On the martingale property in the rough Bergomi model

We consider a class of fractional stochastic volatility models (including the so-called rough Bergomi model), where the volatility is a superlinear function of a fractional Gaussian process. We show that the stock price is a true martingale if and only if the correlation $\rho$ between the driving Brownian motions of the stock and the volatility is nonpositive. We also show that for each $\rho<0$ and $m>\frac{1}{{1-\rho^2}}$, the $m$-th moment of the stock price is infinite at each positive time.


Introduction and main results
We are interested in fractional stochastic volatility models where the dynamics of (discounted) stock price under a risk-neutral measure take the form where Z t = ρW t +ρW t , W,W are two independent Brownian motions on a filtered probability space (Ω, (F t ) t≥0 , P), and ρ 2 +ρ 2 = 1.
A specific example we have in mind is the so-called Rough Bergomi model introduced in [2]. In that model Y is a Riemann-Liouville fractional Brownian motion of Hurst parameter H ∈ (0, 1), i where η > 0 and ζ is a continuous function of t. The rough Bergomi model with H ∈ (0, 1 2 ) is part of a larger class of fractional stochastic volatility models (so-called "rough volatility models") which has been recently observed to reproduce several features of historical [9] and pricing [1,7,2] data, and has been the subject of intense recent academic activity 1 .
The first question we consider in this note is whether the price process S, which is obviously a local martingale (and a supermartingale) is a true martingale. The true martingale property is very important in practice, since using a strict local martingale measure for pricing has some obvious drawbacks. For instance : if S is a strict local martingale then E[S T ] < S 0 for some T > 0, so that already the price given by the model for holding one unit of stock until time T does not coincide with market data (this suggests that the asset price is greater than its actual "fundamental" value The author is grateful to P. K. Friz for related discussions. This work is partially supported by the ANR via the project ANR-16-CE40-0020-01. 1  and for this reason strict local martingale models have been used in the modelisation of bubbles, see [15] and references therein). Note that in the rough Bergomi model, due to the superlinear growth of σ, Novikov's criterion for martingality is never satisfied. Nevertheless we show that if the correlation is nonpositive (which is typically the case in actual applications), the price process is indeed a true martingale. Actually our result does not rely on the specific form of σ in the rough Bergomi model, and only requires a (rather weak) assumption on K and σ. We also show the converse implication in the case of a Riemann-Liouville type kernel, under a more specific assumption of superlinear growth of σ. (2) Assume in addition that there exists T 0 > 0 such that for some α > 1 2 , The second result of this note deals with the moments of the stock price. We show that, under a similar assumption (satisfied by the rough Bergomi model), for each value of ρ ∈ (−1, 0], some of the higher order moments are infinite. Theorem 2. Assume that there exists T 0 > 0 s.t. for some α > 1 2 , K(s, t) = α(t − s) α−1 for all 0 ≤ s ≤ t ≤ T 0 and σ = σ 0 on [0, T 0 ] × R with σ 0 as in Theorem 1 (2). Then if ρ ≤ 0, m > 1 are such that ρ 2 < m−1 m , it holds that for all t > 0, E[S m t ] = +∞. The finiteness of moments is important for instance in Monte Carlo simulation (to know that the Monte Carlo error is ruled by CLT estimates, finite variance is needed) and in asymptotic formulae (to go from stock price large deviations to call price asymptotics, see for instance [6, section 4.2]). It would therefore be very useful to obtain a converse (positive) result.
We remark that in the Brownian case (K ≡ 1), both of the above results are well known, cf. [17,13,14], and in that case the condition ρ 2 > m−1 m is also a sufficient condition for the moments to be finite. The fact that all moments of order greater than 1 are infinite in the fractional case with ρ = 0 was recently obtained in [12] under slightly different assumptions (along with some partial results for ρ = 0). We also point out that in the case of the rough Heston model, [10] have recently obtained some results on moments of the stock price (which are similar in spirit to those for classical Heston, and therefore quite different from ours).
The remainder of this note is devoted to the proofs of Theorems 1 and 2, which we now outline. The proof of Theorem 1 follows the classical argument (found already in the aforementioned [17,13,14], see also [3,16] for additional references) relating the martingale property of stochastic exponentials with explosions of a SDE (in our case, this will be a Volterra SDE). The martingale property (1) is then essentially immediate, while the proof of (2) follows from the fact that the Volterra SDE may blow up in arbitrarily short time with positive probability.
The proof of Theorem 2 relies on the Boué-Dupuis formula, which expresses the expectation of exponentials of Brownian functionals as values of (here : Volterra) stochastic control problems. We then show that for the considered values of the parameters, we may choose a feedback control such that, as in the previous proof, the process (and the value) blow up in arbitrarily small time. This proof is new even in the classical (Markovian) case.

Preliminaries.
2.1.1. Volterra integral equations. In this subsection, we fix and consider the Volterra equation of unknown y. We will use the following results.
(3) Assume that t → z(t) and t → σ(t, x) are nondecreasing (for each x ∈ R). Then so is t → y(t).
Proof. cf. [ We will also use the following lemma which gives an explicit upper bound on blow-up time for solutions to (2.1).

Lemma 1.
In the setting of Proposition 2, assume that t → z(t) is nondecreasing. Then with T ∞ as in (1), it holds that for each T > 0 where h(x) = sup{t : z(t) ≤ x}.
We fix x ≥ 0 such that h(x) and ∞ x w b0(w) 1 α dw w are finite, R > 1, and for each n ≥ 0 we let T n = sup{t : y(t) ≤ xR n } ∈ (0, +∞]. Note that T 0 ≤ h(x). We then have for n ≥ 1, for each t > T n−1 where we have used the monotonicity of y (Proposition 2 (3)). This implies that if T n−1 is finite, so is T n , with We obtain the result by letting R ↓ 1.

Stochastic convolutions.
We consider the following stochastic convolution (recall that (Z t ) t≥0 is a P-Brownian motion). We recall the well-known condition for Y to be continuous (cf. e.g. [8, Theorem 2.1]).

Proposition 3. Assume that
and for each T ≥ 0, letting Then Y admits a version with continuous sample paths.
Note that the assumption above is satisfied for K(t, s) = α(t − s) α−1 , any α > − 1 2 . We will also need a result on the support of the law of Y . Proposition 4. In addition to the assumptions of Proposition 3, assume that for some δ > 0, K(t, s) > 0 for all 0 ≤ s ≤ t with (t − s) ≤ δ. Then for each T ≥ 0, the law of Y has full support in C T 0 := {y ∈ C([0, T ], R); y(0) = 0}. Proof. By a direct application of the Cameron-Martin theorem, the support of the law of Y is the closure in C T 0 of the maps Hence it suffices to show that these maps form a dense set in C T 0 . Let µ be a Radon measure on (0, T ] such that The first term converges to E[S T ] when n goes to infinity, so that On the other hand, we can apply Girsanov's theorem to write whereP n is such thatŴ is a Brownian motion underP n . Note that for t ≤ τ n one has whereẐ (n) is aP n -Brownian motion, and We first treat the case ρ ≤ 0. Since Y t ≤Ŷ t (for t ≤ τ n ) one then has τ n ≥ τ 0 n := inf{t > 0,Ŷ t = n}. In addition, sinceẐ (n) is aP n -Brownian motion, one has We now treat the case ρ > 0. We then have for t < τ n In particular, by Proposition 2 (2) and the fact thatẐ (n) is a Brownian motion underP n , one has where T ∞ is the explosion time of the solutionỸ tõ (which exists and is unique P-a.s. by Proposition 2). Let x, λ > 0 be chosen such that Let z λ (t) = λt − 1 and y λ be the solution to ( (this defines a probability measure by Theorem 1 (1) since ρ ≤ 0), and we have that Note that since Y 0 isP-a.s. continuous, combining Proposition 1 with the a priori bounds On the other hand, if ρ 2 < m−1 m one can find γ such that The idea is then that taking the feedback control v s = γσ(Y s ), using the first inequality, it holds that for each T > 0 Y v has positive probability of blowing up before T . On the other hand, the second inequality ensures that the gain is +∞ in this case, so that the value (and the moment) is infinite.
We now give a rigorous proof. We fix A > 0, n > 0, let where Y n,A is the unique (by Proposition 1) solution to Note that 0 ≤ Y n,A t ≤ Y 0 t +nCt α for some C > 0, which implies that v n,A ≤ C n,A , and in particular v n,A ∈ V.
We therefore have lnÊ [S m T ] ≥Ê Now as in the proof of Theorem 1, we fix x, λ such that let z λ (t) = λt − 1 and for n ∈ N ∪ {∞} let y n λ be the solution to (2.1) with z = z λ and b(t, ·) = (ρm + γ)σ 0 ∧ n. Note that y ∞ λ blows up in time T ∞ < T by Lemma 1. By Proposition 2 (2), y n λ is nondecreasing in n, and therefore for each T ∞ < t < T , y n λ (t) → n↑∞ +∞.