Power law in Sandwiched Volterra Volatility model

In this paper, we present 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, we assess the second-order Malliavin differentiability of the volatility process and investigate the conditions that lead to explosive behavior in the Malliavin derivative. As a supplementary result, we also prove a general Malliavin product rule.


Introduction
One of the well-established benchmarks for evaluating option pricing models is comparing the model-generated Black-Scholes implied volatility surface (τ, κ) → σ(τ, κ) with the empirically observed one (τ, κ) → σ emp (τ, κ).In this context, τ represents the time to maturity and κ := log K e rτ S0 is the log-moneyness with K denoting the strike, S 0 the current price of an underlying asset and r being the instantaneous interest rate.In particular, for any fixed τ , the values of σ emp (τ, κ) plotted against κ are known to produce convex "smiley" patterns with negative slopes at-the-money (i.e. when κ ≈ 0).Furthermore, as reported in e.g.[9,17,21] or [10,Subsection 2.2], the smile at-the-money becomes progressively steeper as τ → 0 with a rule-ofthumb behavior (1.1) The phenomenon (1.1) is known as the power law of the at-the-money implied volatility skew, and if one wants to replicate it, one may look for a model with However, it turns out that the property (1.2) is not easy to obtain: for example, as discussed in [2, Section 7.1] or [24,Remark 11.3.21],classical Brownian diffusion stochastic volatility models fail to produce implied volatilities with power law (1.2).In the literature, (1.2) is usually replicated by introducing a volatility process with very low Hölder regularity within the rough volatility framework popularized by Gatheral, Jaisson and Rosenbaum in their landmark paper [21].The efficiency of this approach can be explained as follows.
• On the one hand, a theoretical result of Fukasawa [18] suggests that the volatility process cannot be Hölder continuous of a high order in continuous non-arbitrage models exhibiting the property (1.2).In other words, the roughness of volatility is, in some sense, a necessary condition to reproduce (1.2) (at least in the fully continuous setting).
• On the other hand, as proved in the seminal 2007 paper [2] of Alòs, León and Vives, the short-term explosion (1.2) of the implied volatility skew can be deduced from the explosion of the Malliavin derivative of volatility.In particular, the latter characteristic is exhibited by fractional Brownian motion with H < 1/2, a common driver in rough volatility literature.
However, despite the ability to reproduce the power law (1.2),rough volatility models are not perfect.In particular, -in the specific context of fractional Brownian motion, roughness contradicts the observations [7,15,16,25,29] of long memory on the market; -in addition, volatility processes with long memory seem to be better in replicating the shape of implied volatility for longer maturities [8,19,20]; -furthermore, there is no guaranteed procedure of transition between physical and pricing measures: it is not always clear whether the volatility process σ = {σ(t), t ∈ [0, T ]} hits zero and therefore the integral ds that is typically present in martingale densities (see e.g.[6]) may be poorly defined; -just like many classical Brownian stochastic volatility models (see e.g.[3]), they may suffer from moment explosions in price, which results in complications with the pricing of some assets, quadratic hedging, and numerical methods.
For more details on rough volatility, we refer the reader to a recent review [10, Subsection 3.3.2]or a regularly updated literature list on the subject [1].
Recently, a series of papers [11,12,13] introduced the Sandwiched Volterra Volatility (SVV) model which accounts for all the problems mentioned above.More precisely, the volatility process driven by a general Hölder continuous Gaussian Volterra process Z(t) = t 0 K(t, s)dB(s).The special part of the equation above is the drift b.It is assumed that there are two continuous functions 0 < ϕ < ψ such that for some ε > 0 Such an explosive nature of the drift resembling the one in SDEs for Bessel processes (see e.g.[28, Chapter XI]) or singular SDEs of [22] ensures that, with probability 1, which immediately solves the moment explosion problem (see e.g.[11,Theorem 2.6]) and allows for a transparent transition between physical and pricing measures [11,Subsection 2.2].In addition, the flexibility in the choice of the kernel K should allow to replicate both the long memory and the power law behavior (1.2).
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 K, the SVV model indeed reproduces (1.2).In order to do that, we employ the fundamental result [2, Theorem 6.3] of Alòs, León and Vives mentioned above and check that the Malliavin derivative DY (t) indeed exhibits explosive behavior.The difficulty of this approach is as follows.While the first-order Malliavin differentiability of Y (t) is established in [11,Section 3] with [2, Theorem 6.3] actually demands the existence of the second-order Malliavin derivative.In principle, it is intuitively clear how this derivative should look like: (1.3)However, the computations in (1.3) are far from straightforward to be justified.For example, the functions y → b ′ y (t, y) and y → b ′′ yy (t, y) demonstrate explosive behavior as y → ϕ(t)+ and y → ψ(t)− for any t ∈ [0, T ].This makes it impossible to use the classical Malliavin chain rules such as [26,Proposition 1.2.3] requiring boundedness of the derivative or [26,Proposition 1.2.4] demanding the Lipschitz condition.In order to overcome this issue, we have to use some special properties of the volatility process established in [13] and tailor a version of the Malliavin chain rule specifically for our needs.
The paper is organized as follows.In Section 2, we provide some necessary details about the sandwiched volatility process Y .In Section 3, we prove second-order Malliavin differentiability of Y (t).Finally, in Section 4, we use [2,Theorem 6.3] to determine conditions on the kernel under which the SVV model reproduces (1.2).In Appendix A, we gather some necessary facts from Malliavin calculus, list some of the notation and, in addition, we prove a general Malliavin product rule to fit our purposes and that we were not able to find in the literature.

Preliminaries on sandwiched processes
In this section, we gather all the necessary details about the main object of our study: the class of sandwiched processes driven by Hölder-continuous Gaussian Volterra noises.
Assumption 1.The kernel K is of Volterra type, i.e.K(t, s) = 0 whenever t ≤ s, and K2) there exists H ∈ (0, 1) such that for all λ ∈ (0, H) and 0 where C λ > 0 is some constant depending on λ. (2.1) Let B = {B(t), t ∈ [0, T ]} be a standard Brownian motion.Assumption 1 allows to define a Gaussian Volterra process and, moreover, Assumption 1(K2) together with [4, Theorem 1 and Corollary 4] imply that Z has a modification with Hölder continuous trajectories of any order λ ∈ (0, H).In what follows, we always use this modification of Z: in other words, with probability 1, for any λ ∈ (0, H) there exists a random variable Λ = Λ(λ) > 0 such that for all 0 Furthermore, as stated in [4, Theorem 1], the random variable Λ from (2.3) can be chosen such that In what follows, we assume that (2.4) always holds.Next, denote Take H ∈ (0, 1) from Assumption 1(K2), consider two H-Hölder continuous functions ϕ, ψ: and define a function b: where the coefficients in (2.6) satisfy the following assumption.
Assumption 2. The constants γ 1 , γ 2 > 0 and functions θ 1 , θ 2 , a are such that (B3) the function a: [0, T ] × R → R is locally Lipschitz in y uniformly in t, i.e. for any N > 0 there exists a constant C N > 0 that does not depend on t such that Finally, fix ϕ(0) < y 0 < ψ(0) and consider a stochastic differential equation of the form In what follows, we will need to analyze the behavior of stochastic processes |b(t, In this regard, the property (2.8) alone is not sufficient: the process Y can, in principle, approach the bounds ϕ and ψ which results in an explosive growth of the processes mentioned above.Luckily, [13,Theorem 4.2] provides a refinement of (2.8) allowing for a more precise control of Y near ϕ and ψ.We give a slightly reformulated version of this result below.
Theorem 2.4.Let Assumptions 1 and 2 hold and λ ∈ (0, H), Λ = Λ(λ) > 0 be from (2.3).Then there exist deterministic constants In particular, since Λ can be chosen to have moments of all orders, for all r ≥ 0 We finalize this section by citing the first-order Malliavin differentiability result for the sandwiched process (2.7) proved in [11,Section 3].
The same is also, in principle, true for the results of the subsequent sections.Namely, it would be sufficient to assume that there exist deterministic constants c > 0, r > 0, γ > 1 H − 1 and • b has an explosive growth to ∞ near ϕ and explosive decay to −∞ near ψ of order γ > 1 • for all (t, y) ∈ D, the partial derivatives b ′ y and b ′′ yy satisfy However, since (2.6) is the most natural choice satisfying these assumptions, we stick to this shape for notational convenience.
Notation.Here and in the sequel, C will denote any positive deterministic constant the exact value of which is not relevant.Note that C may change from line to line (or even within one line).
The main goal of this section is to establish second-order Malliavin differentiability of the sandwiched process (2.7) and compute the corresponding derivative explicitly.As mentioned above, the main difficulty lies in controlling the behavior of b(t, Y (t)), b ′ y (t, Y (t)) and b ′′ yy (t, Y (t)) whenever Y (t) approaces the bounds.Luckily, Theorem 2.4 gives all the necessary tools to do that as summarized in the following proposition.Proposition 3.1.There exists a random variable ξ > 0 such that In particular, for any p ≥ 1, Proof.Fix λ ∈ (0, H) and take the corresponding Λ > 0 from (2.3) and C Y , β > 0 be from Theorem 2.4.Then Note that ξ 0 , ξ 1 and ξ 2 have moments of all orders by the properties of Λ, see (2.4), and hence, putting we obtain the required result.
As noted in Theorem 2.5, Y (t) ∈ D 1,2 for each t ≥ 0. In fact, Proposition 3.1 together with the shape (2.9) of the derivative allow to establish a more general result. Hence By Assumption 1 and Remark 2.1, which ends the proof.
Our next goal is to establish the Malliavin chain rule for the random variables b ′ y (t, Y (t)) and exp 2) Proof. 1) We shall start from proving that b ′ y (t, Y (t)) ∈ D 1,p .Note that b ′ y is not a bounded function itself and it does not have bounded derivatives -hence the classical chain rule from [26, Section 1.2] cannot be applied here in a straightforward manner.In order to overcome this issue, we will use the approach in the spirit of [27,Lemma A.1] or [11,Proposition 3.4].For the reader's convenience, we divide the proof into steps.
Step 0. First of all, observe that b ′ (t, Y (t)) ∈ L 2 (Ω) as a direct consequence of Proposition 3.1.Also, for any p > 1, Indeed, again by Proposition 3.1 together with the proof of Proposition 3.2, we have Therefore, by Lemma A.3, it is sufficient to prove that b ′ y (t, Y (t)) ∈ D 1,2 with (3.3) being the corresponding Malliavin derivative.
Therefore, the function f m satisfies the conditions of the classical Malliavin chain rule [26, Proposition 1.2.3], so f m (Y (t)) ∈ D 1,2 and, with probability 1 for a.a.s ∈ [0, T ], as m → ∞ -then the result will follow immediately from the closedness of the Malliavin derivative operator D.
Step 2: (Ω) and hence the required convergence follows from the dominated convergence theorem.
Step 3: By the definitions of f m and φ, with probability 1, as m → ∞.Moreover, since φ has compact support, max y∈R (φ ′ (y)) 2 < ∞, so we can write Therefore, by the dominated convergence theorem, which proves the first claim of the Proposition.
2) Let us proceed with the second claim and verify that exp and hence it is sufficient to prove that exp Finally, the function x → e x satisfies the conditions of the chain rule from [11,Proposition 3.4] and hence exp (3.5) Proof.For fixed 0 ≤ s < t ≤ T , denote By Proposition 3.3 and Lemma A.4 from the Appendix, it is sufficient to check that for all p ≥ 2 (i) the product X 1 X 2 ∈ L p (Ω), All conditions (i)-(iii) can be checked in a straightforward manner using Proposition 3.1 and the arguments similar to the proof of Proposition 3.2.
We are now ready to formulate the main result of this section.
Proof.By the definition of the • 2,p -norm in (A.1) from Appendix A, it is sufficient to check that and  (3.9).By this, the proof is complete.

Power law in VSV model
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 (2.7) as stochastic volatility.Namely, we consider a (risk-free) market model with the price process S = {S(t), t ∈ [0, T ]} of the form where B 1 , B 2 are two independent Brownian motions, X = {X(t), t ∈ [0, T ]} denotes the (risk-free) log-price of some asset starting from some level x 0 ∈ R, r is a constant instantaneous interest rate, and ρ ∈ (−1, 1) is a correlation coefficient that accounts for the leverage effect.As previously, the drift b and the Volterra kernel K satisfy Assumptions 1 and 2.
Remark 4.1.The model (4.1) was initially introduced in [11] and, given the nature of the volatility process, is called the Sandwiched Volterra Volatility (SVV) model.
To establish the conditions under which (4.1) gives power law of the short-term at-the-money implied volatility, we will apply the fundamental result [2, Theorem 6.3] which connects the shape of the skew with the Malliavin derivative of the volatility.Remark 4.2.In the recent literature (see e.g.[5,9,10,21]), it is typical to characterize the implied volatility skew in terms of ∂ σ ∂κ with κ = log K e rτ +x 0 being the log-moneyness.In [2], a slightly different parametrization σ log-price (τ, x 0 ) is considered with With this parametrization, i.e. the typically negative at-the-money skews for σ are equivalent to positive With Remark 4.2 in mind, let us provide a slightly adjusted version of [2, Theorem 6.3].
Finally, assume that there exists a constant K σ > 0 such that, with probability 1, Then, with probability 1, In particular, if ρK σ < 0, the at-the-money implied volatility skew exhibits the power law behavior with the correct sign of the skew.
Remark 4.4.The original formulation of [2, Theorem 6.3] is slightly more general than Theorem 4.3 above in the sense that 1) in [2, Theorem 6.3], the log-price X is allowed to have jumps; 2) the result in [2] is formulated for the future implied volatility surfaces σ log-price (t 0 , τ, X(t 0 )), t 0 ≥ 0.
Since we are interested in the continuous model (4.1), we removed the jump component in (4.2) and, for the simplicity of notation, we put t 0 = 0.
Proposition 4.6.Let Assumptions 1 and 2 hold and the Volterra kernel K be such that where K Y is some finite constant.Then, with probability 1 and that, by Proposition (3.1), where c := max (t,y)∈D a ′ y (t, y).Then we can write 1 The term 1 Finally, let us deal with (H3).Proposition 4.7.Let Assumptions 1 and 2 hold with H ∈ 1 6 , 1 2 and the Volterra kernel K be such that for any 0 for some constant C > 0. Then the hypothesis (H3) from Theorem 4.3 holds for the volatility process σ = Y .
Proof.Fix 0 < r, s < t.Then, taking into account (4.7), with probability 1, with c := max (t,y)∈D a ′ y (t, y), so we can write Taking into account (4.8) and (4.9), As for the integral All the findings of this Section can now be summarized in the following theorem which should be regarded as the main result of the paper.Lemma A.4.Let X 1 , X 2 ∈ D 1,2 be such that (i) X 1 X 2 ∈ L 2 (Ω);

A.2 Generalized Malliavin product rule
If, in addition, for some p ≥ 2, then X 1 X 2 ∈ D 1,p .
The second claim immediately follows from Lemma A.3.
are strictly positive and continuous;

t u b
′ y (v, Y (v))dv ∈ D 1,p with (3.4) being the corresponding Malliavin derivative.Note that, since b ′ y is bounded from above, exp t u b ′ y (v, Y (v))dv is also bounded from above and hence is an element of L p (Ω) for any p > 1.Moreover, by Proposition 3.1, boundedness of exp t u b ′ y (v, Y (v))dv and (3.1), we can write

t u b Corollary 3 . 4 .
′ y (v, Y (v))dv ∈ D 1,2 and (3.4) holds.Proposition 3.3 and Lemma A.4 together allow us to deduce the following corollary.For any 0 ≤ s < t ≤ T and p > 1, b holds automatically.Next, (3.8) can be easily deduced from (3.2).Finally, using Proposition (3.1) and the boundedness of expt u b ′ y (v, Y (v))dv , it is easy to prove a bound similar to(3

Finally, let us
prove a generalized version of the product rule from [26, Exercise 1.2.12] or[14,  Theorem 3.4].