Modern Stochastics: Theory and Applications

One of the well-established benchmarks for evaluating option pricing models is comparing the model-generated Black–Scholes implied volatility surface $(\tau ,\kappa )\hspace{-0.1667em}\mapsto \hspace{-0.1667em}\widehat{\sigma }(\tau ,\kappa )$ with the empirically observed one $(\tau ,\kappa )\mapsto {\widehat{\sigma }_{\text{emp}}}(\tau ,\kappa )$. In this context, τ represents the time to maturity and $\kappa :=\log \frac{K}{{e^{r\tau }}{S_{0}}}$ 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 ${\widehat{\sigma }_{\text{emp}}}(\tau ,\kappa )$ plotted against κ are known to produce convex “smiley” patterns with negative slopes at-the-money (i.e. when $\kappa \approx 0$). Furthermore, as reported in, e.g., [8, 16, 20] or [12, Subsection 2.2], the smile at-the-money becomes progressively steeper as $\tau \to 0$ with a rule-of-thumb behavior The phenomenon (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 (2) is not easy to obtain: for example, as discussed in [1, Section 7.1] or [23, Remark 11.3.21], classical Brownian diffusion stochastic volatility models fail to produce implied volatilities with power law (2). In the literature, (2) is usually replicated by introducing a volatility process with a very low Hölder regularity within the rough volatility framework popularized by Gatheral, Jaisson and Rosenbaum in their landmark paper [20]. The efficiency of this approach can be explained as follows.

However, despite the ability to reproduce the power law (2), rough volatility models are not perfect. In particular, For more details on rough volatility, we refer the reader to the recent review [12, Subsection 3.3.2] or the regularly updated literature list on the subject [28].

Recently, a series of papers [9–11] introduced the Sandwiched Volterra Volatility (SVV) model which accounts for all the problems mentioned above. More precisely, the volatility process $Y=\{Y(t),\hspace{2.5pt}t\in [0,T]\}$ is assumed to follow the stochastic differential equation driven by a general Hölder continuous Gaussian Volterra process The special part of the equation above is the drift b. It is assumed that there are two continuous functions $0\lt \varphi \lt \psi $ such that for some $\varepsilon \gt 0$ Such an explosive nature of the drift resembling the one in SDEs for Bessel processes (see, e.g., [27, Chapter XI]) or singular SDEs of [21] ensures that, with probability 1, which immediately solves the moment explosion problem (see, e.g., [9, Theorem 2.6]) and allows for a transparent transition between physical and pricing measures [9, Subsection 2.2]. In addition, the flexibility in the choice of the kernel $\mathcal{K}$ should allow to replicate both the long memory and the power law behavior (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 $\mathcal{K}$, the SVV model indeed reproduces (2). In order to do that, we employ the fundamental result [1, Theorem 6.3] by 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 [9, Section 3] with [1, 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: However, justifying the computations in (3) is far from straightforward. For example, the functions $y\mapsto {b^{\prime }_{y}}(t,y)$ and $y\mapsto {b^{\prime\prime }_{yy}}(t,y)$ demonstrate explosive behavior as $y\to \varphi (t)+$ and $y\to \psi (t)-$ for any $t\in [0,T]$. This makes it impossible to use the classical Malliavin chain rules such as [25, Proposition 1.2.3] requiring boundedness of the derivative or [25, 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 [11] 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 [1, Theorem 6.3] to determine conditions on the kernel under which the SVV model reproduces (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.

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.

Fix some $T\in (0,\infty )$ and consider a kernel $\mathcal{K}:{[0,T]^{2}}\to \mathbb{R}$ satisfying the following assumptions.

Let $B=\{B(t),\hspace{2.5pt}t\in [0,T]\}$ be a standard Brownian motion. Assumption 1 allows to define a Gaussian Volterra process and, moreover, Assumption 1(K2) together with [3, Theorem 1 and Corollary 4] implies that Z has a modification with Hölder continuous trajectories of any order $\lambda \in (0,H)$. In what follows, we always use this modification of Z: in other words, with probability 1, for any $\lambda \in (0,H)$ there exists a random variable $\Lambda =\Lambda (\lambda )\gt 0$ such that for all $0\le {t_{1}}\le {t_{2}}\le T$ Furthermore, as stated in [3, Theorem 1], the random variable Λ from (6) can be chosen such that In what follows, we assume that (7) always holds.

Next, denote Take $H\in (0,1)$ from Assumption 1(K2), consider two H-Hölder continuous functions φ, ψ: $[0,T]\to \mathbb{R}$ such that and define a function b: $\mathcal{D}\to \mathbb{R}$ as where the coefficients in (9) satisfy the following assumption.

Finally, fix $\varphi (0)\lt {y_{0}}\lt \psi (0)$ and consider a stochastic differential equation of the form By [11, Theorem 4.1], under Assumptions 1 and 2, the SDE (10) has a unique strong solution $Y=\{Y(t),\hspace{2.5pt}t\in [0,T]\}$. Moreover, with probability 1,

In what follows, we will need to analyze the behavior of the stochastic processes $|b(t,Y(t))|$, $|{b^{\prime }_{y}}(t,Y(t))|$ and $|{b^{\prime\prime }}(t,Y(t))|$, $t\in [0,T]$. In this regard, the property (11) 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, [11, Theorem 4.2] provides a refinement of (11) allowing for a more precise control of Y near φ and ψ. We give a slightly reformulated version of this result below.

We finalize this section by citing the first-order Malliavin differentiability result for the sandwiched process (10) proved in [9, Section 3].

Let Assumptions 1 and 2 hold and $Y=\{Y(t),\hspace{2.5pt}t\in [0,T]\}$ be the sandwiched process defined by (10) with the drift (9).

The main goal of this section is to establish the second-order Malliavin differentiability of the sandwiched process (10) and compute the corresponding derivative explicitly. As mentioned above, the main difficulty lies in controlling the behavior of $b(t,Y(t))$, ${b^{\prime }_{y}}(t,Y(t))$ and ${b^{\prime\prime }_{yy}}(t,Y(t))$ whenever $Y(t)$ approaces the bounds. Luckily, Theorem 1 gives all the necessary tools to do that as summarized in the following proposition.

As noted in Theorem 2, $Y(t)\in {\mathbb{D}^{1,2}}$ for each $t\ge 0$. In fact, Proposition 1 together with the shape (12) of the derivative allows to establish a more general result.

Our next goal is to establish the Malliavin chain rule for the random variables ${b^{\prime }_{y}}(t,Y(t))$ and $\exp \{{\textstyle\int _{u}^{t}}{b^{\prime }_{y}}(v,Y(v))dv\}$.

Proposition 3 and Lemma 2 together allow us to deduce the following corollary.

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

Finally, denote ${\mathbb{L}^{2,p}}:={L^{p}}([0,T];{\mathbb{D}^{2,p}})$. We complete the section with the following result.

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 (10) as stochastic volatility. Namely, we consider a (risk-free) market model with the price process $S=\{S(t),\hspace{2.5pt}t\in [0,T]\}$ of the form where ${B_{1}},{B_{2}}$ are two independent Brownian motions, $X=\{X(t),t\in [0,T]\}$ denotes the (risk-free) log-price of an asset starting from some level ${x_{0}}\in \mathbb{R}$, r is a constant instantaneous interest rate, and $\rho \in (-1,1)$ is a correlation coefficient that accounts for the leverage effect. As previously, the drift b and the Volterra kernel $\mathcal{K}$ satisfy Assumptions 1 and 2.

The goal of this section is to establish conditions under which (22) reproduces the power law (2) of the short-term at-the-money implied volatility. Namely, we have the following result.

To prove Theorem 4, we will apply the fundamental result [1, Theorem 6.3] which connects the shape of the skew with the Malliavin derivative of the volatility.

With Remark 8 in mind, let us provide a slightly adjusted version of [1, Theorem 6.3].

Observe that the SVV model (22) automatically satisfies a number of assumptions of Theorem 5: Therefore, it remains to check (H3), (H5), and (28). Naturally, given the shape of the Malliavin derivative (12), both (H3) and (28) require additional assumptions on the kernel, so let us start with (H5).

Our next step is to handle (28).

Finally, let us deal with (H3).

Having in mind all of the results above, we are ready to prove the main result of this section, namely Theorem 4.