. Finally, keeping the same parameters as those already
tested by Bonnasse–Gahot’s model was considered more practical. Because the
model has already reported good agreement with the data coming from the French
riots of 2005 in Paris, keeping the same parameters also makes it easier to
compare the results between that model and this implementation.

The fraction of the population “willing to join” a riot was determined from
the number of inhabitants located on each grid element, and we multiplied it
by the inverse of the mean income of that grid element. Therefore it was
assumed, in the same way as Davies’ and Bonnasse–Gahot’s models, that
deprivation (which is associated to low income) is a good predictor of the
population’s disposition to participate in a generalized disorder. In
consequence, the initial condition $\sigma(t=0)$ is defined for each grid
element as the total population multiplied by its mean deprivation. It is
important to mention that the deprivation’s highest value is $1$ for the grid
element with the lowest mean income.

The inclusion of transport networks is achieved by changing how the distance
is measured between the different grid elements. In this model, distances are
computed by minimizing the path between two points by using the transport
network. In this case, if the distance is computed over the transport network,
it is multiplied by $0.2$. In the same way as the simplified Latin–American
city pattern by Cartes and DaviesCARTESDavies .

The shortest path between two points is computed over a graph consisting of
the initial and final points and all the subway stations belonging to their
respective lines. Finally, the minimal path is found using the $A^{*}$
algorithm Hart1968 . Only the subway network was included in this work because
the data showed Cartestransport  that a disproportionate large fraction of the
reported disorder activity was clustered around the subway stations.

Two different kernels were tried in the simulations, exponential and power
law. Results are similar in both of them. Therefore the power law kernel was
used to keep the results similar to those of Bonnasse-Gahot’s model.

Finally, Table 1 shows the numerical parameters used for this work. In this
case, $d_{0}$ is chosen to be equivalent to the value $1.6\cdot 10^{-2}$km.
Thus it can be comparable to the work on the French riots of 2005
bonnasse2018epidemiological .

$\omega=0.216$ | $\beta=4.32\cdot 10^{-4}$ | $\delta=0.67$
---|---|---
$\alpha=0.07$ | $dt=0.05$ | $d_{0}=0.016$
Table 1: Parameters used in the simulation. Figure 2:  Temporal evolution of
the total number of rioters, for each transport configuration. Complete
network corresponds to all seven subway lines (1, 2 , 3, 4, 4A, 5 and 6).

In FIG. 2 is shown the total number of rioters as function of time, $R_{\rm
Total}$. for different transport network configurations. It is clear that the
inclusion of a transport network increases the maximum value of $R_{\rm
Total}$, turning the riot activity more intense and potentially more violent.
Nevertheless, this increase in activity is also reflected by a shorter riot
duration. It is a consequence of Burbeck’s model due to the $\omega\lambda$
term in eqns. (1). The magnitude of this increase is related to the extension
of the considered transport network and the characteristics of the different
connected areas of the city from FIG. 2, it is clear that the largest
contributor to the transport network effect is promoted by the subway lines
$1$, $4$, and $5$. This is a consequence of their extension. Those three lines
are composed of $80$ stations and the features of the different neighborhoods
connected by those subway lines. Specifically, lines $1$, $4$ and $5$ connect
the South–West parts of Santiago with the rest of the city, those areas are
characterized by high population density and low–income Cartestransport . The
regions mentioned above are the larger rioter contributors. Thus, the
inclusion of transport networks increases the connected areas’ activation
rate, which, in consequence, turns into the addition of the largest rioting
population.

Figure 3:  Heatmap showing the simulated activity distribution, without
including subway influence, alongside the subway network.

The spatial distribution of rioters, alongside the subway network, is shown on
FIGS. 3 and 4. In both figures, the methodology adopted was to add the rioters
from all the time–steps and divide by the number of time steps. This operation
will provide the average number of rioters for each grid element $k$. This
result was interpolated by a gaussian kernel, with a standard deviation of
$500$ meters. The intensity was normalized so that the maximum value for the
rioting intensity is equal to $1$, therefore making the comparison with the
actual data shown in FIG. 1 direct. In FIG. 3 are shown the results without a
transport network. There is possible to see that the regions with high rioting
activity are located in areas where the population income is low
Cartestransport , and there is a relatively elevated population density. In
addition, the regions with high disorder activity are more extense than those
found in the real data, shown in FIG. 1. Therefore in the absence of a
transport network, most of the disorder is located in the deprived parts of
the city, similar to the results found by Cartes and Davies CARTESDavies  by
using Davies’ model.

On the other hand, the addition of transport networks barely changed the
rioters’ distribution, as shown in FIG. 4, it can be appreciated that some of
the most intense spots at the North of Santiago are slightly larger compared
to FIG. 3, nevertheless, both heatmaps are virtually the same once they are
normalized. Therefore, the inclusion of a working transport network does not
affect the spatial distribution of riots in this model, even when distance
perception and mutual influence are affected by it. The disorder activity
stays concentrated in regions where the population density and deprivation are
higher and comparatively far away from the subway network.

Figure 4:  Heatmap showing the simulated activity distribution, including
subway influence, alongside the subway network.

Finally, a different initial condition was considered, where deprivation is
the same for the whole population, preserving other characteristics like
density and position. Numerical simulations showed a very similar disorder
distribution to those found on FIGS. 3 and 4, where the most intense activity
is placed where population density is higher. The only appreciable difference
is some additional activity in the regions with high income, located in the
city North–East.

##  IV  Discussion

The epidemiological model proposed by Bonnasse–Gahot
bonnasse2018epidemiological  was used to describe the riots of 2005 in France
successfully. The model accurately predicted the rioting activity’s disorder
intensity, time evolution, and spatial distribution. Due to this success and
its relatively simple implementation, the formulation above was used to
replicate the first days of the Chilean riots of October 2019, specifically in
the capital city Santiago. Because the model incorporates long–distance
interaction mechanism, it was used to include the effects of the public
transport system of Santiago on the spatial disorder distribution. In the
present work, only the subway network was included because previous work
Cartestransport  showed a disproportionate clustering of riot activity around
Santiago’s subway stations.

The direct implementation of Bonnasse–Gahot’s model, without including
transport network influence, showed high activity at Santiago’s southwest.
Coincidentally with the location of the most deprived and densely populated
neighborhoods of the city. After the inclusion of transport network effects,
it was found that the rioting activity increments, the peak activity and its
decline occur earlier. Nevertheless, the spatial riot distribution stays
practically the same, with very few changes in intensity at some areas.

It is interesting to compare these results with a similar extension
implemented on Davies model Davies2013  by Cartes and Davies CARTESDavies . In
the later work, the implementation of a working transport network induced
severe changes in the spatial distribution of riot intensity. But also
decreasing its maximum intensity and duration, making the total disorder
easier to control. Apparently, it is possible that the differences are
attributed to the population displacement between different city areas, which
is allowed in Davies’ model but not considered in the Bonnasse–Gahot. It only
permits mutual influence between the different regions. Another critical
difference is that Davies’ model implements targets that are attacked by the
surrounding population. On the other hand, Bonnasse–Gahot’s model assumes that
each city region is equally attractive and prone to be attacked by its
inhabitants. As a final comparison point, Davies’ model showed that deprived
regions presented the highest rioting activity when there was not a working
transport network conducing to the same result as Bonnasse–Gahot’s model.

Apparently, the possibility of displacing the population over long distances
is required to describe the riot’s spatial distribution observed in Santiago
adequately. This behavior is not only limited to Chile’s capital city but also
seems similar to those observed during the "Gilets Jaunes" incidents in french
cities in 2018 boyer2020origins .

A natural extension of this work will be the implementation of Davies’ model
with the previous data of Santiago and try to reproduce the observed disorder
distribution.

## Acknowledgments

The author wishes to acknowledge the support of FONDECYT (CL), project number
1200357, and Universidad de los Andes (CL) through FAI initiatives.

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# eSSVI Surface Calibration

Leo Pasquazzi111email<EMAIL_ADDRESS>ORCID
https://orcid.org/0000-0002-2467-2667
Dipartimento di Statistica e Metodi Quantitativi,
Università degli Studi di Milano - Bicocca,
Piazza dell’Ateneo Nuovo, 1 - 20126, Milano, Italy

## Abstract

In this work I test two calibration algorithms for the eSSVI volatility
surface. The two algorithms are (i) the robust calibration algorithm proposed
in Corbetta et al. (2019) and (ii) the calibration algorithm in Mingone
(2022). For the latter I considered two types of weights in the objective
function. I fitted 108 end-of-month SPXW options chains from the period
2012-2022. The option data come from FactSet. In addition to this empirical
part, this paper contains also a theoretical contribution which is a
sharpening of the Hendriks-Martini proposition about the existence of crossing
points between two eSSVI slices.

Keywords: eSSVI volatility surface, Calibration, Arbitrage free interpolation

##  1 Introduction

At the Global Derivatives & Risk Management conference in Madrid, Gatheral
(2004) presented the SVI parametrization for the implied variance smile $w(k)$
at a fixed maturity. This parametrization reads

$w(k)=a+b\left\\{\rho(k-m)+\sqrt{(k-m)^{2}+\sigma^{2}}\right\\}$

where $k=\ln K/F$ is the natural logarithm of the ratio between the strike and
the forward price of the underlying, and where $a\in\mathbb{R}$, $b\geq 0$,
$|\rho|<1$, $m\in\mathbb{R}$ and $\sigma>0$ are parameters governing the shape
and position of the smile.

The following appealing features of the SVI parametrization are well known:

  1. (i)

the SVI smiles are asymptotically linear in $k$ as $|k|\rightarrow\infty$ and
therefore consistent with Roger Lee’s moment formula (Lee, 2004) and

  2. (ii)

the large maturity limit of the implied variance smile of a Heston model with
correlation parameter $\rho$ is SVI with the same value of $\rho$ (Gatheral
and Jacquier, 2011).

However, it is also well known that the SVI parametrization, in its full
generality, is not arbitrage free. For example, it is not difficult to show
that $\inf_{k}w(k)=a+b\sigma\sqrt{1-\rho^{2}}$ and hence it should always be
required that $a+b\sigma\sqrt{1-\rho^{2}}\geq 0$. However, the latter
condition is not enough to rule out butterfly arbitrage as can be seen from a
well known counterexample of Axel Vogt (see Gatheral and Jacquier 2014).
Moreover, fitting the SVI parametrization to more than a single maturity date
may produce slices that cross over each other which is equivalent to the
existence of calendar spread arbitrage opportunities. To overcome these
issues, Gatheral and Jacquier (2014) introduced the SSVI parametrization which
is a global parametrization for the whole implied total variance surface where
the fixed maturity slices are restricted to a subfamily of the SVI
parametrization (the first ”S” in front of SSVI stands for ”surface”). In the
SSVI parametrization, the implied total variance surface is given by

$w_{t}(k)=\frac{\theta_{t}}{2}\left\\{1+\rho\varphi(\theta_{t})k+\sqrt{(\varphi(\theta_{t})k+\rho)^{2}+(1-\rho^{2})}\right\\}$
(1)

where $\theta_{t}$ is the ATM implied total variance at maturity $t$,
$|\rho|<1$ and $\varphi(\theta_{t})$ is a smooth function from
$\mathbb{R}_{+}^{*}$ to $\mathbb{R}_{+}^{*}$ such that the limit
$\lim_{t\rightarrow\infty}\theta_{t}\varphi(\theta_{t})$ exists in
$\mathbb{R}$. According to Theorems 4.1 and 4.2 in Gatheral and Jacquier
(2014), the SSVI surface (1) is free of calendar spread arbitrage if and only
if

  1. C1)

$\partial_{t}\theta_{t}\geq 0$ for all $t\geq 0$

  2. C2)

and
$0\leq\partial_{\theta}\left(\theta\varphi(\theta)\right)\leq\frac{1}{\rho^{2}}\left(1+\sqrt{1-\rho^{2}}\right)\varphi(\theta)$
for all $\theta>0$ (the upper bound is infinite when $\rho=0$)

and it is free of butterfly arbitrage if for all $\theta>0$ the following two
conditions are both satisfied:

  1. B1)

$\theta\varphi(\theta)(1+|\rho|)<4$,

  2. B2)

$\theta\varphi(\theta)^{2}(1+|\rho|)\leq 4$.

The latter conditions are quite close to necessary as well. In fact, Lemma 4.2
in Gatheral and Jacquier (2014) says that absence of butterfly arbitrage holds
only if $\theta\varphi(\theta)(1+|\rho|)\leq 4$ for all $\theta>0$ (which is
only slightly weaker than B1) and that condition B2 becomes necessary when
$\theta\varphi(\theta)(1+|\rho|)=4$.

In order to make the SSVI surface more flexible, Hendriks and Martini (2019)
made the $\rho$-parameter maturity dependent as well and called the resulting
implied total variance surface model eSSVI surface (the ”e” in front of SSVI
stands for ”extended”). Proposition 3.1 in Hendriks and Martini (2019)
provides necessary and sufficient conditions for the absence of calendar
spread arbitrage between two time slices. In order to state these conditions,
we will indicate the parameters of two slices with $\theta_{i}$,
$\varphi_{i}=\varphi(\theta_{i})$ and $\rho_{i}$, where the subscript $i$
takes on the value $1$ or $2$ according to whether the closer ($i=1$) or
farther ($i=2$) maturity date is referred to. Proposition 3.1 in Hendriks and
Martini (2019) says that two time slices do not cross over each other only if

  * N’)

$\frac{\theta_{2}}{\theta_{1}}\geq 1$ and
$\left(\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}\rho_{2}-\rho_{1}\right)^{2}\leq\left(\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}-1\right)^{2}$

and that condition N along with condition S below is sufficient to rule out
the existence of crossing points:

  * S)

$\frac{\varphi_{2}}{\varphi_{1}}\leq 1$ or
$(\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}\rho_{2}-\rho_{1})^{2}\leq(\frac{\theta_{2}}{\theta_{1}}-1)(\frac{\theta_{2}\varphi_{2}^{2}}{\theta_{1}\varphi_{1}^{2}}-1)$

However, condition N’ and condition S are not jointly sufficient to rule out
the existence of crossing point. In fact, as can be seen from Proposition 4.14
in the appendix of this paper,

  * •

when $\frac{\theta_{2}}{\theta_{1}}=1$ there are no crossing points if and
only if either (i) $\rho_{1}=\rho_{2}=0$ and $\varphi_{2}/\varphi_{1}\geq 1$
or (ii) $\varphi_{2}/\varphi_{1}=\rho_{1}/\rho_{2}$ and
$\rho_{1}^{2}\geq\rho_{2}^{2}$

  * •

and when $\frac{\theta_{2}}{\theta_{1}}\neq 1$ there are no crossing points if
and only if condition S holds jointly with condition

    * N)

$\frac{\theta_{2}}{\theta_{1}}>1$ and
$1-\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}\leq\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}\rho_{2}-\rho_{1}\leq\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}-1$.

Almost all of the proof of Proposition 4.14 in the appendix of this paper is
built on the main ideas of the proof of Proposition 3.1 in Hendriks and
Martini (2019). As far as I know the only novelty are the result about
tangency points in Lemma 4.11 and the two final Lemmas 4.12 and 4.13.

The next section describes the calibration algorithms which I tested: the
robust algorithm of Corbetta et al. (2019) and the algorithm of Mingone
(2022). The test results are summarized in Section 3.

##  2 Two calibration algorithms for the eSSVI surface

###  2.1 The robust algorithm of Corbetta et al. (2019)

The robust calibration algorithm of Corbetta et al. (2019) gives rise to eSSVI
surfaces which satisfy conditions B1, B2, N and the first inequality of
condition S. The key ingredient of this algorithm is a reparametrization of
the SSVI slices which forces them to go through the data point
$(k^{*},\theta^{*})$ which is closest to the ATM forward implied total
variance. To enforce this restriction the parameter $\theta$ is expressed in
terms of the parameters $\rho$, $\varphi$ and the data-driven pair
$(k^{*},\theta^{*})$, i.e. $\theta$ is taken to be

$\theta=\theta^{*}-\rho\theta\varphi k^{*}:=\theta^{*}-\rho\psi k^{*}$

which is a first order approximation to the solution of the equation
$w(k^{*})=\theta^{*}$, where $w(k)$ is defined as in (1).

What are the allowed values for the new parameter $\psi:=\theta\varphi$? Of
course, the non-negativity constraints on $\theta^{*}$, $\theta$ and
$\varphi(\theta)=\varphi$ translate immediately to condition

  * R0)

    * i)

$0\leq\psi$ if $\rho k^{*}\leq 0$

    * ii)

$0\leq\psi\leq\theta^{*}/(\rho k^{*})$ if $\rho k^{*}>0$

and condition B1 translates to

  * R1)

$\psi<\frac{4}{1+|\rho|}$

Moreover, using $\theta=\theta^{*}-\rho\psi k^{*}$ it is easily seen that
condition B2 translates to

  * R2)

$\psi^{-}\leq\psi\leq\psi^{+}$ where

$\psi^{\pm}:=\frac{-2\rho
k^{*}}{1+|\rho|}\pm\sqrt{\frac{4\rho^{2}(k^{*})^{2}}{(1+|\rho|)^{2}}+\frac{4\theta^{*}}{1+|\rho|}}.$

Next, consider the conditions N and S for preventing calendar spread
arbitrage. As in Section 1, the subscript $i=1$ will always refer to the
closer maturity date. Keeping this mind it is not difficult to verify that
condition N translates to

  * R3)

$\theta^{*}_{2}-\rho_{2}\psi_{2}k^{*}_{2}\geq\theta^{*}_{1}-\rho_{1}\psi_{1}k^{*}_{1}$
and
$\psi_{2}\geq\psi_{1}\max\left\\{\frac{1+\rho_{1}}{1+\rho_{2}},\frac{1-\rho_{1}}{1-\rho_{2}}\right\\}$

and that the first inequality in condition S becomes

  * R4)

$\psi_{2}(\theta^{*}_{1}-\rho_{1}\psi_{1}k^{*}_{1})\leq\psi_{1}(\theta^{*}_{2}-\rho_{2}\psi_{2}k^{*}_{2})$.

The algorithm suggested in Corbetta et al. (2019) neglects the second
inequality in S, which could hold when the first one fails and could therefore
allow for more flexibility.

Now, in order to describe the algorithm, we first observe that it is a
sequential algorithm which starts from the closest maturity date and works
forward in time finding optimal values of the parameters $k^{*}_{i}$,
$\theta^{*}_{i}$, $\psi_{i}$ and $\rho_{i}$ for each each maturity date
$i=1,2,\dots,n$ given the optimal values for the previous maturity date. It is
based on the observation that given the parameter values for maturity date
$i-1$, and given an arbitrary value of $\rho_{i}$, one can use the constraints
R0 - R4 to find an interval of admissible values for $\psi_{i}$. The bounds of
this interval depend on the sign of $\rho_{i}k^{*}_{i}$ and in order to write
them down it is convenient to define

$\displaystyle B_{1i}=\frac{4}{1+|\rho_{i}|}\quad\quad
B_{2i}^{\pm}:=\psi^{\pm}_{i}:=\frac{-2\rho
k_{i}^{*}}{1+|\rho_{i}|}\pm\sqrt{\frac{4\rho_{i}^{2}(k_{i}^{*})^{2}}{(1+|\rho_{i}|)^{2}}+\frac{4\theta^{*}_{i}}{1+|\rho_{i}|}},$
$\displaystyle
B_{3i}:=\frac{\theta^{*}_{i}-\theta^{*}_{i-1}+\rho_{i-1}\psi_{i-1}k^{*}_{i-1}}{\rho_{i}k^{*}_{i}}\quad\quad
B_{4i}:=\psi_{i-1}\max\left\\{\frac{1+\rho_{i-1}}{1+\rho_{i}},\frac{1-\rho_{i-1}}{1-\rho_{i}}\right\\},$
$\displaystyle
B_{5i}:=\frac{\psi_{i-1}\theta^{*}_{i}}{\theta^{*}_{i-1}-\psi_{i-1}(\rho_{i-1}k^{*}_{i-1}-\rho_{i}k^{*}_{i})}.$

for $i=1,2,\dots,n$, where $\theta^{*}_{0}$, $k^{*}_{0}$, $\rho_{0}$ and
$\psi_{0}$ are all defined to be zero. Now, it is not difficult to see that
the constraints R0 - R4 are equivalent to the following bounds for $\psi_{i}$
for $i=1,2,\dots,n$:

  * •

If $\rho_{i}k^{*}_{i}>0$, we get the lower bound
$L_{i}:=\max\left\\{B_{2i}^{-},B_{4i}\right\\}$ and the upper bound
$U_{i}:=\min\left\\{B_{1i},B_{2i}^{+},B_{3i},B_{5i}\right\\}$.

  * •

If $\rho_{i}k^{*}_{i}=0$, we get the lower bound $L_{i}:=B_{4i}$ and the upper
bound $U_{i}:=\min\left\\{B_{1i},B_{2i}^{+},B_{5i}\right\\}$; in this case we
must also check whether
$\theta^{*}_{i}\geq\theta^{*}_{i-1}-\rho_{i-1}\psi_{i-1}k^{*}_{i-1}$, in order
to make sure that the first inequality in condition R3 be satisfied.

  * •

If $\rho_{i}k^{*}_{i}<0$, we get the lower bound
$L_{i}:=\max\left\\{B_{2i}^{-},B_{3i},B_{4i}\right\\}$ and the upper bound

$U_{i}:=\begin{cases}\min\left\\{B_{1i},B_{2i}^{+},B_{5i}\right\\}&\text{ if
}B_{5i}>0\\\ \min\left\\{B_{1i},B_{2i}^{+}\right\\}&\text{ if }B_{5i}\leq
0.\end{cases}$

Note that the bounds $L_{i}$ and $U_{i}$ depend on $k^{*}_{i-1}$, $k^{*}_{i}$,
$\theta^{*}_{i}$, $\theta^{*}_{i-1}$, $\psi_{i-1}$, $\rho_{i-1}$ and
$\rho_{i}$. In order to compute these bounds we must therefore know the
optimal values of $\psi_{i-1}$ and $\rho_{i-1}$ which refer to the previous
maturity date, and we must also guess a value for the parameter $\rho_{i}$
which refers to the current maturity date $i$. As we will see in a moment, the
robust algorithm of Corbetta et al. (2019) deals with this problem by
calibrating the SSVI slices sequentially from the closest to farthest maturity
date. As pointed out in Corbetta et al. (2019), one could actually adapt the
algorithm and change the order in which the slices are calibrated, but usually
there are good reasons not to do so.

Of course, if $L_{i}>U_{i}$, there will not exist any value of $\psi_{i}$ such
that $(k^{*}_{i},\theta^{*}_{i},\rho_{i},\psi_{i})$ gives rise to a SSVI slice
which is free of butterfly arbitrage and/or calendar spread arbitrage with
respect to the optimal slice for maturity $i-1$. For the given value of
$\rho_{i}$ it is therefore impossible to avoid arbitrage. In this case the
objective function to be minimized should be set equal to infinity in order to
force the algorithm to consider other $\rho_{i}$ values for which $L_{i}\leq
U_{i}$. For some maturity date $i$ it might also happen that
$\theta^{*}_{i}<\theta^{*}_{i-1}-\rho_{i-1}\psi_{i-1}k^{*}_{i-1}$, in which
case R3 cannot be satisfied with $\rho_{i}=0$. If $k^{*}_{i}\neq 0$, R3 can
however be satisfied for every non zero value of $\rho_{i}$. Of course,
nothing can be done in the remote case (which never occurred with the data I
used) where $k^{*}_{i-1}=k^{*}_{i}=0$ and
$\theta^{*}_{i}<\theta^{*}_{i-1}-\rho_{i-1}\psi_{i-1}k^{*}_{i-1}$: in this
case we must increase the value of $\theta^{*}_{i}$ and make it equal
$\theta^{*}_{i-1}-\rho_{i-1}\psi_{i-1}k^{*}_{i-1}$ or perhaps a little larger.
Note that in the first case the algorithm might produce a slice which is
affected by calendar spread arbitrage (see the case $\Theta=1$ in Proposition
4.14 in the appendix). Keeping in mind these issues, we can now describe the
algorithm:

  * 0)

Choose an objective function to minimize. In this work I used the sum of the
absolute values of the differences between the observed option mid prices and
their theoretical counterparts which would obtain if $w_{i}(k)$ was the Black
and Scholes implied variance. Set $i=1$, i.e. start from the closest maturity
date which is identified by the subscript $i=1$.

  * 1)

Set $\zeta=0$ (the role of $\zeta$ will become clear in a moment) and choose
$r$ values for $\rho_{i}\in(-1,1)$ spaced apart at equal distances. These
values will be denoted by $\rho_{i,j}$, $j=1,2,\dots r$. For each value of
$\rho_{i,j}$ compute the corresponding bounds $L_{i,j}:=L_{i}$ and
$U_{i,j}:=U_{i}$. In my empirical investigation I set $r=100$. At first sight
this may seem a large value, but this choice is motivated by the fact that
with smaller values of $r$ I obtained $L_{i,j}>U_{i,j}$ for every $j$ in some
option chains.

  * 2)

For each value of $\rho_{i,j}$, search the optimal value of $\psi_{i}$ within
the corrseponding interval $(L_{i,j},U_{i,j})$. The optimal value of
$\psi_{i}$ corresponding to $\rho_{i,j}$ will be denoted $\psi_{i,j}$. In my
empirical investigation I used the Brent method as implemented by the
fminbound function from the scipy.optimize library for the minimum search. I
set the maximum number of objective function evaluations to $1000$ and set
xtol to 1e-8.

  * 3)

Compare the minima of the objective function obtained for the considered
$\rho_{i,j}$-values and pick out $\rho_{i}^{*}:=\rho_{i,j}$ and
$\psi_{i}^{*}:=\psi_{i,j}$ which give rise to the smallest minimum.

  * 4)

Increment $\zeta$ by one unit and choose $r$ values for $\rho_{i}$ in the
interval $(\rho_{i}^{*}-1.2/r^{\zeta},\rho_{i}^{*}+1.2/r^{\zeta})$. Denote
these values by $\rho_{i,j}$, $j=1,2,\dots r$ and go back to step 2) until
$2\times 1.2\times r^{-\zeta}$ is smaller than a small value $\varepsilon$. In
this work I used $\varepsilon=10^{-5}$.

  * 5)

Define the parameters of the optimal slice at maturity date $i$ by setting
$\rho_{i}:=\rho_{i}^{*}$ and $\psi_{i}:=\psi_{i}^{*}$. If the current $i$
corresponds to the last maturity date stop. Otherwise, increment the index $i$
and go back to step 1.

Once the algorithm is done we know the parameters
$(\theta_{i},\varphi_{i},\rho_{i})$ which identify an optimal SSVI slice for
each maturity date for which we have observed option prices. In order to
extend these slices to a continuous variance surface we can employ the
interpolation scheme described in Corbetta et al. (2019). Note that the
calibrated surface is free of arbitrage of any kind.

###  2.2 The algorithm of Mingone (2022)

The algorithm proposed in Mingone (2022) is an attempt to overcome the
sequential feature of the robust algorithm of Corbetta et al. (2019) which may
cause a poor fit at large maturity dates. To overcome this problem, Mingone
introduces a reparametrization of the eSSVI surface which leads to a
rectangular parameter space for which the conditions B1, B2, N and the first
inequality in condition S are all satisfied. Actually, Mingone proposes also a
second rectangular reparametrization where conditions B1, N and the first
inequality in condition S are still satisfied, but where only a weaker
condition than B2 holds. The second reparametrization accounts for the
necessary and sufficient no butterfly arbitrage conditions which are given in
Martini and Mingone (2022). However, it is more difficult to implement than
the first one and for this reason I tested only the calibration algorithm
which is based on the first one. In order to describe the latter, it is
convenient to recast conditions B1, B2, N and the first inequality in
condition S in terms of $\theta$, $\psi:=\theta\varphi$ and $\rho$. Doing so
we get the conditions

  * G1)

$\psi<\frac{4}{1+|\rho|}$,

  * G2)

$\psi^{2}\leq\frac{4\theta}{1+|\rho|}$,

  * G3)

$\theta_{2}\geq\theta_{1}$ and
$\psi_{2}\geq\psi_{1}\max\left\\{\frac{1+\rho_{1}}{1+\rho_{2}},\frac{1-\rho_{1}}{1-\rho_{2}}\right\\}$,

  * G4)

$\psi_{2}\leq\psi_{1}\theta_{2}/\theta_{1}$.

Now, in order to get a rectangular parameter space, Mingone introduces the
auxiliary quantities

$\displaystyle p_{i}$
$\displaystyle:=\max\left\\{\frac{1+\rho_{2}}{1+\rho_{1}},\frac{1-\rho_{2}}{1-\rho_{1}}\right\\}$
$\displaystyle\text{ for }i>1,$ $\displaystyle f_{i}$
$\displaystyle:=\min\left\\{\frac{4}{1+|\rho_{i}|},\sqrt{\frac{4\theta_{i}}{1+|\rho_{i}|}}\right\\}$
$\displaystyle\text{ for }i\geq 1,$ $\displaystyle A_{\psi_{1}}$
$\displaystyle:=0$ $\displaystyle\text{ for }i=1,$ $\displaystyle
A_{\psi_{i}}$ $\displaystyle:=\psi_{i-1}p_{i}$ $\displaystyle\text{ for }i>1,$
$\displaystyle C_{\psi_{1}}$
$\displaystyle:=\min\left\\{f_{1},\frac{f_{2}}{p_{2}},\frac{f_{3}}{p_{2}p_{3}},\dots,\frac{f_{n}}{\prod_{j=2}^{n}p_{j}}\right\\}$
$\displaystyle\text{ for }i=1,$ $\displaystyle C_{\psi_{i}}$
$\displaystyle:=\min\left\\{\frac{\psi_{i-1}}{\theta_{i-1}}\theta_{i},f_{i},\frac{f_{i+1}}{p_{i+1}},\frac{f_{i+2}}{p_{i+1}p_{i+2}},\dots,\frac{f_{n}}{\prod_{j=i+1}^{n}p_{j}}\right\\}$
$\displaystyle\text{ for }i>1.$

and substitutes the old parameters $\theta_{2}$, $\theta_{3}$, …, $\theta_{n}$
and $\psi_{1}$, $\psi_{2}$, …, $\psi_{n}$ with the new parameters

$a_{i}:=\theta_{i}-\theta_{i-1}p_{i},\quad i>1,$

and

$c_{i}:=\frac{\psi_{i}-A_{\psi_{i}}}{C_{\psi_{i}}-A_{\psi_{i}}},\quad i\geq
1,$

respectively. Then she shows that for every choice of

$(\rho_{1},\dots,\rho_{n},\theta_{1},a_{2},\dots,a_{n},c_{1},\dots,c_{n})\in(-1,1)^{n}\times(0,\infty)^{n}\times(0,1)^{n}$

the conditions G1 - G4, with strict inequalities in place of weak ones, are
all satisfied.

As emphasized by Mingone, her algorithm aims to overcome the problems caused
by the sequential feature of the robust algorithm of Corbetta et al. (2019).
For this reason she considers a global objective function of the form

$\sum_{i=1}^{n}\sum_{j}[C(K_{j},i)-\widehat{C}(K_{j},i)]^{2}w(k_{j},i),$

where $C(K_{j},i)$ is the observed option mid price at maturity $i$ and strike
$K_{j}$, $j=1,2,\dots,J_{i}$, $\widehat{C}(K_{j},i)$ is the theoretical option
price which is obtained by applying the Black and Scholes formula to the
suitable variance from the SSVI slice for maturity $i$, and where the
$w(k_{j},i)$’s are positive weights. For the latter Mingone suggests to use
the inverses of the squared market Black and Scholes vegas in order to achieve
calibration in implied volatilities at the first order. In this work I
followed this suggestion. For the sake of fair comparison with the robust
algorithm, I also tried to use constant weights. For minimizing the objective
function I used the Trust Region Reflective algorithm as implemented by the
least_squares function from the scipy.optimize library. I set the maximum
number of objective function evaluations to $500$ and used the default value
1e-8 for the termination conditions ftol, xtol and gtol. As initial parameter
vector for the optimization algorithm I used the parameter vector
corresponding to the optimal SSVI slices from the robust algorithm. This
choice is also suggested by Mingone.

##  3 Results

This section describes the results of my empirical investigation. The data I
used refer to SPXW options and are provided by FactSet. In order to cover a
possibly wide range of market situations, I considered the option chains from
the last trading date of each month starting from June 2012 until July 2022
($122$ option chains). As usual in calibration exercises, I excluded some
options from the analysis. To identify the excluded options, I first computed
an implied forward price for the underlying. In order to do so, I considered
all put-call pairs from which I computed an implied dividend yield according
to the formula

$q_{imp}:=\frac{1}{t}\left[\ln(C-P+Ke^{-rt})-\ln S\right]$

where $t$ is time to maturity, $C$ and $P$ are the call and put mid-prices,
$K$ is the strike price, $S$ is the underlying price and $r$ is the risk-free
rate. To determine the latter I interpolated the US treasury yield curve
provided by FactSet. Once I got the implied dividend yields, I averaged them
for each maturity date and used the average values in order to compute forward
prices for the underlying. Then I discarded all forward ITM options and all
options with bid-ask percentage spread larger than $5\%$. From the remaining
options I computed forward ATM implied volatilities for each maturity date
$t$. I did this only for maturity dates where I was left with at least one
option with forward-to-strike ratio larger and smaller than $1$. In this case,
I considered the Black and Scholes implied volatility of the option whose
forward-to-strike ratio is closer to $1$ as forward ATM implied volatility.
Otherwise, I excluded all options with the given maturity date from the
subsequent analysis.

Having applied the above filtering operations, I was left with zero options
for the $14$ chains of 2012-09-28, 2018-01-31, 2013-05-31, 2017-05-31,
2013-06-28, 2012-10-31, 2018-03-29, 2012-11-30, 2012-12-31, 2018-05-31,
2013-01-31, 2013-11-29, 2017-04-28, 2012-08-31. Thus, my analysis refers to
the remaining $122-14=108$ option chains. The two graphs in Figure 1 show the
number of available SPXW options and the number of available maturity dates in
each option chain before and after the filtering operations.

For evaluating the fit of the calibrated eSSVI surfaces I computed four
measures:

  * i)

the ratio

$F_{1}:=\frac{N_{w}}{N_{tot}}=\frac{\begin{array}[]{c}\text{number of
theoretical option prices $\widehat{C}(K_{j},i)$}\\\ \text{which are between
the observed bid and ask prices}\end{array}}{\text{total number of options
used for calibration}}$

  * ii)

the average absolute pricing error

$F_{2}:=\frac{1}{N_{tot}}\sum_{i=1}^{n}\sum_{j}|C(K_{j},i)-\widehat{C}(K_{j},i)|$

  * iii)

the average of the squared pricing errors

$F_{3}:=\frac{1}{N_{tot}}\sum_{i=1}^{n}\sum_{j}|C(K_{j},i)-\widehat{C}(K_{j},i)|^{2}$

  * iv)

the weighted average of squared pricing errors with weights $w(k_{j},i)$ given
by the inverses of the squared market Black and Scholes vegas

$F_{4}:=\frac{1}{\sum_{i=1}^{n}\sum_{j}w(k_{j},i)}\sum_{i=1}^{n}\sum_{j}|C(K_{j},i)-\widehat{C}(K_{j},i)|^{2}w(k_{j},i).$

Of course, none of these measures on its own is perfectly fair for comparing
the algorithms. In fact, one might expect $F_{1}$ to be in favor of the global
algorithm with the quadratic inverse vega weighted objective function, since
bid-ask spreads and vegas are usually positively correlated (see panel (a) in
Figure 2). However, as can be seen from panels (a) and (b) in Figure 4, the
global algorithm with constant weights seems to achieve smaller $F_{1}$ values
on average. Also, in the comparison between the two global algorithms and the
robust algorithm one might expect $F_{1}$ to be more helpful for the latter
since bid-ask spreads and time to maturity are usually positively correlated
(see panel (b) in Figure 2) and the robust algorithm tends to have problems to
fit the larger maturity dates in chains with many different maturity dates. As
can be seen from panel (a) in Figure 4, evidence for the latter conjecture is
ambiguous.

Consider now the $F_{2}$ measure of fit. Also this measure can be expected to
favor the robust algorithm since this algorithm aims to minimize sequentially,
one after another, the inner sums in the definition of $F_{2}$. Moreover, in
the comparison between the two implementations of the global algorithm $F_{2}$
should be in favor of the implementation with constant weights. The results
shown in panel (c) and panel (d) of Figure 4 are slightly supportive for these
conjectures.

Finally, as for the $F_{3}$ and $F_{4}$ measures, the results are as expected.
The global algorithm with constant weights performs best w.r.t. to the $F_{3}$
measure, and the global algorithm with quadratic inverse vega weights is the
best one w.r.t. $F_{4}$.

To complete the picture I will spend some words about computational aspects.
As already mentioned, in both implementations of the global algorithm I used
the optimal parameters from the robust algorithm as initial values. With this
choice of initial values the least_squares function was quite efficient in
finding minima for the objective function. In fact, for the squared inverse
vega weighted objective function the minimization process terminated before
$500$ objective function evaluations for $93$ of the $108$ option chains
because either the ftol, gtol or xtol termination condition was satisfied. For
the objective function with constant weights the least_squares function was
able to find a minimum before $500$ function evaluations in $85$ of the $108$
option chains. However, my implementation of the robust algorithm could be too
slow for some purposes. In fact, on average over all $2023$ calibrated eSSVI
slices, it took about $5523$ objective function evaluations for finding the
optimal parameters of a single slice. Motivated by this fact, I tested also
the less data-driven initial values suggested in Mingone (2022) which are much
faster to compute (these initial values are given by $\rho_{i}=0$,
$a_{i}=\max\\{\theta^{*}_{i}-\theta^{*}_{i-1},0\\}$ and $c_{i}=0.5$ for
$i=1,2,\dots,n$, where the $\theta^{*}_{i}$’s are the closest to ATM forward
implied variances as defined in Section 2.1). As can be seen from Figure 5,
the results of this test are not very encouraging: the less data-driven
initial values often lead to local minima where the $F_{1}$ measure of fit is
much larger than what can be achieved through the robust initial values.

##  4 Appendix: Revised proof of Proposition 3.1 in Hendriks and Martini
(2019)

Consider two eSSVI slices which we shall denote by

$w_{i}(k)=\frac{\theta_{i}}{2}\left\\{1+\rho_{i}\varphi_{i}k+\sqrt{\varphi_{i}^{2}k^{2}+2\varphi_{i}\rho_{i}k+1}\right\\},\quad
i=1,2.$

As in the main text, assume that the subscript $i=1$ refers to the closer
maturity date. Then there is absence of calendar spread arbitrage if and only
if $w_{1}(k)\leq w_{2}(k)$ for all $k\in\mathbb{R}$.

Note that

$\begin{split}w_{i}^{\prime}(k)&=\frac{1}{2}\theta_{i}\varphi_{i}\left(\frac{k\varphi_{i}+\rho_{i}}{\sqrt{\varphi_{i}^{2}k^{2}+2\varphi_{i}\rho_{i}k+1}}+\rho_{i}\right)\\\
w_{i}^{\prime\prime}(k)&=\frac{\theta_{i}\left(1-\rho_{i}^{2}\right)\varphi_{i}^{2}}{2(\varphi_{i}^{2}k^{2}+2\varphi_{i}\rho_{i}k+1)^{3/2}}\end{split}$

so that $w_{i}^{\prime\prime}(k)>0$ for all $k\in\mathbb{R}$. Since
$w_{i}^{\prime}(k)=0$ if and only if
$k=k_{i}^{*}:=-\frac{2\rho_{i}}{\varphi_{i}}$, we conclude that

$\inf_{k}w_{i}(k)=w_{i}(k_{i}^{*})=\theta_{i}(1-\rho_{i}^{2}).$

By combining this result with the fact that $w_{i}(0)=\theta_{i}$, we see that
absence of calendar spread arbitrage implies

$\Theta:=\frac{\theta_{2}}{\theta_{1}}\geq\max\left\\{1,\frac{1-\rho_{1}^{2}}{1-\rho_{2}^{2}}\right\\}.$
(2)

Another necessary condition may be obtained by considering the asymptotes of
the two slices. Since

$2w_{i}(k)\sim\begin{cases}\theta_{i}\varphi_{i}(1+\rho_{i})k&\text{ if
}k\rightarrow\infty,\\\ \theta_{i}\varphi_{i}(1-\rho_{i})k&\text{ if
}k\rightarrow-\infty,\end{cases}$

we conclude that absence of calendar spread arbitrage also implies

$\Theta\Phi:=\frac{\theta_{2}\varphi_{2}}{\theta_{1}\varphi_{1}}\geq\max\left\\{\frac{1+\rho_{1}}{1+\rho_{2}},\frac{1-\rho_{1}}{1-\rho_{2}}\right\\}$
(3)

The latter condition is satisfied if and only if

$\Theta\Phi\geq 1\quad\text{ and
}\quad(\Theta\Phi\rho_{2}-\rho_{1})^{2}\leq(\Theta\Phi-1)^{2}.$ (4)

Of course, in the argument leading to the necessary condition (2) we are
tacitly assuming that $\varphi_{1}$, $\varphi_{2}$ and $\theta_{1}$ are all
strictly positive and in this case it follows from (2) that
$\theta_{2}\geq\theta_{1}$, i.e. that $\Theta\geq 1$. Note that if
$\varphi_{1}=0$ and/or $\theta_{1}=0$, then $w_{1}(k)=\theta_{1}$ for all
$k\in\mathbb{R}$, and in this case we have absence of calendar spread
arbitrage if and only if $\theta_{2}\geq\theta_{1}$ or
$\theta_{2}(1-\rho_{2}^{2})\geq\theta_{1}$ according to whether $\varphi_{2}$
is also zero or not. On the other hand, if $\varphi_{2}=0$, then we have
$w_{2}(k)=\theta_{2}$ for all $k\in\mathbb{R}$, and in this case it follows
from the asymptotic behavior of $w_{1}(k)$ that we have absence of calendar
spread arbitrage if and only if $\varphi_{1}=0$ and
$\theta_{1}\leq\theta_{2}$. In what follows we rule out these trivial cases by
assuming that $\Phi:=\varphi_{2}/\varphi_{1}$ and
$\Theta:=\theta_{2}/\theta_{1}$ are well defined (i.e. that their denominators
are strictly positive) and that $\Phi>0$ and $\Theta\geq 1$.

######  Lemma 4.1.

If $\theta_{1}$, $\varphi_{1}$ and $\varphi_{2}$ are all strictly positive,
then there is absence of calendar spread arbitrage only if conditions (2) and
(3) are both satisfied.

Now it arises the question whether the conditions (2) and (3) are sufficient
as well. To answer this question we look for conditions under which the graphs
of $w_{1}(k)$ and $w_{2}(k)$ have at least one point in common. I will proceed
as in Hendriks and Martini (2019), but I will try to make some steps more
explicit. So let $x:=\varphi_{1}k$,

$\alpha:=\alpha(x)=\Theta-1+(\Theta\Phi\rho_{2}-\rho_{1})x,\quad
z_{1}:=z_{1}(x)=\sqrt{x^{2}+2\rho_{1}x+1},$
$z_{2}:=z_{2}(x)=\sqrt{\Phi^{2}x^{2}+2\rho_{2}\Phi x+1}$

and note that the two eSSVI slices do have points in common if and only if the
equation

$\alpha(x)+\Theta z_{2}(x)=z_{1}(x)$

has real solutions. Squaring twice yields the quartic polynomial

$P(x):=4\alpha^{2}\Theta^{2}z_{2}^{2}-(z_{1}^{2}-\alpha^{2}-\Theta^{2}z_{2}^{2})^{2}$

where we have omitted the independent variable $x$ on the RHS. Note that every
root of $P(x)$ must satisfy one (and only one) of the following conditions:

$\begin{split}\text{a) }&2\alpha\Theta
z_{2}=-(z_{1}^{2}-\alpha^{2}-\Theta^{2}z_{2}^{2})\quad\text{ and
}\quad\alpha-\Theta z_{2}=\pm z_{1},\\\ \text{b) }&2\alpha\Theta
z_{2}=z_{1}^{2}-\alpha^{2}-\Theta^{2}z_{2}^{2}\quad\text{ and
}\quad\alpha+\Theta z_{2}=-z_{1},\\\ \text{c) }&2\alpha\Theta
z_{2}=z_{1}^{2}-\alpha^{2}-\Theta^{2}z_{2}^{2}\quad\text{ and
}\quad\alpha+\Theta z_{2}=z_{1}.\end{split}$ (5)

Of course, a root of $P(x)$ is a point where the two slices intersect if and
only if it satisfies condition c). To explore the existence of such roots we
first observe that $P(x)=x^{2}Q(x)$, where

$\begin{split}Q(x):=&\left[\left(\Theta\Phi\rho_{2}-\rho_{1}\right)^{2}-\left(\Theta\Phi-1\right)^{2}\right]\left[\left(\Theta\Phi+1\right)^{2}-\left(\Theta\Phi\rho_{2}-\rho_{1}\right)^{2}\right]x^{2}\\\
&+4\Theta\left[\rho_{1}\left(-\Theta^{2}\Phi^{2}+(\Theta-2)\Theta\rho_{2}^{2}\Phi^{2}+2\Theta\Phi^{2}-1\right)+\right.\\\
&\quad\quad\quad\quad\left.+\rho_{2}\Phi\left(\Theta^{2}\rho_{2}^{2}\Phi^{2}-\Theta^{2}\Phi^{2}+2\Theta-1\right)+(1-2\Theta)\rho_{2}\rho_{1}^{2}\Phi+\rho_{1}^{3}\right]x\\\
&+4(\Theta-1)\Theta\left(\Theta\Phi^{2}\rho_{2}^{2}-\Theta\Phi^{2}-\rho_{1}^{2}+1\right).\end{split}$
(6)

Note that $x=0$ (which is a root of $P(x)$) is an intersection point if and
only if $\theta_{1}=\theta_{2}$, i.e. if and only if $\Theta=1$ (in fact,
$w_{1}(0)=\theta_{1}=\theta_{2}=w_{2}(0)$ if and only if $\Theta=1$). Assuming
that this is the case, we will now find conditions under which the two slices
do cross over each other. To this aim we consider their derivatives. With
$\theta_{1}=\theta_{2}=\theta$ (i.e. with $\Theta=1$) we obtain

$w_{i}^{\prime}(0)=\theta\varphi_{i}\rho_{i}\quad\text{ and }\quad
w_{i}^{\prime\prime}(0)=\frac{1}{2}\theta\varphi_{i}^{2}(1-\rho_{i}^{2}).$

To rule out the possibility that the two slices cross over in $x=0$, we must
therefore impose

$w_{1}^{\prime}(0)=w_{2}^{\prime}(0)\quad\text{ and }\quad
w_{1}^{\prime\prime}(0)\leq w_{2}^{\prime\prime}(0).$ (7)

If either one of these conditions fails, the two slices cross over in $x=0$.
Since we are assuming that the $\theta_{i}$’s and $\varphi_{i}$’s are all
strictly positive, the conditions (7) can be jointly satisfied only if

  * •

either $\rho_{1}=\rho_{2}=0$ and $\varphi_{2}\geq\varphi_{1}$, in which case
it is easy to verify that $w_{2}(k)\geq w_{1}(k)$ for all $k\in\mathbb{R}$;

  * •

or $\Phi=\rho_{1}/\rho_{2}$ and $\rho_{1}^{2}\geq\rho_{2}^{2}$, in which case
the constant term and the coefficient of $x$ in the polynomial $Q(x)$ do both
vanish, and hence the two slices have no intersection points other than $x=0$.

These considerations prove the following lemma:

######  Lemma 4.2.

Assume that $\Phi$ and $\Theta$ are well defined and that $\Phi>0$. If
$\Theta=1$, there is no calendar spread arbitrage if and only if either (i)
$\rho_{1}=\rho_{2}=0$ and $\Phi\geq 1$ or (ii) $\Phi=\rho_{1}/\rho_{2}$ and
$\rho_{1}^{2}\geq\rho_{2}^{2}$.

Note that conditions (2) and (3) do not imply condition (i) or (ii) of the
previous lemma (take for example $\Phi=1.2$, $\rho_{1}=0.9$ and
$\rho_{2}=0.81$) and the former are therefore not strong enough to rule out
calendar spread arbitrage even if we restrict to the case where $\Theta=1$.

Consider now what happens when $\Theta>1$. In this case
$w_{1}(0)=\theta_{1}<\theta_{2}=w_{2}(0)$ and $x=0$ is therefore not an
intersection point. To investigate the existence of intersection points we
analyze the polynomial $Q(x)$. We begin with the following lemma:

######  Lemma 4.3.

Assume that $\Phi>0$ and $\Theta>1$. Then $Q(x)$ is of second degree if and
only if

$(\Theta\Phi\rho_{2}-\rho_{1})^{2}\neq(\Theta\Phi-1)^{2}.$ (8)

and in this case its discriminant is given by

$\begin{split}D&:=16\Theta(\rho_{1}^{2}-\Theta^{2}\Phi^{2}\rho_{2}^{2}+\Theta^{2}\Phi^{2}-1)^{2}\left[(\Theta\Phi\rho_{2}-\rho_{1})^{2}-(\Theta-1)(\Theta\Phi^{2}-1)\right]\end{split}$
(9)

###### Proof.

The coefficient of $x^{2}$ in $Q(x)$ vanishes if and only if either

$(\Theta\Phi\rho_{2}-\rho_{1})^{2}=(\Theta\Phi-1)^{2}\quad\text{ or
}\quad(\Theta\Phi\rho_{2}-\rho_{1})^{2}=(\Theta\Phi+1)^{2}.$

The second condition implies $\Theta\Phi<0$, and hence we conclude that $Q(x)$
is of second degree if and only if condition (8) holds. In this case the
discriminant of $Q(x)$ can be written as in expression (9). ∎

From the previous lemma we know that $Q(x)$ must have real roots if condition
(8) holds jointly with

$\rho_{1}^{2}-\Theta^{2}\Phi^{2}\rho_{2}^{2}+\Theta^{2}\Phi^{2}-1=0\quad\text{
or }\quad(\Theta-1)(\Theta\Phi^{2}-1)\leq(\Theta\Phi\rho_{2}-\rho_{1})^{2}$
(10)

Since we are only concerned with the case where the necessary conditions (2)
and (4) hold, we must consider the set of $(\rho_{1},\rho_{2})$-pairs where
the equality in (10) holds and the set $(\rho_{1},\rho_{2})$-pairs where

$(\Theta-1)(\Theta\Phi^{2}-1)\leq(\Theta\Phi\rho_{2}-\rho_{1})^{2}\leq(\Theta\Phi-1)^{2}.$

We denote these two sets by $H_{\Theta,\Phi}$ and $R_{\Theta,\Phi}$,
respectively:

$\begin{split}H_{\Theta,\Phi}&:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\rho_{1}^{2}-\Theta^{2}\Phi^{2}\rho_{2}^{2}+\Theta^{2}\Phi^{2}-1=0\\},\\\
R_{\Theta,\Phi}&:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:(\Theta-1)(\Theta\Phi^{2}-1)\leq(\Theta\Phi\rho_{2}-\rho_{1})^{2}\leq(\Theta\Phi-1)^{2}\\}.\end{split}$

It will be useful to visualize these two sets. Since we have already dealt
with the case $\Theta=1$, and since we are assuming the necessary condition
(4), we need to consider only the case where $\Theta>1$ and $\Theta\Phi\geq
1$. Figure 6 shows all possible shapes of $H_{\Theta,\Phi}$ and
$R_{\Theta,\Phi}$. The set $H_{\Theta,\Phi}$ is the graph of a hyperbola. It
is always symmetric with respect to both axes of the
$(\rho_{1},\rho_{2})$-plane and its prolongation always goes through the four
vertices of the square $[-1,1]^{2}\subset\mathbb{R}^{2}$. Moreover,

  * •

if $\Theta\Phi>1$, $H_{\Theta,\Phi}$ does not intersect the $\rho_{1}$ axis
and intersects the $\rho_{2}$ axis in
$\rho_{2}=\pm\sqrt{\frac{\Theta^{2}\Phi^{2}-1}{\Theta^{2}\Phi^{2}}}$;

  * •

if $\Theta\Phi=1$, $H_{\Theta,\Phi}$ reduces to the straight lines
$\rho_{1}=\pm\rho_{2}$.

Also for the set $R_{\Theta,\Phi}$ there are essentially only two possible
shapes:

  * •

If $\Theta\Phi^{2}-1\leq 0$ (since we are assuming $\Theta>1$, this implies
$\Phi<1$), the set $R_{\Theta,\Phi}$ is given by the stripe

$S:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\Theta\Phi\rho_{2}-\Theta\Phi+1\leq\rho_{1}\leq\Theta\Phi\rho_{2}+\Theta\Phi-1\\}.$

The stripe reduces to the line $\rho_{1}=\rho_{2}$ if $\Theta\Phi=1$.

  * •

If $\Theta\Phi^{2}-1>0$ (since we are assuming $\Theta>1$, this implies
$\Theta\Phi>1$), the set $R_{\Theta,\Phi}$ is the union of the two parallel
and disjoint stripes

$S_{1}:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\Theta\Phi\rho_{2}+\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}\leq\rho_{1}\leq\Theta\Phi\rho_{2}+\Theta\Phi-1\\}$

and

$S_{2}:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\Theta\Phi\rho_{2}-\Theta\Phi+1\leq\rho_{1}\leq\Theta\Phi\rho_{2}-\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}\\}.$

The two stripes reduce to the lines $\rho_{1}=\Theta\rho_{2}\pm(\Theta-1)$ if
$\Phi=1$.

From the description of the sets $H_{\Theta,\Phi}$ and $R_{\Theta,\Phi}$ we
see immediately that $H_{\Theta,\Phi}\cap
R_{\Theta,\Phi}=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\rho_{1}=\rho_{2}\\}$ when
$\Theta\Phi=1$. Next we prove that except for this special case the
intersection is empty.

######  Lemma 4.4.

If $\Theta\Phi\neq 1$, then it follows that $S\cap H_{\Theta,\Phi}=\emptyset$.

###### Proof.

If $(\rho_{1},\rho_{2})\in H_{\Theta,\Phi}$ we must have

$\Theta\Phi(1-\rho_{2}^{2})=1-\rho_{1}^{2}\quad\Rightarrow\quad\Theta\Phi=\sqrt{\frac{1-\rho_{1}^{2}}{1-\rho_{2}^{2}}}.$

From the latter equality we obtain

$\begin{split}(\Theta\Phi\rho_{2}-\rho_{1})^{2}-(\Theta\Phi-1)^{2}=2\sqrt{\frac{1-\rho_{1}^{2}}{1-\rho_{2}^{2}}}(1-\rho_{1}\rho_{2})-2(1-\rho_{1}^{2}).\end{split}$

The quantity on the RHS is positive because

$\frac{1-\rho_{1}^{2}}{1-\rho_{2}^{2}}(1-\rho_{1}\rho_{2})^{2}>(1-\rho_{1}^{2})^{2}\quad\Leftrightarrow\quad(\rho_{1}-\rho_{2})^{2}>0$

and because we are assuming that
$\Theta\Phi=\sqrt{\frac{1-\rho_{1}^{2}}{1-\rho_{2}^{2}}}\neq 1$. Hence we
conclude that $(\rho_{1},\rho_{2})\notin S$, because otherwise we should have

$(\Theta\Phi\rho_{2}-\rho_{1})^{2}-(\Theta\Phi-1)^{2}\leq 0,$

∎

From now on we consider roots of $Q(x)$ as functions of $\rho_{1}$ and
$\rho_{2}$. Any root of $Q(x)$ will be denoted by $x_{(\rho_{1},\rho_{2})}$.
Of course the subset of the $(\rho_{1},\rho_{2})$-plane where such a root
exists depends on $\Theta$ and $\Phi$. We denote this set with
$Q_{\Theta,\Phi}$. Since we are only interested in $\rho_{i}$ values within
the interval $(-1,1)$, we consider $Q_{\Theta,\Phi}$ as a subset of the open
square $(-1,1)^{2}$. Note that $x_{(\rho_{1},\rho_{2})}$ must be a continuous
function of $(\rho_{1},\rho_{2})\in Q_{\Theta,\Phi}$ and that
$(\rho_{1},\rho_{2})\in Q_{\Theta,\Phi}$ only if condition (10) holds.

As we have already seen, a root $x_{(\rho_{1},\rho_{2})}$ must be of one of
the three types in (5) and only roots of type c) are intersection points. In
order to determine which type applies, we will need the following functions:

$(\rho_{1},\rho_{2})\mapsto\alpha(x_{(\rho_{1},\rho_{2})}):=\Theta-1+(\Theta\Phi\rho_{2}-\rho_{1})x_{(\rho_{1},\rho_{2})}$

and

$\begin{split}(\rho_{1},\rho_{2})\mapsto
Z(x_{(\rho_{1},\rho_{2})})&:=z_{1}^{2}(x_{(\rho_{1},\rho_{2})})-\Theta^{2}z_{2}^{2}(x_{(\rho_{1},\rho_{2})})-\alpha^{2}(x_{(\rho_{1},\rho_{2})})\\\
&=x_{(\rho_{1},\rho_{2})}^{2}+2\rho_{1}x_{(\rho_{1},\rho_{2})}+1+\\\
&\quad-\Theta^{2}\left(\Phi^{2}x_{(\rho_{1},\rho_{2})}^{2}+2\Phi\rho_{2}x_{(\rho_{1},\rho_{2})}+1\right)+\\\
&\quad-\left[\Theta-1+(\Theta\Phi\rho_{2}-\rho_{1})x_{(\rho_{1},\rho_{2})}\right]^{2}.\end{split}$

Note that these functions must all be continuous functions of
$(\rho_{1},\rho_{2})\in Q_{\Theta,\Phi}$.

######  Lemma 4.5.

Assume that $\Phi>0$ and $\Theta>1$. Then
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})=0$ only if
$(\rho_{1},\rho_{2})\in H_{\Theta,\Phi}$.

###### Proof.

The proof is essentially the same as the proof of Lemma A.2 in Hendriks and
Martini (2019). Since $x_{(\rho_{1},\rho_{2})}$ must satisfy one of the two
equalities

$2\alpha(x_{(\rho_{1},\rho_{2})})\Theta z_{2}(x_{(\rho_{1},\rho_{2})})=\pm
Z(x_{(\rho_{1},\rho_{2})})$

(otherwise $x_{(\rho_{1},\rho_{2})}$ is not a root of $P(x)$ and hence neither
a root of $Q(x)$) and since $z_{2}(x)>0$ for all $x\in\mathbb{R}$, we conclude
that $\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})=0$ if and only
if $\alpha(x_{(\rho_{1},\rho_{2})})$ and $Z(x_{(\rho_{1},\rho_{2})})$ do both
vanish. In this case we must have

$\begin{gathered}\alpha(x_{(\rho_{1},\rho_{2})})=0\quad\Rightarrow\quad\Theta\left(1+\rho_{2}\Phi
x_{(\rho_{1},\rho_{2})}\right)=1+\rho_{1}x_{(\rho_{1},\rho_{2})}\quad\Rightarrow\\\
\Theta^{2}\left(1+2\Phi\rho_{2}x_{(\rho_{1},\rho_{2})}+\Phi^{2}\rho_{2}^{2}x_{(\rho_{1},\rho_{2})}^{2}\right)=1+2\rho_{1}x_{(\rho_{1},\rho_{2})}+\rho_{1}^{2}x_{(\rho_{1},\rho_{2})}^{2}\end{gathered}$
(11)

and we must also have

$\begin{gathered}\Theta^{2}z_{2}^{2}(x_{(\rho_{1},\rho_{2})})=z_{1}^{2}(x_{(\rho_{1},\rho_{2})})\quad\Rightarrow\\\
\Rightarrow\quad\Theta^{2}\left(\Phi^{2}x_{(\rho_{1},\rho_{2})}^{2}+2\Phi\rho_{2}x_{(\rho_{1},\rho_{2})}+1\right)=x_{(\rho_{1},\rho_{2})}^{2}+2\rho_{1}x_{(\rho_{1},\rho_{2})}+1,\end{gathered}$
(12)

Subtracting equation (11) from equation (12) yields

$x_{(\rho_{1},\rho_{2})}^{2}\left(\Theta^{2}\Phi^{2}-\Theta^{2}\Phi^{2}\rho_{2}^{2}-1+\rho_{1}^{2}\right)=0$

Since $x_{(\rho_{1},\rho_{2})}$ must be different from zero (otherwise we
would have $\alpha(x_{(\rho_{1},\rho_{2})})=\Theta-1>0$ contrary to our
assumption), this implies that
$\Theta^{2}\Phi^{2}-\Theta^{2}\Phi^{2}\rho_{2}^{2}-1+\rho_{1}^{2}=0$ which is
equivalent to $(\rho_{1},\rho_{2})\in H_{\Theta,\Phi}$. ∎

######  Corollary 4.6.

Assume that $\Phi>0$, $\Theta>1$ and that $A$ is a connected subset of
$Q_{\Theta,\Phi}$ which does not intersect the set

$H_{\Theta,\Phi}:=\\{(\rho_{1},\rho_{2})\in(-1,1)^{2}:\Theta^{2}\Phi^{2}(1-\rho_{2}^{2})=(1-\rho_{1}^{2})\\}.$

Then it follows that function
$(\rho_{1},\rho_{2})\mapsto\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})$
does not change sign on $A$.

The previous corollary will be useful to distinguish whether a given root
$x_{(\rho_{1},\rho_{2})}$ is of type a) rather than of type b) or c). Once we
know that it is not of type a), we will apply the next lemma in order to find
out whether it is of type b) or c).

######  Lemma 4.7.

Let $A$ be a connected subset of $Q_{\Theta,\Phi}$ such that
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})>0$ for all
$(\rho_{1},\rho_{2})\in A$. Then it follows that the function
$(\rho_{1},\rho_{2})\mapsto\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})$ does not change sign on $A$.

###### Proof.

Under the assumptions of the lemma $x_{(\rho_{1},\rho_{2})}$ must be a root of
either type b) or c) in (5). Hence we must have either

$\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})=z_{1}(x_{(\rho_{1},\rho_{2})})$

or

$\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})=-z_{1}(x_{(\rho_{1},\rho_{2})}).$

The conclusion of the lemma follows now from the fact that
$\alpha(x_{(\rho_{1},\rho_{2})})+\Theta z_{2}(x_{(\rho_{1},\rho_{2})})$ is
continuous and that $z_{1}(x)>0$ for all $x\in\mathbb{R}$. ∎

Now we are finally ready to investigate about the existence of intersection
points. We start from the special cases where $\Phi=1$ or $\Theta\Phi=1$.

######  Lemma 4.8.

Assume that $\Theta>1$ and that either $\Phi=1$ or $\Theta\Phi=1$. Then
$(\rho_{1},\rho_{2})\in R_{\Theta,\Phi}$ implies $w_{1}(k)<w_{2}(k)$ for all
$k\in\mathbb{R}$.

###### Proof.

If $\Theta>1$, we must have $w_{1}(0)<w_{2}(0)$ and thus there exist
intersection points only if the polynomial $Q(x)$ has real roots different
from $x=0$. Now, consider first the case $\Phi=1$. Since we are assuming that
$(\rho_{1},\rho_{2})\in R_{\Theta,\Phi}$, it follows that
$\rho_{1}=\Theta\rho_{2}\pm(\Theta-1)$ (see the description of the set
$R_{\Theta,\Phi}$ for the special case where $\Phi=1$). However, it can be
verified that in this case we must have

$Q(x)=-4\Theta^{2}(\Theta-1)^{2}\left(\rho_{2}\pm 1\right){}^{2}<0$

which has no roots at all.

Next, consider the case $\Theta\Phi=1$. In this case we must have
$\rho_{1}=\rho_{2}$ (see the description of the set $R_{\Theta,\Phi}$ for the
special case where $\Theta\Phi=1$). Substituting $\rho_{1}=\rho_{2}=\rho$ and
$\Phi=1/\Theta$ in the coefficients of $Q(x)$ shows that

$Q(x)=4(\Theta-1)^{2}\left(1-\rho^{2}\right)>0$

which has no roots at all. ∎

Next, we deal with the case where $\Phi$ is strictly smaller than $1$ and
different from $1/\Theta$ (i.e. $\Theta\Phi\neq 1$). The inequality
$\Theta\Phi\geq 1$, which is necessary by condition (4), allows then only for
values of $\Phi$ in the range $1/\Theta<\Phi<1$. Note that for $\Phi\leq 1$
the necessary condition (2) is already implied by condition (4) and therefore
we do not need to assume condition (2) explicitly.

######  Lemma 4.9.

Assume that $\Theta>1$, $\Theta\Phi>1$ and $\Theta\Phi^{2}\leq 1$ (this
implies $\Phi<1$). Then $(\rho_{1},\rho_{2})\in R_{\Theta,\Phi}$ implies
$w_{1}(k)<w_{2}(k)$ for all $k\in\mathbb{R}$.

###### Proof.

Once again, if $\Theta>1$ we must have $w_{1}(0)<w_{2}(0)$ and there exist
points $k\in\mathbb{R}$ where $w_{1}(k)\geq w_{2}(k)$ if and only if
intersection points exist, i.e. if and only if the polynomial $Q(x)$ has at
least one real root which satisfies condition c) in (5). From Lemma 4.3 we
know that $Q(x)$ must have roots if $\Theta>1$, $\Theta\Phi>1$,
$\Theta\Phi^{2}\leq 1$ and if $(\rho_{1},\rho_{2})$ belongs to the interior of
$R_{\Theta,\Phi}$. Since under the present conditions $R_{\Theta,\Phi}$ is
connected and does not intersect $H_{\Theta,\Phi}$ (see Lemma 4.4), we may
apply Corollary 4.6 to check whether the roots in $R_{\Theta,\Phi}$ are of
type a). This will be the case if there exists a single
$(\rho_{1},\rho_{2})\in int(R_{\Theta,\Phi})$ such that
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})<0$. Under the
present conditions the origin belongs to $int(R_{\Theta,\Phi})$. Hence we use
$(\rho_{1},\rho_{2})=(0,0)$ as test point. Of course,
$\alpha(x_{(0,0)})=\Theta-1>0$. Moreover, it is easy to check that

$x_{(0,0)}=2\frac{\sqrt{\Theta(\Theta-1)\left(1-\Theta\Phi^{2}\right)}}{\Theta^{2}\Phi^{2}-1}$

so that

$Z(x_{(0,0)})=-\frac{2(\Theta-1)\Theta\left[\Theta^{2}\Phi^{2}-1+2(1-\Theta\Phi^{2})\right]}{\Theta^{2}\Phi^{2}-1}<0.$

We conclude that $x_{(\rho_{1},\rho_{2})}$ must be of type a) whenever
$(\rho_{1},\rho_{2})\in R_{\Theta,\Phi}$. ∎

In the previous lemma we have assumed that $\Theta\Phi^{2}\leq 1$ which forces
$\Phi<1$. To apply the same method of proof for the case where
$\Theta\Phi^{2}>1$ we must however assume that $\Phi<1$.

######  Lemma 4.10.

Assume $\Theta>1$, $\Theta\Phi^{2}>1$ and $\Phi<1$ (note that $\Theta>1$ and
$\Theta\Phi^{2}>1$ implies $\Theta\Phi>1$). Then $(\rho_{1},\rho_{2})\in
R_{\Theta,\Phi}$ implies $w_{1}(k)<w_{2}(k)$ for all $k\in\mathbb{R}$.

###### Proof.

The proof is similar to the proof of the previous lemma. However, in the
present case we must deal with the fact that the set $R_{\Theta,\Phi}$ is not
connected but only the union of the two connected sets $S_{1}$ and $S_{2}$. In
each one of these two sets we must therefore find a point
$(\rho_{1},\rho_{2})$ such that $\alpha(x_{(\rho_{1},\rho_{2})})$ and
$Z(x_{(\rho_{1},\rho_{2})})$ are of opposite sign. To locate these points,
note that the $\rho_{2}$-axis intersects both sets and hence we choose the
$(\rho_{1},\rho_{2})$-points with $\rho_{1}=0$ and

$\rho_{2}=\rho_{2}^{\pm}:=\pm\frac{\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}}{\Theta\Phi}.$

This choice is convenient because it makes the discriminant of $Q(x)$ vanish.
According to the sign in $\rho_{2}^{\pm}$, it gives rise to the roots

$x_{(0,\rho_{2}^{\pm})}=\pm\frac{2\sqrt{(\Theta-1)\left(\Theta\Phi^{2}-1\right)}}{\Theta(1-\Phi^{2})}$
(13)

which, regardless of the sign, yields

$\alpha\left(x_{(0,\rho_{2}^{\pm})}\right)=\frac{(\Theta-1)(\Theta\Phi^{2}+\Theta-2)}{\Theta(1-\Phi^{2})}$
(14)

and

$Z\left(x_{(0,\rho_{2}^{\pm})}\right)=-\frac{2(\Theta-1)\left(\Theta\Phi^{2}+\Theta-2\right)^{2}}{\Theta\left(\Phi^{2}-1\right)^{2}}.$
(15)

Note that $Z\left(x_{(0,\rho_{2}^{\pm})}\right)<0$ regardless of the value of
$\Phi$ (provided that $\Phi\neq 1$), but to make sure that
$\alpha\left(x_{(0,\rho_{2}^{\pm})}\right)>0$ we need to assume $\Phi<1$.

∎

Lemma 4.8, Lemma 4.9 and Lemma 4.10 show that the necessary condition (4)
along with $\Theta>1$ and $\Phi\leq 1$ are jointly sufficient to rule out
calendar spread arbitrage. The next lemma deals with the condition

$(\Theta-1)(\Theta\Phi^{2}-1)\geq(\Theta\Phi\rho_{2}-\rho_{1})^{2}$ (16)

which allows for values of $\Phi$ larger than $1$.

######  Lemma 4.11.

Assume that $\Theta>1$ and that condition (16) holds (note that these
conditions jointly imply the necessary condition (4)). Then it follows that
$w_{1}(k)\leq w_{2}(k)$ for all $k\in\mathbb{R}$. Under the assumptions of
this lemma there exist tangency points (i.e. values of $k$ where
$w_{1}(k)=w_{2}(k)$) if and only if $\Phi>1$ and condition (16) holds with
equality sign. In that case there must exist exactly one tangency point.

###### Proof.

If $\Theta>1$ and condition (16) holds, we must have $\Theta\Phi^{2}\geq 1$
and hence $\Theta\Phi>1$ (otherwise there would not exist any
$(\rho_{1},\rho_{2})$-pair for which (16) holds). Consider first what happens
when $\Theta\Phi^{2}=1$. In this case we must have $\Phi<1$ and for $\Phi\leq
1$ we have already proved that $w_{1}(k)<w_{2}(k)$ for all $k\in\mathbb{R}$.

Consider now what happens when $\Phi>1$. Since we are assuming that $\Theta>1$
(and hence we must have $\Theta\Phi^{2}>1$), we must have
$R_{\Theta,\Phi}=S_{1}\cup S_{2}$ and the assumed inequality (16) is satisfied
if and only if the $(\rho_{1},\rho_{2})$-pair belongs to the area between the
two stripes $S_{1}$ and $S_{2}$ or to one of the inner boundaries of $S_{1}$
or $S_{2}$. We indicate this set of $(\rho_{1},\rho_{2})$-pairs with $S_{3}$.
Note that $S_{3}$ must be a proper subset of $S$ since we are assuming that
$\Phi>1$. Since $\Phi>1$ implies $\Theta\Phi>1$, we can apply Lemma 4.4 and
conclude that $S_{3}\cap H_{\Theta,\Phi}\subset S\cap
H_{\Theta,\Phi}=\emptyset$. Now it follows from Lemma 4.3 that $Q(x)$ has no
real roots when $(\rho_{1},\rho_{2})$ belongs to the interior of $S_{3}$, i.e.
if the inequality (16) is strict. Since the two slices can intersect only if
$Q(x)$ has real roots, we conclude that $w_{1}(k)<w_{2}(k)$ for all
$k\in\mathbb{R}$ in this case. On the other hand, if the
$(\rho_{1},\rho_{2})$-pair belongs to the boundary of $S_{3}$, then it must
also belong to the inner boundary of one of the two stripes $S_{1}$ or
$S_{2}$. In other words, there must be equality in (16) which means that

$\rho_{2}=\rho_{2}^{\pm}(\rho_{1}):=\frac{1}{\Theta\Phi}\left(\rho_{1}\pm\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}\right)$

and that the discriminant of $Q(x)$ must be zero (see Lemma 4.3). Hence $Q(x)$
must have a root and this root must be unique. As usual we write
$x_{(\rho_{1},\rho_{2})}$ to indicate the root. Since we are assuming that
$\Theta$ and $\Phi$ are both larger than $1$, we can apply Lemma 4.5 and
conclude that $\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})\neq
0$ when the $(\rho_{1},\rho_{2})$-pair belongs to $R_{\Theta,\Phi}=S_{1}\cup
S_{2}$ and hence that
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})\neq 0$ for all
$(\rho_{1},\rho_{2})$-pairs which belong to the boundary of $S_{3}$ where a
root $x_{(\rho_{1},\rho_{2})}$ must exist and must be unique. Since $S_{1}$
and $S_{2}$ are two disjoint and connected sets,
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})$ does not change
sign on each of these two sets. We will now show that the sign of
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})$ is positive on
both sets. This can be done by proving that the sign of
$\alpha(x_{(\rho_{1},\rho_{2})})Z(x_{(\rho_{1},\rho_{2})})$ is positive at a
single point in each of the two sets (see Corollary 4.6). As in the proof of
Lemma (4.10) we use the $(\rho_{1},\rho_{2})$-pairs with $\rho_{1}=0$ and
$\rho_{2}=\rho_{2}^{\pm}:=\pm\frac{1}{\Theta\Phi}\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}$
as test points (note that these points also belong to the boundary of
$S_{3}$). With this choice we still get the expressions in (13), (14) and (15)
for $x_{(0,\rho_{2}^{\pm})}$, $\alpha(x_{(0,\rho_{2}^{\pm})})$ and
$Z(x_{(0,\rho_{2}^{\pm})})$. However, since we are now assuming that $\Phi>1$,
we see that $\alpha(x_{(0,\rho_{2}^{\pm})})<0$ and not
$\alpha(x_{(0,\rho_{2}^{\pm})})>0$ as in the proof of Lemma 4.10 (of course,
$Z(x_{(0,\rho_{2}^{\pm})})$ remains still negative). We conclude that the
roots we are considering now must be either of type b) or c) in (5). Hence we
must have $\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})=\pm z_{1}(x_{(\rho_{1},\rho_{2})})$. In order
to prove that the roots correspond to intersection points, we first note that
the mapping $(\rho_{1},\rho_{2})\mapsto\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})$ does not change sign on each of the two sets
$S_{1}$ and $S_{2}$ (use Lemma 4.7). However, as far as we know by now, the
sign of $\alpha(x_{(\rho_{1},\rho_{2})})+\Theta
z_{2}(x_{(\rho_{1},\rho_{2})})$ might be different according to whether
$(\rho_{1},\rho_{2})$ belongs to $S_{1}$ or to $S_{2}$. Thus, if there exists
a single point $(\rho_{1},\rho_{2})\in S_{i}$ such that the sign of
$\alpha(x_{(\rho_{1},\rho_{2})})+\Theta z_{2}(x_{(\rho_{1},\rho_{2})})$ is
positive, we can conclude that $x_{(\rho_{1},\rho_{2})}$ is of type c) and
hence that
$w_{1}(x_{(\rho_{1},\rho_{2})}/\varphi_{1})=w_{2}(x_{(\rho_{1},\rho_{2})}/\varphi_{1})$
for every $(\rho_{1},\rho_{2})\in S_{i}$ ($i=1,2$). Again, we use the two
$(\rho_{1},\rho_{2})$-pairs with $\rho_{1}=0$ and
$\rho_{2}=\rho_{2}^{\pm}:=\pm\frac{1}{\Theta\Phi}\sqrt{(\Theta-1)(\Theta\Phi^{2}-1)}$
as test points. For these points we get the same expressions of
$x_{(0,\rho_{2}^{\pm})}$ and $\alpha(x_{(0,\rho_{2}^{\pm}})$ as in equations
(13) and (14), while for $z_{2}(x_{(0,\rho_{2}^{\pm})})$ we get the expression

$z_{2}(x_{(0,\rho_{2}^{\pm})})=\frac{\Theta\Phi^{2}+\Theta-2}{\Theta(\Phi^{2}-1)}.$

Hence we conclude that

$\alpha(x_{(0,\rho_{2}^{\pm})})+\Theta
z_{2}(x_{(0,\rho_{2}^{\pm})})=\frac{\Theta\Phi^{2}+\Theta-2}{\Theta(\Phi^{2}-1)}>0.$

This argument shows that every root $x_{(\rho_{1},\rho_{2})}$ with
$(\rho_{1},\rho_{2})\in S_{1}\cup S_{2}$ is an intersection point and that
there must be a unique intersection point when
$(\rho_{1},\rho_{2})\in(S_{1}\cup S_{2})\cap S_{3}$. It is not difficult to
show that in the latter case the intersection point must be a tangency point.
In fact, if it was a crossing point, there should exist one further crossing
point because under our present assumptions the left and right asymptotes of
$w_{2}(k)$ are both steeper than those of $w_{1}(k)$ (recall that we are
assuming $\Theta>1$ and $\Phi>1$: on $(S_{1}\cup S_{2})\cap S_{3}$ condition
(4) must therefore hold with strict inequality sign). ∎

As far as I know, the results in the next two lemmas are new.

######  Lemma 4.12.

If

$\Theta>1,\quad\Phi>1,\quad\text{ and
}\quad(\Theta-1)(\Theta\Phi^{2}-1)<(\Theta\Phi\rho_{2}-\rho_{1})^{2}<(\Theta\Phi-1)^{2},$
(17)

there must exist exactly two points where the slices $w_{1}(k)$ and $w_{2}(k)$
cross over each other.

###### Proof.

From Lemma 4.3 and Lemma 4.4 we know that under condition (17) there must
exist two roots $x_{(\rho_{1},\rho_{2})}$. Moreover, from the proof of Lemma
4.11 we know that both these roots must be intersection points. The chained
inequality in (17) says that both asymptotes of $w_{2}(k)$ are steeper than
those of $w_{1}(k)$. Therefore only two cases can occur: either (i) both
intersection points are tangency points, or (ii) both intersection points are
crossing points. It is not difficult to see that case (i) is impossibile. In
fact, if both intersection points were tangency points, any increase of
$\Theta$ should lead to $w_{1}(k)<w_{2}(k)$ for all $k\in\mathbb{R}$. However,
given fixed values of $\Phi$, $\rho_{1}$ and $\rho_{2}$, a small enough
increase of $\Theta$ does not lead to a violation of condition (17) which
implies the existence of two intersection points. ∎

Now, it remains to see what happens when

$\Theta>1,\quad\Phi>1,\quad\text{ and
}\quad(\Theta\Phi\rho_{2}-\rho_{1})^{2}=(\Theta\Phi-1)^{2}$ (18)

i.e. when the left or right asymptote of $w_{2}(k)$ is the same as the
corresponding asymptote of $w_{1}(k)$.

######  Lemma 4.13.

Assume condition (18) holds. Then there must exist exactly one point where the
slices $w_{1}(k)$ and $w_{2}(k)$ cross over each other.

###### Proof.

Define $B_{\Theta,\Phi}$ as the subset of the $(\rho_{1},\rho_{2})$-plane
where the equality in condition (18) holds. $B_{\Theta,\Phi}$ is then the
boundary of $S$, i.e. the subset of the $(\rho_{1},\rho_{2})$-plane where

$\rho_{1}=\rho_{1}^{+}(\rho_{2}):=\Theta\Phi\rho_{2}+\Theta\Phi-1\quad\text{
and }\quad-1<\rho_{2}<\min\left\\{\frac{2}{\Theta\Phi}-1,1\right\\}$

or

$\rho_{1}=\rho_{1}^{-}(\rho_{2}):=\Theta\Phi\rho_{2}+1-\Theta\Phi\quad\text{
and }\quad\max\left\\{1-\frac{2}{\Theta\Phi},-1\right\\}<\rho_{2}<1.$

On $B_{\Theta,\Phi}$ the polynomial $Q(x)$ reduces to

$Q^{\pm}(x)=-4\Theta^{2}\left(\rho_{2}\pm
1\right)\Phi\left[(\Theta-1)^{2}\Phi\rho_{2}\pm(\Theta-1)(\Theta\Phi+\Phi-2)-2x(\Phi-1)(\Theta\Phi-1)\right]$

and the only root of $Q^{\pm}(x)$ is given by

$x_{(\rho_{1}^{\pm}(\rho_{2}),\rho_{2})}=\frac{(\Theta-1)^{2}\Phi\rho_{2}\pm(\Theta-1)(\Theta\Phi+\Phi-2)}{2(\Phi-1)(\Theta\Phi-1)}.$

We will show that this root must be a crossing point. To this aim note that
$B_{\Theta,\Phi}$ is the union of two disjoint connected sets which we denote
with $B_{\Theta,\Phi}^{\pm}$. From Lemma 4.4 it follows that
$B_{\Theta,\Phi}\cap H_{\Theta,\Phi}=\emptyset$. Hence we may apply Corollary
4.6 and conclude that
$\alpha(x_{(\rho_{1}^{\pm},\rho_{2})})Z(x_{(\rho_{1}^{\pm},\rho_{2})})$ does
not change sign on each one of the two connected components of
$B_{\Theta,\Phi}=B_{\Theta,\Phi}^{+}\cup B_{\Theta,\Phi}^{-}$. In order to
show that $\alpha(x_{(\rho_{1}^{\pm},\rho_{2})})$ and
$Z(x_{(\rho_{1}^{\pm},\rho_{2})})$ are of the same sign, it is therefore
sufficient to find a single point $(\rho_{1},\rho_{2})$ in each of the two
sets $B_{\Theta,\Phi}^{+}$ and $B_{\Theta,\Phi}^{-}$ for which
$\alpha(x_{(\rho_{1}^{\pm},\rho_{2})})$ and $Z(x_{(\rho_{1}^{\pm},\rho_{2})})$
are of the same sign. As test points we choose the points
$(\rho_{1},\rho_{2})=(\rho_{1}^{\pm}(\rho_{2}),\rho_{2})$ where
$\rho_{1}^{\pm}(\rho_{2})=0$. It is easily seen that these points are
$(0,\pm\rho_{*})$ where $\rho_{*}=\frac{1}{\Theta\Phi}-1$. Substituting in the
formula for the root we get

$x_{(0,\pm\rho_{*})}=\pm\frac{2\Theta^{2}\Phi-\Theta^{2}-2\Theta\Phi+1}{2\Theta(\Phi-1)(\Theta\Phi-1)}.$

Regardless of the sign in $\rho_{2}=\rho_{*}$, this root yields

$\alpha(x_{(0,\pm\rho_{*})})=-\frac{(\Theta-1)^{2}}{2\Theta(\Phi-1)}$

and

$Z(x_{(0,\pm\rho_{*})})=-\frac{(\Theta-1)^{2}(\Theta^{2}\Phi+2\Theta\Phi^{2}-4\Theta\Phi-\Phi+2)}{2\Theta(\Phi-1)^{2}(\Theta\Phi-1)}.$

Of course $\alpha(x_{(0,\pm\rho_{*})})<0$. As for $Z(x_{(0,\pm\rho_{*})})$,
its sign depends on the sign of

$\Theta^{2}\Phi+2\Theta\Phi^{2}-4\Theta\Phi-\Phi+2$

which is positive whenever $\Theta>1$ and $\Phi>1$ (we omit the details of the
proof of this assertion). Hence also $Z(x_{(0,\pm\rho_{*})})<0$ and thus we
conclude that $x_{(0,\pm\rho_{*})}$ is a root of either type b) or c) in (5).
To find out which type applies, we must determine the sign of
$\alpha(x_{(0,\pm\rho_{*})})+\Theta z_{2}(x_{(0,\pm\rho_{*})})$. It is not
difficult to verify that

$z_{2}(x_{(0,\pm\rho_{*})})=\frac{\Theta^{2}\Phi+2\Theta\Phi^{2}-4\Theta\Phi-\Phi+2}{2\Theta(\Phi-1)(\Theta\Phi-1)}$

and hence we get

$\alpha(x_{(0,\pm\rho_{*})})+\Theta
z_{2}(x_{(0,\pm\rho_{*})})=\frac{2\Theta^{2}\Phi^{2}-2\Theta^{2}\Phi+\Theta^{2}-2\Theta\Phi+1}{2\Theta(\Phi-1)(\Theta\Phi-1)}.$

The numerator in this expression can be written as

$\Theta^{2}(\Phi-1)^{2}+(\Theta\Phi-1)^{2}$

and therefore we must have $\alpha(x_{(0,\pm\rho_{*})})+\Theta
z_{2}(x_{(0,\pm\rho_{*})})>0$. By Lemma 4.7 we conclude that
$x_{(\rho_{1},\rho_{2})}$ must be an intersection point for every
$(\rho_{1},\rho_{2})\in B_{\Theta,\Phi}$.

To complete the proof it remains to show that for every
$(\rho_{1},\rho_{2})\in B_{\Theta,\Phi}$ the corresponding root
$x_{(\rho_{1},\rho_{2})}$ is a crossing point. To this aim we apply the
argument in the proof of Lemma 4.12 once again: if $x_{(\rho_{1},\rho_{2})}$
was a tangency point, then by increasing $\Theta$ a little bit we should have
no intersection points at all. However, if we increase $\Theta$ a little bit
while leaving $\Phi$, $\rho_{1}$ and $\rho_{2}$ unchanged, we pass from
condition (18) to condition (17) which implies the existence of two crossing
points.

∎

Combining the statements in Lemma 4.1, Lemma 4.2, Lemma 4.8, Lemma 4.9, Lemma
4.10, Lemma 4.11, Lemma 4.12 and Lemma 4.13 yields a corrected and sharper
version of Proposition 3.1 in Hendriks and Martini (2019). The little
corrections concern

  * i)

the special case where $\Theta=1$ and

  * ii)

the fact that Proposition 3.1 in Hendriks and Martini (2019) seems to imply
that with $\Theta>1$,
$(\Theta\Phi\rho_{2}-\rho_{1})^{2}\leq(\Theta\Phi-1)^{2}$ and $\Phi<1$ it
would be possible to have absence of calendar spread arbitrage even if
$\Theta\Phi<1$ which goes against the necessary condition (4). However, the
preprint version of the article contains a slightly different version of the
proposition which is not subject to this problem but where the two necessary
conditions are a little too strong due to strong inequality signs instead of
weak ones (see Proposition 3.5 in Hendriks and Martini (2017)).

The sharper (and corrected) statement of the Hendriks-Martini proposition is
given below. To make it more concise, the necessary condition (3) will be
stated as in (19).

######  Proposition 4.14.

Assume that $\theta_{1}$ and $\varphi_{1}$ are both strictly positive and let
$\Theta:=\theta_{2}/\theta_{1}$ and $\Phi:=\varphi_{2}/\varphi_{1}$. Then,
there is absence of calendar spread arbitrage (i.e. $w_{1}(k)\leq w_{2}(k)$
for all $k\in\mathbb{R}$) only if $\Theta\geq 1$ and

$1-\Theta\Phi\leq\Theta\Phi\rho_{2}-\rho_{1}\leq\Theta\Phi-1$ (19)

Moreover,

  * •

when $\Theta=1$ there is absence of calendar spread arbitrage if and only if
either (i) $\rho_{1}=\rho_{2}=0$ and $\Phi\geq 1$ or (ii)
$\Phi=\rho_{1}/\rho_{2}$ and $\rho_{1}^{2}\geq\rho_{2}^{2}$;

  * •

when $\Theta>1$ there is absence of calendar spread arbitrage if and only if
condition (19) holds jointly with

$\Phi\leq 1\quad\text{ or
}\quad(\Theta\Phi\rho_{2}-\rho_{1})^{2}\leq(\Theta-1)(\Theta\Phi^{2}-1);$

  * •

when $\Theta>1$ and condition (19) holds jointly with

$\Phi\leq 1\quad\text{ or
}\quad(\Theta\Phi\rho_{2}-\rho_{1})^{2}<(\Theta-1)(\Theta\Phi^{2}-1)$

there are no intersection points (i.e. $w_{1}(k)<w_{2}(k)$ for all
$k\in\mathbb{R}$)

  * •

when $\Theta>1$, $\Phi>1$ and
$(\Theta\Phi\rho_{2}-\rho_{1})^{2}=(\Theta-1)(\Theta\Phi^{2}-1)$ the two
slices have exactly one intersection point which is a tangency point;

  * •

when $\Theta>1$, $\Phi>1$ and
$(\Theta-1)(\Theta\Phi^{2}-1)<(\Theta\Phi\rho_{2}-\rho_{1})^{2}<(\Theta\Phi-1)^{2}$
there must exist exactly two points where the slices $w_{1}(k)$ and $w_{2}(k)$
cross over each other.

  * •

when $\Theta>1$, $\Phi>1$ and
$(\Theta\Phi\rho_{2}-\rho_{1})^{2}=(\Theta\Phi-1)^{2}$ there must exist
exactly one point where the slices $w_{1}(k)$ and $w_{2}(k)$ cross over each
other.

## References

  * Corbetta et al. (2019) Corbetta, J., Cohort, P., Laachir, I., Martini, C., Robust calibration and arbitrage-free interpolation of SSVI slices. Decisions in Economics and Finance, 2019, 42, 665–677.
  * Gatheral (2004) Gatheral, J., A parsimonious arbitrage-free implied volatility parameterization with application to the valuation of volatility derivatives [online]. Global Derivatives & Risk Management, Madrid May 26, 2004. Available online at: https://faculty.baruch.cuny.edu/jgatheral/madrid2004.pdf (accessed 2 February 2023).
  * Gatheral and Jacquier (2011) Gatheral, J. and Jacquier, A., Convergence of Heston to SVI. Quantitative Finance, 2011, 118, 1129–1132.
  * Gatheral and Jacquier (2014) Gatheral, J. and Jacquier, A., Arbitrage-free SVI volatility surfaces. Quantitative Finance, 2014, 141, 59-71.
  * Martini and Mingone (2022) Martini, C., Mingone, A., No arbitrage SVI. SIAM Journal on Financial Mathematics, 2022, 13:1, 227-261.
  * Mingone (2022) Mingone, A., No arbitrage global parametrization for the eSSVI volatility surface. Quantitative Finance, 2022, 22:12, 2205–2217.
  * Hendriks and Martini (2019) Hendriks, S., Martini, C., The Extended SSVI Volatility Surface. Journal of Computational Finance, 2019, 22:5, 25–39.
  * Hendriks and Martini (2017) Hendriks, S., Martini, C., The Extended SSVI Volatility Surface [online]. Available at SSRN: https://ssrn.com/abstract=2971502 or http://dx.doi.org/10.2139/ssrn.2971502
  * Lee (2004) Lee, R., The moment formula for implied volatility at extreme strikes. Mathematical Finance, 2004, 143, 469–480.

## Figures

(a) Total number of available SPXW options before (continuous) and after
(dashed) the filtering procedure.

(b) Total number of available maturity dates before (continuous) and after
(dashed) the filtering procedure.

Figure 1: The graphs show the total number of available options and available
maturity dates in each option chain before and after filtering.

(a) Correlation coefficients vega versus bid-ask-spread.

(b) Correlation coefficients DTE versus bid-ask-spread

Figure 2: Histograms based on $108$ correlation coefficients (one coefficient
for each option chain).

(a) $F_{1}$ measure of fit.

(b) $F_{2}$ measure of fit.

(c) $F_{3}$ measure of fit.

(a) $F_{4}$ measure of fit. Figure 4: Scatterplots and boxplots for the
measures of fit obtained on the $108$ calibrated eSSVI surfaces. ”robust”
refers to the algorithm of Corbetta et al. (2019) as described in Section 2.1.
”global 1” and ”global 2” refer to the algorithm of Mingone (2022) with
inverse quadratic vega weights and with constant weights, respectively (see
Section 2.2). The abscissa values in the scatterplots are the $F_{i}$ values
for the ”robust” algorithm; the ordinate values are the $F_{i}$ values for the
other two algorithms: circles refer to ”global 1”, triangles to ”global 2”.

(a) Squared Inverse vega weights

(b) Constant weights

Figure 5: Scatterplots of the $F_{1}$ measure of fit for the global algorithm.
The $x$-axes refer to the implementation with initial values from robust
algorithm, the $y$-axes to the implementation with the less data-driven
initial values given by $\rho_{i}=0$,
$a_{i}=\max\\{\theta^{*}_{i}-\theta^{*}_{i-1},0\\}$ and $c_{i}=0.5$ for
$i=1,2,\dots,n$. Dots refer to cases where both choices for the initial values
led to convergence in less than 500 objective function evaluations. Triangles
refer to cases where the robust initial values led to convergence in less than
500 objective function evaluations, but the less data-driven initial values
did not. Squares refer to cases where the robust initial values did not lead
to convergence in less than 500 objective function evaluations, but the less
data-driven initial values did. Circles refer to cases where none of the two
choices for the initial values led to convergence in less than 500 objective
function evaluations.

(a) $\Theta\Phi=1$ ($\Rightarrow$ $\Theta\Phi^{2}<1$)

(b) $\Theta\Phi>1$ and $\Theta\Phi^{2}<1$

(c) $\Theta\Phi>1$ and $\Theta\Phi^{2}=1$

(d) $\Theta\Phi>1$ and $\Theta\Phi^{2}>1$

(e) $\Theta\Phi>1$ and $\Phi=1$

Figure 6: The red areas show all possible shapes of the $R_{\Theta,\Phi}$ when
$\Theta>1$ and $\Theta\Phi\geq 1$. The black line is the graph of the
hyperbola $H_{\Theta,\Phi}$. The two dashed lines are the asymptotes
$\rho_{1}=\pm\Theta\Phi\rho_{2}$ of $H_{\Theta,\Phi}$.