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.
SVV modelstochastic volatilitysandwiched processGaussian Volterra noiseMalliavin calculus91G3060H1060H3560G22Research Council of Norway274410The present research is carried out within the frame and support of the ToppForsk project nr. 274410 of the Research Council of Norway with the title STORM: Stochastics for Time-Space Risk Models. Introduction
One of the well-established benchmarks for evaluating option pricing models is comparing the model-generated Black–Scholes implied volatility surface (τ,κ)↦σˆ(τ,κ) with the empirically observed one (τ,κ)↦σˆemp(τ,κ). In this context, τ represents the time to maturity and κ:=logKerτS0 is the log-moneyness with K denoting the strike, S0 the current price of an underlying asset and r being the instantaneous interest rate. In particular, for any fixed τ, the values of σˆemp(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., [8, 16, 20] or [12, Subsection 2.2], the smile at-the-money becomes progressively steeper as τ→0 with a rule-of-thumb behavior
|σˆemp(τ,κ)−σˆemp(τ,κ′)κ−κ′|∝τ−12+H,κ,κ′≈0,H∈(0,12).
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
|∂σˆ∂κ(τ,κ)|κ=0=O(τ−12+H),τ→0.
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.
On the one hand, a theoretical result of Fukasawa [17] suggests that the volatility process cannot be Hölder continuous of a high order in continuous nonarbitrage models exhibiting the property (2). In other words, the roughness of volatility is, in some sense, a necessary condition to reproduce (2) (at least in the fully continuous setting).
On the other hand, as proved in the seminal 2007 paper [1] by Alòs, León and Vives, the short-term explosion (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 withH<1/2, a common driver in the rough volatility literature.
However, despite the ability to reproduce the power law (2), rough volatility models are not perfect. In particular,
in the specific context of a fractional Brownian motion, roughness contradicts the observations [6, 14, 15, 24, 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 [7, 18, 19];
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 ∫0t1σ2(s)ds that is typically present in martingale densities (see, e.g., [5]) may be poorly defined;
just like many classical Brownian stochastic volatility models (see, e.g., [2]), 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 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),t∈[0,T]} is assumed to follow the stochastic differential equation
Y(t)=y0+∫0tb(s,Y(s))ds+Z(t)
driven by a general Hölder continuous Gaussian Volterra process
Z(t)=∫0tK(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>0b(t,y)≥C(y−φ(t))γ,y∈(φ(t),φ(t)+ε),b(t,y)≤−C(ψ(t)−y)γ,y∈(ψ(t)−ε,ψ(t)).
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,
0<φ(t)<Y(t)<ψ(t),
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 Kshould 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 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
DsY(t)=K(t,s)+∫stK(u,s)by′(u,Y(u))exp{∫utby′(v,Y(v))dv}du,
[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:
DrDsY(t)=Dr∫stK(u,s)by′(u,Y(u))exp{∫utby′(v,Y(v))dv}du=∫stK(u,s)Dr[by′(u,Y(u))exp{∫utby′(v,Y(v))dv}]du=∫stK(u,s)exp{∫utby′(v,Y(v))dv}Dr[by′(u,Y(u))]du+∫stK(u,s)by′(u,Y(u))Dr[exp{∫utby′(v,Y(v))dv}]du=∫stK(u,s)byy″(u,Y(u))exp{∫utby′(v,Y(v))dv}Dr[Y(u)]du+∫stK(u,s)by′(u,Y(u))exp{∫utby′(v,Y(v))dv}××∫utbyy″(v,Y(v))Dr[Y(v)]dvdu.
However, justifying the computations in (3) is far from straightforward. For example, the functions y↦by′(t,y) and y↦byy″(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 [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.
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.
Fix some T∈(0,∞) and consider a kernel K:[0,T]2→R satisfying the following assumptions.
The kernel K is of Volterra type, i.e. K(t,s)=0 whenever t≤s, and
K is square-integrable, i.e.
∫0T∫0TK2(t,s)dsdt<∞,
there exists H∈(0,1) such that for all λ∈(0,H) and 0≤t1≤t2≤T∫0T(K(t2,s)−K(t1,s))2ds≤Cλ|t2−t1|2λ,
where Cλ>0 is some constant depending on λ.
Note that items (K1) and (K2) of Assumption 1 jointly imply that
supt∈[0,T]∫0TK2(t,s)ds<∞.
Let B={B(t),t∈[0,T]} be a standard Brownian motion. Assumption 1 allows to define a Gaussian Volterra processZ(t):=∫0tK(t,s)dB(s),t∈[0,T],
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 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≤t1≤t2≤T|Z(t2)−Z(t1)|≤Λ|t2−t1|λ.
Furthermore, as stated in [3, Theorem 1], the random variable Λ from (6) can be chosen such that
E[Λr]<∞for allr∈R.
In what follows, we assume that (7) always holds.
Next, denote
D:={(t,y)∈[0,T]×R|φ(t)<y<ψ(t)},D‾:={(t,y)∈[0,T]×R|φ(t)≤y≤ψ(t)}.
Take H∈(0,1) from Assumption 1(K2), consider two H-Hölder continuous functions φ, ψ: [0,T]→R such that
0<φ(t)<ψ(t)for allt∈[0,T],
and define a function b: D→R as
b(t,y):=θ1(t)(y−φ(t))γ1−θ2(t)(ψ(t)−y)γ2+a(t,y),
where the coefficients in (9) satisfy the following assumption.
The constants γ1, γ2>0 and functions θ1, θ2, a are such that
γ1>1H−1, γ2>1H−1 with H∈(0,1) being from Assumption 1(K2);
the functions θ1, θ2: [0,T]→R are strictly positive and continuous;
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 CN>0 that does not depend on t such that
|a(t,y2)−a(t,y1)|≤CN|y2−y1|,t∈[0,T],y1,y2∈[−N,N];
a: [0,T]×R→R is two times differentiable w.r.t. the spatial variable y with a, ay′, ayy″ all being continuous on [0,T]×R.
Note that by′ is bounded from above on D: indeed,
by′(t,y)=−γ1θ1(t)(y−φ(t))γ1+1−γ2θ2(t)(ψ(t)−y)γ2+1+ay′(t,y)<max(t,y)∈D‾ay′(t,y)<∞.
Finally, fix φ(0)<y0<ψ(0) and consider a stochastic differential equation of the form
Y(t)=y0+∫0tb(s,Y(s))ds+Z(t),t∈[0,T].
By [11, Theorem 4.1], under Assumptions 1 and 2, the SDE (10) has a unique strong solution Y={Y(t),t∈[0,T]}. Moreover, with probability 1,
φ(t)<Y(t)<ψ(t)for allt∈[0,T].
Motivated by the property (11), we will call the solution Y of (10) a sandwiched process.
In what follows, we will need to analyze the behavior of the stochastic processes |b(t,Y(t))|, |by′(t,Y(t))| and |b″(t,Y(t))|, t∈[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.
Let Assumptions1and2hold andλ∈(0,H),Λ=Λ(λ)>0be from (6). Then there exist deterministic constantsCY=CY(λ)>0andβ=β(λ)>0such thatφ(t)+CY(1+Λ)β≤Y(t)≤ψ(t)−CY(1+Λ)βfor allt∈[0,T].In particular, since Λ can be chosen to have moments of all orders, for allr≥0E[supt∈[0,T]1(Y(t)−φ(t))r]<∞,E[supt∈[0,T]1(ψ(t)−Y(t))r]<∞.
We finalize this section by citing the first-order Malliavin differentiability result for the sandwiched process (10) proved in [9, Section 3].
Let Assumptions1and2hold and Y be the sandwiched process given by (10). Then, for anyt∈[0,T],Y(t)∈D1,2and, with probability 1, for a.a.s∈[0,T]DsY(t)=K(t,s)+∫stK(u,s)by′(u,Y(u))exp{∫utby′(v,Y(v))dv}du.
The result above actually holds for more general drifts than the one given in (9). 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>1H−1 and 0<y∗<maxt∈[0,T]|ψ(t)−φ(t)| such that
b: D→R is continuous on D and has continuous partial derivatives by′, byy″;
for any 0<ε<12maxt∈[0,T]|ψ(t)−φ(t)|,
|b(t,y2)−b(t,y1)|≤cεr|y2−y1|,t∈[0,T],φ(t)+ε≤y1≤y2≤ψ(t)−ε;
b has an explosive growth to ∞ near φ and explosive decay to −∞ near ψ of order γ>1H−1, i.e.
b(t,y)≥c(y−φ(t))γ,y∈(φ(t),φ(t)+y∗),b(t,y)≤−c(ψ(t)−y)γ,y∈(ψ(t)−y∗,ψ(t));
for all (t,y)∈D, the partial derivatives by′ and byy″ satisfy
−C(1+c(y−φ(t))r+c(ψ(t)−y)r)<by′(t,y)<C
and
|byy″|≤C(1+c(y−φ(t))r+c(ψ(t)−y)r).
However, since (9) is the most natural choice satisfying these assumptions, we stick to this shape for notational convenience.
Second-order Malliavin differentiability
Let Assumptions 1 and 2 hold and Y={Y(t),t∈[0,T]} be the sandwiched process defined by (10) with the drift (9).
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 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)), by′(t,Y(t)) and byy″(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.
There exists a random variableξ>0such that
for anyp≥1,E[ξp]<∞;
for anyt∈[0,T],|b(t,Y(t))|+|by′(t,Y(t))|+|byy″(t,Y(t))|<ξ.
In particular, for anyp≥1,E[supt∈[0,T](|b(t,Y(t))|p+|by′(t,Y(t))|p+|byy″(t,Y(t))|p)]<∞.
Fix λ∈(0,H) and take the corresponding Λ>0 from (6) and CY,β>0 being from Theorem 1. Then
|b(t,Y(t))|=|θ1(t)|(Y(t)−φ(t))γ1+|θ2(t)|(ψ(t)−Y(t))γ2+|a(t,Y(t))|≤supt∈[0,T]|θ1(t)|(1+Λ)βγ1CYγ1+supt∈[0,T]|θ2(t)|(1+Λ)βγ2CYγ2+sup(t,y)∈D|a(t,y)|:=ξ0,|by′(t,Y(t))|=γ1|θ1(t)|(Y(t)−φ(t))γ1+1+γ2|θ2(t)|(ψ(t)−Y(t))γ2+1+|ay′(t,Y(t))|≤γ1supt∈[0,T]|θ1(t)|(1+Λ)β(γ1+1)CYγ1+1+γ2supt∈[0,T]|θ2(t)|(1+Λ)β(γ2+1)CYγ2+1+sup(t,y)∈D|ay′(t,y)|:=ξ1,|byy″(t,Y(t))|=γ1(γ1+1)|θ1(t)|(Y(t)−φ(t))γ1+2+γ2(γ2+1)|θ2(t)|(ψ(t)−Y(t))γ2+2+|ayy″(t,Y(t))|≤γ1(γ1+1)supt∈[0,T]|θ1(t)|(1+Λ)β(γ1+2)CYγ1+2+γ2(γ2+1)supt∈[0,T]|θ2(t)|(1+Λ)β(γ2+2)CYγ2+2+sup(t,y)∈D|ayy″(t,y)|:=ξ2.
Note that ξ0, ξ1 and ξ2 have moments of all orders by the properties of Λ, see (7), and hence, putting
ξ:=ξ0+ξ1+ξ2,
we obtain the required result. □
As noted in Theorem 2, Y(t)∈D1,2 for each t≥0. In fact, Proposition 1 together with the shape (12) of the derivative allows to establish a more general result.
For anyt∈[0,T]andp>1,Y(t)∈D1,p.
Note that, by (11), E[|Y(t)|p]<∞ for any p>1, so, by Lemma 1 from the Appendix, it is sufficient to prove that
E[(∫0T(DsY(t))2ds)p2]<∞
for any p>1. Note that, by Remark 2,
exp{∫stby′(v,Y(v))dv}<exp{cT},
where
c:=max(t,y)∈D‾ay′(t,y),
and, by Proposition 1, there exists a random variable ξ having all moments such that
sups∈[0,T]|by′(s,Y(s))|≤ξ.
Hence
|DsY(t)|≤|K(t,s)|+∫st|K(u,s)||by′(u,Y(u))|exp{∫utby′(v,Y(v))dv}du≤|K(t,s)|+ξexp{cT}∫st|K(u,s)|du.
By Assumption 1 and Remark 1,
(∫0TK2(t,s)ds)p2<∞,
therefore
E[(∫0T(DsY(t))2ds)p2]≤C(∫0TK2(t,s)ds)p2+CE[(∫0T∫0tK2(u,s)(by′(u,Y(u)))2exp{2∫utby′(v,Y(v))dv}duds)p2]≤C(∫0TK2(t,s)ds)p2+CE[ξp]exp{pcT}(∫0T∫0tK2(u,s)duds)p2<∞,
which ends the proof. □
Our next goal is to establish the Malliavin chain rule for the random variables by′(t,Y(t)) and exp{∫utby′(v,Y(v))dv}.
1) We shall start from proving that by′(t,Y(t))∈D1,p. Note that by′ is not a bounded function itself and it does not have bounded derivatives – hence the classical chain rule from [25, 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 [26, Lemma A.1] or [9, 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))∈L2(Ω) as a direct consequence of Proposition 1. Also, for any p>1,
E[(∫0T(byy″(t,Y(t))DsY(t))2ds)p2]<∞.
Indeed, again by Proposition 1 together with the proof of Proposition 2, we have
E[(∫0T(byy″(t,Y(t))DsY(t))2ds)p2]≤E[ξp(∫0T(DsY(t))2ds)p2]≤(E[ξ2p])12(E[(∫0T(DsY(t))2ds)p])12<∞.
Therefore, by Lemma 1, it is sufficient to prove that by′(t,Y(t))∈D1,2 with (15) being the corresponding Malliavin derivative.
Step 1. Let ϕ∈C1(R) be a compactly supported function such that ϕ(x)=x whenever |x|≤1 and |ϕ(x)|≤|x| for all |x|>1. Fix t∈[0,T] and, for m≥1, put
fm(y):=mϕ(by′(t,y)m).
Observe that
fm′(y)=byy″(t,y)ϕ′(by′(t,y)m)
is bounded. Indeed, let 0<εm<ψ(t)−φ(t) be such that
−γ1θ1(t)εmγ1+1+maxφ(t)≤x≤ψ(t)ay′(t,x)<minfsuppϕ
and
−γ2θ2(t)εmγ2+1+maxφ(t)≤x≤ψ(t)ay′(t,x)<minfsuppϕ.
Then,
if y∈(φ(t),φ(t)+εm), then
by′(t,y)=−γ1θ1(t)(y−φ(t))γ1+1−γ2θ2(t)(ψ(t)−y)γ2+1+ay′(t,y)≤−γ1θ1(t)εmγ1+1+maxφ(t)≤x≤ψ(t)ay′(t,x)<minfsuppϕ,
so by′(t,y)m∉suppϕ, fm(y)=0 and fm′(y)=0;
if y∈(ψ(t)−εm,ψ(t)), then, similarly,
by′(t,y)=−γ1θ1(t)(y−φ(t))γ1+1−γ2θ2(t)(ψ(t)−y)γ2+1+ay′(t,y)≤−γ2θ2(t)εmγ2+1+maxφ(t)≤x≤ψ(t)ay′(t,x)<minfsuppϕ,
so by′(t,y)m∉suppϕ, fm(y)=0 and fm′(y)=0;
on the compact set [φ(t)+εm,ψ(t)−εm], both fm and its derivative fm′ are continuous and hence bounded.
Therefore, the function fm satisfies the conditions of the classical Malliavin chain rule [25, Proposition 1.2.3], so fm(Y(t))∈D1,2 and, with probability 1 for a.a. s∈[0,T],
Dsfm(Y(t))=byy″(t,Y(t))ϕ′(by′(t,Y(t))m)DsY(t).
Now it remains to prove that
fm(Y(t))→b′(t,Y(t))
in L2(Ω) and
Dfm(Y(t))→byy″(t,Y(t))DY(t)
in L2(Ω×[0,T]) as m→∞; then the result will follow immediately from the closedness of the Malliavin derivative operator D.
Step 2:fm(Y(t))→b′(t,Y(t)) in L2(Ω) as m→∞. By the definitions of fm and ϕ, fm(Y(t))→b′(t,Y(t)) a.s. as m→∞. Moreover, with probability 1, |fm(Y(t))|≤|by′(t,Y(t))|∈L2(Ω) and hence the required convergence follows from the dominated convergence theorem.
Step 3:Dfm(Y(t))→byy″(t,Y(t))DY(t) in L2(Ω×[0,T]) as m→∞. By the definitions of fm and ϕ, with probability 1,
(byy″(t,Y(t))ϕ′(by′(t,Y(t))m))2∫0T(DsY(t))2ds→(byy″(t,Y(t)))2∫0T(DsY(t))2ds
as m→∞. Moreover, since ϕ has compact support, maxy∈R(ϕ′(y))2<∞, so we can write
∫0T(Dsfm(Y(t)))2ds=(byy″(t,Y(t))ϕ′(by′(t,Y(t))m))2∫0T(DsY(t))2ds≤maxy∈R(ϕ′(y))2(byy″(t,Y(t)))2∫0T(DsY(t))2ds∈L2(Ω).
Therefore, by the dominated convergence theorem,
E[∫0T(Dsfm(Y(t))−byy″(t,Y(t))DsY(t))2ds]→0,m→∞,
which proves the first claim of the Proposition.
2) Let us proceed with the second claim and verify that
exp{∫utby′(v,Y(v))dv}∈D1,p
with (16) being the corresponding Malliavin derivative. Note that, since by′ is bounded from above, exp{∫utby′(v,Y(v))dv} is also bounded from above and hence is an element of Lp(Ω) for any p>1. Moreover, by Proposition 1, boundedness of exp{∫utby′(v,Y(v))dv} and (13), we can write
E[(∫0T(exp{∫utby′(v,Y(v))dv}∫utbyy″(v,Y(v))DsY(v)dv)2ds)p2]≤CE[ξp(∫0T∫ut(DsY(v))2dvds)p2]≤CE[ξp(∫0T∫utK2(v,s)dvds)p2]+Cexp{pcT}E[ξ2p](∫0T∫ut∫svK2(u,s)dudvds)p2<∞,
and hence it is sufficient to prove that exp{∫utby′(v,Y(v))dv}∈D1,2.
Since the Malliavin derivative operator D is closed and the expression ∫utbyy″(v,Y(v))DsY(v)dv is well-defined by Proposition 1, Step 1 of the current proof and Hille’s theorem [22, Theorem 1.2.4] guarantee that
∫utby′(v,Y(v))Y(v)dv∈D1,2
and
Ds∫utby′(v,Y(v))Y(v)dv=∫utbyy″(v,Y(v))DsY(v)dv.
Finally, the function x↦ex satisfies the conditions of the chain rule from [9, Proposition 3.4] and hence exp{∫utby′(v,Y(v))dv}∈D1,2 and (16) holds. □
Proposition 3 and Lemma 2 together allow us to deduce the following corollary.
For any0≤s<t≤Tandp>1,by′(s,Y(s))exp{∫stby′(v,Y(v))dv}∈D1,pandDu[by′(s,Y(s))exp{∫stby′(v,Y(v))dv}]=byy″(s,Y(s))exp{∫stby′(v,Y(v))dv}DuY(s)+by′(s,Y(s))exp{∫stby′(v,Y(v))dv}∫stbyy″(v,Y(v))DuY(v)dv.
For fixed 0≤s<t≤T, denote
X1:=by′(s,Y(s)),X2:=exp{∫stby′(v,Y(v))dv}.
By Proposition 3 and Lemma 2 from the Appendix, it is sufficient to check that for all p≥2
the product X1X2∈Lp(Ω),
E[(∫0T(X2DuX1)2)p2]<∞ and
E[(∫0T(X1DuX2)2)p2]<∞.
All conditions (i)–(iii) can be checked in a straightforward manner using Proposition 1 and the arguments similar to the proof of Proposition 2. □
We are now ready to formulate the main result of this section.
For anyt∈[0,T]andp≥2,
Y(t)∈D2,p,
with probability 1 and for a.a.r,s∈[0,T],DrDsY(t)=∫stK(u,s)F1(t,u)(∫utbyy″(v,Y(v))DrY(v)dv)du+∫stK(u,s)F2(t,u)DrY(u)du,whereF1(t,u):=by′(u,Y(u))exp{∫utby′(v,Y(v))dv},F2(t,u):=byy″(u,Y(u))exp{∫utby′(v,Y(v))dv}.
Our goal is to prove that Y(t)∈D2,p and
DrDsY(t)=∫stK(u,s)Dr[by′(u,Y(u))exp{∫utby′(v,Y(v))dv}]du=∫stK(u,s)Dr[F1(t,u)]du,
since, in such case, (18) follows immediately from Corollary 1. Recall that
DsY(t)=K(t,s)+∫stK(u,s)F1(t,u)du.
Clearly, for any 0≤r,s<t≤T,
DrK(t,s)=0,
so, by closedness of D and Hille’s theorem [22, Theorem 1.2.4], it is enough to show that
for a.a. 0≤s≤u<t≤T, K(u,s)F1(t,u)∈D1,p and
for a.a. 0≤s<t≤T,
∫0T(E[(∫0T(Dr[K(u,s)F1(t,u)])2dr)p2])1pdu=∫0TK(u,s)(E[(∫0T(Dr[F1(t,u)])2dr)p2])1pdu<∞.
Item (i) above follows immediately from Corollary 1. As for item (ii), observe that, by Proposition 1, (13) as well as the boundedness of exp{∫utby′(v,Y(v))dv}, we have
(Dr[F1(t,u)])2≤C((byy″(u,Y(u)))2exp{2∫utby′(v,Y(v))dv}(DrY(u))2+(by′(u,Y(u)))2exp{2∫utby′(v,Y(v))dv}××∫ut(byy″(v,Y(v))DrY(v))2dv)≤C(ξ2(DrY(u))2+ξ4∫ut(DrY(v))2dv)≤Cξ2(K2(u,r)+∫ruK2(z,r)dz)+Cξ4(∫utK2(v,r)dv+∫ut∫rvK2(z,r)dzdv).
Hence, for any p≥2, Remark 1 implies
∫0T(Dr[F1(t,u)])2dr≤Cξ2(∫0TK2(u,r)dr+∫0T∫ruK2(z,r)dzdr)+Cξ4∫0T∫utK2(v,r)dvdr+Cξ4∫0T∫ut∫rvK2(z,r)dzdvdr≤C(ξ2+ξ4),
so, since ξ has moments of all orders, (ii) holds, which finalizes the proof. □
Finally, denote L2,p:=Lp([0,T];D2,p). We complete the section with the following result.
For anyp≥2,Y∈L2,p.
By the definition of the ‖·‖2,p-norm in (32) from Appendix A, it is sufficient to check that ∫0TE[|Y(t)|p]<∞,∫0TE[(∫0T(DsY(t))2ds)p2]dt<∞ and
∫0TE[(∫0T∫0T(DrDsY(t))2dsdr)p2]dt<∞.
By (11), (19) holds automatically. Next, (20) can be easily deduced from (14). Finally, using Proposition (1) and the boundedness of exp{∫utby′(v,Y(v))dv}, it is easy to prove a bound similar to (14) for
E[(∫0T∫0T(DrDsY(t))2dsdr)p2],
which implies (21). By this, the proof is complete. □
Power law in SVV 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 (10) as stochastic volatility. Namely, we consider a (risk-free) market model with the price process S={S(t),t∈[0,T]} of the form
S(t)=eX(t),X(t)=x0+rt−12∫0tY2(s)ds+∫0tY(s)(ρdB1(s)+1−ρ2dB2(s)),Y(t)=y0+∫0tb(s,Y(s))ds+∫0tK(t,s)dB1(s),
where B1,B2 are two independent Brownian motions, X={X(t),t∈[0,T]} denotes the (risk-free) log-price of an asset starting from some level x0∈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.
The model (22) was initially introduced in [9] and, given the nature of the volatility process, is called the Sandwiched Volterra Volatility (SVV) model.
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.
Let Assumptions1and2hold withH∈(16,12). Assume that the Volterra kernelKis such that, for any0≤s<t≤T,|K(t,s)|≤C|t−s|−12+Hfor some constantC>0, and1τ32+H∫0τ∫sτK(t,s)dtds→KY,τ→0+,for some finite constantKY. Then, with probability 1, the SVV implied volatilityσˆexhibits the propertylimτ→0τ12−H∂σˆ∂κ(τ,κ)|κ=0=ρy0KY.In particular, ifρKY≠0, the SVV model (22) reproduces the power law (2) of the at-the-money implied volatility skew.
The behavior of empirically observed implied volatilities (see, e.g., [12]) shows that realistic market models should produce σˆ with
∂σˆ∂κ(τ,κ)|κ=0<0.
In the SVV setting (22), Theorem 4 guarantees that (24) holds for all small enough τ provided that ρKY<0.
The condition H>16 in Theorem 4 is consistent with the recent empirical estimate H≈0.19 for the SPX implied volatility obtained in [8].
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.
In the recent literature (see, e.g., [4, 8, 12, 20]), it is typical to characterize the implied volatility skew in terms of ∂σˆ∂κ with κ=logKerτ+x0 being the log-moneyness. In [1], a slightly different parametrization σˆlog-price(τ,x0) is considered with
σˆlog-price(τ,x)=σˆ(τ,logKerτ−x).
With this parametrization,
∂σˆlog-price(τ,x)∂x=−∂σˆ(τ,logKerτ−x)∂κ
and the power law (2) is equivalent to
|∂σˆlog-price∂x(τ,x)|x=logKerτ=O(τ−12+H),τ→0.
With Remark 8 in mind, let us provide a slightly adjusted version of [1, Theorem 6.3].
Consider a risk-free log-priceX(t)=x0+rt−12∫0tσ2(s)ds+∫0tσ(s)(ρdB1(s)+1−ρ2dB2(s)),whereB1,B2are two independent Brownian motions,x0∈Ris a deterministic initial value, r is an instantaneous interest rate,ρ∈(−1,1)is a correlation coefficient andσ={σ(t),t∈[0,T]}is a square-integrable stochastic process with right-continuous trajectories adapted to the filtrationF={Ft,t∈[0,T]}generated byB1.
Assume that
σ∈L2,4with respect toB1;
there exists a constantφ∗>0such that, with probability 1,σ(t)>φ∗for allt∈[0,T];
there exists a constantH∈(0,12)such that, with probability 1, for any0<s<t<T,E[(Dsσ(t))2]≤C(t−s)1−2H,E[(DrDsσ(t))2]≤C(t−rt−s)1−2H,whereC>0is some constant;
σ has a.s. right-continuous trajectories;
supr,s,t∈[0,τ]E[(σ(s)σ(t)−σ2(r))2]→0whenτ→0+.
Finally, assume that there exists a constantKσ>0such that, with probability 1,1τ32+H∫0τ∫sτE[Dsσ(t)]dtds−Kσ→0,τ→0+.Then, with probability 1,limτ→0τ12−H∂σˆlog-price∂x(τ,x)|x=logKerτ=−ρσ(0)Kσ.
The original formulation of [1, Theorem 6.3] is slightly more general than Theorem 5 above in the sense that
in [1, Theorem 6.3], the log-price X is allowed to have jumps;
the result in [1] is formulated for the future implied volatility surfaces σˆlog-price(t0,τ,X(t0)), t0≥0.
Since we are interested in the continuous model (22), we removed the jump component in (25) and, for the simplicity of notation, we put t0=0.
Observe that the SVV model (22) automatically satisfies a number of assumptions of Theorem 5:
assumption (H2) with φ∗:=mint∈[0,T]φ(t)>0;
assumption (H4) since Y is continuous a.s.;
assumption (H1) by the results of Section 3 above.
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).
Let Assumptions1and2hold. Then with probability 1,supr,s,t∈[0,τ]E[(Y(s)Y(t)−Y2(r))2]→0,τ→0.
By [10, Lemma 3.6], there exists a positive random variable Υ=ΥT such that, for all t1,t2∈[0,T],
|Y(t1)−Y(t2)|≤Υ|t1−t2|λ
and, for any r>0,
E[Υr]<∞.
Therefore, given that maxt∈[0,T]Y(t)<maxt∈[0,T]ψ(t) by (11),
E[(Y(s)Y(t)−Y2(r))2]=E[(Y(s)(Y(t)−Y(r))+Y(r)(Y(s)−Y(r)))2]≤2E[(Y2(s)(Y(t)−Y(r))2]+2E[Y2(r)(Y(s)−Y(r))2]≤2|t−r|2λmaxs∈[0,T]ψ2(s)E[Υ2]+2|s−r|2λmaxs∈[0,T]ψ2(s)E[Υ2]
and hence, with probability 1,
supr,s,t∈[0,τ]E[(Y(s)Y(t)−Y2(r))2]≤4τ2λmaxs∈[0,T]ψ2(s)E[Υ2]→0
as τ→0+. □
Our next step is to handle (28).
Let Assumptions1and2hold and the Volterra kernelKsatisfy (23) for some finite constantKY. Then, with probability 1,1τ32+H∫0τ∫sτE[DsY(t)]dtds−KY→0,τ→0+.
Recall that
F1(t,u):=by′(u,Y(u))exp{∫utby′(v,Y(v))dv}
and that, by Proposition 1,
|F1(t,u)|≤ecTξ,
where c:=max(t,y)∈D‾ay′(t,y). Then we can write
1τ32+H∫0τ∫sτE[DsY(t)]dtds=1τ32+H∫0τ∫sτK(t,s)dtds+1τ32+H∫0τ∫sτ∫stK(u,s)E[F1(t,u)]dudtds=1τ32+H∫0τ∫sτK(t,s)dtds+1τ32+H∫0τ∫sτK(u,s)(∫uτE[F1(t,u)]dt)duds.
The term 1τ32+H∫0τ∫sτK(t,s)dtds converges to KY by (23). As for the second term, note that, with probability 1, for any u∈[0,τ],
∫uτ|E[F1(t,u)]|dt≤CE[ξ]τ
and hence, given (23), with probability 1,
1τ32+H∫0τ∫sτK(u,s)(∫uτE[F1(t,u)]dt)duds→0,τ→0+,
which ends the proof. □
Finally, let us deal with (H3).
Let Assumptions1and2hold withH∈(16,12)and the Volterra kernelKbe such that for any0≤s<t≤T|K(t,s)|≤C|t−s|−12+Hfor some constantC>0. Then the hypothesis (H3) from Theorem5holds for the sandwiched volatility processσ=Y.
Fix 0<r,s<t. Then, taking into account (29), with probability 1,
|DsY(t)|≤|K(t,s)|+∫st|K(u,s)||F1(t,u)|du≤C(|t−s|−12+H+ξ∫st|u−s|−12+Hdu)≤C(1+Tξ)|t−s|−12+H=:ζ|t−s|−12+H,
which immediately implies (26). Next, by Proposition 1,
|byy″(v,Y(v))|≤ξ
for any v∈[0,T] and, for any 0≤u≤t≤T,
|F2(t,u)|=|byy″(u,Y(u))exp{∫utby′(v,Y(v))dv}|≤ecTξ
with c:=max(t,y)∈D‾ay′(t,y), so we can write
|DrDsY(t)|≤∫st|K(u,s)||F1(t,u)|(∫ut|byy″(v,Y(v))||DrY(v)|dv)du+∫st|K(u,s)||F2(t,u)||DrY(u)|du≤C(ξ2∫st|K(u,s)|(∫ut|DrY(v)|dv)du+ξ∫st|K(u,s)||DrY(u)|du)=C(ξ2∫st|K(u,s)|(∫u∨rt|DrY(v)|dv)du+ξ∫r∨st|K(u,s)||DrY(u)|du).
Taking into account (30) and (31),
|DrDsY(t)|≤C(ξ2ζ∫st|u−s|−12+H(∫u∨rt|v−r|−12+Hdv)du+ξζ∫r∨st|u−s|−12+H|u−r|−12+Hdu)≤C(ξ2ζ∫st|u−s|−12+H|t−r|12+Hdu+ξζ∫r∨st|u−s|−12+H|u−r|−12+Hdu).
Note that
∫st|u−s|−12+H|t−r|12+Hdu≤C|t−r|12+H|t−s|12+H≤C(t−rt−s)12−H.
As for the integral ∫r∨st|u−s|−12+H|u−r|−12+Hdu, we have two cases:
if 0<r≤s<t, we can write
∫st|u−s|−12+H|u−r|−12+Hdu≤∫st|u−s|−1+2Hdu≤C(t−s)2H≤C(t−rt−s)12−H;
similarly, if 0<s<r<t and given that H>16, we have
∫rt|u−s|−12+H|u−r|−12+Hdu≤∫rt|u−r|−1+2Hdu≤C(t−r)2H≤C(t−rt−s)12−H.
In any case,
|DrDsY(t)|≤Cξζ(ξ+1)(t−rt−s)12−H,
where ξ and ζ are random variables having all moments, and hence (27) holds. □
Having in mind all of the results above, we are ready to prove the main result of this section, namely Theorem 4.
The results above show that the SVV model satisfies conditions (H1)–(H5) of Theorem 5. Therefore, taking into account the reparametrization described in Remark 8, Theorem 4 follows immediately from Theorem 5. □
Let 16<H0<H1<⋯<Hn<1 be such that H0<12 and αk>0, k=0,…,n. Then the kernel
K(t,s)=(∑k=0nαk(t−s)Hk−12)1s<t
satisfies the assumptions of Theorem 4, so the corresponding SVV model generates power law (2) with H=H0 provided that ρ<0 in (22).
Selected results from the Malliavin calculusThe Malliavin derivative and the space <inline-formula id="j_vmsta246_ineq_266"><alternatives><mml:math>
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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 [25] or [13].
Denote by Cp(∞)(Rn) the space of all infinitely differentiable functions with the derivatives of at most polynomial growth. Let B={B(t),t∈[0,T]} be a standard Brownian motion. For any h∈L2([0,T]), denote
B(h):=∫0Th(t)dB(t).
The random variables X of the form
X=f(B(h1),…,B(hn)),
where n≥1, f∈Cp(∞)(Rn) and h1,…,hn∈L2([0,T]) are called smooth. The set of all smooth random variables is denoted by S.
Let X∈S. The Malliavin derivative of X (with respect to B) is the L2([0,T])-valued random variable of the form
DX:=∑k=1n∂f∂xk(B(h1),…,B(hn))hk.
By [25, Proposition 1.2.1], the operator D is closable from Lp(Ω) to Lp(Ω×[0,T]) for any p≥1, and we use the same notation D for the closure. The domain of this closure D in Lp(Ω), i.e. the closure of the class S with respect to the norm
‖X‖1,p:=(E[|X|p]+E[(∫0T(DsX)2ds)p2])1p,
is traditionally denoted by D1,p. This definition can be iterated as described in [25, p. 27] to introduce the iterated derivative DkX as a random variable with values in (L2([0,T]))⊗k∼L2([0,T]k). One can also define Dk,p as the completion of S with respect to the seminorm
‖X‖k,p:=(E[|X|p]+∑j=1kE[‖DjX‖L2([0,T]k)p])1p.
Throughout the paper, we often use the following lemma which is essentially a simplified version of [25, Proposition 1.5.5].
Letp>1andX∈D1,2be such thatE[|X|p]<∞andE[(∫0T(DsX)2ds)p2]<∞.ThenX∈D1,p.
Generalized Malliavin product rule
Finally, let us prove a generalized version of the product rule from [25, Exercise 1.2.12] or [13, Theorem 3.4].
LetX1,X2∈D1,2be such that
X1X2∈L2(Ω);
X2DX1,X1DX2∈L2(Ω×[0,T]).
ThenX1X2∈D1,2andD[X1X2]=X2DX1+X1DX2.If, in addition,E[|X1X2|p]<∞,E[(∫0T(X2DuX1+X1DuX2)2du)p2]<∞for somep≥2, thenX1X2∈D1,p.
Let ϕ∈C∞(R) be a compactly supported function such that ϕ(x)=x whenever |x|≤1 and |ϕ(x)|≤|x| for all |x|>1. For m≥1, put
fm(x1,x2):=m2ϕ(x1m)ϕ(x2m)
and observe that both partial derivatives
∂fm∂x1(x1,x2)=mϕ′(x1m)ϕ(x2m),∂fm∂x2(x1,x2)=mϕ(x1m)ϕ′(x2m)
are bounded. Therefore, by the classical chain rule [25, Proposition 1.2.3],
Dfm(X1,X2)=m(ϕ′(X1m)ϕ(X2m)DX1+ϕ(X1m)ϕ′(X2m)DX2).
Now it is sufficient to prove that
fm(X1,X2)→X1X2
in L2(Ω) and
m(ϕ′(X1m)ϕ(X2m)DX1+ϕ(X1m)ϕ′(X2m)DX2)→X2DX1+X1DX2
in L2(Ω×[0,T]) as m→∞.
Observe that |fm(X1,X2)|→X1X2 a.s. as m→∞ and
|fm(X1,X2)|≤X1X2∈L2(Ω),
so (33) holds by the dominated convergence theorem. Next, since ϕ′ is bounded, we have that, with probability 1,
m|ϕ′(X1m)ϕ(X2m)DX1+ϕ(X1m)ϕ′(X2m)DX2|≤maxx∈R|ϕ′(x)|(|X2DX1|+|X1DX2|)∈L2(Ω×[0,T]).
Therefore, since mϕ′(X1m)ϕ(X2m)→X2 a.s. and mϕ(X1m)ϕ′(X2m)→X1 a.s. as m→∞, (34) holds by the dominated convergence, which ends the proof of the first claim.
The second claim immediately follows from Lemma 1. □
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