Moment Explosions in the Rough Heston Model

We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arxiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment.


Introduction
It has long been known that the marginal distributions of a realistic asset price model should not feature tails that are too thin (as, e.g., in the Black-Scholes model). In many models that have been proposed, the tails are of power law type. Consequently, not all moments of the asset price are finite. Existence of the moments has been thoroughly investigated for classical models; in particular, we mention here Keller-Ressel's work [20] on affine stochastic volatility models. Precise information on the critical moments -the exponents where the stock price ceases to be integrable, depending on maturity -is of interest for several reasons. It allows to approximate the wing behavior of the volatility smile, to assess the convergence rate of some numerical procedures, and to identify models that would assign infinite prices to certain financial products. We refer to [3,20] and the article Moment Explosions in [8] for further details and references on these motivations. Moreover, when using the Fourier representation to price options, choosing a good integration path (equivalently, a good damping parameter) to avoid highly oscillatory integrands requires knowing the strip of analyticity of the characteristic function. Its boundaries are described by the critical moments [24,26].
In recent years, attention has shifted in financial modeling from the classical (jump-)diffusion and Lévy models to rough volatility models. Since the pioneering work by Gatheral et al. [15], the literature on these non-Markovian stochastic volatility models, inspired by fractional Brownian motion, has grown rapidly. We refer, e.g., to Bayer et al. [4] for many references. In the present paper we provide some results on the explosion time and the critical moments of the rough Heston model. While there are several "rough" variants of the Heston model, we work with the one proposed by El Euch and Rosenbaum [9]. The dynamics of this model are Date: April 5, 2018. 2010 Mathematics Subject Classification. 91G20,45D05. Financial support from the Austrian Science Fund (FWF) under grants P 24880 and P 30750 is gratefully acknowledged. We thank Omar El Euch, Antoine Jacquier, and Martin Keller-Ressel for helpful comments.
given by where W and Z are correlated Brownian motions, ρ ∈ (−1, 1), and λ, ξ,v are positive parameters. The smoothness parameter α is in ( 1 2 , 1). (For α = 1, the model clearly reduces to the classical Heston model.) Besides having a microstructural foundation, this model features a characteristic function that can be evaluated numerically in an efficient way, by solving a fractional Riccati equation (equivalently, a non-linear Volterra integral equation; see Section 2). Its tractability makes the rough Heston model attractive for practical implementation, and at the same time facilitates our analysis.
We first analyze the explosion time, i.e., the maturity at which a fixed moment explodes. While the explosion time of the classical Heston model has an explicit formula, for rough Heston we arrive at a well-known hard problem: Computing the explosion time of the solution of a non-linear Volterra integral equation (VIE) of the second kind. There is no general algorithm known, and in most cases that have been studied in the literature, only bounds are available. See Roberts [30] for an overview. Using the specific structure of our case, we show that the explosion time is finite if and only if it is finite for the classical Heston model, and we provide a lower and an upper bound (Sections 3-5). As a byproduct, the validity of the fractional Riccati equation, respectively the VIE, for all moments is established, which culminates in Section 6. In Section 7 we derive an algorithm to compute the explosion time, under some restrictions on the parameters. The critical moments are finite for all maturities (Section 8) and can be computed by numerical root finding (Section 9). Our approach has two limitations: First, to compute the critical moments, maturity must not be too high. Second, our algorithm can compute the upper critical moment only for ρ > 0, and the lower critical moment for ρ < 0. As the latter is the more important case in practice, we focus on the left wing of implied volatility when recalling the relation between critical moments and strike asymptotics (Lee's moment formula; see Section 10).
Corollary 3.1 in [10] is related to our results. For each maturity, it gives explicit lower and upper bounds for the critical moments. Inverting them yields a lower bound for the explosion time; the latter is not comparable to our bounds.

Preliminaries
El Euch and Rosenbaum [9] established a semi-explicit representation of the moment generating function (mgf) of the log-price X t = log(S t /S 0 ) in the rough Heston model. The mgf is given by where ψ satisfies a fractional Riccati differential equation (see below). The constraint ρ ∈ (−1/ √ 2, 1/ √ 2] from [9] was removed recently in [10]. The paper [2] contains an alternative derivation of the fractional Riccati equation, and [1] has more general results, embedding the rough Heston model into the new class of affine Volterra processes. Recall the following definition (see e.g. [21]): Definition 2.1. The (left-sided) Riemann-Liouville fractional integral I α t of order α ∈ (0, ∞) of a function f is given by whenever the integral exists, and the (left-sided) Riemann-Liouville fractional derivative D α t of order α ∈ [0, 1) of f is given by whenever this expression exists.
(The fractional derivative D α t can be defined for α > 1 as well, but this is not needed in our context.) The function ψ from (2.1) is the unique continuous solution of the fractional Riccati initial value problem where R is defined as with coefficients For α = 1, this becomes a standard Riccati differential equation, which admits a well-known explicit solution [14,Chapter 2]. The roots of R(u, ·) are located at the points 1 c3 (−e 0 (u) ± e 1 (u)) with e 0 (u) := 1 2 c 2 (u) = 1 2 (ρξu − λ), (2.7) The following result, relating fractional differential equations and Volterra integral equations, is a special case of Theorem 3.10 in [21]. Theorem 2.2. Let α ∈ (0, 1), T > 0 and suppose that ψ ∈ C[0, T ] and H ∈ C(R). Then ψ satisfies the fractional differential equation Using Theorem 2.2, the Riccati differential equation (2.4) with initial value (2.5) can be transformed into the non-linear Volterra integral equation This integral equation was used in [9] to compute ψ numerically. The function (2.10) where e 0 (u) and e 1 (u) are defined in (2.7) and (2.8). Equation (2.9) is a nonlinear Volterra integral equation with weakly singular kernel; it will be used to analyze the blow-up behavior of f (and thus of ψ) in Section 3. We quote the following standard existence and uniqueness result for equations of this kind. Theorem 2.3. Let α ∈ (0, 1), g ∈ C[0, ∞), and suppose that H : R → R is locally Lipschitz continuous. Then there is T * ∈ (0, ∞] such that the Volterra integral equation Proof. For existence and uniqueness on a sufficiently small interval [0, T 0 ] with T 0 > 0 see, e.g., Theorem 3.1.4 in Brunner's recent monograph [6]. The continuation to a maximal right-open interval is discussed there as well (p. 107; see also Section 12 of Gripenberg et al. [19]).
Note that cases (A) and (B) combined are exactly the cases in which the moment explosion time T * 1 (u) in the classical Heston model is finite, by (2.13). We can now state our first main result. The proof of Theorem 2.4 consists of two main parts. First, Propositions 3.2, 3.4, 3.6, and 3.7 discuss the blow-up behavior of the solution of (2.9) in cases (A)-(D), and Lemma 3.8 shows that blow-up of f leads (unsurprisingly) to blow-up of the right-hand side of (2.1). Second, we show in Section 5 that the explosion time of f (u, ·) (the solution of (2.9)) agrees with T * α (u) (the explosion time of the rough Heston model) for all u ∈ R. As mentioned after Theorem 2.3, this is not obvious from the results in the existing literature.

Explosion time of the Volterra integral equation
We begin by citing a result from Brunner and Yang [7] which characterizes the blow-up behavior of non-linear Volterra integral equations defined by positive and increasing functions. We note that some arguments in our subsequent proofs (from Proposition 3.2 onwards) are similar to arguments used in [7]. Alternatively, it should be possible to extend the arguments in Appendix A of [16]; there, u is in [0, 1].
Proof. This is a special case of Corollary 2.22 in Brunner and Yang [7], with G not depending on time.
In case (A), all assumptions of Proposition 3.1 are satisfied and only the integrability condition (3.1) has to be checked to determine whether the solution f of (2.9) blows up in finite time. Proof. Fix u ∈ R such that c 1 (u) > 0 and e 0 (u) ≥ 0. Note that e 2 0 − e 1 > 0 in this case. (Here and in the following, we will often suppress u in the notation.) If we write the Volterra integral equation (2.9) in the form with non-linearityḠ(w) = w 2 + 2e 0 w and φ(t) =  [7], or from Lemma 3.2.11 in [6].) It is easy to check that all the assumptions (G1), (G2), (P) and (K) of Proposition 3.1 are satisfied. Moreover, lim t→∞ φ(t) = ∞ and for all U > 0. By Proposition 3.1, the solution f blows up in finite time.
In case (B), Proposition 3.1 cannot be applied directly to the solution f of (2.9). Hence, the Volterra integral equation has to be modified in order to satisfy the assumptions of Proposition 3.1 in a way that f is still a subsolution of the modified equation, i.e. f satisfies (2.9) with "≥" instead of "=". First, we provide a comparison lemma for solutions and subsolutions.
be a strictly increasing, continuous function and T > 0. Suppose that g is the unique continuous solution of the Volterra integral equation where k satisfies condition (K) from Proposition 3.1. If f is a continuous subsolution, Since c ∈ (0, T ) was arbitrary, the result follows easily. Proof. Fix u ∈ R such that c 1 (u) > 0, e 0 (u) < 0 and e 1 (u) < 0. Note that in this case, the non-linearity G is obviously positive by (2.10). However, G is strictly decreasing on [0, −e 1 ]. To deal with this problem, let 0 < a < −e 1 and define the modified non-linearityḠ a as ThenḠ a is a positive, strictly increasing, Lipschitz continuous function that starts at a andḠ a ≤ G. Letf be the unique continuous solution (recall Theorem 2.3) of the Volterra integral equation Note that the second equality in (3.3) follows from (7.3). Due to the positivity of φ andḠ on (0, ∞), the solutionf is positive on (0, ∞) as well. The functions φ,Ḡ and k satisfy the assumptions (G1), (G2), (P) and (K) in Proposition 3.1. Furthermore, lim t→∞ φ(t) = ∞ andḠ satisfies (3.1). By Proposition 3.1,f blows up in finite time. Because f satisfies (2.9) andḠ a ≤ G, it follows that f is a subsolution of the modified Volterra integral equation, i.e., Since G is non-negative and k is decreasing, which is a contradiction. Therefore, f satisfies 0 ≤ f (t) ≤ a for all t ≥ 0. Proposition 3.6. In case (C), the solution f of (2.9) is non-negative and bounded, and exists globally.
Proof. Fix u ∈ R such that c 1 (u) > 0, e 0 (u) < 0 and e 1 (u) ≥ 0. Note that the inequality 0 ≤ e 1 = e 2 0 − c 1 c 3 < e 2 0 implies a := −e 0 − √ e 1 > 0. Moreover, from (2.10), it follows that a is the smallest positive root of G. Define the non-linearityḠ asḠ ThenḠ is a non-negative, Lipschitz continuous function that starts at e 2 0 − e 1 > 0. Therefore, Lemma 3.5 yields that the unique continuous solutionf of , the functionf solves the original Volterra integral equation and from the uniqueness of the solution we obtain f =f .
Proposition 3.7. In case (D), the solution f of (2.9) is non-positive and bounded, and exists globally.
If we define the non-linearityḠ as thenḠ is a non-negative, Lipschitz continuous function that starts at e 1 − e 2 0 > 0. With Lemma 3.5 we obtain that the unique continuous solutionf of The uniqueness of the solution yields We have shown that (A) and (B) are exactly the cases in which the solution f of the Volterra integral equation (2.9), and thus the solution ψ of the fractional Riccati differential equation (2.4) with initial value (2.5), blows up in finite time.
The following lemma shows that blow-up of ψ is equivalent to blow-up of the righthand side of (2.1).
Proof. First, suppose that the non-negative, continuous function f explodes at T and let M > 0. Then we can find ε ∈ (0,T /2) such that f (t) ≥ M for all t ∈ (T − ε,T ). Hence, for all t ∈ (T − ε,T ). For the second assertion, suppose that f is continuous and bounded with M > 0. Then we have

Bounds for the explosion time
We now establish lower and upper bounds forT α (u), valid whenever it is finite (cases (A) and (B)). We denote byT α (u) the explosion time of the solution f (u, ·) of (2.9). As we will see later, it agrees with T * α (u), and so both bounds of this section hold for the explosion time of the rough Heston model. We prove them first, because we will apply the lower bound in the proof of T * α (u) =T α (u). of (2.9) satisfies where a(u) = 0 in case (A) and a(u) = −e 0 (u) > 0 in case (B).
Proof. Fix u satisfying the requirements of case (A) or (B). It follows from Propositions 3.2 and 3.4 that in either case the solution f is non-negative, starts at 0 and lim t↑Tα f (t) = ∞. For any n ∈ N 0 choose Thus, we obtain for n ∈ N Finally,T Maximization over c > 0, then r > 1, and the substitution w = s α + a yield the inequality (4.1).
For α ↑ 1, the right-hand side of (4.1) simplifies to In case (A), the lower bound (4.1) is sharp in the limit α ↑ 1: We have a(u) = 0 then, and therefore Thus, we obtain for n ∈ N Therefore,T Note that from the definition of t 0 , it depends on c > 0 and r > 1. The fact that f is only zero at t = 0 implies that t 0 → 0 as c ↓ 0. Taking the limit c ↓ 0, then minimizing over r > 1 and substitution w = s α yieldŝ In case (A), we are finished. In case (B), we haveḠ =Ḡ a . Then the dominated convergence theorem for a ↑ −e 1 yields the inequality (4.3).
See Figures 1-3 for numerical examples of these bounds.

Explosion time in the rough Heston model
In Section 3, we established that the right-hand side of (2.1), defined using the solution f of the VIE (2.9), explodes if and only if u satisfies the conditions of cases (A) or (B). As before, we writeT α (u) for the explosion time of f (u, ·). Recall that T * α (u) denotes the explosion time of the rough Heston model, as defined in (2.11). The goal of the present section is to show thatT α (u) = T * α (u), and that (2.1) holds for all u ∈ R and 0 < t < T * α (u). The following result from [10] was already mentioned at the end of the introduction.
Proof. We check the requirements of Theorem 13.1.2 in Gripenberg et al. [19]. The polynomial G(u, w) is differentiable. The kernel (t − s) α−1 /Γ(α) =: k(t − s) is of continuous type in the sense of [19]; see the remark to Theorem 12.1.1 there, which states local integrability of k as a sufficient condition for this property. Proof. We only discuss the case u < 0, because u > 0 is analogous.
(i) Note that (5.1) is a "linear VIE" that can be written as where we define to bring the notation close to that of Section 6.1.2 in [5]. Clearly, (5.2) is not really a linear VIE, because the unknown function f appears in g and K (u) . But as our aim is not to solve it, but to control the sign of ∂ 1 f , this viewpoint is good enough.
(ii) Recall that we assume that u < 0, because u > 0 is analogous. We have to show that satisfies τ (u) <T α (u). We use the following facts: ∂ 1 G(u, w) < 0 for w large, ∂ 2 G(u, w) > 0 for w large, and f (u, t) explodes as t ↑T α (u). Thus, g from (5.3) satisfies (5.9) lim t↑Tα(u) and K (u) satisfies lim t↑Tα(u) K (u) (t) = +∞. We can therefore pick ε > 0 such that For z ∈ [0, 1] and any s, t <T α (u) satisfyingT α (u) − ε ≤ s ≤ t, we have Using this observation in (5.7), we see from a straightforward induction proof that The same then holds for the resolvent kernel (5.6), By (5.5), we obtain where the right-hand side is positive. Indeed, (5.12) follows from (5.9) and (5.10), as g(s) on the left-hand side of (5.12) is O(1). Thus, letting t ↑T α (u), we find that the negative terms g(t) + t t−ε on the right-hand side of (5.11) dominate. This completes the proof.
Proof. According to Section 3.1.1 in [6], the solution can be constructed by successive iteration and continuation. We just show that the first iteration step leads to an analytic function, because the finitely many further steps needed to arrive at arbitrary t <T α (u) can be dealt with analogously. Define the iterates f 0 = 0 and On a sufficiently small time interval, f n (v, ·) converges uniformly to f (v, ·), and the solution can then be continued by solving an updated integral equation and so on (see [6]), until we hitT α (v). Now fix u and t as in the statement of the lemma. For a sufficiently small open complex neighborhood U u, it is easy to see that t <T α (v) holds for v ∈ U . Define γ := 1 ∨ sup v∈U |v| and η := 1 ∨ t α Γ(α + 1) .
Then there is c ≥ 1 such that, for arbitrary v ∈ U and w ∈ C, By the definition of f n , a trivial inductive proof then shows that By a standard result on parameter integrals (Theorem IV.5.8 in [11]), the bound (5.13) implies that each function f n (·, t) is analytic in U . From the bounds in Section 3.1.1 of [6], it is very easy to see that the convergence f n (v, t) → f (v, t) is locally uniform w.r.t. v for fixed t. It is well known (see Theorem 3.5.1 in [18]) that this implies that the limit function f (·, t) is analytic.
Lemma 5.5. The function u →T α (u) increases for u ≤ 0 and decreases for u ≥ 1.
Proof. We assume that u < 0, as u ≥ 0 is handled analogously. By Lemma 5.5, u →T α (u) increases. In this proof, we write M (u, t) for the right-hand side of (2.1), andM (u, t) = E[e uXt ] for the mgf. Now fix u < 0 and 0 < t <T α (u) such that (u, t) has positive distance from the graph of the increasing functionT α (·). Clearly, it suffices to consider pairs (u, t) with this property. By Lemma 5.1, there are v − < v + such that We now show that (5.15) extends to u ≤ v ≤ v + by analytic continuation. From general results on characteristic functions (Theorems II.5a and II.5b in [34]), v → M (v, t) is analytic in a vertical strip w − < Re(v) < w + of the complex plane, and has a singularity at v = w − . If we suppose that w − > u, then Lemma 5.4 leads to a contradiction: The left-hand side of (5.15) would then be analytic at v = w − , and the right-hand side singular. This shows that (5.15) can be extended to the left up to u by analytic continuation.
The following theorem completes the proof of Theorem 2.4.
Proof. In the light of Lemma 5.6, it only remains to show thatT α (u) ≥ T * α (u). (Obviously, Lemma 5.6 implies thatT α (u) ≤ T * α (u).) But this is clear from the continuity of the map t →M (u, t) = E[e uXt ] on the interval (0, T * α (u)). This continuity follows from the continuity of t → X t , Doob's submartingale inequality, and dominated convergence.

Validity of the fractional Riccati equation for complex u
Although the focus of this paper is on real u, the mgf needs to be evaluated at complex arguments when used for option pricing. The following result fully justifies using the fractional Riccati equation (2.4), respectively the VIE (2.9), to do so. As above, we write T * α (u) for the moment explosion time of S, andT α (u) for the explosion time of the VIE (2.9). Theorem 6.1. Let u ∈ C. Then T * α (u) = T * α (Re(u)), and (2.1) holds for 0 < t < T * α (u).
Proof of Theorem 6.1. The first statement is clear from |e uXt | = e Re(u)Xt . Now let t > 0 be arbitrary. As above, we writeM for the mgf and M for the right-hand side of (2.1). By Theorem 5.7, we have M (v, t) =M (v, t) for v in the real interval The functionM (·, t) is analytic on the strip (6.1) {v ∈ C : Re(v) ∈ I} = {v ∈ C : T * α (v) ≥ t}. By the same argument as in Lemma 5.4, the function M (·, t) is analytic on the set {v ∈ C :T α (v) ≥ t}, which contains the strip (6.1) by Lemma 6.2. Therefore, M (·, t) andM (·, t) agree on (6.1) by analytic continuation. This implies the assertion.

Computing the explosion time
Recall that, for fixed u ∈ R, the explosion time T * α (u) of the rough Heston model is the blow-up time of f (t) = f (u, t) = c 3 ψ(u, t), where ψ solves the fractional Riccati initial value problem (2.4)-(2.5). We know from Theorem 2.4 that T * α (u) < ∞ exactly in the cases (A) and (B), defined in Section 2. We now develop a method (Algorithm 7.5) to compute T * α (u) for u satisfying the conditions of case (A). In case (B), a lower bound can be computed, which is sometimes sharper than the explicit bound (4.1). The function f satisfies the fractional Riccati equation where d 1 (u) := c 1 (u)c 3 and d 2 (u) := c 2 (u), with initial condition I 1−α f (0) = 0.
(Recall that we often suppress the dependence on u in the notation.) We try a fractional power series ansatz a n (u)t αn with unknown coefficients a n = a n (u).
Note that v n is an increasing sequence; this follows easily from the fact that log • Γ is convex (see Example 11.14 in [32]). By Stirling's formula, v n ∼ (αn) α for n → ∞. From (7.5), we obtain a convolution recurrence for a n = a n (u): a k (u)a n−k (u) , n ≥ 1.
The function f can thus be expressed as f (u, t) = F (u, t α ), where F (u, z) := ∞ n=1 a n (u)z n . Lemma 7.2. Let u ∈ R, satisfying case (A) (recall the definition in Section 2). Then F (u, ·) is analytic at zero, with a positive and finite radius of convergence R(u).
Proof. To see that the radius of convergence is positive, we show that there is A = A(u) > 0 such that (7.8) |a n | ≤ A n n α−1 , n ≥ 1.
(Adding the factor n α−1 to this geometric bound facilitates the inductive proof.) We have Choose n 0 such that the left-hand side is ≤ 3α −α Γ(α) 2 /Γ(2α) for all n ≥ n 0 , and such that 2v n ≥ (αn) α for all n ≥ n 0 . The latter is possible because v n ∼ (αn) α . Fix a number A with A ≥ 3α −α Γ(α) 2 /Γ(2α) and such that A n n α−1 ≥ |a n | holds for 1 ≤ n ≤ n 0 . Let n ≥ n 0 and assume, inductively, that |a k | ≤ A k k α−1 holds for 1 ≤ k ≤ n. From the recurrence (7.7), we then obtain Since x α−1 (n − x) α−1 is a strictly convex function of x on (0, n) with minimum at n/2, it is easy to see that where the last equality follows from the well-known representation of the beta function in terms of the gamma function (see 12.41 in [33]). We conclude This completes the inductive proof of (7.8).
The finiteness of the radius of convergence will follow from the existence of a number B = B(u) > 0 such that (7.9) a n ≥ B n , n ≥ 1.
To this end, define By Stirling's formula, we have r n /r n−1 = 1 + (1 − α)/n + O(n −2 ) as n → ∞, and so r n eventually increases. Let n 0 ≥ 2 be such that r n increases for n ≥ n 0 , and define B := min{r n0 , a 1 , a 1/2 2 , . . . , a 1/n0 n0 }. This number satisfies a n ≥ B n for n ≤ n 0 by definition. Let us fix some n ≥ n 0 and assume, inductively, that a k ≥ B k holds for 1 ≤ k ≤ n. By (7.7) Thus, (7.9) is proved by induction.
From the estimates in Lemma 7.2, it is clear that termwise fractional derivation of the series (7.2) is allowed, and so the right-hand side of (7.2) really represents the solution f of (7.1) with initial condition I 1−α f (0) = 0, as long as t satisfies 0 ≤ t < R(u) 1/α . We proceed to show how the explosion time T * α (u) can be computed from the coefficients a n (u). The essential fact is that there is no gap between R(u) 1/α and T * α (u). For this, we require the following classical result from complex analysis ( [29], p. 235). Theorem 7.3 (Pringsheim's theorem, 1894). Suppose that the power series F (z) = ∞ n=0 a n z n has positive finite radius of convergence R, and that all the coefficients are non-negative real numbers. Then F has a singularity at R. Theorem 7.4. Suppose that u ∈ R satisfies case (A). Define the sequence a n (u) by the recurrence (7.7) with initial value (7.6). Then we have (7.10) lim sup n→∞ a n (u) −1/(αn) = T * α (u).
Note that, in case (B), we can argue similarly as in the preceding proof. However, the coefficients a n are no longer positive, and so Pringsheim's theorem is not applicable. Then, the inequality R(u) 1/α ≤ T * α (u) need not be an equality. Still, we can compute a lower bound for the explosion time: (7.11) lim sup n→∞ |a n (u)| −1/(αn) ≤ T * α (u).
Now assume that we are in case (A) again. We now discuss how to speed up the convergence in (7.10). Roberts and Olmstead [31] studied the blow-up behavior of solutions of nonlinear Volterra integral equations with (asymptotically) fractional kernel. Their arguments hinge on the asymptotic behavior of the nonlinearity for large argument. In particular, in our situation, with G(u, w) from (2.10) satisfying G(u, w) ∼ w 2 for w → ∞, formula (3.2) in [31] yields We write (?) ∼ for two reasons: First, our integral equation (2.9) does not quite satisfy the technical assumptions in [31]. Second, not all steps in [31] are rigorous. We proceed, heuristically, to infer refined asymptotics of a n (u) from (7.12). Define Φ(z) := ∞ n=1 a n (u)R(u) n z n , a power series with radius of convergence 1, by the definition of R(u) in Lemma 7.2. Its asymptotics for z ↑ 1 can be derived from (7.12). Recall that the explosion time and the radius of convergence of F are related by T * The method of singularity analysis (see Section VI in [12]) allows to transfer the asymptotics of Φ to asymptotics of its Taylor coefficients a n R n . Sweeping some analytic conditions under the rug, we arrive at and thus (7.13) a n (u) Numerical tests confirm (7.13), and we have little doubt that it is true (in case (A)). Summing up, T * α (u) can be computed by the following algorithm, which converges much faster than the simpler approximation lim sup n→∞ a −1/(αn) n : Algorithm 7.5. Let u be a real number satisfying case (A).
We stress that, while the arguments leading to (7.13) are heuristic, we have rigorously shown in Theorem 7.4 that T * α (u) is the lim sup of the left-hand side of (7.14). The heuristic part is that the subexponential factor n 1−α × const improves the relative error of the approximation from O( log n n ) to O( 1 n 2 ). Note that our approach to compute the blow-up time can of course be extended to more general fractional Riccati equations. Finally, as mentioned above (see (7.11)), we can compute a lower bound for T * α (u) if it is finite, but u is outside of case (A): Algorithm 7.6. Let u be a real number satisfying case (B).
Remark 7.7. As for the applicability of Algorithm 7.5, suppose that ρ < 0 (with analogous comments applying to the less common case ρ > 0). From (2.7), we have e 0 (u) ∼ 1 2 ρξu > 0 for u ↓ −∞, and so we are in case (A) for large enough |u|. More precisely, case (A) corresponds to the interval u ∈ (−∞, λ/(ξρ)]. For u from that interval, the explosion time can be computed by Algorithm 7.5. To the right of u = λ/(ξρ), there is a (possibly empty) interval corresponding to case (B), where T * α (u) is still finite, but Algorithm 7.5 cannot be applied. Still, a lower bound can be computed by (7.11), and we have the bounds from Theorems 4.1 and 4.2, which can be easily evaluated numerically. Proceeding further to the right on the u-axis, we encounter an interval containing [0, 1], on which T * α (u) = ∞ (cases (C) and (D)). Afterwards, T * α (u) becomes finite again, but these u belong to case (B), leaving us with bounds for T * α (u) only.  To conclude this section, we note that f can be approximated by replacing the coefficients in (7.2) by the right-hand side of (7.13). Let us write b n (u) for the latter. Retaining the first N exact coefficients, this leads to the approximation a n (u) − b n (u) t αn , (7.15) where Li ν (z) := ∞ n=1 z n /n ν denotes the polylogarithm. While this approximation seems to be very accurate even for small N (see [17]), it is limited to real u satisfying case (A), and thus not applicable to option pricing. are of interest. Using the upper bound for the moment explosion time T * α in Theorem 4.2, we will now show the finiteness of the critical moments for every maturity T > 0. Computing u + (T ) and u − (T ) is discussed in Section 9. Proof. Only the finiteness of u + (T ) is proven, as the proof for u − (T ) is very similar. Denote the upper bound of T * α (u) in (4.3) by B(u) for all u ∈ R in the cases (A) and (B). First, we show that for sufficiently large u, we are always in case (A) or (B), depending on the sign of the correlation parameter ρ. From (2.7) and (2.8), it is easy to see that (8.2) e 0 (u) ∼ 1 2 ξρu and e 1 (u) ∼ − 1 4 ξ 2ρ2 u 2 as u → ∞, whereρ 2 = 1 − ρ 2 . Thus, eventually e 1 (u) < 0 for sufficiently large u. In the next step, we show that the upper bound B(u) converges to 0 as u → ∞. Indeed, in for all T > 0, which suffices for numerical computations (under the above restriction on T ). The validity of (9.2) and (9.3) is clear from (5.16): If T * α (·) is constant on some interval, lying to the left of zero, say, then the mgf blows up as u approaches the interval's right endpoint from the right.

Application to asymptotics
In the introduction we mentioned several potential applications of our work. In this section, we give some details on one of them: Knowing the critical moments gives first order asymptotics for the implied volatility for large and small strikes. We writeσ(k) for the implied volatility, where k = log(K/S 0 ) is the log-moneyness. According to Lee's moment formula [23], the left wing of implied volatility satisfies We focus on negative log-moneyness, because then the slope depends on the lower critical moment, which Algorithm 7.5 computes in the important case ρ < 0. As in any model with finite critical moments, the marginal densities of the rough Heston model have power-law tails. More precisely, if we write f T for the density of S T , then f T (x) = x −u + (T )−1+o(1) , x → ∞, and Our approach (see Section 9) allows to evaluate the right-hand sides of (10.1) and (10.2) numerically for the rough Heston model, if T is not too large.
In [13], (10.1)-(10.2) were considerably sharpened for the classical Heston model. We expect that such a refined smile expansion can be done for rough Heston, too, with density asymptotics of the form f T (x) ∼ c 1 x −u + (T )−1 e c2(log x) 1−1/(2α) (log x) c3 , x → ∞, where the c i depend on T and α. In the classical Heston model, the factor e c2(log x) 1−1/(2α) becomes e c2 √ log x , in line with [13]. Extending the analysis of [13] to 1 2 < α < 1 will require a detailed study of the blow-up behavior of the Volterra integral equation (2.9). Among other things, (a special case of) the heuristic analysis in [31], which we already mentioned in Section 7, would have to be made rigorous, and extended to ensure uniformity w.r.t. the parameter u. We postpone this to future work. Note that the approximation (7.15) might be useful in this context.