The paper presents an analytical proof demonstrating that the Sandwiched Volterra Volatility (SVV) model is able to reproduce the power-law behavior of the at-the-money implied volatility skew, provided the correct choice of the Volterra kernel. To obtain this result, the second-order Malliavin differentiability of the volatility process is assessed and the conditions that lead to explosive behavior in the Malliavin derivative are investigated. As a supplementary result, a general Malliavin product rule is proved.

One of the well-established benchmarks for evaluating option pricing models is comparing the model-generated Black–Scholes implied volatility surface

On the one hand, a theoretical result of Fukasawa [

On the other hand, as proved in the seminal 2007 paper [

However, despite the ability to reproduce the power law (

in the specific context of a fractional Brownian motion, roughness contradicts the observations [

in addition, volatility processes with long memory seem to be better in replicating the shape of implied volatility for longer maturities [

furthermore, there is no guaranteed procedure of transition between physical and pricing measures: it is not always clear whether the volatility process

just like many classical Brownian stochastic volatility models (see, e.g., [

Recently, a series of papers [

The main goal of this paper is to give the theoretical justification to the latter claim: we prove that, with the correct choice of the Volterra kernel

The paper is organized as follows. In Section

In this section, we gather all the necessary details about the main object of our study: the class of

Fix some

The kernel

there exists

Note that items (K1) and (K2) of Assumption

Let

Next, denote

The constants

the functions

the function

Note that

Finally, fix

Motivated by the property (

In what follows, we will need to analyze the behavior of the stochastic processes

We finalize this section by citing the first-order Malliavin differentiability result for the sandwiched process (

The result above actually holds for more general drifts than the one given in (

for any

for all

Let Assumptions

Here and in the sequel,

The main goal of this section is to establish the second-order Malliavin differentiability of the sandwiched process (

Fix

As noted in Theorem

Note that, by (

Our next goal is to establish the Malliavin chain rule for the random variables

1) We shall start from proving that

if

if

on the compact set

2) Let us proceed with the second claim and verify that

Since the Malliavin derivative operator

Proposition

For fixed

the product

We are now ready to formulate the main result of this section.

Our goal is to prove that

for a.a.

for a.a.

Finally, denote

By the definition of the

Having the second-order Malliavin differentiability in place, we now possess all the necessary tools to analyze the behavior of implied volatility skew of a model with the sandwiched process (

The model (

The goal of this section is to establish conditions under which (

The behavior of empirically observed implied volatilities (see, e.g., [

The condition

To prove Theorem

In the recent literature (see, e.g., [

With Remark

The original formulation of [

in [

the result in [

Observe that the SVV model (

assumption (H2) with

assumption (H4) since

assumption (H1) by the results of Section

By [

Our next step is to handle (

Recall that

Finally, let us deal with (H3).

Fix

if

similarly, if

Having in mind all of the results above, we are ready to prove the main result of this section, namely Theorem

The results above show that the SVV model satisfies conditions (H1)–(H5) of Theorem

Let

Hereafter, we summarize the essentials of the Malliavin derivative with respect to the classical Brownian motion. For more details, we refer the reader to the classical books [

Denote by

The random variables

Let

By [

Throughout the paper, we often use the following lemma which is essentially a simplified version of [

Finally, let us prove a generalized version of the product rule from [

Let

Observe that

The second claim immediately follows from Lemma