# On the implied volatility of European and Asian call options under the
stochastic volatility Bachelier model

Elisa Alòs∗, Eulalia Nualart∗, and Makar Pravosud  Universitat Pompeu Fabra
and Barcelona School of Economics, Department of Economics and Business, Ramón
Trias Fargas 25-27, 08005, Barcelona, Spain. EN acknowledges support from the
Spanish MINECO grant PGC2018-101643-B-I00 and Ayudas Fundacion BBVA a Equipos
de Investigación Científica 2017.

###### Abstract

In this paper we study the short-time behavior of the at-the-money implied
volatility for European and arithmetic Asian call options with fixed strike
price. The asset price is assumed to follow the Bachelier model with a general
stochastic volatility process. Using techniques of the Malliavin calculus such
as the anticipating Itô’s formula we first compute the level of the implied
volatility when the maturity converges to zero. Then, we find a short maturity
asymptotic formula for the skew of the implied volatility that depends on the
roughness of the volatility model. We apply our general results to the SABR
and fractional Bergomi models, and provide some numerical simulations that
confirm the accurateness of the asymptotic formula for the skew.

Keywords: Bachelier model, Stochastic volatility, European options, Asian
options, Malliavin calculus, implied volatility

##  1 Introduction

The life of quantitative finance and derivatives pricing has started when
Louis Bachelier presented an option pricing model in his Ph.D. thesis [6].
However, the suggested model is arithmetic and allows negative asset prices,
as a result, the Bachelier model has not gained popularity. Until recently the
model has not been widely studied in the literature and applied in practice.
Nowadays option pricing models are strongly based on the Black-Scholes model.
Local, stochastic and even fractional volatility models (see Alòs and Lorite
[2] for an introduction to these topics) are extensions of the Black-Scholes
model, where the volatility is allowed to be a function of the price of the
stock price (local), a diffusion process (stochastic) or a function of a
fractional Brownian motion (fractional). As the Black-Scholes model is a
geometric Brownian motion (following a log-normal distribution), all the above
models are also exponentials of random variables, and they share the
positivity of the stock price. This plausible and intuitive property has been
assumed for a long time for the prices of the underlying asset of an option.

From a practical perspective, the Bachelier model has started to gain more
attention when negative prices have been registered, for example, on April
20th 2020, when crude oil futures crossed the zero mark and took negative
values. As a prompt response to enable markets to continue to function
normally, the CME Clearing started to use the Bachelier model to cope with
negative underlying prices. The empirical paper by Galeeva and Ehud [11]
studies the turbulent period on the oil-futures options markets which was
observed in the middle of February 2020. The authors analyzed the observed
implied volatilities by applying the Bachelier and Black models to prevail
oil-futures option prices.

Apart from the energy market, the Bachelier model is also used in modelling
interest rate products, for example, Euro denominated caps. The detailed
analysis on the reasons to use a normal model rather than a log-normal one in
the case of interest rates is presented in Goodman [12]. The recent paper by
Choi et al. [7] is a comprehensive review of various topics related to the
Bachelier model for both researchers and practitioners. In particular, they
cover topics such as the implied volatility inversion problem, volatility
conversion between related models, Greeks and hedging, as well as pricing of
exotic options.

Concerning exact analytical results, in the paper by Schachermayer and
Teichmann [15], the authors compare the option pricing problem under the
Bachelier and Black-Scholes models. They find that prices coincide very well
and explain by the means of chaos expansion theory why the Bachelier model
yields good short-time approximations of prices and volatilities. In addition
to that, the computation of prices and implied volatilities under the
Bachelier model model has been presented in, for example, Terkado [16].
However, the existing literature lacks the rigorous analysis of the behaviour
of the ATM level and skew of the implied volatility under the Bachelier model
with stochastic volatility. By contrast, in the case of a regular Black-
Scholes framework, there are numerous papers that study the short end maturity
behaviour of the implied volatility under a variety of models such as the
Heston, the SABR and fractional Begomi. For example, see Alòs [4], Figueroa-
López et al.[8], Fukasawa [10], Fouque et al. [9].

The goal of this paper is to fill the gap in the existing literature and to
analyze the option pricing problem when we allow the volatility of the
Bachelier model to be a stochastic process. That is, to study option prices
and implied volatilities for local, stochastic and fractional Bachelier
models, both for vanilla and for exotic, in particular, Asian options.
Specifically, we present analytical results for the behaviour of the level and
the skew of the implied volatility for a general stochastic volatility
Bachelier model. Our main tool for proving these results is Malliavin
calculus, see Appendix A for an introduction to this topic. Additionally,
using the findings from the theoretical part of the paper we do an extensive
numerical simulations to investigate the quality of the first order
approximations of the implied volatility across different moneyness and
maturity scenarios. The presented results are extension of the papers Alòs
[1], Alòs et.al [3] and Alòs, Nualart, Pravosud [5].

The paper is organized as follows. Section 2 is devoted to the statement of
the problem and main results. Intermediary steps and the proofs of the main
results are presented in Sections 3 and 4. Finally, in Section 5 we present
and discuss the results of the numerical study.

##  2 Statement of the problem and main results

Let $T>0$ and consider the following model for asset prices $S$ (without lost
of generality, we take the interest rate equal to zero for the sake of
simplicity) in a time interval $[0,T]$

$\begin{split}dS_{t}&=\sigma_{t}dW_{t}\\\ W_{t}&=\rho
W_{t}^{\prime}+\sqrt{(1-\rho^{2})}B_{t},\end{split}$ (1)

where $S_{0}>0$ is fixed, $W_{t}$, $W_{t}^{\prime}$, and $B_{t}$ are three
standard Brownian motions on $[0,T]$ defined on the same risk-neutral complete
probability space $(\Omega,\mathcal{G},\mathbb{P})$. We assume that
$W_{t}^{\prime}$ and $B_{t}$ are independent and $\rho\in[-1,1]$ is the
correlation coefficient between $W_{t}$ and $W_{t}^{\prime}$, and that
$\sigma_{t}$ is an a.s. continuous and square integrable process adapted to
the filtration generated by $W^{\prime}_{t}$. We denote by $\mathbb{E}_{t}$
the conditional expectation with respect to the filtration generated by
$W_{t}$.

Observe that when the volatility $\sigma$ is constant, equation (1) is the
classical Bachelier model. Notice that, in this case, asset prices are
normally distributed.

We consider the following hypotheses on the volatility of the asset price.
These hypotheses have been taken for the sake of simplicity and can be
replaced by adequate integrability conditions.

######  Hypothesis 1.

There exist positive constants $c_{1}$ and $c_{2}$ such that for all
$t\in[0,T]$,

$c_{1}\leq\sigma_{t}\leq c_{2}.$

######  Hypothesis 2.

For $p\geq 2$, $\sigma\in\mathbb{L}^{2,p}_{W^{\prime}}$.

######  Hypothesis 3.

There exists $H\in(0,1)$ and for all $p\geq 1$ there exist constants
$c_{1},c_{2}>0$ such that for all $0\leq s\leq r\leq u\leq T\leq 1$

$\\{\mathbb{E}(|D_{r}^{W^{\prime}}\sigma_{u}|^{p})\\}^{1/p}\leq
c_{1}(u-r)^{H-\frac{1}{2}}$ (2)

and

$\\{\mathbb{E}(|D_{s}^{W^{\prime}}D_{r}^{W^{\prime}}\sigma_{u}|^{p})\\}^{1/p}\leq
c_{2}(u-r)^{H-\frac{1}{2}}(u-s)^{H-\frac{1}{2}}.$ (3)

We denote by $(V^{E}_{t})_{t\in[0,T]}$ the value of a European call option
with fixed strike $k$. In particular,

$V^{E}_{T}=\mathbb{E}(S_{T}-k)_{+}.$

In a similar way, we denote by $(V^{A}_{t})_{t\in[0,T]}$ the price of an
arithmetic Asian call option with fixed strike $k$. Then

$V^{A}_{T}=\mathbb{E}(A_{T}-k)_{+},$

where $A_{T}=\frac{1}{T}\int_{0}^{T}S_{t}dt$.

In oder to to deal with Asian options we follow the same approach as in [3],
that is, we consider the martingale $M_{t}=\mathbb{E}_{t}(A_{T})$. Applying
the stochastic Fubini’s theorem we get that

$\displaystyle\begin{split}A_{T}=\frac{1}{T}\int_{0}^{T}S_{t}dt&=\frac{1}{T}\int_{0}^{T}\left(S_{0}+\int_{0}^{t}\sigma_{u}dW_{u}\right)dt=\\\
&=S_{0}+\frac{1}{T}\int_{0}^{T}\sigma_{u}\left(\int_{u}^{T}dt\right)dW_{u}.\end{split}$

This implies that

$dM_{t}=\frac{\sigma_{t}(T-t)}{T}dW_{t}=\phi_{t}dW_{t},$ (4)

where

$\phi_{t}:=\frac{\sigma_{t}(T-t)}{T}.$

Notice that, under the classical Bachelier model, $A_{T}$ is a Gaussian random
variable with mean $S_{0}$ and variance $\frac{\sigma^{2}T}{3}$.

We denote by $B_{E}(t,S_{t},k,\sigma)$ the classical Bachelier price of a
European call option with time to maturity $T-t$, current stock price $S_{t}$,
strike price $k$ and constant volatility $\sigma$. That is,

$B_{E}(t,S_{t},k,\sigma)=(S_{t}-k)N(d_{E}(k,\sigma))+n(d_{E}(k,\sigma))\sigma\sqrt{T-t},$

where

$d_{E}(k,\sigma)=\frac{S_{t}-k}{\sigma\sqrt{T-t}}.$

Here $N$ and $n$ denote the cumulative distribution function and the
probability density function of a standard normal random variable,
respectively. Additionally, we recall that the Bachelier price satisfies the
following PDE

$\displaystyle\partial_{t}B_{E}(t,x,k,\sigma)+\frac{1}{2}\sigma^{2}\partial^{2}_{xx}B_{E}(t,x,k,\sigma)=0.$
(5)

Similarly, we denote by $B_{A}(t,S_{t},y_{t},k,\sigma)$ the Bachelier price of
an arithmetic Asian call option with constant volatility $\sigma$, where
$y_{t}=\int_{0}^{t}S_{u}du$. A direct computation shows that (see the Appendix
for the details)

$B_{A}(t,S_{t},y_{t},k,\sigma)=\left(S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}-k\right)N(d_{A}(k,\sigma))+\left(\frac{\sigma(T-t)\sqrt{T-t}}{T\sqrt{3}}\right)n(d_{A}(k,\sigma)),$

where

$d_{A}(k,\sigma)=\frac{S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}-k}{\left(\frac{\sigma(T-t)\sqrt{T-t}}{T\sqrt{3}}\right)}.$

Notice that we have the relation

$B_{A}(t,S_{t},y_{t},k,\sigma)=B_{E}\left(t,M_{t},k,\frac{\sigma(T-t)}{T\sqrt{3}}\right).$

We next define the implied volatility (IV) of a European call option as the
quantity $I_{E}(t,k)$ such that

$V_{t}^{E}=B_{E}(t,S_{t},k,I_{E}(t,k)),$

and we denote by $I_{E}(t,k^{*}_{t})$, where $k^{*}_{t}=S_{t}$, the
corresponding at-the-money implied volatility (ATMIV) which, in the case of
zero interest rates, takes the form $B_{E}^{-1}(t,S_{t},S_{t},V_{t}^{E})$.
Similarly, we define the implied volatility of an Asian call option
$I_{A}(t,k)$ as the quantity such that

$V^{A}_{t}=B_{A}(t,S_{t},y_{t},k,I_{A}(t,k)),$

and we denote by $I_{A}(t,k_{t}^{*})$ the corresponding ATMIV which, in the
case of zero interest rates, takes the form
$B_{A}^{-1}(t,S_{t},y_{t},S_{t}\frac{T-t}{T}+\frac{y_{t}}{T},V^{A}_{t})$. We
set $k^{\ast}=k^{\ast}_{0}$.

The aim of this paper is to apply the Malliavin calculus techniques developed
in Alòs [4] and Muguruza et al. [3] in order to obtain formulas for the ATMIV
level and skew as $T\rightarrow 0$ under the general stochastic volatility
model (1).

The main result of this paper is the following theorem.

######  Theorem 1.

Assume Hypotheses 1-3. Then,

$\displaystyle\lim_{T\to
0}I_{E}(0,k^{*})=\sigma_{0}\quad\text{and}\quad\lim_{T\to
0}I_{A}(0,k^{*})=\sigma_{0}.$ (6)

Moreover,

$\begin{split}&\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\partial_{k}I_{E}(0,k^{*})\\\ &=\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\frac{\rho}{\sigma_{0}T^{2}}\int_{0}^{T}\int_{r}^{T}\mathbb{E}(D_{r}^{W^{\prime}}\sigma_{u})dudr\\\
\end{split}$ (7)

and

$\begin{split}&\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\partial_{k}I_{A}(0,k^{*})\\\ &=\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\frac{9\rho}{\sigma_{0}T^{5}}\int_{0}^{T}(T-r)\int_{r}^{T}(T-u)^{2}\mathbb{E}(D_{r}^{W^{\prime}}\sigma_{u})dudr,\end{split}$
(8)

Observe that when prices and volatilities are uncorrelated then the short-time
skew equals to zero, which coincides with the constant volatility case. Notice
also that if the term $\mathbb{E}(D_{r}^{W^{\prime}}\sigma_{u})$ is of order
$(u-r)^{H-\frac{1}{2}}$, the limit of the right hand side of (7) will be $0$
if $H>1/2$ and it will converge to a constant when $H=\frac{1}{2}$. When
$H<\frac{1}{2}$ we need to multiply by $T^{\frac{1}{2}-H}$ in order to obtain
a finite limit.

The results of Theorem 1 can be used in order to derive approximation formulas
for the price of European and Asian call options. Notice that

$V^{E}_{0}=B_{E}(0,S_{0},k,I_{E}(0,k))\quad\text{and}\quad
V^{A}_{0}=B_{A}(0,S_{0},y_{0},k,I_{A}(0,k)).$

Then, using Taylor’s formula we can use the approximations

$\begin{split}I_{E}(0,k)&\approx
I_{E}(0,k^{*})+\partial_{k}I_{E}(0,k^{*})(k-k^{*}),\\\ I_{A}(0,k)&\approx
I_{A}(0,k^{*})+\partial_{k}I_{A}(0,k^{*})(k-k^{*}).\end{split}$ (9)

The great utility of these relations is that we can use them to approximate
the price of European and Asian call options for a wide range of stochastic
and fractional volatility models. We numerically investigate the quality of
this approximation for the SABR and fractional Bergomi models in Section 5.3.

##  3 Preliminary results: decomposition formulas

In this section we provide closed form decomposition formulas for the prices
and for the ATM implied volatility skew of European and Asian call options
under the stochastic volatility model (1) that will be crucial for the proof
of the main results.

We start with a preliminary lemma.

######  Lemma 1.

Consider the model (1). Let $0\leq t\leq s\leq T$, $\rho\in(-1,1)$ and
$\mathcal{G}_{t}:=\mathcal{F}_{t}\lor\mathcal{F}^{W^{\prime}}_{T}$. Then for
every $n\geq 0$, there exists $C=C(n,\rho)$ such that

$\begin{split}\left|\mathbb{E}\left(\frac{\partial^{n}}{\partial
x^{n}}\left(\frac{\partial^{2}}{\partial
x^{2}}B_{E}(s,M_{s},k,v_{s})\right)|\mathcal{G}_{t}\right)\right|&=\left|\mathbb{E}\left(\frac{\partial^{n}}{\partial
x^{n}}\left(\frac{\partial^{2}}{\partial x\partial
k}B_{E}(s,M_{s},k,v_{s})\right)|\mathcal{G}_{t}\right)\right|\\\ &\leq
C\left(\int_{t}^{T}\phi_{r}^{2}dr\right)^{-\frac{n+1}{2}},\end{split}$

where $v_{t}=\sqrt{\frac{1}{T-t}\int_{t}^{T}\phi_{r}^{2}dr}$

###### Proof.

The proof follows the same steps as the proof of Lemma 6.3.1 in [2]. ∎

The main result of this section is the following theorem.

######  Theorem 2.

Assume Hypotheses 1-3. Then, the following relations hold for all $t\in[0,T]$,

$\begin{split}V^{E}_{t}=\mathbb{E}_{t}\left(B_{E}(t,S_{t},k,v^{\prime}_{t})\right)+\mathbb{E}_{t}\left(\int_{t}^{T}H(s,S_{s},k,v^{\prime}_{s})\sigma_{s}\left(\int_{s}^{T}D_{s}^{W}\sigma_{r}^{2}dr\right)ds\right),\\\
V^{A}_{t}=\mathbb{E}_{t}\left(B_{E}(t,M_{t},k,v_{t})\right)+\mathbb{E}_{t}\left(\int_{t}^{T}H(s,M_{s},k,v_{s})\phi_{s}\left(\int_{s}^{T}D_{s}^{W}\phi_{r}^{2}dr\right)ds\right),\end{split}$

where $H(s,x,k,\sigma)=\frac{1}{2}\partial_{xxx}^{3}B_{E}(s,x,k,\sigma)$,
$v_{t}^{\prime}=\sqrt{\frac{1}{T-t}\int_{t}^{T}\sigma_{s}^{2}ds}$, and
$v_{t}=\sqrt{\frac{1}{T-t}\int_{t}^{T}\phi_{s}^{2}ds}$.

###### Proof.

The proof follows similar ideas as the proof of Theorem 25 in [1]. See also
Theorem 6.3.2 in [2]. Notice that, as $B_{E}(T,x,k,\sigma)=(x-k)_{+}$ for
every $\sigma>0$, the prices of our European and Asian call options can be
written as

$V_{t}^{E}=\mathbb{E}_{t}(B_{E}(T,S_{T},k,v^{\prime}_{T}))\quad\text{and}\quad
V_{t}^{A}=\mathbb{E}_{t}(B_{E}(T,M_{T},k,v_{T})).$

We provide the proof of the decomposition formula for $V_{t}^{A}$. A similar
argument applies for $V_{t}^{E}$ and we safely skip it. Applying Theorem 3 to
the function $B_{E}(t,M_{t},k,v_{t})$ and $Y_{t}=\int_{t}^{T}\phi^{2}_{s}ds$
noticing that $v_{t}=\sqrt{\frac{Y_{t}}{T-t}}$, we obtain

$\displaystyle B_{E}(T,M_{T},k,v_{T})=B_{E}(t,M_{t},k,v_{t})$
$\displaystyle\qquad+\int_{t}^{T}\left(\partial_{s}B_{E}(s,M_{s},k,v_{s})+\partial_{\sigma}B_{E}(s,M_{s},k,v_{s})\frac{v_{s}^{2}}{2(T-s)v_{s}}\right)ds$
$\displaystyle\qquad+\int_{t}^{T}\partial_{x}B_{E}(s,M_{s},k,v_{s})\phi_{s}dW_{s}-\int_{t}^{T}\partial_{\sigma}B_{E}(s,M_{s},k,v_{s})\frac{\phi_{s}^{2}}{2(T-s)v_{s}}ds$
$\displaystyle\qquad+\int_{t}^{T}\partial^{2}_{\sigma
x}B_{E}(s,M_{s},k,v_{s})\frac{\phi_{s}}{2(T-s)v_{s}}\left(\int_{s}^{T}D_{s}^{W}\phi_{r}^{2}dr\right)ds$
$\displaystyle\qquad+\frac{1}{2}\int_{t}^{T}\partial_{xx}^{2}B_{E}(s,M_{s},k,v_{s})\phi_{s}^{2}ds.$

Notice that the following relation holds

$\displaystyle\partial_{xx}^{2}B_{E}(s,M_{s},k,\sigma)=\frac{\partial_{\sigma}B_{E}(s,M_{s},k,\sigma))}{\sigma(T-s)}.$
(10)

Then we get the following

$\displaystyle B_{E}(T,M_{T},k,v_{T})=B_{E}(t,M_{t},k,v_{t})$
$\displaystyle\qquad+\int_{t}^{T}\left(\partial_{s}B_{E}(s,M_{s},k,v_{s})+\frac{1}{2}v_{s}^{2}\partial_{xx}^{2}B_{E}(s,M_{s},k,v_{s})\right)ds$
$\displaystyle\qquad+\int_{t}^{T}\partial_{x}B_{E}(s,M_{s},k,v_{s})\phi_{s}dW_{s}-\frac{1}{2}\int_{t}^{T}\partial_{xx}^{2}B_{E}(s,M_{s},k,v_{s})\phi_{s}^{2}ds$
$\displaystyle\qquad+\frac{1}{2}\int_{t}^{T}\partial^{3}_{xxx}B_{E}(s,M_{s},k,v_{s})\phi_{s}\left(\int_{s}^{T}D_{s}^{W}\phi_{r}^{2}dr\right)ds$
$\displaystyle\qquad+\frac{1}{2}\int_{t}^{T}\partial_{xx}^{2}B_{E}(s,M_{s},k,v_{s})\phi_{s}^{2}ds.$

The first integral in the above expression is equal to zero due to equation
(5). Finally, taking conditional expectations we conclude that

$\displaystyle\mathbb{E}_{t}\left(B_{E}(T,M_{T},k,v_{T})\right)=\mathbb{E}_{t}\left(B_{E}(t,M_{t},k,v_{t})\right)$
$\displaystyle\qquad+\mathbb{E}_{t}\left(\frac{1}{2}\int_{t}^{T}\partial_{xxx}^{3}B_{E}(s,M_{s},k,v_{s})\phi_{s}\left(\int_{s}^{T}D_{s}^{W}\phi_{r}^{2}dr\right)ds\right).$

Observe that by Lemma 1 and Hypotheses 1 and 3(2), the last conditional
expectation is finite. This completes the desired proof.∎

Based on the result of Theorem 2, we derive an expression for the ATMIV skew
of European and Asian call options under the stochastic volatility model (1).

######  Proposition 1.

Assume Hypotheses 1-3. Then, for every $t\in[0,T]$ the following holds

$\displaystyle\begin{split}\partial_{k}I_{E}(t,k_{t}^{*})=\frac{\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{k}H(s,S_{s},k_{t}^{*},v_{s}^{\prime})\Lambda_{s}^{\prime}ds\right)}{\partial_{\sigma}B_{E}(t,S_{t},k_{t}^{*},I_{E}(t,k_{t}^{*}))},\\\
\partial_{k}I_{A}(t,k^{*}_{t})=\frac{\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{k}H(s,M_{s},k_{t}^{*},v_{s})\Lambda_{s}ds\right)}{\partial_{\sigma}B_{A}(t,S_{t},y_{t},k_{t}^{*},I_{A}(t,k_{t}^{*}))},\end{split}$

where $\Lambda_{s}^{\prime}=\sigma_{s}\int_{s}^{T}D_{s}^{W}\sigma_{r}^{2}dr$
and $\Lambda_{s}=\phi_{s}\int_{s}^{T}D_{s}^{W}\phi_{r}^{2}dr$.

###### Proof.

We provide the proof for the second part of the theorem. The first part
follows by similar arguments. Since
$V_{t}^{A}=B_{A}(t,S_{t},y_{t},k,I_{A}(t,k))$, differentiating we obtain that

$\partial_{k}V^{A}_{t}=\partial_{k}B_{A}(t,S_{t},y_{t},k,I_{A}(t,k))+\partial_{\sigma}B_{A}(t,S_{t},y_{t},k,I_{A}(t,k))\partial_{k}I_{A}(t,k).$

On the other hand, using Theorem 2, we get that

$\partial_{k}V^{A}_{t}=\partial_{k}\mathbb{E}_{t}\left(B_{E}(t,M_{t},k,v_{t})\right)+\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{k}H(s,M_{s},k,v_{s})\Lambda_{s}ds\right).$

Combining both equations, we obtain that the volatility skew
$\partial_{k}I_{A}(t,k)$ is equal to

$\frac{\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{k}H(s,M_{s},k,v_{s})\Lambda_{s}ds\right)+\mathbb{E}_{t}\left(\partial_{k}B_{E}(t,M_{t},k,v_{t})\right)-\partial_{k}B_{A}(t,S_{t},y_{t},k,I_{A}(t,k))}{\partial_{\sigma}B_{A}(t,S_{t},y_{t},k,I_{A}(t,k))}.$

Finally, using the fact that

$\partial_{k}B_{E}(t,M_{t},k_{t}^{*},\sigma)=\partial_{k}B_{A}(t,S_{t},y_{t},k_{t}^{*},\sigma)=-\frac{1}{2}$

we complete the desired proof. ∎

In order to compute the limit of ATMIV skew, we need to identify the leading
order terms in the numerator of the formulas obtained in Proposition 1, for
which the next result will be crucial.

######  Proposition 2.

Assume Hypotheses 1-3. Then, for all $t\leq T$,

$\displaystyle\begin{split}\mathbb{E}_{t}\left(\int_{t}^{T}G(s,S_{s},k,v^{\prime}_{s})\Lambda_{s}^{\prime}ds\right)&=\mathbb{E}_{t}\left(G(t,S_{t},k,v^{\prime}_{t})J_{t}^{\prime}\right)\\\
&+\mathbb{E}_{t}\left(\frac{1}{2}\int_{t}^{T}\partial_{xxx}^{3}G(s,S_{s},k,v^{\prime}_{s})J_{s}^{\prime}\Lambda_{s}^{\prime}ds\right)\\\
&+\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{x}G(s,S_{s},k,v^{\prime}_{s})\sigma_{s}D^{-}J_{s}^{\prime}ds\right),\\\
\mathbb{E}_{t}\left(\int_{t}^{T}G(s,M_{s},k,v_{s})\Lambda_{s}ds\right)&=\mathbb{E}_{t}\left(G(t,M_{t},k,v_{t})J_{t}\right)\\\
&+\mathbb{E}_{t}\left(\frac{1}{2}\int_{t}^{T}\partial_{xxx}^{3}G(s,M_{s},k,v_{s})J_{s}\Lambda_{s}ds\right)\\\
&+\mathbb{E}_{t}\left(\int_{t}^{T}\partial_{x}G(s,M_{s},k,v_{s})\phi_{s}D^{-}J_{s}ds\right),\end{split}$

where $G(t,x,k,\sigma)=\partial_{k}H(t,x,k,\sigma)$,
$J_{t}=\int_{t}^{T}\Lambda_{s}ds$,
$J^{\prime}_{t}=\int_{t}^{T}\Lambda^{\prime}_{s}ds$,
$D^{-}J^{\prime}_{s}=\int_{s}^{T}D_{s}^{W}\Lambda^{\prime}_{r}dr$ and
$D^{-}J_{s}=\int_{s}^{T}D_{s}^{W}\Lambda_{r}dr$.

###### Proof.

We only prove the second part of the theorem since the first part follows by
similar arguments. Applying Theorem 3 to the function
$\partial_{k}H(t,M_{t},k,v_{t})\int_{t}^{T}\Lambda_{s}ds$, we obtain that

$\displaystyle\int_{t}^{T}G(s,M_{s},k,v_{s})\Lambda_{s}ds=G(t,M_{t},k,v_{t})J_{t}$
$\displaystyle\qquad+\int_{t}^{T}\left(\partial_{s}G(s,M_{s},k,v_{s})+\frac{v_{s}^{2}}{2(T-s)v_{s}}\partial_{v}G(s,M_{s},k,v_{s})\right)J_{s}ds$
$\displaystyle\qquad+\int_{t}^{T}\partial_{x}G(s,M_{s},k,v_{s})J_{s}\phi_{s}dW_{s}$
$\displaystyle\qquad-\int_{t}^{T}\partial_{v}G(s,M_{s},k,v_{s})J_{s}\frac{\phi_{s}^{2}}{2(T-s)v_{s}}ds+\int_{t}^{T}\partial_{vx}^{2}G(s,M_{s},k,v_{s})J_{s}\Lambda_{s}\frac{1}{2(T-s)v_{s}}ds$
$\displaystyle\qquad+\int_{t}^{T}\partial_{x}G(s,M_{s},k,v_{s})\phi_{s}D^{-}J_{s}ds+\frac{1}{2}\int_{t}^{T}\phi_{s}^{2}\partial_{xx}^{2}G(s,M_{s},k,v_{s})J_{s}ds.$

Next, equations (5) and (10) imply that

$\displaystyle\partial_{s}G(s,M_{s},k,v_{s})+\frac{1}{2}v_{s}^{2}\partial^{2}_{xx}G(s,M_{s},k,v_{s})=0,$
$\displaystyle\partial_{xx}^{2}G(s,M_{s},k,v_{s})=\frac{\partial_{v}G(s,M_{s},k,v_{s})}{v_{s}(T-s)}.$

Using the above equations we get that

$\displaystyle\begin{split}&\int_{t}^{T}G(s,M_{s},k,v_{s})\Lambda_{s}ds=G(t,M_{t},k,v_{t})J_{t}+\int_{t}^{T}\partial_{x}G(s,M_{s},k,v_{s})J_{s}\phi_{s}dW_{s}\\\
&\qquad+\frac{1}{2}\int_{t}^{T}\partial_{xxx}^{3}G(s,M_{s},k,v_{s})J_{s}\Lambda_{s}ds+\int_{t}^{T}\partial_{x}G(s,M_{s},k,v_{s})\phi_{s}D^{-}J_{s}ds.\end{split}$

Finally, taking conditional expectations and noticing that by Lemma 1 and
Hypotheses 1 and 3, all conditional expectations are finite, we complete the
desired proof.∎

##  4 Proof of Theorem 1

###  4.1 Proof of (6) in Theorem 1: ATM implied volatility level

This section is devoted to the proof of (6) in Theorem 1. We only show the
result for $I_{A}(0,k^{*})$ since the proof for $I_{E}(0,k^{*})$ follows along
the same lines. We start proving the result for the implied volatility in the
uncorrelated case ($\rho=0$) that we denote by $I_{A}^{0}(t,k)$.

####  4.1.1 The uncorrelated case

We aim to apply the decomposition for the option price obtained in Theorem 2.
Observe that

$D_{s}^{W}\phi_{r}^{2}dr=\frac{(T-r)^{2}}{T^{2}}D_{s}^{W}\sigma_{r}^{2}=\frac{(T-r)^{2}}{T^{2}}2\sigma_{r}D_{s}^{W}\sigma_{r}=\frac{(T-r)^{2}}{T^{2}}2\sigma_{r}\rho
D_{s}^{W^{\prime}}\sigma_{r}.$

Thus, if $\rho=0$, the decomposition formula give us that
$V_{0}^{A}=\mathbb{E}(B_{E}(0,M_{0},k,v_{0}))$. Then the ATMIV satisfies that

$\begin{split}&I_{A}^{0}(0,k^{*})=(B_{A})^{-1}(0,S_{0},y_{0},k^{*},V_{0}^{A})=\mathbb{E}\left(B_{A}^{-1}(0,S_{0},y_{0},k^{*},\mathbb{E}B_{E}(0,M_{0},k^{*},v_{0}))\right)\\\
&\quad=\mathbb{E}\left(B_{A}^{-1}(0,S_{0},y_{0},k^{*},\mathbb{E}B_{E}(0,M_{0},k^{*},v_{0}))\pm
B_{A}^{-1}(0,S_{0},y_{0},k^{*},B_{E}(0,M_{0},k^{*},v_{0}))\right)\\\
&\quad=\mathbb{E}\left(B_{A}^{-1}(0,S_{0},y_{0},k^{*},\Phi_{0})-B_{A}^{-1}(0,S_{0},y_{0},k^{*},\Phi_{T})\right)+\sqrt{3}\mathbb{E}(v_{0})\\\
&\quad=\sqrt{3}\mathbb{E}\left(B_{E}^{-1}(0,M_{0},k^{*},\Phi_{0})-B_{E}^{-1}(0,M_{0},k^{*},\Phi_{T})\right)+\sqrt{3}\mathbb{E}(v_{0}),\end{split}$

where
$\Phi_{r}:=\mathbb{E}_{r}\left(B_{E}\left(0,M_{0},k^{*},v_{0}\right)\right)$.
The last two lines follow from the observation that
$B_{A}^{-1}(t,S_{t},y_{t},k^{*}_{t},B_{E}(t,M_{t},k^{*}_{t},\sigma))=\frac{T\sqrt{3}}{T-t}\sigma$.

Observe that as $\rho=0$, the Brownian motions $W$ and $W^{\prime}$ are
independent. Thus,
$\Phi_{r}=\mathbb{E}\left(B_{E}\left(0,M_{0},k^{*},v_{0}\right)|\mathcal{F}_{r}^{W^{\prime}}\right)$
and $(\Phi_{r})_{r\geq 0}$ is a martingale wrt to the filtration
$(\mathcal{F}^{W^{\prime}}_{r})_{r\geq 0}$. By the martingale representation
theorem, there exists a square integrable and
$\mathcal{F}^{W^{\prime}}$-adapted process $(U_{r})_{r\geq 0}$ such that

$\Phi_{r}=\Phi_{0}+\int_{0}^{r}U_{s}dW_{s}^{\prime}.$

A direct application of the classical Itô’s formula gives

$\begin{split}&\mathbb{E}\left(B_{E}^{-1}(0,M_{0},k^{*},\Phi_{0})-B_{E}^{-1}(0,M_{0},k^{*},\Phi_{T})\right)\\\
&=-\mathbb{E}\left(\int_{0}^{T}(B_{E}^{-1})^{\prime}(0,M_{0},k^{*},\Phi_{s})U_{s}dW_{s}^{\prime}+\frac{1}{2}\int_{0}^{T}(B_{E}^{-1})^{\prime\prime}(0,M_{0},k^{*},\Phi_{s})U_{s}^{2}ds\right),\end{split}$

where $(B_{E}^{-1})^{\prime}$ and $(B_{E}^{-1})^{\prime\prime}$ denote,
respectively, the first and second derivative of $B_{E}^{-1}$ with respect to
$\sigma$. A direct computation gives

$B_{E}^{-1}(0,M_{0},k^{*},\Phi_{s})=\frac{\Phi_{s}\sqrt{2\pi}}{\sqrt{T}}\quad\text{and}\quad((B_{E})^{{}^{-}1})^{\prime\prime}(0,M_{0},k^{*},\Phi_{s})=0,$

which leads to

$I_{A}^{0}(0,k^{*})=\sqrt{3}\mathbb{E}(v_{0}).$

By continuity, we conclude that

$\lim_{T\rightarrow 0}I_{A}^{0}(0,k^{*})=\sigma_{0},$ (11)

which proves (6) in the uncorrelated case.

####  4.1.2 The correlated case

Using similar ideas as in the uncorrelated case we get that

$\begin{split}I_{A}(0,k^{*})&=(B_{A})^{-1}(0,S_{0},y_{0},k^{*},V_{0}^{A})\\\
&=\mathbb{E}\left((B_{A})^{-1}(0,S_{0},y_{0},k^{*},\Gamma_{T})\pm(B_{A})^{-1}(0,S_{0},y_{0},k^{*},\Gamma_{0})\right)\\\
&=\mathbb{E}\left((B_{A})^{-1}(0,S_{0},y_{0},k^{*},\Gamma_{T})-(B_{A})^{-1}(0,S_{0},y_{0},k^{*},\Gamma_{0})\right)+I_{A}^{0}(0,k^{*})\\\
&=\sqrt{3}\mathbb{E}\left(B_{E}^{-1}(0,M_{0},k^{*},\Gamma_{T})-B_{E}^{-1}(0,M_{0},k^{*},\Gamma_{0})\right)+I_{A}^{0}(0,k^{*}),\end{split}$

where
$\Gamma_{s}:=\mathbb{E}[B_{E}(0,M_{0},k^{*},v_{0})]+\frac{\rho}{2}\mathbb{E}\left(\int_{0}^{s}H(r,M_{r},k^{*}_{t},v_{r})\Lambda_{r}dr\right)$.

A direct application of Itô’s formula gives

$\begin{split}I_{A}(0,k^{*})&=I_{A}^{0}(0,k^{*})+\mathbb{E}\left(\int_{0}^{T}(B_{E}^{-1})^{\prime}(0,M_{0},k^{\ast},\Gamma_{s})H(s,M_{s},k^{*}_{t},v_{s})\Lambda_{s}ds\right)\\\
&=I_{A}^{0}(0,k^{*})+\frac{\sqrt{2\pi}}{\sqrt{T}}\mathbb{E}\left(\int_{0}^{T}H(s,M_{s},k^{*}_{t},v_{s})\Lambda_{s}ds\right).\end{split}$

Next, Hypotheses 1 and 3 together with Lemma 1 imply that

$\begin{split}&\bigg{|}\mathbb{E}\left(\int_{0}^{T}H(s,M_{s},k^{*},v_{s})\Lambda_{s}ds\right)\bigg{|}\leq
C\mathbb{E}\left(\left(\int_{0}^{T}\phi_{r}^{2}dr\right)^{-1}\left(\int_{0}^{T}|\Lambda_{s}|ds\right)\right)\\\
&\qquad\leq
C\left(\int_{0}^{T}\frac{(T-r)^{2}}{T^{2}}dr\right)^{-1}\int_{0}^{T}\frac{(T-s)^{3}}{T^{3}}\int_{s}^{T}\mathbb{E}(|D_{s}^{W^{\prime}}\sigma_{r}|)drds\\\
&\qquad\leq
C\frac{1}{T^{4}}\int_{0}^{T}(T-s)^{3}\int_{s}^{T}(r-s)^{H-\frac{1}{2}}drds\\\
&\qquad=CT^{H+\frac{1}{2}}.\end{split}$

Thus, we conclude that
$\frac{1}{\sqrt{T}}\mathbb{E}\left(\int_{0}^{T}H(s,M_{s},k^{*}_{t},v_{s})\Lambda_{s}ds\right)\to
0$ as $T\to 0$. Finally, using (11), we get that

$I_{A}(0,k^{*})\to\sigma_{0}$

as $T\to 0$, which concludes the proof of (6).

###  4.2 Proof of (7) in Theorem 1: ATMIV skew

We provide the proof for the Asian call option case. The result for the
European call follows by the same argument.

Appealing to Propositions 1 and 2 we have that

$\begin{split}\partial_{k}I_{A}(0,k^{*})&=\frac{1}{\partial_{\sigma}B_{A}(0,S_{0},y_{0},k^{*},I_{A}(0,k^{*}))}\bigg{\\{}\mathbb{E}\left(G(0,M_{0},k^{*},v_{0})J_{0}\right)\\\
&\qquad+\mathbb{E}\left(\frac{1}{2}\int_{0}^{T}\partial_{xxx}^{3}G(s,M_{s},k^{*},v_{s})J_{s}\Lambda_{s}ds\right)\\\
&\qquad+\mathbb{E}\left(\int_{0}^{T}\partial_{x}G(s,M_{s},k^{*},v_{s})\phi_{s}D^{-}J_{s}ds\right)\bigg{\\}}.\end{split}$
(12)

We start bounding the second term in (12). Using Hypotheses 1 and 3 together
with Cauchy-Schwarz inequality we get that

$\begin{split}&\mathbb{E}(|J_{s}\Lambda_{s}|)\\\ &\leq
C\frac{(T-s)}{T^{6}}\mathbb{E}\left(\left(\int_{s}^{T}(T-r)^{2}|D_{s}^{W^{\prime}}\sigma_{r}|dr\right)\left(\int_{s}^{T}(T-u)^{3}\int_{u}^{T}|D_{u}^{W^{\prime}}\sigma_{r}|drdu\right)\right)\\\
&\leq
C\frac{(T-s)}{T^{6}}\sqrt{\mathbb{E}\left(\left(\int_{s}^{T}(T-r)^{2}|D_{s}^{W^{\prime}}\sigma_{r}|dr\right)^{2}\right)\mathbb{E}\left(\left(\int_{s}^{T}(T-u)^{3}\int_{u}^{T}|D_{u}^{W^{\prime}}\sigma_{r}|drdu\right)^{2}\right)}\\\
&\leq
C\frac{(T-s)}{T^{6}}\sqrt{(T-s)^{5}\int_{s}^{T}\mathbb{E}\left(|D_{s}^{W^{\prime}}\sigma_{r}|^{2}\right)dr(T-s)^{8}\int_{s}^{T}\int_{u}^{T}\mathbb{E}\left(|D_{u}^{W^{\prime}}\sigma_{r}|^{2}\right)drdu}\\\
&\leq C\frac{(T-s)^{2H+8}}{T^{6}}.\end{split}$

Then, using Lemma 1 we conclude that

$\begin{split}&\bigg{|}\mathbb{E}\left(\frac{1}{2}\int_{0}^{T}\partial_{xxx}^{3}G(s,M_{s},k,v_{s})J_{s}\Lambda_{s}ds\right)\bigg{|}\\\
&\qquad\leq
C\frac{1}{T^{3}}\int_{0}^{T}\frac{(T-s)^{2H+8}}{T^{6}}dt=CT^{2H}.\end{split}$

We next bound the third term in (12). We have that

$\displaystyle\begin{split}\int_{s}^{T}D_{s}^{W}\Lambda_{r}dr&=\int_{s}^{T}D_{s}^{W}\left(\phi_{r}\int_{r}^{T}D_{r}^{W}\phi_{u}^{2}du\right)dr\\\
&=\int_{s}^{T}\left(\left(D_{s}^{W}\phi_{r}\right)\int_{r}^{T}D_{r}^{W}\phi_{u}^{2}du+\phi_{r}\int_{r}^{T}D_{s}^{W}D_{r}^{W}\phi_{u}^{2}du\right)dr,\end{split}$

where

$D_{s}^{W}D_{r}^{W}\phi_{u}^{2}=2(D_{s}^{W}\phi_{u}D_{r}^{W}\phi_{u}+\phi_{u}D_{s}^{W}D_{r}^{W}\phi_{u}).$

Hypothesis 1 implies that

$|D_{s}^{W}D_{r}^{W}\phi_{u}^{2}|\leq
C\frac{(T-u)^{2}}{T^{2}}\left(|D_{s}^{W^{\prime}}\sigma_{u}|\,|D_{r}^{W^{\prime}}\sigma_{u}|+|D_{s}^{W^{\prime}}D_{r}^{W^{\prime}}\sigma_{u}|\right).$

Then, appealing to Hypothesis 3, we get that

$\begin{split}\mathbb{E}\left(\bigg{|}\phi_{r}\int_{r}^{T}D_{s}^{W}D_{r}^{W}\phi_{u}^{2}du\bigg{|}\right)&\leq
C\frac{(T-r)^{3}}{T^{3}}\int_{r}^{T}(u-r)^{H-\frac{1}{2}}(u-s)^{H-\frac{1}{2}}du\\\
&\leq C\frac{(T-r)^{3}}{T^{3}}(T-s)^{2H}\end{split}$

and

$\begin{split}\mathbb{E}\left(\bigg{|}D_{s}^{W}\phi_{r}\int_{r}^{T}D_{r}^{W}\phi_{u}^{2}du\bigg{|}\right)&\leq
C\frac{(T-r)^{3}}{T^{3}}\mathbb{E}\left(|D_{s}^{W^{\prime}}\sigma_{r}|\int_{r}^{T}|D_{r}^{W^{\prime}}\sigma_{u}|du\right)\\\
&\leq
C\frac{(T-r)^{3}}{T^{3}}(r-s)^{H-\frac{1}{2}}(T-r)^{H+\frac{1}{2}}.\end{split}$

Therefore, we conclude that

$\begin{split}\mathbb{E}\left(\bigg{|}\int_{s}^{T}D_{s}^{W}\Lambda_{r}dr\bigg{|}\right)&\leq
C\int_{s}^{T}\frac{(T-r)^{3}}{T^{3}}\left((r-s)^{H-\frac{1}{2}}(T-s)^{H+\frac{1}{2}}+(T-s)^{2H}\right)dr\\\
&\leq C\frac{(T-s)^{2H+4}}{T^{3}}.\end{split}$

Finally, using Lemma 1 we obtain that the third term in (12) satisfies that

$\begin{split}\bigg{|}\mathbb{E}\left(\int_{0}^{T}\partial_{x}G(s,M_{s},k,v_{s})\phi_{s}D^{-}J_{s}ds\right)\bigg{|}&\leq
C\frac{1}{T^{2}}\int_{0}^{T}\frac{(T-s)^{5+2H}}{T^{4}}ds\\\
&=CT^{2H}.\end{split}$

Taking into account the relation
$\partial_{\sigma}B_{A}(0,S_{0},y_{0},k^{*},\sigma)=\frac{1}{\sqrt{3}}\partial_{\sigma}B_{E}(0,S_{0},k^{*},\sigma)$
we get that

$\partial_{\sigma}B_{A}(0,S_{0},y_{0},k^{*},\sigma)=\frac{\sqrt{T}}{\sqrt{6\pi}}.$

Thus, as

$\lim_{T\rightarrow 0}T^{\max(\frac{1}{2}-H,0)}\frac{1}{\sqrt{T}}T^{2H}=0,$

the second and third terms in (12) will contribute as $0$ in the limit (8).

We finally study the first term in (12). Direct differentiation give us that

$G(0,M_{0},k^{*},v_{0})=\frac{1}{2\sqrt{2\pi}v_{0}^{3}T^{3/2}}.$

Thus, by (12), we get that

$\begin{split}&\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\partial_{k}I_{A}(0,k^{*})\\\
&=\sqrt{3}\rho\lim_{T\to
0}T^{\max(\frac{1}{2}-H,0)}\frac{1}{T^{5}}\mathbb{E}\left(v_{0}^{-3}\int_{0}^{T}\sigma_{r}(T-r)\int_{r}^{T}\sigma_{u}(T-u)^{2}D_{r}^{W^{\prime}}\sigma_{u}dudr\right).\end{split}$

By continuity, it holds that $v_{0}^{-3}$ converges towards
$\sigma_{0}^{-3}\sqrt{3}^{3}$ a.s. as $T\rightarrow 0$. On the other hand, by
Hypotheses 1 and (2), we have that

$\begin{split}&\bigg{|}\mathbb{E}\left(\int_{0}^{T}\sigma_{r}(T-r)\int_{r}^{T}\sigma_{u}(T-u)^{2}D_{r}^{W^{\prime}}\sigma_{u}dudr\right)\bigg{|}\\\
&\leq C\int_{0}^{T}(T-r)^{3}\int_{r}^{T}(u-r)^{H-\frac{1}{2}}dudr\\\ &\leq
CT^{4+H+\frac{1}{2}}.\end{split}$

Thus, by dominated convergence, we conclude the proof of (8).

##  5 Numerical analysis

In this section we apply the theoretical results presented earlier to some
examples of stochastic volatility models. We justify our findings with
numerical simulations.

###  5.1 The SABR model

We consider the SABR stochastic volatility model with skewness parameter 1,
which is the most common case from a practical point of view. This corresponds
to equation (1), where $S_{t}$ denotes the forward price of the underlying
asset and

$d\sigma_{t}=\alpha\sigma_{t}dW_{t}^{\prime},\qquad\sigma_{t}=\sigma_{0}e^{\alpha
W^{\prime}_{t}-\frac{\alpha^{2}}{2}t}.$

where $\alpha>0$ is the volatility of volatility.

Notice that this model does not satisfy Hypothesis 1, so a truncation argument
is needed in order to check that Theorem 1 is true for this model. We skip it
here for the sake of simplicity since it is identical to the one presented in
Alós, Nualart, Pravosud [5].

For $r\leq t$, we have that $D_{r}^{W^{\prime}}\sigma_{t}=\alpha\sigma_{t}$
and $\mathbb{E}\left(D_{r}^{W^{\prime}}\sigma_{t}\right)=\alpha\sigma_{0}$.
Therefore, applying Theorem 1 we conclude that

$\displaystyle\begin{split}\lim_{T\to
0}\partial_{k}I_{A}(0,k^{*})&=\frac{3}{5}\rho\alpha,\\\ \lim_{T\to
0}\partial_{k}I_{E}(0,k^{*})&=\frac{1}{2}\rho\alpha.\end{split}$

We next proceed with some numerical simulations using the following parameters

$S_{0}=10,\,T=\frac{1}{252},\,dt=\frac{T}{50},\,\alpha=0.5,\,\rho=-0.3,\,\sigma_{0}=(0.1,0.2,\dots,1.4).$

In order to get estimates of an Asian call option we use antithetic variates.
The estimator of the price is defined as follows

$\displaystyle\begin{split}\hat{V}_{sabr}&=\frac{\frac{1}{N}\sum_{i=1}^{N}V_{T}^{i}+\frac{1}{N}\sum_{i=1}^{N}V_{T}^{i,A}}{2},\end{split}$
(13)

where $N=2000000$ and the sub-index $A$ denotes the value of an Asian call
option computed on the antithetic trajectory of a Monte Carlo path.

In order to retrieve the implied volatility we use method presented in Jaeckel
[13]. For the estimation of the skew, we use the following expression which
allows us to avoid a finite difference method

$\displaystyle\begin{split}\partial_{k}\hat{I}^{A}(0,k^{*})&=\frac{-\partial_{k}B_{A}(0,X_{0},y_{0},k^{*},I_{A}(0,k^{*}))-\mathbb{E}\left(1_{A_{T}\geq
k^{*}}\right)}{\partial_{\sigma}B_{A}(0,X_{0},y_{0},k^{*},I_{A}(0,k^{*}))}=\frac{\frac{1}{2}-\mathbb{E}\left(1_{A_{T}\geq
k^{*}}\right)}{\sqrt{\frac{T}{6\pi}}},\\\
\partial_{k}\hat{I}^{E}(0,k^{*})&=\frac{-\partial_{k}B_{E}(0,S_{0},k^{*},I_{E}(0,k^{*}))-\mathbb{E}\left(1_{S_{T}\geq
k^{*}}\right)}{\partial_{\sigma}B_{E}(0,S_{0},k^{*},I_{E}(0,k^{*}))}=\frac{\frac{1}{2}-\mathbb{E}\left(1_{S_{T}\geq
k^{*}}\right)}{\sqrt{\frac{T}{2\pi}}}.\end{split}$ (14)

We use equation (14) in order to get estimates of the skew. In Figure 1 we
present the results of a Monte Carlo simulation which aims to evaluate
numerically the level and the skew of the at-the-money implied volatility of
an Asian call option under the SABR model. And in Figure 2 we do the same, but
for the European call option. We observe that all the numerical results fit
the theoretical ones.

|
---|---
Figure 1: At-the-money level and the skew of the Implied Volatility of an Asian call under the SABR model. |
---|---
Figure 2: At-the-money level and the skew of the Implied Volatility of the
European call option under the SABR model.

###  5.2 The fractional Bergomi model

The fractional Bergomi stochastic volatility model asssumes equation (1) with

$\begin{split}\sigma_{t}^{2}&=\sigma_{0}^{2}e^{v\sqrt{2H}Z_{t}-\frac{1}{2}v^{2}t^{2H}},\\\
Z_{t}&=\int_{0}^{t}(t-s)^{H-\frac{1}{2}}dW^{\prime}_{s},\end{split}$

where $H\in(0,1)$ and $v>0$.

As for the SABR model, a truncation argument as in Alós, Nualart, Pravosud [5]
is needed in order to apply the results in the previous sections, as
Hypothesis 1 is not satisfied. Moreover, for $r\leq u$, we have

$\begin{split}D_{r}^{W^{\prime}}\sigma_{u}&=\frac{1}{2}\sigma_{u}v\sqrt{2H}(u-r)^{H-\frac{1}{2}},\\\
\mathbb{E}(D_{r}^{W^{\prime}}\sigma_{u})&=e^{-\frac{1}{8}v^{2}u^{2H}}\frac{1}{2}\sigma_{0}v\sqrt{2H}(u-r)^{H-\frac{1}{2}},\end{split}$

which gives

$\begin{split}\lim_{T\to
0}\partial_{k}I_{A}(0,k^{*})&=\begin{cases}0\quad&\text{if}\quad
H>\frac{1}{2}\\\ \frac{3\rho v}{10}\quad&\text{if}\quad
H=\frac{1}{2},\end{cases},\\\ \lim_{T\to
0}\partial_{k}I_{E}(0,k^{*})&=\begin{cases}0\quad&\text{if}\quad
H>\frac{1}{2}\\\ \frac{\rho v}{4}\quad&\text{if}\quad
H=\frac{1}{2}.\end{cases}\end{split}$ (15)

and for $H<\frac{1}{2}$

$\begin{split}\lim_{T\to
0}T^{\frac{1}{2}-H}\partial_{k}I_{A}(0,k^{*})&=\frac{288\rho
v\sqrt{2H}}{(2H+9)\left(8H^{3}+36H^{2}+46H+15\right)},\\\ \lim_{T\to
0}T^{\frac{1}{2}-H}\partial_{k}I_{E}(0,k^{*})&=\frac{2\rho
v\sqrt{2H}}{\left(3+4H(2+H)\right)}.\end{split}$ (16)

The parameters used for the Monte Carlo simulation are the following

$S_{0}=100,\,T=0.001,\,dt=\frac{T}{50},\,H=(0.4,0.7),\,v=0.5,\,\rho=-0.3,\,\sigma_{0}=(0.1,0.2,\dots,1.4).$

In order to get estimates of the price of an Asian call option under the
fractional Bergomi model we use antithetic variates presented in equation
(13). Then, the level of at-the-money implied volatility of an Asian and
European call options are presented on Figures 3 and 4, respectively. One can
see that the result is independent of $H$.

|
---|---
(a) H=0.4 | (b) H=0.7
Figure 3: At-the-money level of the IV of an Asian call under fractional Bergomi model. |
---|---
(a) H=0.4 | (b) H=0.7
Figure 4: At-the-money level of the IV of the European call under fractional
Bergomi model.

On Figures 5 and 6 we present the ATM implied volatility skew as a function of
maturity of an Asian and European call options, respectively, for two
different values of $H$, where we observe the blow up to $-\infty$ for the
case $H=0.4$. Equation (15) suggests that the theoretical values of the slope
of the at-the-money skew (as a function $\alpha T^{\beta}$) in the cases of
Asian and European call options with $H<\frac{1}{2}$ are $-0.0497$ and
$-0.0336$, respectively. We conclude that theoretical results are in line with
the observed numbers.

|
---|---
(a) H=0.4, $\sigma_{0}=0.3$ | (b) H=0.7, $\sigma_{0}=0.3$
Figure 5: At-the-money IV skew of an Asian call as a function of $T$ under fractional Bergomi model. |
---|---
(a) H=0.4, $\sigma_{0}=0.3$ | (b) H=0.7, $\sigma_{0}=0.3$
Figure 6: At-the-money IV skew of the European call as a function of $T$ under
fractional Bergomi model.

Due to the blow up of the at-the-money implied volatility skew of an Asian and
European call options when $H<\frac{1}{2}$ we also plot (as a function of
$\sigma_{0}$) the quantities
$T^{\frac{1}{2}-H}\partial_{k}\hat{I}^{A}(0,k^{*})$ and
$T^{\frac{1}{2}-H}\partial_{k}\hat{I}^{E}(0,k^{*})$ for $H=0.4$ as well as
$\partial_{k}\hat{I}^{A}(0,k^{*})$ and $\partial_{k}\hat{I}^{E}(0,k^{*})$ for
$H=0.7$ for fixed value of $T=0.001$. The result is presented at Figures 7 and
8. We see that numerical estimates agree with the presented theoretical
findings.

|
---|---
(a) H=0.4 | (b) H=0.7
Figure 7: At-the-money IV skew of an Asian call as a function of $\sigma_{0}$ under fractional Bergomi model |
---|---
(a) H=0.4 | (b) H=0.7
Figure 8: At-the-money IV skew of the European call as a function of
$\sigma_{0}$ under fractional Bergomi model

####  5.2.1 Local volatility model

We consider the local volatility model

$dS_{t}=\sigma(S_{t})dW_{t},$

where $\sigma\in C^{2}_{b}$ (bounded with bounded first and second
derivatives) and $\sigma(x)\geq c>0$, for all $x$. We apply Theorem 1 when
$S_{t}$ follows this model with $\rho=0$ and $\sigma_{t}=\sigma(S_{t})$. Under
the above assumptions it is easy to see that all the hypotheses are satisfied
with $H=\frac{1}{2}$. Thus, for the ATMIV level, we directly see that the
limit is equal to $\sigma(S_{0})$. For the ATMIV skew, we need to compute
$D_{r}\sigma(S_{t})$. We have for $r\leq u$,

$\begin{split}D^{W}_{r}\sigma(S_{u})=\sigma^{\prime}(S_{u})D^{W}_{r}S_{u}=\sigma^{\prime}(S_{u})\left(\sigma(S_{r})+\int_{r}^{u}D_{r}^{W}\sigma(S_{s})dW_{s}\right).\end{split}$

In particular,

$\mathbb{E}\left(D_{r}\sigma(S_{u})\right)=\mathbb{E}\left(\sigma^{\prime}(S_{u})\sigma(S_{r})\right).$

This can be written as

$\begin{split}\mathbb{E}\left(D_{r}\sigma(S_{u})\right)=\sigma^{\prime}(S_{0})\sigma(S_{0})&+\mathbb{E}\left((\sigma^{\prime}(S_{u})-\sigma^{\prime}(S_{0}))\sigma(S_{r})\right)\\\
&+\sigma^{\prime}(S_{0})\mathbb{E}\left((\sigma(S_{r})-\sigma(S_{0}))\right).\end{split}$

Then, using the mean value theorem and the fact that $S_{t}$ has Hölder
continuous sample paths of any order $\gamma<\frac{1}{2}$, we see that the
last two terms of the last display will not contribute in the limit (7) and
(8). Thus, we get

$\displaystyle\begin{split}\lim_{T\to
0}\partial_{k}I_{A}(0,k^{*})=\frac{3}{5}\sigma^{\prime}(S_{0}),\\\ \lim_{T\to
0}\partial_{k}I_{E}(0,k^{*})=\frac{1}{2}\sigma^{\prime}(S_{0}).\end{split}$

###  5.3 Approximations of implied volatility

In this last section we study numerically the adequacy of the linear
approximation of the implied volatility of an Asian and European call options
given by equation (9) in the case of the SABR and fractional Bergomi models.

####  5.3.1 Asian call option

We start our analysis by presenting the results for the case of an Asian call
option.

##### The SABR Model

We proceed with the following numerical experiment. We randomly sample the
parameters as $\sigma_{0}\sim U(0.2,0.8)$, $\alpha\sim U(0.3,1.5)$ and
$\rho\sim U(-0.9,0.9)$, where $U$ stands for the uniform distribution. We fix
$S_{0}=10$ and consider the following strikes

$K=(9.97,9.9743,9.9786,9.9829,9.9872,9.9915,9.9958,10.0001)$

and maturities $T=(0.01,0.1,0.5,1,2)$. We consider the narrow OTM range around
ATM region because due to the law of large numbers deep OTM quickly loose the
value and prices become indistinguishable from 0. An Asian call option is
priced using Monte Carlo with 100000 paths and discretization step 0.01 and
the IV is estimated using the same approach as discussed in Section 5.1. The
approximation accuracy of the Monte Carlo for the IV is computed using the
pointwise relative error with respect to the 95$\%$ Monte Carlo confidence
interval and is presented in Table 1.

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0
---|---|---|---|---|---|---|---|---
0.01 | 0.94 | 0.92 | 0.91 | 0.91 | 0.91 | 0.92 | 0.93 | 0.94
0.1 | 0.94 | 0.94 | 0.94 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95
0.50 | 1.02 | 1.02 | 1.02 | 1.02 | 1.02 | 1.0 | 1.0 |
1.0 | 1.1 | 1.1 | 1.10 | 1.1 | 1.1 | 1.1 | 1.1 | 1.0
2.0 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3 | 1.3
Table 1: Median percentage error wrt the 95% Monte Carlo confidence interval
for an Asian call IV under the SABR model.

We then compare the estimated IV with the approximation formula (9). We
compute in Table 2 the median relative percentage error, the 90$\%$ quantile
and the maximum of the relative percentage error of the Monte Carlo prices
computed across 2000 random parameter combinations. We consider the 90$\%$
quantile in order to take into account the fact that we might generate ’bad’
parameter combinations that may require more Monte Carlo samples to converge.
In order to help the visualization of these quantities we also plot the heat
map in Figure 9.

Overall, we can see that the suggested implied volatility approximation
performs well in the ATM region for short dated options. We achieve implied
volatility approximation accuracy comparable to the original Monte Carlo
simulation. However, the quality of it impairs as soon as we move in the
maturity ($T>0.5$) and moneyness dimension.

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.24 | 0.23 | 0.22 | 0.23 | 0.23 | 0.24 | 0.24 | 0.72
0.10 | 0.26 | 0.25 | 0.25 | 0.25 | 0.24 | 0.24 | 0.24 | 0.29
0.50 | 0.89 | 0.89 | 0.87 | 0.87 | 0.87 | 0.87 | 0.86 | 0.81
1.00 | 1.95 | 1.93 | 1.92 | 1.91 | 1.90 | 1.89 | 1.89 | 1.84
2.00 | 4.04 | 4.04 | 4.03 | 4.01 | 4.02 | 4.05 | 4.02 | 4.00
(a) Median % error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.57 | 0.54 | 0.51 | 0.51 | 0.51 | 0.53 | 0.53 | 1.18
0.10 | 0.71 | 0.69 | 0.66 | 0.64 | 0.63 | 0.62 | 0.61 | 0.63
0.50 | 3.53 | 3.50 | 3.47 | 3.45 | 3.44 | 3.41 | 3.41 | 3.36
1.00 | 6.94 | 6.92 | 6.91 | 6.90 | 6.90 | 6.86 | 6.82 | 6.79
2.00 | 12.76 | 12.75 | 12.71 | 12.67 | 12.67 | 12.58 | 12.55 | 12.54
(b) 90th quantile % Error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 5.17 | 1.87 | 1.19 | 0.95 | 1.07 | 1.13 | 1.20 | 1.93
0.10 | 2.26 | 2.04 | 1.84 | 1.67 | 1.53 | 1.42 | 1.36 | 1.35
0.50 | 6.17 | 5.94 | 5.80 | 5.68 | 5.64 | 5.62 | 5.61 | 5.56
1.00 | 10.78 | 10.66 | 10.63 | 10.61 | 10.60 | 10.59 | 10.58 | 10.55
2.00 | 18.65 | 18.58 | 18.52 | 18.46 | 18.40 | 18.36 | 18.31 | 18.25
(c) Maximum % error

Table 2: Approximation error of Asian call IV under the SABR model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 9: Accuracy of the approximation (9) for an Asian call under the SABR
model.

In order to explain why the approximation impairs with the maturity of an
option, we show a "typical" implied volatility surface in the simulated data
set at Figure 10. One can see that ATM level of the implied volatility
$I_{A}(0,k^{*})$ changes considerably with the increase of maturity leading to
the decrease in the quality of the linear approximation.

Figure 10: Typical surface of the IV of an Asian call under the SABR model.

##### fractional Bergomi Model

In the case of the fractional Bergomi model we randomly sample the parameters
of the model as $\sigma_{0}\sim U(0.2,0.8)$, $v\sim U(0.3,1.5)$ and $\rho\sim
U(-0.9,0.9)$. We keep the values of $S_{0}$, $K$ and $T$ identical to the one
in the SABR case. In order to investigate the influence of fractionalness of
the volatility process we consider two values of $H=\\{0.2,0.7\\}$. We price
the Asian call option using Monte Carlo with 100000 paths and discretization
step 0.01.

We start with the case $H=0.2$. The implied volatility approximation accuracy
using Monte Carlo is computed using the corresponding pointwise median
relative error with respect to the 95$\%$ Monte Carlo confidence interval. The
result is presented at Table 3.

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 1.07 | 1.01 | 0.97 | 0.94 | 0.93 | 0.93 | 0.94 | 0.95
0.10 | 0.96 | 0.96 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95
0.50 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | 0.97 | 0.97
1.00 | 1.00 | 1.00 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99
2.00 | 1.30 | 1.29 | 1.29 | 1.29 | 1.28 | 1.28 | 1.28 | 1.28
Table 3: Median percentage error wrt the 95% Monte Carlo confidence interval
for an Asian call IV under the fractional Bergomi (H=0.2) model.

Next, we look at the behaviour of the linear approximation of the implied
volatility. The methodology is the same as in the case of the SABR. The
results are presented in Table 4 and Figure 11. Overall, the conclusion is the
same as in the case of the SABR model. However, the approximation performs
very well at short maturity ($T\leq 0.1$) around ATM region and gets worse
with the increase of maturity. This impairment of the approximation is not
drastic, but the error is bigger than in Monte Carlo simulation. One can see
that the median error is bounded by 3.5 percent over the whole surface under
consideration. This behaviour becomes clear by looking at Figure 12

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.49 | 0.50 | 0.50 | 0.54 | 0.59 | 0.64 | 0.67 | 1.19
0.10 | 1.17 | 1.21 | 1.23 | 1.25 | 1.26 | 1.28 | 1.29 | 1.46
0.50 | 2.16 | 2.18 | 2.19 | 2.19 | 2.19 | 2.20 | 2.20 | 2.26
1.00 | 2.80 | 2.82 | 2.83 | 2.83 | 2.84 | 2.85 | 2.85 | 2.90
2.00 | 3.68 | 3.68 | 3.69 | 3.69 | 3.69 | 3.69 | 3.69 | 3.73
(a) Median % error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 1.41 | 1.28 | 1.25 | 1.24 | 1.29 | 1.33 | 1.34 | 1.93
0.10 | 2.61 | 2.64 | 2.68 | 2.67 | 2.69 | 2.72 | 2.72 | 2.93
0.50 | 4.92 | 4.93 | 4.96 | 4.98 | 5.00 | 5.00 | 5.02 | 5.07
1.00 | 6.50 | 6.51 | 6.52 | 6.52 | 6.52 | 6.52 | 6.51 | 6.57
2.00 | 8.72 | 8.74 | 8.74 | 8.74 | 8.74 | 8.74 | 8.74 | 8.77
(b) 90th quantile % Error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 12.39 | 6.28 | 2.45 | 2.19 | 2.02 | 2.06 | 2.11 | 2.95
0.10 | 4.37 | 4.36 | 4.35 | 4.34 | 4.34 | 4.34 | 4.34 | 4.68
0.50 | 7.68 | 7.67 | 7.67 | 7.66 | 7.66 | 7.67 | 7.68 | 7.86
1.00 | 10.67 | 10.65 | 10.63 | 10.62 | 10.61 | 10.59 | 10.57 | 10.69
2.00 | 13.55 | 13.55 | 13.54 | 13.53 | 13.52 | 13.51 | 13.51 | 13.57
(c) Maximum % error

Table 4: Approximation error of an Asian call IV under the fractional Bergomi (H=0.2) model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 11: Accuracy of the approximation (9) for an Asian call IV under
fractional Bergomi with $H=0.2$.  Figure 12: Typical surface of the IV of an
Asian call under the fractional Bergomi (H=0.2) model.

Finally, we repeat our analysis in the case of the fractional Bergomi model
with $H=0.7$. The results are presented in Tables 5 and 6, Figures 13 and 14.
One can see that the "typical" implied volatility surface in the sample is
very close to the flat one. The level changes quite slowly as we move in the
direction of moneyness and maturity. The curvature seems to be minimal if any.
As a result, the linear approximation given by equation (9) works quite well
in this case across different moneyness with maturity $T\leq 1$. This is quite
different from the behaviour that we observed in the case of the SABR and
fractional Bergomi with $H=0.2$.

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.91 | 0.90 | 0.90 | 0.90 | 0.91 | 0.92 | 0.93 | 0.94
0.10 | 0.92 | 0.92 | 0.92 | 0.93 | 0.93 | 0.93 | 0.94 | 0.94
0.50 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94 | 0.94
1.00 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95 | 0.95
2.00 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99 | 0.99
Table 5: Median percentage error wrt the 95% Monte Carlo confidence interval
for an Asian call IV under the fractional Bergomi (H=0.7) model.

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.32 | 0.30 | 0.28 | 0.27 | 0.27 | 0.26 | 0.27 | 0.74
0.10 | 0.39 | 0.36 | 0.33 | 0.31 | 0.31 | 0.31 | 0.30 | 0.46
0.50 | 0.61 | 0.62 | 0.64 | 0.65 | 0.68 | 0.72 | 0.74 | 0.82
1.00 | 1.34 | 1.36 | 1.37 | 1.42 | 1.42 | 1.42 | 1.45 | 1.50
2.00 | 3.28 | 3.27 | 3.25 | 3.25 | 3.25 | 3.27 | 3.32 | 3.38
(a) Median % error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 0.84 | 0.77 | 0.70 | 0.65 | 0.62 | 0.59 | 0.56 | 1.19
0.10 | 1.08 | 0.97 | 0.88 | 0.81 | 0.74 | 0.68 | 0.63 | 0.79
0.50 | 2.06 | 1.93 | 1.79 | 1.68 | 1.55 | 1.45 | 1.47 | 1.54
1.00 | 3.56 | 3.44 | 3.32 | 3.26 | 3.22 | 3.17 | 3.15 | 3.16
2.00 | 7.59 | 7.60 | 7.56 | 7.65 | 7.64 | 7.63 | 7.62 | 7.51
(b) 90th quantile % Error

Maturity/Strike | 9.9700 | 9.9743 | 9.9786 | 9.9829 | 9.9872 | 9.9915 | 9.9958 | 10.0001
---|---|---|---|---|---|---|---|---
0.01 | 2.10 | 1.69 | 1.32 | 1.20 | 1.04 | 0.99 | 0.94 | 1.85
0.10 | 3.77 | 3.37 | 2.97 | 2.56 | 2.15 | 1.75 | 1.35 | 1.35
0.50 | 6.16 | 5.58 | 5.01 | 4.45 | 3.89 | 3.34 | 2.79 | 2.54
1.00 | 9.04 | 8.37 | 7.72 | 7.07 | 6.43 | 5.80 | 5.17 | 5.08
2.00 | 17.74 | 16.94 | 16.15 | 15.37 | 14.60 | 13.84 | 13.08 | 12.42
(c) Maximum % error

Table 6: Approximation error of an Asian call IV under the fractional Bergomi (H=0.7) model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 13: Accuracy of the approximation (9) for an Asian call IV under
fractional Bergomi with $H=0.7$.  Figure 14: Typical surface of the IV of an
Asian call under the fractional Bergomi (H=0.7) model.

####  5.3.2 European call option

We continue our analysis by repeating the calculations from the previous
section in the case of the European call option. By contrast to the Asian
case, now we are using Monte Carlo with 200000 paths and discretization step
0.02, $K=(9.,9.1,9.2,\dots,10)$. All the other parameters and set up in
general are preserved.

##### The SABR Model

We start by looking at the European call option under the Bachelier model with
the SABR volatility. The implied volatility approximation accuracy using Monte
Carlo is presented in Table 7.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 5.01 | 5.02 | 5.11 | 5.13 | 5.33 | 5.39 | 5.51 | 5.83 | 6.10 | 1.26 | 0.66
0.10 | 7.13 | 7.47 | 7.59 | 7.70 | 7.32 | 6.10 | 4.13 | 2.06 | 1.02 | 0.71 | 0.68
0.50 | 4.93 | 4.22 | 3.57 | 2.78 | 2.12 | 1.63 | 1.28 | 1.05 | 0.89 | 0.81 | 0.78
1.00 | 2.73 | 2.37 | 2.09 | 1.80 | 1.58 | 1.39 | 1.23 | 1.10 | 1.01 | 0.94 | 0.91
2.00 | 2.16 | 2.03 | 1.92 | 1.80 | 1.68 | 1.56 | 1.47 | 1.40 | 1.33 | 1.28 | 1.24
Table 7: Median percentage error wrt the 95% Monte Carlo confidence interval
for the European call IV under the SABR model.

Next, we look at the behaviour of the implied volatility approximation. The
detailed results of the behaviour of different error types are presented in
Table 8 and Figure 15. Overall, the approximation performs very well around
ATM region and gets slightly worse with the increase of maturity. However, the
approximation becomes very poor at the short end as we move away from the ATM
region. This happens due to considerable curvature of the implied volatility
in this region. This behaviour can be clearly seen at Figure 16, where we
present "typical" surface for our simulated data set. As maturity increases,
the curvature considerably decreases leading to the improvement of the
approximation (9). Notice, that, by contrast to an Asian call option, ATM
level of the implied volatility is much more stable making the approximation
to work quite well as we increase the maturity of the option.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 79.81 | 78.02 | 75.87 | 72.65 | 68.73 | 63.17 | 54.73 | 41.38 | 15.11 | 0.50 | 0.12
0.10 | 45.08 | 40.44 | 34.55 | 26.61 | 17.64 | 10.22 | 5.74 | 3.15 | 1.56 | 0.68 | 0.36
0.50 | 18.00 | 15.85 | 13.73 | 11.79 | 9.59 | 7.63 | 5.95 | 4.34 | 2.98 | 2.01 | 1.64
1.00 | 17.40 | 15.76 | 14.42 | 12.59 | 10.99 | 9.32 | 7.54 | 6.04 | 4.56 | 3.62 | 3.27
2.00 | 18.97 | 17.86 | 16.57 | 15.07 | 13.58 | 11.97 | 10.36 | 8.97 | 7.64 | 6.76 | 6.31
(a) Median % error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 93.11 | 92.30 | 91.62 | 89.5 | 87.3 | 84.13 | 79.02 | 70.72 | 55.43 | 14.58 | 0.30
0.10 | 81.61 | 79.17 | 76.51 | 71.2 | 65.8 | 57.31 | 44.07 | 23.27 | 7.53 | 2.52 | 1.18
0.50 | 64.29 | 62.99 | 58.69 | 51.7 | 43.1 | 34.75 | 25.43 | 16.92 | 10.34 | 6.54 | 5.44
1.00 | 63.31 | 59.60 | 56.81 | 50.3 | 44.7 | 36.31 | 28.17 | 20.67 | 14.55 | 11.11 | 10.05
2.00 | 64.19 | 61.34 | 58.16 | 53.6 | 47.7 | 40.42 | 33.17 | 26.61 | 21.58 | 18.42 | 17.45
(b) 90th quantile % Error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 99.9 | 99.9 | 99.9 | 99.9 | 99.8 | 99.6 | 99.1 | 96.6 | 84.89 | 108.34 | 0.60
0.10 | 99.8 | 99.8 | 99.8 | 99.8 | 99.4 | 98.8 | 112.3 | 91.5 | 62.06 | 8.82 | 2.10
0.50 | 99.79 | 99.6 | 99.6 | 99.7 | 98.9 | 98.2 | 94.5 | 81.98 | 34.07 | 14.22 | 8.1
1.00 | 99.75 | 99.7 | 99.6 | 99.7 | 99.1 | 97.9 | 94.8 | 80.43 | 38.08 | 19.46 | 14.27
2.00 | 99.76 | 99.7 | 99.7 | 99.8 | 99.19 | 98.1 | 95.0 | 83.84 | 44.11 | 27.61 | 24.3
(c) Maximum % error

Table 8: Approximation error of the European call IV under the SABR model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 15: Accuracy of the approximation (9) for the European call under the
SABR model.  Figure 16: Typical surface of the IV of the European call under
the SABR model.

##### fractional Bergomi Model

We continue our analysis of the behaviour of the implied volatility
approximation of the European call under the fractional Bergomi model. We
start with the case of $H=0.2$ and present the accuracy of the Monte Carlo
pricing at Table 9.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 5.70 | 5.81 | 5.86 | 5.99 | 6.05 | 6.24 | 6.46 | 6.71 | 7.15 | 2.00 | 0.67
0.10 | 7.10 | 7.49 | 7.74 | 7.72 | 7.87 | 6.76 | 4.68 | 2.37 | 1.13 | 0.74 | 0.68
0.50 | 6.16 | 5.36 | 4.18 | 3.19 | 2.28 | 1.58 | 1.21 | 0.95 | 0.80 | 0.73 | 0.70
1.00 | 3.31 | 2.71 | 2.15 | 1.69 | 1.42 | 1.17 | 0.97 | 0.85 | 0.78 | 0.74 | 0.72
2.00 | 1.87 | 1.64 | 1.43 | 1.26 | 1.10 | 0.97 | 0.88 | 0.83 | 0.78 | 0.76 | 0.74
Table 9: Median percentage error wrt the 95% Monte Carlo confidence interval
for the European call IV under the fractional Bergomi (H=0.2) model.

Next, we move to the analysis of the behaviour of the implied volatility
approximation (9). The detailed results regarding different error types are
presented in Table 10 and Figure 18. To sum up, the behaviour of the implied
volatility approximation is close to the SABR model. Similarly, the curvature
of the implied volatility decreases considerably as options maturity
increases. The stability of the ATM implied volatility level gives the
improvement (except $T\leq 0.1$) in the quality of the linear approximation
for a wide variety of options across all strikes and maturities. This
behaviour can be clearly seen at Figure 17, where we present "typical" surface
in our data set.

Figure 17: Typical surface of the IV of the European call under the fractional
Bergomi (H=0.2) model.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 76.52 | 74.74 | 72.48 | 70.59 | 67.12 | 61.92 | 54.90 | 42.06 | 17.64 | 1.09 | 0.64
0.10 | 44.01 | 39.03 | 33.18 | 25.90 | 16.32 | 9.26 | 4.73 | 1.97 | 0.98 | 1.06 | 1.53
0.50 | 9.72 | 7.38 | 5.62 | 4.08 | 2.77 | 2.04 | 1.76 | 1.78 | 2.17 | 2.75 | 2.99
1.00 | 5.39 | 4.22 | 3.30 | 2.64 | 2.24 | 2.19 | 2.35 | 2.74 | 3.15 | 3.80 | 3.96
2.00 | 3.37 | 2.90 | 2.86 | 2.83 | 2.95 | 3.28 | 3.64 | 3.98 | 4.82 | 5.15 | 5.25
(a) Median % error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 94.49 | 93.43 | 92.8 | 91.17 | 90.03 | 87.50 | 83.44 | 76.84 | 61.84 | 17.52 | 1.5
0.1 | 81.38 | 78.6 | 75.2 | 71.6 | 66.11 | 57.02 | 43.55 | 21.90 | 5.21 | 3.30 | 3.8
0.5 | 57.95 | 51.8 | 42.5 | 34.3 | 23.15 | 13.6 | 8.73 | 6.95 | 6.28 | 7.02 | 7.3
1.00 | 39.11 | 30.8 | 22.8 | 15.8 | 12.06 | 9.58 | 8.80 | 8.09 | 8.60 | 9.4 | 9.6
2.00 | 20.47 | 16.7 | 14.4 | 12.9 | 11.63 | 10.87 | 10.67 | 11.20 | 11.98 | 12.57 | 12.9
(b) 90th quantile % Error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 99.9 | 99.9 | 99.95 | 99.86 | 99.97 | 99.97 | 99.81 | 99.73 | 98.30 | 77.47 | 2.8
0.10 | 99.8 | 99.8 | 113.99 | 99.81 | 99.74 | 97.15 | 97.02 | 96.83 | 63.78 | 6.46 | 5.8
0.50 | 99.8 | 98.8 | 95.42 | 96.90 | 92.12 | 98.48 | 79.16 | 48.12 | 11.34 | 11.06 | 11.4
1.00 | 94.7 | 99.2 | 96.14 | 95.76 | 95.07 | 72.91 | 32.94 | 15.38 | 14.98 | 14.8 | 14.7
2.00 | 97.4 | 99.4 | 88.06 | 88.41 | 67.31 | 38.39 | 19.24 | 19.54 | 19.58 | 19.50 | 19.3
(c) Maximum % error

Table 10: Approximation error of the European call IV under the fractional Bergomi (H=0.2) model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 18: Accuracy of the approximation (9) for the European call under
fractional Bergomi (H=0.2) model.

We conclude this section by looking at the approximation of the European call
implied volatility under the fractional Bergomi model with $H=0.7$. We present
the accuracy of the Monte Carlo at Table 11.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 4.26 | 4.35 | 4.27 | 4.24 | 4.89 | 4.39 | 4.87 | 4.92 | 3.79 | 0.86 | 0.66
0.10 | 5.88 | 5.99 | 6.68 | 6.41 | 5.67 | 4.25 | 2.30 | 1.03 | 0.71 | 0.64 | 0.66
0.50 | 4.93 | 3.95 | 2.91 | 2.10 | 1.45 | 1.07 | 0.86 | 0.74 | 0.68 | 0.67 | 0.67
1.00 | 2.76 | 2.23 | 1.74 | 1.37 | 1.14 | 0.95 | 0.83 | 0.75 | 0.72 | 0.70 | 0.70
2.00 | 1.86 | 1.62 | 1.41 | 1.22 | 1.07 | 0.96 | 0.89 | 0.84 | 0.80 | 0.78 | 0.77
Table 11: Median percentage error wrt the 95% Monte Carlo confidence interval
for the European call IV under the fractional Bergomi (H=0.7) model.

The detailed results regarding different error types are presented in Table 12
and Figure 20. Due to the equation (15) the implied volatility approximation
in this case is a horizontal plane leveled at $\sigma_{0}$. Visual
justification of the flat skew can be clearly seen at Figure 19, where we
present "typical" surface observed in our data set.

Figure 19: Typical surface of the IV of the European call under the fractional
Bergomi (H=0.7) model.

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 78.96 | 76.27 | 73.63 | 70.05 | 65.67 | 59.42 | 49.13 | 34.11 | 5.86 | 0.69 | 0.11
0.10 | 42.65 | 36.66 | 29.48 | 20.49 | 11.22 | 6.98 | 4.47 | 3.02 | 1.95 | 0.98 | 0.16
0.50 | 14.28 | 12.76 | 10.69 | 9.22 | 7.60 | 6.29 | 5.17 | 3.80 | 2.47 | 1.27 | 1.21
1.00 | 13.16 | 11.85 | 10.62 | 9.17 | 8.05 | 6.64 | 5.35 | 3.88 | 2.94 | 2.69 | 3.13
2.00 | 12.59 | 11.37 | 10.41 | 9.18 | 7.85 | 6.89 | 6.35 | 6.23 | 6.70 | 7.92 | 8.26
(a) Median % error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 88.6 | 87.53 | 85.8 | 83.93 | 81.58 | 78.05 | 73.05 | 64.31 | 47.62 | 5.37 | 0.2
0.10 | 69.5 | 66.35 | 62.5 | 57.49 | 50.85 | 41.63 | 28.34 | 10.56 | 5.41 | 2.83 | 0.4
0.50 | 40.1 | 34.09 | 28.9 | 23.43 | 19.74 | 16.83 | 14.13 | 11.52 | 8.27 | 5.03 | 2.9
1.00 | 31.4 | 28.42 | 26.1 | 24.08 | 22.49 | 20.28 | 17.70 | 15.02 | 11.88 | 8.7 | 7.6
2.00 | 31.2 | 29.55 | 29.4 | 29.37 | 29.09 | 28.46 | 26.44 | 24.34 | 21.68 | 19.55 | 19.5
(b) 90th quantile % Error

Maturity/Strike | 9.0 | 9.1 | 9.2 | 9.3 | 9.4 | 9.5 | 9.6 | 9.7 | 9.8 | 9.9 | 10.0
---|---|---|---|---|---|---|---|---|---|---|---
0.01 | 91.6 | 90.9 | 89.5 | 88.3 | 86.4 | 83.8 | 80.4 | 73.6 | 73.50 | 64.55 | 0.62
0.10 | 77.4 | 75.1 | 92.8 | 98.2 | 63.7 | 57.3 | 47.0 | 33.02 | 33.19 | 8.9 | 0.83
0.50 | 57.6 | 54.9 | 58.8 | 52.6 | 56.8 | 68.4 | 51.4 | 42.07 | 31.86 | 17.3 | 4.40
1.00 | 60.3 | 53.4 | 53.4 | 74.7 | 68.6 | 63.6 | 64.1 | 53.6 | 44.64 | 27.1 | 11.12
2.00 | 98.7 | 135.5 | 100.9 | 85.9 | 109.2 | 116.8 | 98.0 | 78.1 | 68.54 | 48.6 | 28.58
(c) Maximum % error

Table 12: Approximation error of the European call IV under the fractional Bergomi (H=0.7) model. |  |
---|---|---
(a) Median | (b) $90\%$ Quantile | (c) Maximum
Figure 20: Accuracy of the approximation (9) for the European call under
fractional Bergomi (H=0.7) model.

##  Appendix A A primer on Malliavin Calculus

We introduce the elementary notions of the Malliavin calculus used in this
paper (see Nualart and Nualart [14]). Let us consider a standard Brownian
motion $Z=(Z_{t})_{t\in[0,T]}$ defined on a complete probability space
$(\Omega,\mathcal{F},\mathbb{P})$ and the corresponding filtration
$\mathcal{F}_{t}$ generated by $Z_{t}$. Let ${\cal S}^{Z}$ be the set of
random variables of the form

$F=f(Z(h_{1}),\ldots,Z(h_{n})),$ (17)

with $h_{1},\ldots,h_{n}\in L^{2}([0,T])$, $Z(h_{i})$ denotes the Wiener
integral of the function $h_{i}$, for $i=1,..,n$, and $f\in
C_{b}^{\infty}(\mathbb{R}^{n})$ (i.e., $f$ and all its partial derivatives are
bounded). Then the Malliavin derivative of $F$, $D^{Z}F$, is defined as the
stochastic process given by

$D_{s}^{Z}F=\sum_{j=1}^{n}{\frac{\partial f}{\partial
x_{j}}}(Z(h_{1}),\ldots,Z(h_{n}))h_{j}(s),\quad s\in[0,T].$

This operator is closable from $L^{p}(\Omega)$ to $L^{p}(\Omega;L^{2}([0,T])$,
for all $p\geq 1$, and we denote by ${\mathbb{D}}_{Z}^{1,p}$ the closure of
${\cal S}^{Z}$ with respect to the norm

$||F||_{1,p}=\left(\mathbb{E}\left|F\right|^{p}+\mathbb{E}||D^{Z}F||_{L^{2}([0,T])}^{p}\right)^{1/p}.$

We also consider the iterated derivatives $D^{Z,n}$ for all integers $n>1$
whose domains will be denoted by $\mathbb{D}^{n,p}_{Z}$, for all $p\geq 1$. We
will use the notation
$\mathbb{L}_{Z}^{n,p}:=L^{p}([0,T];{\mathbb{D}}_{Z}^{n,p})$.

The following theorem is an extension of the classical Itô’s Lemma for non-
anticipating processes, see Proposition 4.3.1 in Alòs and García-Lorite [2].

######  Theorem 3 (Anticipating Itô’s Formula).

Consider a process of the form

$X_{t}=X_{0}+\int_{0}^{t}u_{s}dZ_{s}+\int_{0}^{t}v_{s}ds,$

where $X_{0}$ is an $\mathcal{F}_{0}$-measurable random variable and $u$ and
$v$ are $\mathcal{F}_{t}$-adapted processes in $L^{2}([0,T]\times\Omega)$.
Consider also a process $Y_{t}=\int_{t}^{T}\theta_{s}ds$, for some
$\theta\in\mathbb{L}_{Z}^{1,2}$. Let
$F:[0,T]\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a
$C^{1,2}([0,T]\times\mathbb{R}^{2})$ function such that there exists a
positive constant $C$ such that, for all $t\in[0,T]$, $F$ and its derivatives
evaluated in $(t,X_{t},Y_{t})$ are bounded by $C$. Then it follows that for
all $t\in[0,T]$,

$\displaystyle F(t,X_{t},Y_{t})$
$\displaystyle=F(0,X_{0},Y_{0})+\int_{0}^{t}\partial_{s}F(s,X_{s},Y_{s})ds+\int_{0}^{t}\partial_{x}F(s,X_{s},Y_{s})u_{s}dZ_{s}$
$\displaystyle\qquad+\int_{0}^{t}\partial_{x}F(s,X_{s},Y_{s})v_{s}ds+\int_{0}^{t}\partial_{y}F(s,X_{s},Y_{s})dY_{s}$
$\displaystyle\qquad+\int_{0}^{t}\partial_{xy}^{2}F(s,X_{s},Y_{s})u_{s}D^{-}Y_{s}ds+\frac{1}{2}\int_{0}^{t}\partial_{xx}^{2}F(s,X_{s},Y_{s})u_{s}^{2}ds,$

where $D^{-}Y_{s}=\int_{s}^{T}D^{Z}_{s}\theta_{r}dr$ and the integral
$\int_{0}^{t}\partial_{x}F(s,X_{s},Y_{s})u_{s}dZ_{s}$ is defined in the
Skorohod sense since the process $\partial_{x}F(s,X_{s},Y_{s})u_{s}$ is not
adapted.

##  Appendix B The Price of an Asian Call Option under the Bachelier Model

In this section by prove the closed form formula for the price of an
arithmetic Asian option under the constant volatility Bachelier model.

######  Theorem 4.

Consider the model (1) in the case of constant volatility $\sigma$. Then, the
price of an arithmetic Asian call option for $t\in[0,T]$ satisfies

$B_{A}(t,S_{t},y_{t},k,\sigma)=\left(S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}-k\right)N(d_{A}(k,\sigma))+\left(\frac{\sigma(T-t)\sqrt{T-t}}{T\sqrt{3}}\right)n(d_{A}(k,\sigma)),$

where

$d_{A}(k,\sigma)=\frac{S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}-k}{\left(\frac{\sigma(T-t)\sqrt{T-t}}{T\sqrt{3}}\right)},$

$T$ is the maturity, $S_{t}$ is the stock price at time $t$, $k$ is the strike
price, and $y_{t}=\int_{0}^{t}S_{u}du$ is the state variable.

###### Proof.

Firstly, notice the following representation

$\displaystyle\begin{split}\frac{1}{T}\int_{0}^{T}S_{t}dt&=\frac{1}{T}\int_{0}^{T}\left(S_{0}+\int_{0}^{t}\sigma
dW_{u}\right)dt=S_{0}+\frac{\sigma}{T}\int_{0}^{T}W_{s}ds=\\\
&=S_{0}+\frac{\sigma}{T}\int_{0}^{t}(T-s)dW_{s}+\frac{\sigma}{T}\int_{t}^{T}(T-s)dW_{s}.\end{split}$

The last term on the right hand side is normally distributed with mean zero
and variance equal to

$\displaystyle\begin{split}\mathbb{E}\left(\frac{\sigma}{T}\int_{t}^{T}(T-s)dW_{s}\right)^{2}&=\frac{\sigma^{2}}{T^{2}}\int_{t}^{T}(T-s)^{2}ds=\frac{\sigma^{2}(T-t)^{3}}{3T^{2}}.\end{split}$

Due to the risk neutral pricing argument we know that

$B_{A}(t,S_{t},y_{t},k,\sigma)=\mathbb{E}_{t}\left(\frac{1}{T}\int_{0}^{T}S_{t}dt-k\right)_{+}.$

Therefore, using the above formulas this can be equivalently written as

$\displaystyle\begin{split}\mathbb{E}_{t}\left(\frac{1}{T}\int_{0}^{T}S_{t}dt-k\right)_{+}&=\mathbb{E}_{t}\left(S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}-k+\frac{\sigma}{T}\int_{t}^{T}(T-s)dW_{s}\right)_{+}=\\\
&=\frac{1}{\sqrt{2\pi}}\int_{\frac{k-u_{t}}{z_{t}}}^{\infty}\left(u_{t}-k+z_{t}x\right)e^{-\frac{x^{2}}{2}}dx=\\\
&=\left(u_{t}-k\right)N\left(\frac{u_{t}-k}{z_{t}}\right)+z_{t}n\left(\frac{u_{t}-k}{z_{t}}\right),\end{split}$

where $u_{t}=S_{t}\frac{T-t}{T}+\frac{y_{t}}{T}$ and
$z_{t}=\frac{\sigma(T-t)\sqrt{T-t}}{T\sqrt{3}}$.

The last step allows us to complete the proof. ∎

## References

  * [1] Elisa Alòs.  A generalization of the Hull and White formula with applications to option pricing approximation.  Finance and Stochastics, 10(3):353–365, 2006.
  * [2] Elisa Alòs, David García Lorite, and Dariusz Gatarek.  Malliavin Calculus in Finance.  Chapman and Hall/CRC, 2021.
  * [3] Elisa Alòs, David García-Lorite, and Aitor Muguruza Gonzalez.  On Smile Properties of Volatility Derivatives: Understanding the VIX Skew.  SIAM Journal on Financial Mathematics, 13(1):32–69, 2022.
  * [4] Elisa Alòs, Jorge A León, and Josep Vives.  On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility.  Finance and Stochastics, 11(4):571–589, 2007.
  * [5] Elisa Alós, Eulalia Nualart, and Makar Pravosud.  On the implied volatility of Asian options under stochastic volatility models.  2022\.
  * [6] L Bachelier.  Théorie de la spéculation.  Annales scientifiques de l’École Normale Supérieure, 3e série,:21–86, 1900.
  * [7] Jaehyuk Choi, Minsuk Kwak, Chyng Wen Tee, and Yumeng Wang.  A Black–Scholes user’s guide to the Bachelier model.  Journal of Futures Markets, 42(5):959–980, 2022.
  * [8] José E Figueroa-López and Sveinn Ólafsson.  Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps.  Finance and Stochastics, 20(4):973–1020, 2016.
  * [9] Jean-Pierre Fouque, George Papanicolaou, and K Ronnie Sircar.  From the implied volatility skew to a robust correction to Black-Scholes American option prices.  International Journal of Theoretical and Applied Finance, 04(04):651–675, 2001.
  * [10] Masaaki Fukasawa.  Short-time at-the-money skew and rough fractional volatility.  Quantitative Finance, 17(2):189–198, 2017.
  * [11] Roza Galeeva and Ehud Ronn.  Oil futures volatility smiles in 2020: Why the bachelier smile is flatter.  Review of Derivatives Research, 25(2):173–187, 2022.
  * [12] Jeffrey Ho Goodman, Laurie S.  Interest Rates—Normal or Lognormal?, 2003.
  * [13] Peter Jaeckel.  Implied Normal Volatility.  Wilmott, 2017:54–57, 2017.
  * [14] David Nualart and Eulalia Nualart.  Introduction to Malliavin Calculus.  Cambridge University Press, 2018.
  * [15] Walter Schachermayer and Josef Teichmann.  How close are the option pricing formulas of Bachelier and Black–Merton–Scholes?  Mathematical Finance, 18(1):155–170, 2008.
  * [16] Satoshi Terakado.  On the Option Pricing Formula Based on the Bachelier Model.  SSRN Electronic Journal, 2019.