###### Abstract

This paper generalizes the local variance gamma model of Carr and Nadtochiy,
to a piecewise quadratic local variance function. The formulation encompasses
the piecewise linear Bachelier and piecewise linear Black local variance gamma
models. The quadratic local variance function results in an arbitrage-free
interpolation of class $\mathcal{C}^{3}$. The increased smoothness over the
piecewise-constant and piecewise-linear representation allows to reduce the
number of knots when interpolating raw market quotes, thus providing an
interesting alternative to regularization while reducing the computational
cost.

###### keywords:

volatility surface; interpolation; risk-neutral density; arbitrage;
quantitative finance; European options

xx 1 1 The Quadratic Local Variance Gamma Model: an arbitrage-free
interpolation of class $\mathcal{C}^{3}$ for option prices Fabien Le Floc’h
Fabien Le Floc’h

##  1 Introduction

The financial markets provide option prices for a discrete set of strike
prices and maturity dates. In order to price over-the-counter vanilla options
with different strikes, or to hedge complex derivatives with vanilla options,
it is useful to have a continuous arbitrage-free representation of the option
prices, or equivalently of their implied volatilities. For example, the
variance swap replication of Carr and Madan consists in integrating a specific
function over a continuum of vanilla put and call option prices (Carr and
Madan, 2001; Carr and Lee, 2008). More generally, Breeden and Litzenberger
(1978) have shown that any path-independent claim can be valued by integrating
over the probability density implied by market option prices. An arbitrage-
free representation is also particularly important for the Dupire local
volatility model (Dupire, 1994), where arbitrage will translate to a negative
local variance. A option price representation of class $\mathcal{C}^{2}$ is
also key to guarantee the second-order convergence of numerical schemes
applied to the Dupire partial differential equation, commonly used to price
exotic financial derivative contracts.

A rudimentary, but popular representation is to interpolate market implied
volatilities with a cubic spline across option strikes. Unfortunately this may
not be arbitrage-free as it does not preserve the convexity of option prices
in general. A typical convex interpolation of the call option prices by
quadratic splines or rational splines is also not satisfactory in general
since it may generate unrealistic oscillations in the corresponding implied
volatilities, as evidenced in (Jäckel, 2014). Kahalé (2004) designs an
arbitrage-free interpolation of the call option prices, which however requires
convex input quotes, employs two embedded non-linear minimizations, and it is
not proven that the algorithm for the interpolation function of class
$\mathcal{C}^{2}$ converges.

More recently, Andreasen and Huge (2011) have proposed to calibrate the
discrete piecewise constant local volatility corresponding to a single-step
finite difference discretization of the forward Dupire equation. In their
representation of the local volatility, the authors use as many constants as
the number of market option strikes for an optimal fit. It is thus sometimes
considered to be "non-parametric". Their technique works well in general but
requires some care around the choice of discretization grid: it must be
sufficiently dense so that two market strikes do not fall in between the same
consecutive grid nodes, and sufficiently wide to properly model the boundary
behaviour. Those two requirements complicate, and slow down the non-linear
optimization involved in the technique. Furthermore the output is a discrete
set of option prices, which, while relatively dense, must still be
interpolated carefully to obtain the price of options whose strike falls in
between grid nodes.

Le Floc’h and Oosterlee (2019b) derived a specific B-spline collocation to fit
the market option prices, while ensuring the arbitrage-free property at the
same time. While the fit is quite good in general, it may not be applicable to
interpolate the original quotes with high accuracy. For example, input quotes
may already be smoothed out if they stem from a prior model, or from a market
data broker, or from another system in the bank. In those cases, it is
desirable to use a nearly exact interpolation.

Le Floc’h (2021) extends the local variance gamma model of Carr and Nadtochiy
(2017), which relies on a piecewise-constant representation of the local
variance function, to a piecewise-linear Bachelier representation. This paper
generalizes the model to a piecewise-quadratic function. It encompasses the
piecewise-linear Bachelier and piecewise-linear Black representations. The
full piecewise-quadratic model results in an arbitrage-free interpolation of
class $\mathcal{C}^{3}$ for the option prices. The smoother implied
probability density allows for the use of a sparser set of interpolation
knots, thus providing an alternative to regularization in order to avoid
overfitting. In addition, a sparser set of knots reduces the computational
cost of the technique.

##  2 Dupire’s PDDE in the local variance gamma model

We recall Dupire’s partial difference differential equation (PDDE) for a call
option price $C(T,x)$ of strike $x$ and maturity $T$ (Carr and Nadtochiy,
2017):

$\frac{C(T,x)-\max(X(0)-x,0)}{T}=\frac{1}{2}a^{2}(x)\frac{\partial^{2}C(T,x)}{\partial
x^{2}}\,,$ (1)

for a Martingale asset price process $X(t)$ of expectation
$\mathbb{E_{\mathbb{Q}}}[X(t)]=X(0)$.

Let $\\{x_{0},x_{1},...,x_{m},x_{m+1}\\}$ be a increasing set of the strike
prices, such that $x_{0}=L$, $x_{m+1}=U$ with the interval $(L,U)$ being the
spatial interval where the asset $X$ lives. Furthermore, we require the
following to hold

$\exists s\in[1,m]|x_{s}=X(0)\,.$

The $(x_{1},...,x_{m})$ may correspond to the strike prices of the options of
maturity $T$ we want to calibrate against, along with the forward price as in
(Carr and Nadtochiy, 2017; Le Floc’h, 2021). This choice allows for a nearly
exact fit. It may also be some specific discretization of size $m$ with $m$
lower or equal to the number of market strike prices.

We consider $a$ to be a piecewise-quadratic function of class
$\mathcal{C}^{0}$ on $[x_{0},x_{m}]$.

Let $V$ be the function defined by $V(x)=C(x,T)-\max(X(0)-x,0)$. $V$ is
effectively the price of an out-of-the-money option (the price of a call
option for $x>X(0)$ and of a put option for $x<X(0)$). The Dupire PDDE leads
to

$V(x)=\frac{1}{2}a^{2}(x)T\left[V^{\prime\prime}(x)+\delta(x=X(0))\right]\,,$
(2)

on the interval $(L,U)$, where $\delta$ is the Dirac delta function. Instead
of solving Equation 2 directly, we look for a solution $V$ on the two
intervals $(L,X(0))$ and $(X(0),U)$ separately. On each interval, we have

$V(x)=\frac{1}{2}a^{2}(x)TV^{\prime\prime}(x)\,,$ (3)

Then, the continuity of $\frac{\partial C}{\partial x}$ at $x=X(0)$ implies

$\lim\limits_{x\to X(0)-}V^{\prime}(x)=1+\lim\limits_{x\to
X(0)^{+}}V^{\prime}(x)\,.$ (4)

In order to define a unique $V$, we also impose the absorbing boundary
conditions

$V(L)=0=V(U)\,.$ (5)

The continuity of the second derivative of $V$ at $x=X(0)$ follows from the
continuity of $a(x)$ at $x=X(0)$. We may further impose a $\mathcal{C}^{3}$
continuity relation at $x=x_{s}$:

$\lim\limits_{x\to
X(0)-}\left(\frac{V}{a^{2}}\right)^{\prime}(x)=\lim\limits_{x\to
X(0)+}\left(\frac{V}{a^{2}}\right)^{\prime}(x)\,.$ (6)

##  3 Explicit solution

Let $a(x)=\alpha_{i}x^{2}+\beta_{i}x+\gamma_{i}$ on $[x_{i},x_{i+1})$ with
$(\alpha_{i},\beta_{i},\gamma_{i})\in\mathbb{R}^{3}$. Being a quadratic, $a$
may also be expressed as
$a(x)=\alpha_{i}(x-\tilde{x}_{i,1})(x-\tilde{x}_{i,2})$ with

$\displaystyle\tilde{x}_{i,1}=\frac{-\beta_{i}+\sqrt{\delta_{i}}}{2\alpha_{i}}\,,$
$\displaystyle\quad\tilde{x}_{i,2}=\frac{-\beta_{i}-\sqrt{\delta_{i}}}{2\alpha_{i}}\,,\quad\textmd{
with }\delta_{i}=\beta_{i}^{2}-4\alpha_{i}\gamma_{i}\,,\quad\textmd{ for
}\alpha\neq 0\,.$

In particular, $\tilde{x}_{i,1}$ and $\tilde{x}_{i,2}$ may be complex numbers.
When $\alpha_{i}=0$ and $\beta_{i}\neq 0$, we may define
$\delta_{i}=\beta_{i}^{2}$ and we have $a(x)=\alpha_{i}(x-\tilde{x}_{i,1})$
with $\tilde{x}_{i,1}=-\gamma_{i}/\beta_{i}$.

The solutions of Equation 3 on $[x_{i},x_{i+1})$ read

$\displaystyle V(x)$
$\displaystyle=\frac{\chi_{i}(x)}{\chi_{i}(x_{i})}\left[\Theta_{i}^{c}\cosh\left(\omega_{i}\left(z_{i}(x)-z_{i}(x_{i})\right)\right)+\Theta_{i}^{s}\sinh\left(\omega_{i}\left(z_{i}(x)-z_{i}(x_{i})\right)\right)\right]\,,$
(7)

with111See Appendix A on how to avoid the use of complex numbers.

$\displaystyle z_{i}(x)$
$\displaystyle=\ln\left(\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}\right)\,,$
$\displaystyle\quad\omega_{i}$
$\displaystyle=\frac{1}{2}\sqrt{1+\frac{8}{\delta_{i}T}}\,,$
$\displaystyle\quad\chi_{i}$
$\displaystyle=\sqrt{\left(x-\tilde{x}_{i,1}\right)\left(x-\tilde{x}_{i,2}\right)}\,,$
$\displaystyle\quad\textmd{ for }\alpha_{i}\neq 0\,,$ $\displaystyle z_{i}(x)$
$\displaystyle=\ln\left|x-\tilde{x}_{i,1}\right|\,,$
$\displaystyle\quad\omega_{i}$
$\displaystyle=\frac{1}{2}\sqrt{1+\frac{8}{\delta_{i}T}}\,,$
$\displaystyle\quad\chi_{i}$
$\displaystyle=\sqrt{\left|x-\tilde{x}_{i,1}\right|}\,,$
$\displaystyle\quad\textmd{for }\alpha_{i}=0,\textmd{ and }\beta_{i}\neq 0\,,$
$\displaystyle z_{i}(x)$ $\displaystyle=x\,,$ $\displaystyle\quad\omega_{i}$
$\displaystyle=\frac{1}{\gamma_{i}}\sqrt{\frac{2}{T}}\,,$
$\displaystyle\quad\chi_{i}$ $\displaystyle=1\,,$
$\displaystyle\quad\textmd{for }\alpha_{i}=0,\textmd{ and }\beta_{i}=0\,.$

where $(\Theta_{i}^{c},\Theta_{i}^{s})\in\mathbb{C}^{2}$. The normalization
makes $V(x_{i})=\Theta_{i}^{c}$.

The derivative of $V$ reads

$\displaystyle V^{\prime}(x)$
$\displaystyle=\frac{\chi_{i}(x)}{\chi_{i}(x_{i})}z_{i}^{\prime}(x)\left[(\kappa_{i}\Theta_{i}^{c}+\omega_{i}\Theta_{i}^{s})\cosh\left(\omega_{i}(z_{i}(x)-z_{i}(x_{i}))\right)+(\kappa_{i}\Theta_{i}^{s}+\omega_{i}\Theta_{i}^{c})\sinh\left(\omega_{i}(z_{i}(x)-z_{i}(x_{i}))\right)\right]\,,$
(8)

with

$\displaystyle z_{i}^{\prime}(x)$
$\displaystyle=\frac{1}{x-\tilde{x}_{i,1}}-\frac{1}{x-\tilde{x}_{i,2}}\,,$
$\displaystyle\quad\kappa_{i}$
$\displaystyle=\frac{1}{2z_{i}^{\prime}(x)}\left(\frac{1}{x-\tilde{x}_{i,1}}+\frac{1}{x-\tilde{x}_{i,2}}\right)\,,$
$\displaystyle\quad\textmd{ for }\alpha_{i}\neq 0\,,$ $\displaystyle
z_{i}^{\prime}(x)$ $\displaystyle=\frac{1}{x-\tilde{x}_{i,1}}\,,$
$\displaystyle\quad\kappa_{i}$ $\displaystyle=\frac{1}{2}\,,$
$\displaystyle\quad\textmd{ for }\alpha_{i}=0\textmd{ and }\beta_{i}\neq 0\,,$
$\displaystyle z_{i}^{\prime}(x)$ $\displaystyle=1\,,$
$\displaystyle\quad\kappa_{i}$ $\displaystyle=0\,,$
$\displaystyle\quad\textmd{ for }\alpha_{i}=0\textmd{ and }\beta_{i}=0\,.$

The conditions to impose continuity of $V$ and its derivative at $x=x_{i+1}$
results in the following linear system

$\displaystyle\cosh_{i}\Theta_{i}^{c}+\sinh_{i}\Theta_{i}^{s}$
$\displaystyle=\frac{\Theta_{i+1}^{c}}{\chi_{i}(x_{i+1})}\,,$ (9)
$\displaystyle(\kappa_{i}\cosh_{i}+\omega_{i}\sinh_{i})\Theta_{i}^{c}+(\omega_{i}\cosh_{i}+\kappa_{i}\sinh_{i})\Theta_{i}^{s}$
$\displaystyle=\frac{\left(\kappa_{i+1}\Theta_{i+1}^{c}+\omega_{i+1}\Theta_{i+1}^{s}\right)z_{i+1}^{\prime}(x_{i+1})}{\chi_{i}(x_{i+1})z_{i}^{\prime}(x_{i+1})}$
(10)

for $i=0,...,s-2$, with

$\displaystyle\cosh_{i}=\cosh\left(\omega_{i}(z_{i}(x_{i+1})-z_{i}(x_{i}))\right)\,,$
$\displaystyle\quad\sinh_{i}=\sinh\left(\omega_{i}(z_{i}(x_{i+1})-z_{i}(x_{i}))\right)\,.$

The boundary condition at $x=x_{0}=L$ translates to $\Theta_{0}^{c}=0$. At
$x=x_{m+1}=U$, the boundary condition translates to
$\Theta_{m}^{c}=-\Theta_{m}^{s}\frac{\sinh_{m}}{\cosh_{m}}$. The jump
condition at $x=s$ reads

$\displaystyle V_{s-1}(x_{s})$ $\displaystyle=V_{s}(x_{s})\,,$ $\displaystyle
V_{s-1}^{\prime}(x_{s})$ $\displaystyle=1+V_{s}^{\prime}(x_{s})\,,$

with

$\displaystyle V_{s-1}(x_{s})$
$\displaystyle=\chi_{s-1}(x_{s})(\Theta_{s-1}^{c}\cosh_{s-1}+\Theta_{s-1}^{s}\sinh_{s-1})\,,$
$\displaystyle V_{s}(x_{s})$ $\displaystyle=\Theta_{s}^{c}\,,$ $\displaystyle
V_{s-1}^{\prime}(x_{s})$
$\displaystyle=\chi_{s-1}(x_{s})z_{s-1}^{\prime}(x_{s})\left[(\kappa_{s-1}\Theta_{s-1}^{c}+\omega_{s-1}\Theta_{s-1}^{s})\cosh_{s-1}+(\omega_{s-1}\Theta_{s-1}^{c}+\kappa_{s-1}\Theta_{s-1}^{s})\sinh_{s-1}\right]\,,$
$\displaystyle V_{s}^{\prime}(x_{s})$
$\displaystyle=(\kappa_{s}\Theta_{s}^{c}+\omega_{s}\Theta_{s}^{s})z_{s}^{\prime}(x_{s})\,.$

From the above equations, we deduce that the coefficients
$\Theta_{i}^{c},\Theta_{i}^{s}$ are solutions of the following tridiagonal
system

$\begin{pmatrix}B_{0}&C_{0}&&0\\\ A_{1}&\ddots&\ddots\\\
&\ddots&\ddots&C_{2m}\\\
0&&A_{2m+1}&B_{2m+1}\end{pmatrix}\begin{pmatrix}\Theta_{0}^{s} \\\
\Theta_{0}^{c}\\\ \vdots\\\ \Theta_{m}^{s}\\\
\Theta_{m}^{c}\end{pmatrix}=\begin{pmatrix}D_{0} \\\ \vdots\\\
D_{2m+1}\end{pmatrix}\,,$ (11)

with $D_{i}=0$ for $i\notin\left\\{2s-1,2s\right\\}$, $D_{2s-1}=D_{2s}=1$,

$\displaystyle\begin{cases}A_{2i+1}&=(\omega_{i}\cosh_{i}+\kappa_{i}\sinh_{i})\chi_{i}z_{i}^{\prime}(x_{i+1})-\kappa_{i+1}\sinh_{i}\chi_{i}z_{i+1}^{\prime}(x_{i+1})\,,\\\
B_{2i+1}&=(\kappa_{i}\cosh_{i}+\omega_{i}\sinh_{i})\chi_{i}z_{i}^{\prime}(x_{i+1})-\kappa_{i+1}\cosh_{i}\chi_{i}z_{i+1}^{\prime}(x_{i+1})\,,\\\
C_{2i+1}&=-\omega_{i+1}z_{i+1}^{\prime}(x_{i+1})\,,\\\
A_{2i+2}&=\left(\kappa_{i}\cosh_{i}+\omega_{i}\sinh_{i}-\frac{\omega_{i}\cosh_{i}+\kappa_{i}\sinh_{i}}{\sinh_{i}}\cosh_{i}\right)\chi_{i}z_{i}^{\prime}(x_{i+1})\,,\\\
B_{2i+2}&=-\omega_{i+1}z_{i+1}^{\prime}(x_{i+1})\,,\\\
C_{2i+2}&=\frac{\omega_{i}\cosh_{i}+\kappa_{i}\sinh_{i}}{\sinh_{i}}z_{i}^{\prime}(x_{i+1})-\kappa_{i+2}z_{i+1}^{\prime}(x_{i+1})\,,\end{cases}$

for $i=0,...,m-1$, and
$B_{0}=0,C_{0}=1,A_{2m+1}=\sinh_{m},B_{2m+1}=\cosh_{m}$.

Using the continuity of $V(x_{s})=\Theta^{c}_{s}$, the jump condition of
$V^{\prime}$ at $x_{s}$ and the continuity of $a(x_{s})$, the
$\mathcal{C}^{3}$ condition (Equation 6) reads

$\displaystyle 1-2\Theta^{c}_{s}\frac{\lim\limits_{x\to
x_{s}-}a^{\prime}(x)}{a(x_{s})}=-2\Theta^{c}_{s}\frac{\lim\limits_{x\to
x_{s}+}a^{\prime}(x)}{a(x_{s})}\,.$ (12)

Equation 12 implies that $a^{\prime}$ is not continuous at $x_{s}$, unless
$a(x_{s})=0$. The condition can not be imposed as an additional constraint on
$\Theta_{s}^{c}$ since its value is already fully determined by the
tridiagonal system. It may however be imposed by choosing the correct model
parameter to adjust the value of $a$ at $x_{s}$ along with its left and right
derivative values.

##  4 Parameterizations

###  4.1 Linear Bachelier

The linear Bachelier local variance consists in $\alpha_{i}=0$ and may be
rewritten using values at the knots $\sigma_{i}$ as

$\displaystyle a(x)$
$\displaystyle=\frac{x-x_{i}}{x_{i+1}-x_{i}}(\sigma_{i+1}-\sigma_{i})+\sigma_{i}\quad\textmd{
for }x_{i}\leq x<x_{i+1}\,,\quad i=0,...,m\,,$ (13)

where the parameters $\sigma_{i}>0$.

It corresponds to the parameterization studied in (Le Floc’h, 2021), where it
is shown that the local variance function must not be $\mathcal{C}^{1}$ at
$x=x_{s}$ but must follow $\mathcal{C}^{3}$ condition (Equation 12) in order
to avoid a spurious spike at $x=x_{s}$. Under the linear Bachelier local
variance, the condition reads

$1-2\Theta_{s}^{c}\frac{\sigma_{s}-\sigma_{s-1}}{(x_{s}-x_{s-1})\sigma_{s}}=-2\Theta_{s}^{c}\frac{\sigma_{s+1}-\sigma_{s}}{(x_{s+1}-x_{s})\sigma_{s}}\,,$

or equivalently

$\sigma_{s}=\frac{2\Theta_{s}^{c}\left(\frac{\sigma_{s-1}}{x_{s}-x_{s-1}}+\frac{\sigma_{s+1}}{x_{s+1}-x_{s}}\right)}{2\Theta_{s}^{c}\left(\frac{1}{x_{s}-x_{s-1}}+\frac{1}{x_{s+1}-x_{s}}\right)-1}\,.$
(14)

This is not a linear problem, as $\Theta_{s}^{c}$ depends on $\sigma_{s}$
through $\Theta_{s+1}^{c},\Theta_{s+1}^{s}$ in a non-linear way (Equations 9
and 10). Starting with the algorithm described in Section 3 to compute
$\Theta^{c},\Theta^{s}$, using Equation 14 with $\Theta_{s}^{c}\approx
V_{\textsf{market}}(x_{s})$ as initial guess for $\sigma_{s}$, we may however
apply the following iteration

  * •

Update $\sigma_{s}$ through Equation 14.

  * •

Recalculate $\Theta_{i}^{c}$ and $\Theta_{i}^{s}$ for $i=0,...,m$ by solving
the updated tridiagonal system.

Three iterations are enough in practice.

###  4.2 Linear Black

The linear Black local variance model is defined by $\gamma=0$. The local
variance function may be rewritten using values at the knots $\sigma_{i}$ as

$\displaystyle a(x)$
$\displaystyle=\left(\frac{x-x_{i}}{x_{i+1}-x_{i}}(\sigma_{i+1}-\sigma_{i})+\sigma_{i}\right)x\quad\textmd{
for }x_{i}\leq x<x_{i+1}\,,\quad i=0,...,m\,,$ (15)

where the parameters $\sigma_{i}>0$. Interestingly, the $\mathcal{C}^{3}$
condition (Equation 12) is also given by Equation 14.

###  4.3 Positive quadratic B-spline

A B-spline parameterization with positive coefficients implies $a$ positive.
Furthermore, Equation 12 imposes a double knot at $x=x_{s}$ (because the
derivative of $a$ is not continuous there). We thus consider

$a(x)=\sum_{i=1}^{m+3}\lambda_{i}B_{i,3}(x)$ (16)

where $\lambda_{i}>0$ and $B_{i,3}$ is the quadratic basis spline with knots
$\bm{t}=(L,L,L,x_{1},x_{2},...,X(0),X(0),...,x_{m},U,U,U)$. In particular, we
have $t_{s+2}=t_{s+3}=X(0)$. Using the B-spline derivative identity (De Boor,
1978) and the fact that the order of the B-spline is 3, we obtain

$a^{\prime}(x)=2\sum_{i}\frac{\lambda_{i}-\lambda_{i-1}}{t_{i+2}-t_{i}}B_{i-1,2}(x)\,,$

and the $\mathcal{C}^{3}$ condition reads

$\displaystyle\sum_{i}\lambda_{i}B_{i,3}(x_{s})=4\Theta^{c}_{s}\left(\frac{\lambda_{s+1}-\lambda_{s}}{t_{s+3}-t_{s+1}}B_{s,2}(x_{s}^{-})-\frac{\lambda_{s+2}-\lambda_{s+1}}{t_{s+4}-t_{s+2}}B_{s+1,2}(x_{s}^{+})\right)\,.$

Using the definitions of $B_{i,3}$ and $B_{i,2}$ we obtain

$\lambda_{s+1}=4\Theta^{c}_{s}\left(\frac{\lambda_{s+1}-\lambda_{s}}{t_{s+3}-t_{s+1}}-\frac{\lambda_{s+2}-\lambda_{s+1}}{t_{s+4}-t_{s+2}}\right)\,,$

or equivalently

$\lambda_{s+1}=\frac{4\Theta^{c}_{s}\left(\frac{\lambda_{s}}{t_{s+3}-t_{s+1}}+\frac{\lambda_{s+2}}{t_{s+4}-t_{s+2}}\right)}{4\Theta^{c}_{s}\left(\frac{1}{t_{s+3}-t_{s+1}}+\frac{1}{t_{s+4}-t_{s+2}}\right)-1}\,.$
(17)

As an illustration, we consider the same example as in (Le Floc’h, 2021): we
fit the quadratic LVG model to 10 option prices of strikes (0.85, 0.90, 0.95,
1, 1.05, 1.1, 1.15, 1.2, 1.3, 1.4), obtained by the Black-Scholes model with
constant volatility $\sigma_{B}=20\%$, time to maturity $T=0.25$ and forward
price $1.025$. We know that the theoretical distribution is a lognormal
distribution. A straightforward $\mathcal{C}^{1}$ quadratic B-spline leads to
a large spike in the probability density implied from the calibrated LVG model
(Figure 1). Adding the $\mathcal{C}^{3}$ condition through an additional
B-spline knot recovers a smooth implied probability density.

Figure 1: Implied probability density for the quadratic LVG model with or
without the $\mathcal{C}^{3}$ condition (Equation 6), fitted to a Black-
Scholes model with constant volatility $\sigma_{B}=20\%$, time to expiry
$T=0.25$ and forward $1.025$.

##  5 Calibration

###  5.1 Error measure

The calibration of a single maturity consists in finding the parameters (the
$\bm{\alpha}$ for the linear models, or $\bm{\lambda}$ for the quadratic
B-spline) such that the function $C(x,T)$, solution of the Dupire PDDE fits
the market option prices $(\hat{C}_{i})_{i=1,...n}$ of respective strikes
$(K_{i})_{i=1,...,n}$ according to an appropriate measure. A common practice
is to perform a least-squares minimization of the error measure $E$ defined by

$E=\sum_{i=1}^{m}\mu_{i}^{2}\left(\sigma(\alpha,x_{i})-\hat{\sigma}_{i}\right)^{2}\,,$
(18)

with $\alpha_{i}>0$ for $i=1,...,m$ and where $\sigma(\alpha,x)$ is the
implied volatility corresponding to the option prices obtained with the
piecewise-linear local gamma variance model and $\hat{\sigma}_{i}$ is the
market implied volatility at strike $x_{i}$, $(\mu_{i})_{i=1,...,m}$ are
weights associated to the accuracy of the fit at each point.

In order to solve this non-linear least-squares problem, we will use the
Levenberg-Marquardt algorithm as implemented by Klare and Miller (2013). The
box constraints $\alpha_{i}>0$ can be added in a relatively straightforward
manner to any Levenberg-Marquardt algorithm, through the projection technique
described in (Kanzow et al., 2004), or through a variable transform from
$\mathbb{R}$ to a subset of $\mathbb{R^{+}}$ (for example through the function
$x\to x^{2}+\epsilon$ with some small positive $\epsilon$).

The implied volatility for a given option price may be found efficiently and
accurately through the algorithm of Jäckel (2015). In general, we prefer to
solve an almost equivalent formulation in terms of option prices, using the
error measure $E_{V}$ defined by

$E_{V}=\sum_{i=1}^{m}w_{i}^{2}\left(C(\alpha,x_{i})-\hat{C}_{i}\right)^{2}\,,$
(19)

with $C(\alpha,x)$ being the local variance gamma option price with parameter
$\alpha$ and strike $x$, and the capped inverse Vega weights $w_{i}$ given by

$w_{i}=\min\left(\frac{1}{\nu_{i}},\frac{10^{6}}{X(0)}\right)\mu_{i}\,,$ (20)

where $\nu_{i}=\frac{\partial\hat{C}_{i}}{\partial\sigma}$ is the Black-
Scholes Vega corresponding the market option price $\hat{C}_{i}$, and $10^{6}$
is a cap applied to avoid numerical issues related to the limited machine
accuracy (see Le Floc’h and Oosterlee (2019b); Le Floc’h (2021) for the
justification).

###  5.2 Exact interpolation

Sometimes, it is desirable to interpolate a given set of reference prices
nearly exactly. This is typically the case when the reference prices come from
some prior model. We apply the same least-square minimization but choose the
number of free parameters to be equal to the number of reference prices.

####  5.2.1 Linear models

For the linear models, this means to use $m=n$ and set $\alpha_{0}=\alpha_{1}$
and $\alpha_{m+1}=\alpha_{m}$ to model a flat extrapolation. In general, the
market strikes will not include $X(0)$. In this case, $X(0)$ must be added to
the knots $\\{x_{i}\\}_{i=1,...,m}$ used in the local variance gamma
representation. This adds one more parameter $\alpha_{s}$ to the
representation, where $s$ is the index corresponding to $X(0)$ in the set of
knots. The value of $\alpha_{s}$ is not free, it is given by the
$\mathcal{C}^{3}$ condition (Equation 6) and enforced through the iterative
procedure described in the previous sections.

####  5.2.2 B-spline knots locations

For the linear Bachelier and Black parameterization, choosing the knots at the
market strikes works well. The situation is more complex for the quadratic
B-spline parameterization.

Let $(K_{i})_{i=1,...,n}$ be the market options strikes. Let the index $i_{F}$
be such that $K_{i_{F}}\leq F<K_{i_{F}+1}$. We may:

  * •

place the knots at the market strikes (labeled "Strikes" in the figures)

$\displaystyle\bm{t}=\begin{cases}\left(L,L,L,K_{1},...,K_{i_{F}},F,F,K_{i_{F}+1},...,K_{n},U,U,U\right)&\textmd{
if }K_{i_{F}}\neq F\\\
\left(L,L,L,K_{1},...,K_{i_{F}},F,K_{i_{F}+1},...,K_{n},U,U,U\right)&\textmd{
if }K_{i_{F}}=F\end{cases}\,,$

The dimension of $\bm{\lambda}$ is then $n_{\lambda}=n+5$ if $F\neq K_{i_{F}}$
and $n+4$ if $F=K_{i_{F}}$. The change in the number of dimensions suggests
that the interpolation may change significantly when the forward price moves
across a market strike.

  * •

place the knots in the middle of market strikes. According to (De Boor, 1978,
p. 61), the $\mathcal{C}^{1}$ quadratic spline is then solution to a
diagonally dominant tridiagonal system, which increases the stability and
reduce oscillations of the interpolation. There are however several ways to do
it:

    * –

choose the direct mid-points (labeled "Mid-Strikes")

$\displaystyle\bm{t}=\begin{cases}\left(L,L,L,\frac{K_{1}+K_{2}}{2},...,\frac{K_{i_{F}-1}+K_{i_{F}}}{2},F,F,\frac{K_{i_{F}}+K_{i_{F}+1}}{2},...,\frac{K_{n-1}+K_{n}}{2},U,U,U\right)&\textmd{
if }F<\frac{K_{i_{F}}+K_{i_{F}+1}}{2}\\\
\left(L,L,L,\frac{K_{1}+K_{2}}{2},...,\frac{K_{i_{F}-1}+K_{i_{F}}}{2},\frac{K_{i_{F}}+K_{i_{F}+1}}{2},F,F,...,\frac{K_{n-1}+K_{n}}{2},U,U,U\right)&\textmd{
if }F>\frac{K_{i_{F}}+K_{i_{F}+1}}{2}\\\
\left(L,L,L,\frac{K_{1}+K_{2}}{2},...,\frac{K_{i_{F}-1}+K_{i_{F}}}{2},F,F,\frac{K_{i_{F}+1}+K_{i_{F}+2}}{2},...,\frac{K_{n-1}+K_{n}}{2},U,U,U\right)&\textmd{
if }F=\frac{K_{i_{F}}+K_{i_{F}+1}}{2}\end{cases}\,,$

The dimension of $\bm{\lambda}$ is then $n_{\lambda}=n+4$ if $F\neq K_{i_{F}}$
and $n+3$ if $F=K_{i_{F}}$.

    * –

choose the mid-points, excluding the point closest to the forward price $F$
(labeled "Mid-X")

$\bm{t}=\left(L,L,L,\frac{K_{1}+K_{2}}{2},...,\frac{K_{i_{F}-1}+K_{i_{F}}}{2},F,F,\frac{K_{i_{F}+1}+K_{i_{F}+2}}{2},...,\frac{K_{n-1}+K_{n}}{2},U,U,U\right)\,,$

The dimension of $\bm{\lambda}$ is then $n_{\lambda}=n+3$.

    * –

choose the mid-points, excluding the point closest to the forward price and
placing the first and last strike in the middle of two knots (labeled "Mid-
XX")

$\bm{t}=\left(L,L,L,\frac{3K_{1}-K_{2}}{2},\frac{K_{1}+K_{2}}{2},...,\frac{K_{i_{F}-1}+K_{i_{F}}}{2},F,F,\frac{K_{i_{F}+1}+K_{i_{F}+2}}{2},...,\frac{K_{n-1}+K_{n}}{2},\frac{3K_{n}-K_{n-1}}{2},U,U,U\right)\,,$

The dimension of $\bm{\lambda}$ is then $n_{\lambda}=n+5$.

  * •

use a uniform discretization of $[K_{1},K_{n}]$ composed of $n+1$ points, and
shift it such that the forward is exactly part of the knots and we have
$n_{\lambda}=n+5$.

In each of those case, we make sure to add the forward price as a double knot,
as well as the boundaries $L,U$. The dimension of $\bm{\lambda}$ implied by
the knots is larger than the number of market strikes. We choose the extra
parameters as such:

  * •

if $n_{\lambda}=n+5$, we set $\lambda_{1}=\lambda_{2}=\lambda_{3}$,
$\lambda_{n+3}=\lambda_{n+4}=\lambda_{n+5}$ and $\lambda_{i_{F}+3}$ is
obtained from $\lambda_{i_{F}+2}$ and $\lambda_{i_{F}+4}$.

  * •

if $n_{\lambda}=n+4$, we set $\lambda_{1}=\lambda_{2}=\lambda_{3}$,
$\lambda_{n+3}=\lambda_{n+4}$ and $\lambda_{i_{F}+3}$ is obtained from
$\lambda_{i_{F}+2}$ and $\lambda_{i_{F}+4}$.

  * •

if $n_{\lambda}=n+3$, we set $\lambda_{1}=\lambda_{2}$,
$\lambda_{n+2}=\lambda_{n+3}$ and $\lambda_{i_{F}+2}$ is obtained from
$\lambda_{i_{F}+1}$ and $\lambda_{i_{F}+3}$.

In order to assess the various knots candidates, we consider the same example
as in the previous section, but using a few different random sets of 10
strikes in the interval [85,140] and a forward price $F=101$ (Table 1).

Table 1: Sets of market strikes with a Black-Scholes volatility of 20% for a maturity $T=0.25$ and forward $F=101$.  Set | $K_{1}$ | $K_{2}$ | $K_{3}$ | $K_{4}$ | $K_{5}$ | $K_{6}$ | $K_{7}$ | $K_{8}$ | $K_{9}$ | $K_{10}$
---|---|---|---|---|---|---|---|---|---|---
A | 88.77 | 92.85 | 93.38 | 99.37 | 107.99 | 120.29 | 122.03 | 123.9 | 134.71 | 135.43
B | 85.02 | 101.92 | 103.55 | 114.45 | 121.85 | 123.69 | 125.07 | 125.58 | 131.63 | 133.86
C | 98.07 | 100.93 | 101.06 | 106.88 | 109.12 | 110.93 | 119.76 | 119.83 | 132.19 | 138.27
D | 85.00 | 90.00 | 95.00 | 100.00 | 101.00 | 105.00 | 110.00 | 115.00 | 120.00 | 130.00
Table 2: Root mean square error in implied volatilities % for various choices of B-spline knots. The reference volatilities are flat 20%. Set | Strikes | Mid-Strikes | Mid-X | Mid-XX | Uniform
---|---|---|---|---|---
A | 9.4e-8 | 6.0e-3 | 5.2e-3 | 4.1e-8 | 4.8e-9
B | 9.9e-9 | 2.8e-3 | 5.8e-1 | 2.9e-6 | 9.9e-3
C | 1.0e-6 | 1.9e-3 | 1.0e-2 | 1.1e-8 | 1.4e-3
D | 4.1e-4 | 8.1e-2 | 4.1e-2 | 2.6e-5 | 5.0e-7

The uniform discretization may222In practice, market strikes are not randomly
distributed, but according to multiples of a minimum strike width, with more
strikes near the money. The uniform discretization may still be relevant if
some regularization is added to the objective of the minimizer. result in
strong oscillations due to overfitting in places where no market strike is
quoted as in the set A (Figure 2(a)).

The "Mid-Strikes" choice leads to a strong oscillation around the forward in
set B (Figure 2(b)). The "Mid-X" produces a somewhat awkward shape on the set
B. When the forward is very close to some of the knots as in set C, the
"Strikes", "Mid-Strikes" choices lead to a density with a sharp gradient near
the forward, a feature not desirable (Figure 2(c)). When the forward is part
of the market strikes, a small wiggle is visible at the forward for "Strikes"
and "Mid-Strikes" (Figure 2(d)).

(a) Set A. (b) Set B.

(c) Set C. (d) Set D.

Figure 2: Implied probability density of the calibrated quadratic LVG model
using different sets of knots.

Finally it is also interesting to look at the overall root mean square error
in implied volatilities for the different choices (Table 2). The "Strikes" and
"Mid-XX" choices consistently result in a near-perfect333The error in
volatility is always below one basis point. fit.

Overall, the "Mid-XX" knots lead to the most stable probability density along
with an excellent fit.

####  5.2.3 Many quotes, few parameters

In (Le Floc’h, 2021), regularization is employed to ensure a smooth implied
probability density when fitting to many, eventually noisy, market option
quotes. An interesting simpler alternative is to use few knots/few parameters
instead of as many parameters as market quotes: by limiting the number of
free-parameters, we may avoid overfitting issues, and at the same time we
reduce the number of dimensions of the problem, thus increasing stability and
performance. Where to place the knots then? Based on the previous
observations, we may choose knots such that the market strikes are
equidistributed. Concretely, we use

$\tilde{\bm{K}}=\left\\{K_{1},K_{1+j},K_{1+2j},...,K_{n}\right\\}\,,$

where $j=n/m$ with $m\leq n$ and use the "Mid-XX" knots on top of
$\tilde{\bm{K}}$. It may happen that many market strikes are quoted in a
narrow range, in which case the set could be adjusted with a minimum strike
width, although we did not need this tweak on the market examples presented
below.

We consider several different underlying assets: SPX500 expiring on March 24,
2017 as March 16, 2017 (one week - 1w) and on March 7, 2018 as of February 5,
2018 (one month - 1m), TSLA of maturity July 20, 2018 as of June 15, 2018
(1m), AAPL with expiry in 4 days as of October 28, 2013 (4d) from Alexiou et
al. (2021), as as well as the AUD/NZD currency pair of maturity July 9, 2014
as of July 2, 2014 (1w) from Wystup (2018). There is an apparent focus on
short maturities as those are more difficult to capture, with a variety of
smile shapes, with some exhibiting multi-modality. The latter are particularly
challenging for parametric models such as SVI (Gatheral, 2006), SABR (Hagan et
al., 2002), or even polynomial stochastic collocation (Le Floc’h and
Oosterlee, 2019a, b). As the implied volatility smile flattens for long
maturities, those are much easier to fit to.

(a) Implied volatility. (b) Implied probability density in log scale.

Figure 3: Quadratic LVG model with 10 points calibrated to SPX500 options of
maturity 1w.

(a) Implied volatility. (b) Implied probability density.

Figure 4: Quadratic LVG model with 10 points calibrated to SPX500 options of
maturity 1m.

(a) Implied volatility. (b) Implied probability density.

Figure 5: Quadratic LVG model with 10 points calibrated to TSLA options of
maturity 1m.

(a) Implied volatility. (b) Implied probability density.

Figure 6: Quadratic LVG model with 10 points calibrated to AAPL options of
maturity 4d.

The AUD/NZD foreign exchange smile from Wystup (2018) is useful to see how the
local variance gamma model behaves on a minimalistic example: indeed, as is
usual on the foreign exchange options market, it involves only five options
quotes. In this case we use as many parameters as market quotes. Figure 7(b)
shows that the density implied by the quadratic LVG model is smooth, but the
one implied by the linear Bachelier LVG model exhibits some sharp unnatural
gradients near the money.

(a) Implied volatility. (b) Implied probability density.

Figure 7: Quadratic LVG model with 5 points calibrated to AUD/NZD options of
maturity 1w.

Indeed, the linear LVG model leads only to a $\mathcal{C}^{0}$ probability
density. Such gradients are then inevitable when the number of quotes is
small. On this example, the (unconstrained) SVI model is known to lead to some
negative probability density.

In all the examples considered so far, the fit in terms of implied
volatilities is excellent, and the implied probability density is smooth,
without spurious peaks, although the number of points considered ($n=10$) is
somewhat arbitrary.

####  5.2.4 Challenging examples of exact interpolation

We consider the manufactured examples of Jäckel (2014) presented in Table 8.
In the first example, a cubic spline interpolation of option prices is known
to produce oscillations in the implied volatility, while a cubic spline on the
volatilities introduces spurious arbitrages. In the second example, some of
the quotes are at the limit of arbitrage.

On those examples, some care need to be taken in the choice of the boundaries
$L$ and $U$: they must be far away enough. We pick $L=K_{1}/2$ and $U=2K_{m}$
where $K_{1}$ is the smallest quoted strike and $K_{m}$ the largest.

On the example case I, the fit is nearly exact for the linear Bachelier, Black
and quadratic LVG models and there is no oscillation or wiggles in the implied
volatility interpolation (Figure 8(a)). The corresponding implied probability
density is of course smoothest with the quadratic LVG model (Figure 8(b)).

(a) Implied volatility. (b) Implied probability density in logarithmic scale.

Figure 8: LVG models calibrated to the example case I of Jäckel (2014).

On the example case II, the quadratic LVG model does not allow for an exact
fit. The root mean square error is around 4 basis points (Table 3), the fit is
qualitatively good (Figure 9(a)). The near-arbitrages impose the probability
density to go to almost zero, which conflicts with the $\mathcal{C}^{1}$
continuity constraints of parameterized B-spline density. The density stays
however smooth, and looks more natural than the clear overfit of the linear
LVG models (Figure 9(b)).

(a) Implied volatility. (b) Implied probability density in logarithmic scale.

Figure 9: LVG models calibrated to the example case II of Jäckel (2014).

Our choice of range for the $\bm{\alpha}$ parameter does not allow the linear
Bachelier model to fit as well as the linear Black model. Increasing the range
would make the two implied probability densities even more similar.

Table 3: Root mean square error (RMSE) in implied volatilities of the LVG models calibrated to the market data of Table 8. Model | Case I | Case II
---|---|---
Linear Bachelier | 5.00e-13 | 4.54e-6
Linear Black | 3.64e-12 | 8.04e-8
Quadratic | 2.25e-12 | 4.02e-4

##  6 Conclusion

The quadratic local variance gamma model with a small number of knots lead to
smooth implied probability densities on a variety of market option quotes,
while providing an excellent fit in terms of implied volatilities, even on
challenging examples of non convex implied volatilities or multi-modal
probability densities.

It thus constitutes an interesting alternative to the non-parametric
approaches of the linear or quadratic local variance gamma models or of the
Andreasen-Huge one-step local volatility model, which all require a non-
trivial choice of regularization constant to produce a smooth implied
probability density.

In similar fashion as Falck and Deryabin (2017), it may be used as specific
generic parameterization with a fixed, small number of parameters using only 3
or 5 points. In contrast to Falck and Deryabin (2017), where the linear
parameterization leads to edges in the implied probability density and where
the problem of the discontinuity at the forward is not dealt with, the
probability density of the quadratic LVG model with a reduced number of points
will be smooth.

Further research may explore the use of Fourier transforms to recover
efficient pricing formulae for European options under the local variance gamma
model with a more general local variance function.

yes

## References

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no

##  Appendix A Avoiding the use of complex numbers

We have $\chi_{i}=\sqrt{\frac{a(x)}{\alpha_{i}}}$ and we know that $a(x)\geq
0$ since all the B-spline coefficients are positive, thus $\chi_{i}$ is either
real or imaginary if the sign of $\alpha_{i}$ is positive (respectively
strictly negative). Since $V(x)$ involves the ratio
$\chi_{i}(x)/\chi_{i}(x_{i})$, we may equivalently calculate $\chi_{i}$ via
$\chi_{i}=\sqrt{\frac{a(x)}{|\alpha_{i}|}}$.

From its definition, we know that $\omega_{i}$ is real if $\delta_{i}T+8\geq
0$ and imaginary otherwise.

For $z_{i}$ we need to consider the two cases where the roots
$\tilde{x}_{i,1},\tilde{x}_{i,2}$ are both real or both complex.

If they are real, we have
$\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}=\frac{a(x)}{\alpha_{i}(x-\tilde{x}_{i,2})^{2}}$.
We know that $a(x)\geq 0$ since the B-spline coefficients are all positive. If
$\alpha_{i}>0$ then $z_{i}$ is real, otherwise
$z_{i}=\ln\left|\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}\right|+\mathrm{i}\mkern
1.0mu\pi$.

If the roots are complex with non null imaginary parts,
$\tilde{x}_{i,1},\tilde{x}_{i,2}$ are conjugates of each other. We may write
$\tilde{x}_{i,1}=u+\mathrm{i}\mkern 1.0muv$ and
$\tilde{x}_{i,2}=u-\mathrm{i}\mkern 1.0muv$ with
$(u,v)\in\mathbb{R}\times\mathbb{R}^{\star}$. We have then
$\left|\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}\right|=\left|\frac{x-u-\mathrm{i}\mkern
1.0muv}{x-u+\mathrm{i}\mkern 1.0muv}\right|=1$ and thus $z_{i}$ is imaginary.

The term $\cosh\left(\omega_{i}\left(z_{i}(x)-z_{i}(x_{i})\right)\right)$ is
thus always real. When $\omega_{i}\left(z_{i}(x)-z_{i}(x_{i})\right)$ is
imaginary, the term
$\sinh\left(\omega_{i}\left(z_{i}(x)-z_{i}(x_{i})\right)\right)$ is imaginary.

To evaluate the expression $V(x)$, there are four cases two consider:

  * •

$\delta_{i}\geq 0$ and $\delta_{i}T+8\geq 0$. All the variables are real.

  * •

$\delta_{i}\geq 0$ and $\delta_{i}T+8<0$. Take the absolute value in
$\omega_{i}$,
$\bar{\omega}_{i}=\frac{1}{2}\sqrt{\left|1+\frac{8}{\delta_{i}T}\right|}$.
Then use $\sin$ instead of $\sinh$. $\Theta_{i}^{s}$ will then be real instead
of complex.

  * •

$\delta_{i}<0$ and $\delta_{i}T+8\geq 0$. The logarithm in $z_{i}$ may be
calculated using
$\Im\left(z_{i}(x)\right)=\arcsin\left[\Im\left(\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}\right)\right]$.
The imaginary part may be directly calculated (without complex numbers). Then
use $\cos\left[\bar{\omega}_{i}\Im\left(z_{i}(x)\right)\right]$,
$\sin\left[-\bar{\omega}_{i}\Im\left(z_{i}(x)\right)\right]$ instead of
$\cosh$ and $\sinh$ terms. $\Theta_{i}^{s}$ will then be real instead of
complex.

  * •

$\delta_{i}<0$ and $\delta_{i}T+8<0$. The logarithm in $z_{i}$ may be
calculated using
$\Im\left(z_{i}(x)\right)=\arcsin\left[\Im\left(\frac{x-\tilde{x}_{i,1}}{x-\tilde{x}_{i,2}}\right)\right]$.
The imaginary part may be directly calculated (without complex numbers). Then
use $\cosh\left[\bar{\omega}_{i}\Im\left(z_{i}(x)\right)\right]$,
$\sinh\left[-\bar{\omega}_{i}\Im\left(z_{i}(x)\right)\right]$.

##  Appendix B Market data

Table 4: Implied volatilities SPX500 European options expiring on March 24, 2017 (one week, $T=0.021918$) assuming proportional dividends, with $r=0.9\%$ and implied forward $F(T)=2385.103981$. $y$ represents the log-moneyness $\ln\frac{K}{F}$ taken from Healy (2019). Strike | $y$ | Volume | Vol | Call Bid | Call Ask | Put Bid | Put Ask
---|---|---|---|---|---|---|---
1800 | -0.2815 | 0 | 0.58 | 0.00 | 1.24 | 0.00 | 0.61
1850 | -0.2541 | 3300 | 0.53 | 0.00 | 1.14 | 0.00 | 0.56
1875 | -0.2406 | 3402 | 0.50 | 0.00 | 1.09 | 0.00 | 0.53
1880 | -0.2380 | 1551 | 0.54 | 0.00 | 1.08 | 0.52 | 0.56
1890 | -0.2327 | 200 | 0.53 | 0.00 | 1.06 | 0.51 | 0.55
1900 | -0.2274 | 200 | 0.52 | 0.00 | 1.04 | 0.50 | 0.53
1910 | -0.2221 | 0 | 0.49 | 0.00 | 1.02 | 0.00 | 0.52
1980 | -0.1861 | 3 | 0.43 | 0.00 | 0.89 | 0.42 | 0.45
1990 | -0.1811 | 2 | 0.43 | 0.00 | 0.87 | 0.41 | 0.45
2000 | -0.1761 | 12 | 0.42 | 0.00 | 0.86 | 0.40 | 0.44
2010 | -0.1711 | 1 | 0.41 | 0.00 | 0.84 | 0.39 | 0.43
2030 | -0.1612 | 11 | 0.39 | 0.00 | 0.80 | 0.37 | 0.41
2035 | -0.1587 | 1 | 0.38 | 0.00 | 0.79 | 0.36 | 0.40
2050 | -0.1514 | 10 | 0.38 | 0.00 | 0.76 | 0.37 | 0.40
2055 | -0.1490 | 10 | 0.38 | 0.00 | 0.75 | 0.36 | 0.39
2085 | -0.1345 | 11 | 0.36 | 0.00 | 0.69 | 0.34 | 0.36
2090 | -0.1321 | 1 | 0.35 | 0.00 | 0.69 | 0.34 | 0.36
2100 | -0.1273 | 105 | 0.34 | 0.00 | 0.67 | 0.33 | 0.35
2105 | -0.1249 | 1 | 0.33 | 0.00 | 0.66 | 0.32 | 0.34
2120 | -0.1178 | 6 | 0.32 | 0.00 | 0.63 | 0.32 | 0.33
2125 | -0.1155 | 111 | 0.32 | 0.00 | 0.62 | 0.31 | 0.33
2130 | -0.1131 | 1 | 0.31 | 0.00 | 0.61 | 0.30 | 0.32
2135 | -0.1108 | 11 | 0.31 | 0.00 | 0.61 | 0.30 | 0.31
2140 | -0.1084 | 10 | 0.31 | 0.00 | 0.60 | 0.30 | 0.31
2150 | -0.1038 | 6610 | 0.29 | 0.00 | 0.53 | 0.29 | 0.30
2155 | -0.1015 | 1 | 0.29 | 0.00 | 0.52 | 0.28 | 0.30
2160 | -0.0991 | 8 | 0.29 | 0.00 | 0.51 | 0.28 | 0.30
2165 | -0.0968 | 2 | 0.28 | 0.00 | 0.50 | 0.28 | 0.29
2175 | -0.0922 | 3 | 0.27 | 0.00 | 0.48 | 0.27 | 0.28
2180 | -0.0899 | 66 | 0.27 | 0.00 | 0.47 | 0.27 | 0.28
2185 | -0.0876 | 15 | 0.27 | 0.00 | 0.46 | 0.26 | 0.27
2190 | -0.0853 | 26 | 0.26 | 0.00 | 0.45 | 0.26 | 0.27
2195 | -0.0831 | 13 | 0.26 | 0.00 | 0.44 | 0.25 | 0.26
2200 | -0.0808 | 113 | 0.25 | 0.00 | 0.43 | 0.25 | 0.25
2205 | -0.0785 | 13 | 0.25 | 0.00 | 0.42 | 0.25 | 0.25
2210 | -0.0762 | 14 | 0.24 | 0.00 | 0.41 | 0.24 | 0.25
2215 | -0.0740 | 102 | 0.24 | 0.00 | 0.41 | 0.23 | 0.24
2220 | -0.0717 | 2 | 0.23 | 0.00 | 0.40 | 0.23 | 0.24
2225 | -0.0695 | 43 | 0.23 | 0.00 | 0.39 | 0.22 | 0.23
2230 | -0.0672 | 93 | 0.22 | 0.00 | 0.38 | 0.22 | 0.23
2235 | -0.0650 | 2 | 0.22 | 0.00 | 0.37 | 0.21 | 0.22
2240 | -0.0628 | 134 | 0.21 | 0.00 | 0.36 | 0.21 | 0.22
2245 | -0.0605 | 11 | 0.21 | 0.00 | 0.35 | 0.21 | 0.21
2250 | -0.0583 | 104 | 0.20 | 0.00 | 0.34 | 0.20 | 0.21
2255 | -0.0561 | 205 | 0.20 | 0.00 | 0.34 | 0.20 | 0.20
2260 | -0.0539 | 184 | 0.19 | 0.00 | 0.33 | 0.19 | 0.20
2265 | -0.0517 | 89 | 0.19 | 0.00 | 0.32 | 0.19 | 0.19
2270 | -0.0495 | 74 | 0.18 | 0.00 | 0.28 | 0.18 | 0.19
2275 | -0.0473 | 377 | 0.18 | 0.00 | 0.27 | 0.18 | 0.18
2280 | -0.0451 | 207 | 0.17 | 0.00 | 0.26 | 0.17 | 0.17
2285 | -0.0429 | 478 | 0.17 | 0.00 | 0.25 | 0.17 | 0.17
2290 | -0.0407 | 295 | 0.16 | 0.00 | 0.24 | 0.16 | 0.17
2295 | -0.0385 | 280 | 0.16 | 0.00 | 0.18 | 0.16 | 0.16
2300 | -0.0363 | 1106 | 0.15 | 0.00 | 0.18 | 0.15 | 0.16
2305 | -0.0342 | 377 | 0.15 | 0.00 | 0.17 | 0.15 | 0.15
2310 | -0.0320 | 793 | 0.14 | 0.00 | 0.17 | 0.14 | 0.15
2315 | -0.0298 | 567 | 0.14 | 0.09 | 0.16 | 0.14 | 0.14
2320 | -0.0277 | 864 | 0.14 | 0.09 | 0.15 | 0.13 | 0.14
2325 | -0.0255 | 569 | 0.13 | 0.10 | 0.15 | 0.13 | 0.13
2330 | -0.0234 | 567 | 0.13 | 0.10 | 0.14 | 0.13 | 0.13
2335 | -0.0212 | 260 | 0.12 | 0.10 | 0.14 | 0.12 | 0.12
2340 | -0.0191 | 407 | 0.12 | 0.10 | 0.13 | 0.12 | 0.12
2345 | -0.0170 | 506 | 0.12 | 0.10 | 0.13 | 0.11 | 0.12
2350 | -0.0148 | 1850 | 0.11 | 0.10 | 0.12 | 0.11 | 0.11
2355 | -0.0127 | 528 | 0.11 | 0.10 | 0.11 | 0.11 | 0.11
2360 | -0.0106 | 1134 | 0.10 | 0.10 | 0.11 | 0.10 | 0.11
2365 | -0.0085 | 436 | 0.10 | 0.10 | 0.10 | 0.10 | 0.10
2370 | -0.0064 | 1063 | 0.10 | 0.09 | 0.10 | 0.10 | 0.10
2375 | -0.0042 | 1420 | 0.09 | 0.09 | 0.10 | 0.09 | 0.10
2380 | -0.0021 | 1248 | 0.09 | 0.09 | 0.09 | 0.09 | 0.09
2385 | 0.0000 | 229 | 0.09 | 0.09 | 0.09 | 0.09 | 0.09
2390 | 0.0021 | 2252 | 0.09 | 0.09 | 0.09 | 0.09 | 0.09
2395 | 0.0041 | 912 | 0.09 | 0.09 | 0.09 | 0.08 | 0.09
2400 | 0.0062 | 2650 | 0.09 | 0.08 | 0.09 | 0.08 | 0.09
2405 | 0.0083 | 2523 | 0.08 | 0.08 | 0.09 | 0.08 | 0.09
2410 | 0.0104 | 789 | 0.08 | 0.08 | 0.09 | 0.08 | 0.10
2415 | 0.0125 | 863 | 0.08 | 0.08 | 0.08 | 0.07 | 0.10
2420 | 0.0145 | 905 | 0.08 | 0.08 | 0.09 | 0.00 | 0.13
2425 | 0.0166 | 1178 | 0.09 | 0.08 | 0.09 | 0.00 | 0.13
2430 | 0.0187 | 815 | 0.09 | 0.08 | 0.09 | 0.00 | 0.14
2435 | 0.0207 | 267 | 0.09 | 0.09 | 0.09 | 0.00 | 0.15
2440 | 0.0228 | 402 | 0.09 | 0.09 | 0.09 | 0.00 | 0.16
2445 | 0.0248 | 50 | 0.09 | 0.09 | 0.09 | 0.00 | 0.17
2450 | 0.0268 | 2027 | 0.09 | 0.09 | 0.10 | 0.00 | 0.17
2455 | 0.0289 | 59 | 0.10 | 0.10 | 0.10 | 0.00 | 0.18
2460 | 0.0309 | 159 | 0.10 | 0.10 | 0.10 | 0.00 | 0.19
2465 | 0.0330 | 26 | 0.10 | 0.10 | 0.11 | 0.00 | 0.20
2470 | 0.0350 | 53 | 0.10 | 0.10 | 0.11 | 0.00 | 0.21
2475 | 0.0370 | 229 | 0.11 | 0.10 | 0.11 | 0.00 | 0.22
2495 | 0.0450 | 1 | 0.12 | 0.11 | 0.13 | 0.00 | 0.25
2550 | 0.0669 | 800 | 0.15 | 0.00 | 0.16 | 0.00 | 0.37
Table 5: Implied volatility quotes for SPX500 options expiring on March 7, 2018, as of February, 5, 2018. In ACT/365, the option maturity is $T=0.082192$. The implied forward price is $F(0,T)=2629.80$. The interest rate is $r=0.97\%$. Strike | Logmoneyness | Implied vol. |  | Strike | Logmoneyness | Implied vol.
---|---|---|---|---|---|---
1900 | -0.325055 | 0.684883 |  | 2650 | 0.007651 | 0.280853
1950 | -0.299079 | 0.6548 |  | 2655 | 0.009536 | 0.277035
2000 | -0.273762 | 0.627972 |  | 2660 | 0.011417 | 0.273715
2050 | -0.249069 | 0.604067 |  | 2665 | 0.013295 | 0.270891
2100 | -0.224971 | 0.576923 |  | 2670 | 0.01517 | 0.267889
2150 | -0.201441 | 0.551253 |  | 2675 | 0.017041 | 0.264533
2200 | -0.178451 | 0.526025 |  | 2680 | 0.018908 | 0.262344
2250 | -0.155979 | 0.500435 |  | 2685 | 0.020772 | 0.258598
2300 | -0.134 | 0.474137 |  | 2690 | 0.022632 | 0.2555
2325 | -0.123189 | 0.461716 |  | 2695 | 0.024489 | 0.25219
2350 | -0.112493 | 0.445709 |  | 2700 | 0.026343 | 0.249534
2375 | -0.101911 | 0.433661 |  | 2705 | 0.028193 | 0.246659
2400 | -0.09144 | 0.42016 |  | 2710 | 0.03004 | 0.243553
2425 | -0.081077 | 0.407463 |  | 2715 | 0.031883 | 0.240202
2450 | -0.070821 | 0.393168 |  | 2720 | 0.033723 | 0.236588
2470 | -0.062691 | 0.381405 |  | 2725 | 0.03556 | 0.234574
2475 | -0.060668 | 0.3793 |  | 2730 | 0.037393 | 0.230407
2480 | -0.05865 | 0.377109 |  | 2735 | 0.039223 | 0.227866
2490 | -0.054626 | 0.372471 |  | 2740 | 0.041049 | 0.223049
2510 | -0.046626 | 0.360294 |  | 2745 | 0.042872 | 0.219888
2520 | -0.04265 | 0.354671 |  | 2750 | 0.044692 | 0.218498
2530 | -0.03869 | 0.350533 |  | 2755 | 0.046509 | 0.214702
2540 | -0.034745 | 0.34419 |  | 2760 | 0.048322 | 0.210506
2550 | -0.030815 | 0.339273 |  | 2765 | 0.050132 | 0.208175
2560 | -0.026902 | 0.333069 |  | 2770 | 0.051939 | 0.205508
2570 | -0.023003 | 0.328206 |  | 2775 | 0.053742 | 0.199967
2575 | -0.021059 | 0.324314 |  | 2780 | 0.055542 | 0.199007
2580 | -0.019119 | 0.322041 |  | 2785 | 0.057339 | 0.195062
2590 | -0.015251 | 0.3168 |  | 2790 | 0.059133 | 0.190547
2600 | -0.011397 | 0.310914 |  | 2795 | 0.060923 | 0.188427
2610 | -0.007559 | 0.305042 |  | 2800 | 0.062711 | 0.185893
2615 | -0.005645 | 0.302416 |  | 2805 | 0.064495 | 0.182878
2620 | -0.003734 | 0.299488 |  | 2810 | 0.066276 | 0.179292
2625 | -0.001828 | 0.29609 |  | 2815 | 0.068053 | 0.175001
2630 | 7.5E-05 | 0.292378 |  | 2835 | 0.075133 | 0.185751
2635 | 0.001974 | 0.289516 |  | 2860 | 0.083913 | 0.207173
2640 | 0.00387 | 0.28584 |  | 2900 | 0.097802 | 0.225248
2645 | 0.005762 | 0.283342 |  |  |  |
Table 6: Implied volatilities of TSLA European options expiring on July 20, 2018 as of June 15, 2018 ($T=0.095890$) with implied forward $F(T)=357.755926$. Strike | Mid Vol in % | Weight | Strike | Mid Vol in % | Weight
---|---|---|---|---|---
150.0 | 102.7152094560499 | 1.7320508075688772 | 335.0 | 47.11478482467228 | 7.365459931328131
155.0 | 99.05195749900227 | 1.0 | 340.0 | 46.78835216769108 | 7.0992957397195395
160.0 | 96.57262376591365 | 1.224744871391589 | 345.0 | 46.56217516966071 | 7.628892449104261
165.0 | 94.05597986379826 | 1.0 | 350.0 | 46.299652559206564 | 7.461009761866454
175.0 | 91.81603362313814 | 2.738612787525831 | 355.0 | 45.939930288424485 | 8.706319543871567
180.0 | 90.19382314978117 | 1.558387444947959 | 360.0 | 45.856510564386596 | 8.78635305459552
185.0 | 88.46745842549402 | 1.9999999999999998 | 365.0 | 45.79048747963794 | 7.000000000000021
190.0 | 86.5754243981787 | 1.2602520756252087 | 370.0 | 45.52139844132191 | 7.745966692414834
195.0 | 84.561554924342 | 1.3301243435223526 | 375.0 | 45.3447302139774 | 8.093207028119338
200.0 | 82.45634579529838 | 2.273030282830976 | 380.0 | 45.04013827012644 | 6.16441400296897
205.0 | 80.28174604214972 | 1.3944333775567928 | 385.0 | 44.800472164335794 | 4.974937185533098
210.0 | 78.053958851195 | 1.2089410496539776 | 390.0 | 44.91995553643971 | 4.650268809434567
215.0 | 76.36802684802436 | 1.9999999999999998 | 395.0 | 44.78840707248649 | 4.315669125408015
220.0 | 74.54192306685303 | 2.0976176963403033 | 400.0 | 45.00659311379786 | 4.636809247747854
225.0 | 72.60651215584285 | 3.500000000000001 | 405.0 | 45.17530880150887 | 4.732863826479693
230.0 | 70.58414693439228 | 3.286335345030995 | 410.0 | 44.99007489879635 | 3.1144823004794873
235.0 | 68.49143304434797 | 2.6692695630078282 | 415.0 | 44.8814967685824 | 2.8809720581775857
240.0 | 66.3409356115238 | 2.7838821814150116 | 420.0 | 45.16047756853698 | 2.8284271247461894
245.0 | 64.6230979973991 | 3.1622776601683804 | 425.0 | 45.63938928347205 | 2.7718093060793882
250.0 | 63.0129173926189 | 3.605551275463988 | 430.0 | 46.002220642176724 | 4.092676385936223
255.0 | 61.30540004186168 | 3.3541019662496834 | 435.0 | 46.102443173801966 | 2.7041634565979926
260.0 | 59.469230763484425 | 3.0 | 440.0 | 46.40646817026155 | 2.652259934210953
265.0 | 58.11921286363728 | 2.9742484506432634 | 445.0 | 47.09795491400157 | 3.710691413905333
270.0 | 56.87314890047378 | 3.6469165057620923 | 450.0 | 47.625950451280104 | 3.777926319123662
275.0 | 55.39815904720001 | 3.8729833462074152 | 455.0 | 48.10009989573377 | 3.929942040850535
280.0 | 54.226712926697765 | 4.183300132670376 | 460.0 | 48.559069655772966 | 3.921096785339529
285.0 | 53.38887990387771 | 3.7505555144093887 | 465.0 | 49.06446878461756 | 3.70809924354783
290.0 | 52.34154661207794 | 4.1918287860346295 | 470.0 | 49.60612773473766 | 3.517811819867573
295.0 | 51.68510552270313 | 3.7670248460125917 | 475.0 | 50.11170526132832 | 3.3354160160315844
300.0 | 50.72806473672073 | 4.795831523312714 | 480.0 | 50.59204240563133 | 3.1622776601683777
305.0 | 49.979731599616564 | 4.527692569068711 | 500.0 | 51.591022062492634 | 1.3483997249264843
310.0 | 48.96563997378466 | 3.482097069296032 | 520.0 | 55.056251469410256 | 1.8929694486000912
315.0 | 48.23975850368014 | 3.2333489534143167 | 540.0 | 57.838819666460616 | 1.914854215512676
320.0 | 47.93681836406913 | 3.687817782917155 | 560.0 | 59.92609035805611 | 1.699673171197595
325.0 | 48.00058538405501 | 6.3245553203367555 | 580.0 | 62.59792014943735 | 1.8708286933869707
330.0 | 47.57525564073338 | 6.837397165588683 |  |  |
Table 7: Quotes for options on AUD/NZD currency pair of maturity July 9th, 2014 as of July 2nd, 2014. $S=1.0784,F(0,T)=1.07845,B(0,T)=0.999712587139$, 10-$\Delta$ RR = 0.35%, 25-$\Delta$ RR = 0.40%, ATM vol = 5.14%, 25-$\Delta$ BF = 0.25%, 10-$\Delta$ BF = 1.175%. Maturity | 10$\Delta$-Put | 25$\Delta$-Put | ATM | 25$\Delta$-Call | 10$\Delta$-Call
---|---|---|---|---|---
1w | 6.14% | 5.19% | 5.14% | 5.59% | 6.49%
Table 8: Black-Scholes implied volatilities against moneyness $\frac{x}{X(0)}$ for an option of maturity $T=5.0722$, examples 1 and 2 of Jäckel (2014). Moneyness | Volatility (Case I) | Volatility (Case II)
---|---|---
0.035123777453185 | 0.642412798191439 | 0.649712512502887
0.049095433048156 | 0.621682849924325 | 0.629372247414191
0.068624781300891 | 0.590577891369241 | 0.598339248024188
0.095922580089594 | 0.553137221952525 | 0.560748840467284
0.134078990076508 | 0.511398042127817 | 0.518685454812697
0.18741338653678 | 0.466699250819768 | 0.473512707134552
0.261963320525776 | 0.420225808661573 | 0.426434688827871
0.366167980681693 | 0.373296313420122 | 0.378806875802102
0.511823524787378 | 0.327557513727855 | 0.332366264644264
0.715418426368358 | 0.285106482185545 | 0.289407658380454
1 | 0.249328882881654 | 0.253751752243855
1.39778339939642 | 0.228967051575314 | 0.235378088110653
1.95379843162821 | 0.220857187809035 | 0.235343538571543
2.73098701349666 | 0.218762825294675 | 0.260395028879884
3.81732831143284 | 0.218742183617652 | 0.31735041252779
5.33579814376678 | 0.218432406892364 | 0.368205175099723
7.45829006788743 | 0.217198426268117 | 0.417582432865276
10.4250740447762 | 0.21573928902421 | 0.46323707706565
14.5719954372667 | 0.214619929462215 | 0.504386489988866
20.3684933182917 | 0.2141074555437 | 0.539752566560924
28.4707418310251 | 0.21457985392644 | 0.566370957381163