# Moment explosions in the rough Heston model

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## Abstract

We show that the moment explosion time in the rough Heston model, introduced by El Euch and Rosenbaum in 2016, 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.

## Keywords

Option pricing Rough volatility Rough Heston model Moment explosion Volterra integral equation## Mathematics Subject Classification

91G20 45D05## JEL Classification

G13 C65## 1 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 (Keller-Ressel 2011) 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 Andersen and Piterbarg (2007), Keller-Ressel (2011) and the article *Moment Explosions* in Cont (2014) 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 (Lee 2004b; Lord and Kahl 2007). See Sect. 10 for some details.

*W*and

*Z*are correlated Brownian motions, \(\rho \in (-1,1)\) and \(\lambda \), \(\xi \), \(\bar{v}\) are positive parameters. The smoothness parameter \(\alpha \) is in \((\tfrac{1}{2},1)\). (For \(\alpha =1\), the model clearly reduces to the classical Heston model.) W.l.o.g. we assume zero interest rate. 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 nonlinear Volterra integral equation; see Sect. 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 nonlinear 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 (1998) 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 (Sects. 3–5). As a by-product, the validity of the fractional Riccati equation, respectively, the VIE, for all moments (including complex exponents) is established in Sect. 6. In Sect. 7, we derive an algorithm to compute the explosion time, under some restrictions on the parameters.

The critical moments are finite for all maturities (Sect. 8) and can be computed by numerical root finding (Sect. 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 \(\rho >0\) and the lower critical moment for \(\rho <0\). 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 Sect. 11).

Corollary 3.1 in El Euch and Rosenbaum (2018) 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. Further bounds are contained in the recent preprint (Keller-Ressel and Majid 2019).

## 2 Notation and preliminaries

We assume in most of our analysis that \(\alpha \in (\tfrac{1}{2},1)\). Some of the subsequent definitions and results are valid for larger intervals, but then the range of \(\alpha \) is always explicitly stated. The letter \(k(\cdot )\) will always denote the fractional convolution kernel \( k(t - s)=\frac{1}{\varGamma (\alpha )}(t - s)^{\alpha -1},\) unless explicitly stated otherwise.

### Definition 1

*f*is given by

*f*is given by

*R*is defined as

### Theorem 1

*f*(and thus of \(\psi \)) in the following sections. We quote the following standard existence and uniqueness result for equations of this kind.

### Theorem 2

### Proof

For existence and uniqueness on a sufficiently small interval \([0,T_0]\) with \(T_0>0\), see, for example, Theorem 3.1.4 in Brunner’s recent monograph (Brunner 2017). The continuation to a maximal right-open interval is discussed there as well [p. 107; see also Sect. 12 of Gripenberg et al. (1990)].

In El Euch and Rosenbaum (2019), the fractional Riccati equation was established for \(u\in i\mathbb R\), whereas we are interested in \(u\in \mathbb R\). The justification of (4)–(5), and thus of (9), for \(u\in \mathbb R\) hinges on a result from El Euch and Rosenbaum (2018) and the analytic dependence of *f*(*u*, *t*) on *u*. See Sects. 5 and 6 for details.

*rough*Heston model. We distinguish between the following cases for \(u\in \mathbb {R}\):

- (A)
\(c_1(u)>0\), \(e_0(u)\ge 0,\)

- (B)
\(c_1(u)>0\), \(e_0(u)<0\) and \(e_1(u)<0,\)

- (C)
\(c_1(u)>0\), \(e_0(u)<0\) and \(e_1(u)\ge 0,\)

- (D)
\(c_1(u)\le 0.\)

*u*satisfying case (A), we explicitly note

### Theorem 3

For \(u\in \mathbb R\), the moment explosion time \(T_\alpha ^*(u)\) of the rough Heston model is finite if and only if *u* satisfies (A) or (B). This is equivalent to \(T_1^*(u)\) (explosion time of the classical Heston model) being finite.

The proof of Theorem 3 consists of two main parts. First, Propositions 2–5 discuss the blowup behavior of the solution of (9) in cases (A)–(D), and Lemma 3 in Sect. 4 shows that blowup of *f* leads to blowup of the right-hand side of (1). Second, we show in Sect. 5 that the explosion time of \(f(u,\cdot )\) [the solution of (9)] agrees with \(T_\alpha ^*(u)\) (the explosion time of the rough Heston model) for all \(u\in \mathbb R\). As mentioned after Theorem 2, this is not obvious from the results in the existing literature.

## 3 Explosion time of the Volterra integral equation

We begin by citing a result from Brunner and Yang (2012) which characterizes the blowup behavior of nonlinear Volterra integral equations defined by positive and increasing functions. “Blowup in finite time” means that there is a finite number \(\hat{T}>0\) such that *h* is defined on \([0,\hat{T})\) and satisfies \(\lim _{t\uparrow \hat{T}}|h(t)|=\infty .\) We note that some arguments in our subsequent proofs (from Proposition 2 onward) are similar to arguments used in Brunner and Yang (2012). Alternatively, it should be possible to extend the arguments in Appendix A of Gatheral and Keller-Ressel (2019); there, *u* is in [0, 1].

### Proposition 1

- (H1)
\(H(0)=0\) and

*H*is strictly increasing, - (H2)
\(\lim _{w\rightarrow \infty } H(w)/w = \infty \),

- (P)
\(\phi :[0,\infty )\rightarrow (0,\infty )\) is a positive, nondecreasing, continuous function,

- (K)
\(k:(0,\infty )\rightarrow [0,\infty )\) is locally integrable and \(K(t):=\int _0^t k(z)\,\mathrm{d}z>0\) is a nondecreasing function.

*h*of the Volterra integral equation

### Proof

This is a special case of Corollary 2.22 in Brunner and Yang (2012), with *H* (in their notation, *G*) not depending on time.

In case (A), all assumptions of Proposition 1 are satisfied and only the integrability condition (14) has to be checked to determine whether the solution *f* of (9) blows up in finite time.

### Proposition 2

In case (A), the solution *f* of (9) starts at 0, is positive thereafter and blows up in finite time.

### Proof

*u*in the notation.) If we write the Volterra integral equation (9) in the form

*f*is positive for positive values, and \(f(0)=0\). (Positivity follows from Lemma 2.4 in Brunner and Yang (2012), or from Lemma 3.2.11 in Brunner (2017).) It is easy to check that all the assumptions (H1), (H2), (P) and (K) of Proposition 1 are satisfied. Moreover, \(\lim _{t\rightarrow \infty }\phi (t)=\infty \) and

*f*blows up in finite time.

In case (B), Proposition 1 cannot be applied directly to the solution *f* of (9). Hence, the Volterra integral equation has to be modified in order to satisfy the assumptions of Proposition 1 in a way that *f* is still a subsolution of the modified equation, i.e., *f* satisfies (9) with “\(\ge \)” instead of “\(=\).” First, we provide a comparison lemma for solutions and subsolutions.

### Lemma 1

*g*is the unique continuous solution of the Volterra integral equation

*k*satisfies condition (K) from Proposition 1. If

*f*is a continuous subsolution,

### Proof

*H*, it follows that \(f_c(0)=f(c)>0\) and

*H*is strictly increasing, we have

### Proposition 3

In case (B), the solution *f* of (9) starts at 0, is positive thereafter and blows up in finite time.

### Proof

*G*is obviously positive by (10). However,

*G*is strictly

*decreasing*on \([0,-e_0]\). To deal with this problem, let \(0<a<-e_1\) and define the modified nonlinearity \(\bar{G}_a\) as

*a*and \(\bar{G}_a\le G\). Let \(\bar{f}\) be the unique continuous solution (recall Theorem 2) of the Volterra integral equation

*k*satisfy the assumptions (H1), (H2), (P) and (K) in Proposition 1. Furthermore, \(\lim _{t\rightarrow \infty }\phi (t)=\infty \) and \(\bar{G}\) satisfies (14). By Proposition 1, \(\bar{f}\) blows up in finite time. Because

*f*satisfies (9) and \(\bar{G}_a\le G\), it follows that

*f*is a subsolution of the modified Volterra integral equation, i.e.,

*f*blows up as well.

Cases (C) and (D) are the cases where the solution *f* of (9) does not blow up in finite time. In fact, *f* does not blow up at all, as we will see. The following lemma provides the key argument for both cases.

### Lemma 2

*a*) and satisfies \(H\equiv 0\) on \([a,\infty )\) for an \(a>0\). Then, the unique continuous solution

*f*of the Volterra integral equation

### Proof

*H*implies \(f \ge 0\). Suppose \(t>0\) exists such that \(f(t)>a\). By the continuity of

*f*, there exists \(0<t_0<t\) that satisfies \(f(t_0) = a\) and \(f(s)>a\) for all \(s\in (t_0,t)\). From \(H\equiv 0\) on \([a,\infty )\), we have

*H*is nonnegative and

*k*is decreasing,

*f*satisfies \(0\le f(t)\le a\) for all \(t\ge 0\).

### Proposition 4

In case (C), the solution *f* of (9) is nonnegative and bounded and exists globally.

### Proof

*a*is the smallest positive root of

*G*. Define the nonlinearity \(\bar{G}\) as

*a*], the function \(\bar{f}\) solves the original Volterra integral equation

### Proposition 5

In case (D), the solution *f* of (9) is nonpositive and bounded and exists globally.

### Proof

*a*is the smallest positive root of

*G*. Define \(f_-:=-\,f\), which satisfies

*f*is bounded with \(-a\le f(t)\le 0\).

We have shown that (A) and (B) are exactly the cases in which the solution *f* of the Volterra integral equation (9), and thus the solution \(\psi \) of the fractional Riccati differential equation (4) with initial value (5), blows up in finite time.

## 4 Bounds for the explosion time

We denote by \(\hat{T}_\alpha (u)\) the explosion time of the solution \(f(u,\cdot )\) of (9). We now establish lower and upper bounds for \(\hat{T}_\alpha (u)\), valid whenever it is finite [cases (A) and (B)]. As we will see later, \(\hat{T}_\alpha (u)\) agrees with \(T^*_\alpha (u)\), and so both bounds of this section hold for the explosion time \(T^*_\alpha (u)\) of the rough Heston model.

### Theorem 4

### Proof

*u*satisfying the requirements of case (A) or (B). It follows from Propositions 2 and 3 that in either case the solution

*f*is nonnegative, starts at 0 and \(\lim _{t\uparrow \hat{T}_\alpha }f(t)=\infty \). For any \(n\in \mathrm {N}_0\), choose

*G*and that

*G*is strictly increasing on \([a,\infty )\), we have for \(n\in \mathrm {N}\)

Another lower bound for \(\hat{T}_\alpha \) can be obtained from Corollary 3.1 in El Euch and Rosenbaum (2018). Numerical examples show that it is not comparable to the bound from our Theorem 4.

### Theorem 5

### Proof

*u*satisfying the requirements of case (A) or (B). From Propositions 2 and 3, in either case the solution

*f*is positive on \((0,\infty )\), starts at 0 and \(\lim _{t\uparrow \hat{T}_\alpha }f(t)=\infty \). For any \(n\in \mathrm {N}_0\), choose

*f*is only zero at \(t=0\) implies that \(t_0\rightarrow 0\) as \(c\downarrow 0\). Taking the limit \(c\downarrow 0\), then minimizing over \(r>1\) and substituting \(w=s^\alpha \) yield

*f*blows up fast enough for \(I^{1-\alpha }f\) to blow up, too. This is important for inferring blowup of the right-hand side of (1); see the remark after the proof.

### Lemma 3

### Proof

*r*and

*c*) such that \(\hat{T}_\alpha - t_n\ge C_1r^{-n}\). Assume for contradiction that there is a sequence \(0<s_n\uparrow \hat{T}_\alpha \) with

*G*increases on \([0,\infty )\), and so

*f*increases as well (see, e.g., Theorem 3.1 in Banaś and Martinon (2004)). Therefore,

*h*satisfying

To obtain blowup of the right-hand side of (1), recall that \(f=c_3 \psi \). By Lemma 3, the blowup of *f* at \(\hat{T}_\alpha \) is at least of order \((\hat{T}_\alpha -t)^{-\alpha }\), and so we have a lower bound of order \((\hat{T}_\alpha -t)^{1-2\alpha }\) for \(I^{1-\alpha }f\).

## 5 Explosion time in the rough Heston model

In Sects. 3 and 4, we established that the right-hand side of (1), defined using the solution *f* of the VIE (9), explodes if and only if *u* satisfies the conditions of cases (A) or (B). As before, we write \(\hat{T}_\alpha (u)\) for the explosion time of \(f=f(u,\cdot )\). Recall that \(T^*_\alpha (u)\) denotes the explosion time of the rough Heston model, as defined in (11). The goal of the present section is to show that \(\hat{T}_\alpha (u)=T^*_\alpha (u)\) and that (1) holds for all \(u\in \mathbb R\) and \(0<t<T^*_\alpha (u)\). The following result from El Euch and Rosenbaum (2018) was already mentioned at the end of the introduction.

### Lemma 4

(Corollary 3.1 in El Euch and Rosenbaum 2018) For each \(t>0\), there is an open interval such that (1) holds for all *u* from that interval.

### Lemma 5

*f*of the Volterra integral equation (9) is differentiable w.r.t.

*u*, and its derivative satisfies

### Proof

We check the requirements of Theorem 13.1.2 in Gripenberg et al. (1990).

The polynomial *G*(*u*, *w*) is differentiable. The kernel \((t - s)^{\alpha -1} / \varGamma (\alpha ) =: k(t - s) \) is of *continuous type* in the sense of Gripenberg et al. (1990); see the remark to Theorem 12.1.1 there, which states local integrability of *k* as a sufficient condition for this property.

### Lemma 6

- (i)
In case (A), we have \( \partial _1 f(u, t) < 0 \) for \( u < 0 \) and \( \partial _1 f(u, t) > 0 \) for \( u > 0 \).

- (ii)
If

*u*satisfies case (B), then the same holds if \(\hat{T}_\alpha (u)-t\) is sufficiently small.

### Proof

- (i)Note that (25) is a “
*linear*VIE” that can be written aswhere we define$$\begin{aligned} \partial _1 f(u, t) = g(t) + \int _{0}^{t} (t - s)^{\alpha - 1} K^{(u)}(t, s) \partial _1 f(u, s)\, \mathrm{d}s, \end{aligned}$$(26)$$\begin{aligned} g(t):= & {} \int _{0}^{t} \frac{(t - s)^{\alpha - 1}}{\varGamma (\alpha )} \partial _1 G(u, f(u, s))\, \mathrm{d}s, \end{aligned}$$(27)to bring the notation close to that of Sect. 6.1.2 in Brunner (2004). Clearly, (26) is not really a linear VIE, because the unknown function$$\begin{aligned} K^{(u)}(t, s)= & {} K^{(u)}(s):=\frac{\partial _2 G(u, f(u, s))}{\varGamma (\alpha )} \end{aligned}$$(28)*f*appears in*g*and \(K^{(u)}\). But as our aim is not to solve it, but to control the sign of \(\partial _1 f\), this viewpoint is good enough.As we are in case (A), we get from \( c_1(u) > 0 \) and \( e_0(u) \ge 0 \) that \( u \le \lambda / (\rho \xi ) < 0 \). Furthermore, we have \( e_0'(u) = \xi \rho / 2 < 0 \), \( c_1'(u) = u - 1/2 < 0 \) and thereforesince \( c_3 = \xi ^2 / 2 > 0 \) and \( f(u, s) \ge 0 \) by Proposition 2. From this, we obtain \( \partial _1 G(u, f(u, s)) < 0 \), and hence \( g(t) < 0 \) for all \( t \in [0, T_\alpha ^*(u)) \).$$\begin{aligned} f(u, s) e_0'(u) + \frac{1}{2} c_3 c_1'(u) < 0, \end{aligned}$$By Theorem 6.1.2 of Brunner (2004), we can express the solution of (25) with the resolvent kernel \( R_{\alpha }(\cdot ,\cdot )\),The resolvent kernel has the explicit representation (see Brunner 2004)$$\begin{aligned} \partial _1 f(u, t) = g(t) + \int _{0}^{t} R_{\alpha }(t, s) g(s)\, \mathrm{d}s. \end{aligned}$$(29)where$$\begin{aligned} R_\alpha (t,s)= (t-s)^{\alpha -1}\sum _{n=1}^\infty Q_{\alpha ,n}(t,s), \end{aligned}$$(30)From this representation of the resolvent kernel, and the fact that (28) is nonnegative in case (A), it is obvious that \( R_{\alpha } \ge 0 \). Since \(g<0,\) we thus conclude from (29) that \( \partial _1 f(u, t) < 0 \) for all \( t \in [0, \hat{T}_\alpha (u)) \).$$\begin{aligned} Q_{\alpha ,1}(t,s)&:= K^{(u)}(t, s) = K^{(u)}(s), \nonumber \\ Q_{\alpha ,n}(t,s)&:= (t-s)^{(n-1)\alpha } \Phi _{\alpha ,n}(t,s), \quad n\ge 2, \nonumber \\ \Phi _{\alpha ,n}(t,s)&:= K^{(u)}(s) \int _0^1 (1-z)^{\alpha -1} z^{(n-1)\alpha -1} Q_{\alpha ,n-1}(t,s+(t-s)z)dz, \quad n\ge 2. \end{aligned}$$(31) - (ii)Recall that we assume that \(u<0\), because \(u>0\) is analogous. We have to show thatsatisfies \(\tau (u)<\hat{T}_\alpha (u)\). We use the following facts: \(\partial _1 G(u,w)<0\) for$$\begin{aligned} \tau (u) := \inf \{ 0<t<\hat{T}_\alpha (u) : \partial _1 f(u,\cdot )<0\ \text {on}\ (t,\hat{T}_\alpha (u)) \} \end{aligned}$$(32)
*w*large, \(\partial _2 G(u,w)>0\) for*w*large, and*f*(*u*,*t*) explodes as \(t\uparrow \hat{T}_\alpha (u)\). Thus,*g*from (27) satisfiesand \(K^{(u)}\) satisfies \(\lim _{t\uparrow \hat{T}_\alpha (u)} K^{(u)}(t) = +\infty \). We can therefore pick \(\varepsilon >0\) such that$$\begin{aligned} \lim _{t\uparrow \hat{T}_\alpha (u)} g(t) = -\infty , \end{aligned}$$(33)For \(z\in [0,1]\) and any \(s,t<\hat{T}_\alpha (u)\) satisfying \(\hat{T}_\alpha (u)-\varepsilon \le s\le t\), we have$$\begin{aligned} g(t)<0,\ K^{(u)}(t) > 0 \quad \text {for} \quad \hat{T}_\alpha (u)-\varepsilon \le t<\hat{T}_\alpha (u). \end{aligned}$$Using this observation in (31), we see from a straightforward induction proof that$$\begin{aligned} s+(t-s)z \ge s \ge \hat{T}_\alpha (u)-\varepsilon . \end{aligned}$$The same then holds for the resolvent kernel (30),$$\begin{aligned} Q_{\alpha ,n}(t,s) > 0, \quad n\ge 1,\ \hat{T}_\alpha (u)-\varepsilon \le s\le t < \hat{T}_\alpha (u). \end{aligned}$$By (29), we obtain$$\begin{aligned} R_{\alpha }(t,s) > 0, \quad \hat{T}_\alpha (u)-\varepsilon \le s\le t < \hat{T}_\alpha (u). \end{aligned}$$(34)Now note that$$\begin{aligned} \partial _1 f(u,t) = g(t) + \int _{0}^{t-\varepsilon } R_{\alpha }(t,s)g(s)\mathrm{d}s + \int _{t-\varepsilon }^t R_{\alpha }(t,s)g(s)\mathrm{d}s. \end{aligned}$$(35)where the right-hand side is positive. Indeed, (36) follows from (33) and (34), as$$\begin{aligned} \Big | \int _{0}^{t-\varepsilon } R_{\alpha }(t,s)g(s)\mathrm{d}s \Big | \ll - \int _{t-\varepsilon }^t R_{\alpha }(t,s)g(s)\mathrm{d}s \quad \text {as}\ t\uparrow \hat{T}_\alpha (u), \end{aligned}$$(36)*g*(*s*) on the left-hand side of (36) is*O*(1). Thus, letting \(t\uparrow \hat{T}_\alpha (u)\), we find that the negative terms \(g(t)+\int _{t-\varepsilon }^t\) on the right-hand side of (35) dominate. This completes the proof.

### Lemma 7

Let \(u\in \mathbb R\) and \(0<t<\hat{T}_{\alpha }(u)\). Then, \(f(\cdot ,t)\) is analytic at *u*.

### Proof

*u*and

*t*as in the statement of the lemma. For a sufficiently small open complex neighborhood \(U\ni u\), it is easy to see that \(t<\hat{T}_\alpha (v)\) holds for \(v\in U\). Define

*U*. From the bounds in Sect. 3.1.1 of Brunner (2017), it is very easy to see that the convergence \(f_n(v,t)\rightarrow f(v,t)\) is locally uniform w.r.t.

*v*for fixed

*t*. It is well known [see Theorem 3.5.1 in Greene and Krantz (2006)] that this implies that the limit function \(f(\cdot ,t)\) is analytic.

### Lemma 8

The function \(u\mapsto \hat{T}_\alpha (u)\) increases for \(u\le 0\) and decreases for \(u\ge 1\).

### Proof

*u*satisfy case (B), where again we assume w.l.o.g. that \(u<0\). Suppose that \(\hat{T}_\alpha (\cdot )\) does not increase. Then, we can pick \(u_0<0\) such that any left neighborhood of \(u_0\) contains a point

*u*with \(\hat{T}_\alpha (u)>\hat{T}_\alpha (u_0)\). From the continuity of \(\partial _1 f\) (see Lemma 7), part (ii) of Lemma 6 and the continuity of \(\tau \) from (32), there are \(u_1<u_0\) satisfying \(\hat{T}_\alpha (u_1)>\hat{T}_\alpha (u_0)\) and \(t_1<\hat{T}_\alpha (u_0)\) such that \(\partial _1 f(u,t)<0\) in the rectangle

### Lemma 9

Let \(u\in \mathbb R\) and \(0<t<\hat{T}_{\alpha }(u)\). Then, (1) holds, where (as above) \(f=c_3 \psi \) and \(f(u,\cdot )\) is the solution of (9).

### Proof

*M*(

*u*,

*t*) for the right-hand side of (1), and \(\tilde{M}(u,t)=\mathbb {E}[\hbox {e}^{uX_t}]\) for the mgf. Now fix \(u< 0\) and \(0<t<\hat{T}_{\alpha }(u)\) such that (

*u*,

*t*) has positive distance from the graph of the increasing function \(\hat{T}_\alpha (\cdot )\). Clearly, it suffices to consider pairs (

*u*,

*t*) with this property. By Lemma 4, there are \(v^-<v^+\) such that

*u*by analytic continuation.

In light of Sect. 3 and Lemma 3 from Sect. 4, the following theorem completes the proof of Theorem 3 from Sect. 2.

### Theorem 6

Let \(u\in \mathbb R\). Then, \(\hat{T}_\alpha (u)= T^*_\alpha (u)\), and (1) holds for \(0<t<T^*_\alpha (u)\).

### Proof

By Lemma 9, it only remains to show that \(\hat{T}_\alpha (u)\ge T^*_\alpha (u)\). (Obviously, Lemma 9 implies that \(\hat{T}_\alpha (u)\le T^*_\alpha (u)\).) But this is clear from the continuity of the map \(t\mapsto \tilde{M}(u,t)=\mathbb {E}[\hbox {e}^{uX_t}]\) on the interval \((0,T^*_\alpha (u))\). This continuity follows from the continuity of \(t\mapsto X_t\), Doob’s submartingale inequality and dominated convergence.

For later use (Sect. 9), we give the following alternative argument:

### Proof

(*Another proof that*\(\hat{T}_\alpha (u)\ge T^*_\alpha (u)\)) Let us suppose that there is \(u_0\) with \(\hat{T}_\alpha (u_0) < T^*_\alpha (u_0)\). From Theorem 4, it is easy to see that the continuous function \(u\mapsto \hat{T}_\alpha (u)\) tends to \(+\infty \) as \(u<0\) approaches the region where \(\hat{T}_\alpha (u)=\infty \). Thus, there is \(u_1>u_0\) with \(\hat{T}_\alpha (u_0)<\hat{T}_\alpha (u_1)<T^*_\alpha (u_0)\).

*t*. But it is also analytic w.r.t.

*t*for fixed

*u*: From Theorem 1 in Lubich (1983), itself based on earlier work by Miller and Feldstein (1971), it follows that \(f(u,\cdot )\) is analytic on the whole interval \((0,\hat{T}_\alpha (u))\). By Hartogs’s theorem [Theorem 1.2.5 in Krantz (1992)], we conclude that the bivariate function \(f(\cdot ,\cdot )\) is continuous. Thus, the blowup of \(f(u_1,\cdot )\) at \(\hat{T}_\alpha (u_1)\) implies that

*M*for the right-hand side of (1) and \(\tilde{M}\) for the mgf.) By Lemma 9, \(\tilde{M}(\cdot ,\hat{T}_\alpha (u_1))\) also blows up there and thus has a singularity at \(u_1\). Since \(u_0<u_1\), we conclude from Corollary II.1b in Widder (1941) that \(\tilde{M}(u_0,\hat{T}_\alpha (u_1))=\infty \). As \(S=\hbox {e}^X\) is a martingale, this implies that \(\tilde{M}(u_0,t)=\infty \) for all \(t\ge \hat{T}_\alpha (u_1)\). In particular, it contradicts \(\hat{T}_\alpha (u_1)<T^*_\alpha (u_0)\).

## 6 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 (see Sect. 10). The following result fully justifies using the fractional Riccati equation (4), respectively, the VIE (9), to do so. As above, we write \(T^*_\alpha (u)\) for the moment explosion time of *S* and \(\hat{T}_\alpha (u)\) for the explosion time of the VIE (9).

### Theorem 7

Let \(u\in \mathbb C\). Then, \(T^*_\alpha (u)=T^*_\alpha ({ Re}(u))\), and (1) holds for \(0<t<T^*_\alpha (u)\).

### Lemma 10

Let \(u\in \mathbb C\). Then, \(\hat{T}_\alpha (u) \ge T^*_\alpha (u)\).

### Proof

Suppose that \(\hat{T}_\alpha (u) < T^*_\alpha (u)\). The VIE (9) translates into a two-dimensional real VIE for \(({ Re}(f),{ Im}(f))\). As \(\hat{T}_\alpha (u)<\infty \), we get from Theorem 12.1.1 in Gripenberg et al. (1990) that \(({ Re}(f),{ Im}(f))\) explodes as \(t\uparrow \hat{T}_\alpha (u)\). This contradicts the continuity of \(t\mapsto \mathbb {E}[\hbox {e}^{uX_t}]\), where the latter is shown as in the proof of Theorem 6.

### Proof of Theorem 7

*M*for the right-hand side of (1). By Theorem 6, we have \(M(v,t)=\tilde{M}(v,t)\) for

*v*in the real interval

## 7 Computing the explosion time

*u*satisfying the conditions of case (A). In case (B), a lower bound can be computed, which is sometimes sharper than the explicit bound (18). We note that the recent preprint (Callegaro et al. 2019) extends and complements our computational approach. The function

*f*satisfies the fractional Riccati equation

*u*in the notation.) We try a fractional power series ansatz

### Lemma 11

*f*can thus be expressed as \(f(u,t) = F(u,t^\alpha )\), where

### Lemma 12

Let \(u\in \mathbb {R}\), satisfying case (A) or (B) (recall the definition in Sect. 2). Then, \(F(u,\cdot )\) is analytic at zero, with a positive and finite radius of convergence *R*(*u*).

### Proof

*A*with \(A\ge 3\alpha ^{-\alpha }\varGamma (\alpha )^2/\varGamma (2\alpha )\) and such that \(A^n n^{\alpha -1}\ge |a_n|\) holds for \(1\le n\le n_0\). Let \(n\ge n_0\) and assume, inductively, that \(|a_k| \le A^k k^{\alpha -1}\) holds for \(1\le k\le n\). From the recurrence (48), we then obtain

*x*on (0,

*n*) with minimum at

*n*/ 2, it is easy to see that

*f*blows up in finite time in cases (A) and (B). In case (A), we can alternatively show in an elementary way that there is a number \(B=B(u)>0\) such that

From the estimates in Lemma 12, it is clear that termwise fractional derivation of the series (43) is allowed, and so the right-hand side of (43) really represents the solution *f* of (42) with initial condition \(I^{1-\alpha }f(0)=0\), as long as *t* satisfies \(0\le t<R(u)^{1/\alpha }\). We proceed to show how the explosion time \(T_\alpha ^*(u)\) can be computed from the coefficients \(a_n(u)\). The essential fact is that there is no gap between \(R(u)^{1/\alpha }\) and \(T_\alpha ^*(u)\). For this, we require the following classical result from complex analysis (Remmert 1991, p. 235).

### Theorem 8

(Pringsheim’s theorem, 1894) Suppose that the power series \(F(z)=\sum _{n=0}^\infty a_n z^n\) has positive finite radius of convergence *R* and that all the coefficients are nonnegative real numbers. Then, *F* has a singularity at *R*.

### Theorem 9

### Proof

Recall that \(f(u,\cdot )\), the solution of (42), also solves the Volterra integral equation (9). From the references on smoothness cited before (40), it follows that \(f(u,\cdot )\) is analytic on the whole interval \((0,T_\alpha ^*(u))\). As *f*(*u*, *t*) blows up for \(t\uparrow T_\alpha ^*(u)\) by Proposition 2, and \(t\mapsto F(u,t^\alpha )\) is analytic on \((0,R(u)^{1/\alpha })\), we must have \(R(u)^{1/\alpha } \le T_\alpha ^*(u)\).

*f*is analytic at \(R(u)^{1/\alpha }\). But since \(z\mapsto z^{1/\alpha }\) is analytic at \(R(u)>0\), the composition \(F(u,z)=f(u,z^{1/\alpha })\) would be analytic at \(z=R(u)\) as well, which contradicts Theorem 8. Therefore,

*G*(

*u*,

*w*) from (10) satisfying \(G(u,w)\sim w^2\) for \(w\rightarrow \infty \), formula (3.2) in Roberts and Olmstead (1996) yields

*R*(

*u*) in Lemma 12. Its asymptotics for \(z\uparrow 1\) can be derived from (53). Recall that the explosion time and the radius of convergence of

*F*are related by \(T_\alpha ^*(u)=R(u)^{1/\alpha }\).

*singularity analysis*[see Sect. VI in Flajolet and Sedgewick (2009)] allows to transfer the asymptotics of \(\Phi \) to asymptotics of its Taylor coefficients \(a_n R^n\). Sweeping some analytic conditions under the rug, we arrive at

### Algorithm 1

*u*be a real number satisfying case (A).

Fix \(n_{\max } \in \mathbb N\) (e.g., \(n_{\max }=100\)),

compute \(a_1(u),\dots ,a_{n_{\max }}(u)\) by the recursion (48),

- compute the approximationfor the explosion time.$$\begin{aligned} \bigg ( a_n(u) n^{1-\alpha } \frac{\varGamma (\alpha )^2}{\alpha ^\alpha \varGamma (2\alpha )} \bigg )^{\frac{-1}{\alpha (n+1)}} \bigg |_{n=n_{\max }} \approx T_\alpha ^*(u) \end{aligned}$$(55)

We stress that while the arguments leading to (54) are heuristic, we have rigorously shown in Theorem 9 that \(T_\alpha ^*(u)\) is the \(\limsup \) of the left-hand side of (55). The heuristic part is that the subexponential factor \(n^{1-\alpha }\times const \) improves the relative error of the approximation from \(O(\frac{\log n}{n})\) to \(O(\frac{1}{n^2})\). Note that our approach to compute the blowup time can of course be extended to more general fractional Riccati equations. Finally, as mentioned above (see (52)), we can compute a *lower bound* for \(T_\alpha ^*(u)\) if it is finite, but *u* is outside of case (A):

### Algorithm 2

*u*be a real number satisfying case (B).

Fix \(n_{\max } \in \mathbb N\) (e.g., \(n_{\max }=200\)),

compute \(a_1(u),\dots ,a_{n_{\max }}(u)\) by the recursion (48),

- compute the approximate lower boundof the explosion time.$$\begin{aligned} |a_n(u)|^{-1/(\alpha n)}\big |_{n=n_{\max }}\, \lessapprox T_\alpha ^*(u) \end{aligned}$$(56)

### Remark 1

As for the applicability of Algorithm 1, suppose that \(\rho <0\) (with analogous comments applying to the less common case \(\rho >0\)). From (7), we have \(e_0(u)\sim \tfrac{1}{2} \rho \xi u>0\) for \(u\downarrow -\infty \), and so we are in case (A) for large enough |*u*|. More precisely, case (A) corresponds to the interval \(u\in (-\infty ,\lambda /(\xi \rho )]\). For *u* from that interval, the explosion time can be computed by Algorithm 1. To the right of \(u=\lambda /(\xi \rho )\), there is a (possibly empty) interval corresponding to case (B), where \(T_\alpha ^*(u)\) is still finite, but Algorithm 1 cannot be applied. Still, a lower bound can be computed by (52), and we have the bounds from Theorems 4 and , which can be easily evaluated numerically. Proceeding further to the right on the *u*-axis, we encounter an interval containing [0, 1], on which \(T_\alpha ^*(u)=\infty \) (cases (C) and (D)). Afterward, \(T_\alpha ^*(u)\) becomes finite again, but these *u* belong to case (B), leaving us with bounds for \(T_\alpha ^*(u)\) only.

*f*can be approximated by replacing the coefficients in (43) by the right-hand side of (54). Let us write \(b_n(u)\) for the latter. Retaining the first

*N*

*exact*coefficients, this leads to the approximation

*N*(see Gerstenecker 2018), it is limited to

*real*

*u*satisfying case (A) and thus not applicable to option pricing (see Sect. 10). Possibly, (57) can be applied in a saddle point approximation of the rough Heston call price for large or small strike; see the end of Sect. 11.

## 8 Finiteness of the critical moments

*explosion time*of the rough Heston model so far, in most applications of moment explosions (see the introduction), the

*critical moments*

### Theorem 10

In the rough Heston model, the critical moments \(u^+(T)\) and \(u^-(T)\) are finite for every \(T>0\).

### Proof

*B*(

*u*) for all \(u\in \mathbb {R}\) 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 \(\rho \). From (7) and (8), it is easy to see that

*u*. In the next step, we show that the upper bound

*B*(

*u*) converges to 0 as \(u\rightarrow \infty \). Indeed, in case (A) the integral in (21) satisfies

*u*. Note that in this case \(G(u,\cdot )\) attains its global minimum at \(-e_0(u)\) and the minimum value is \(-e_1(u)\). Thus, the integral in (21) satisfies

Altogether, we have \(\lim _{u\rightarrow \infty } B(u)=0\). Since \(0\le T_\alpha ^*(u)\le B(u)\), the same is true for the moment explosion time \(T_\alpha ^*\), i.e., \(\lim _{u\rightarrow \infty } T_\alpha ^*(u) = 0\). Now let \(T>0\) be arbitrary. Then, there exists \(u_0\in \mathbb {R}\) such that \(T_\alpha ^*(u)<T\) for all \(u\ge u_0\). This inequality implies \(\mathbb {E}[\hbox {e}^{uX_T}]=\infty \) for all \(u\ge u_0\), and therefore \(u_+(T)\le u_0\). \(\square \)

From the preceding proof, it easily follows that \(u^+(T)\) and \(u^-(T)\) are of order \(T^{-\alpha }\) as \(T\downarrow 0\). This is consistent with the classical Heston model (\(\alpha =1\)), where the decay order is \(T^{-1}\), by inverting (13).

## 9 Computing the critical moments

We first collect some simple facts that apparently have not been made explicit in the literature on moment explosions. Moment explosion time and critical moments are defined as in (11) resp. (58).

### Lemma 13

- (i)
\(T^*(u)\) increases for \(u\le 0\) and decreases for \(u\ge 0\).

- (ii)
If

*S*is a martingale, then \(u^+(T)\) decreases, and \(u^-(T)\) increases. - (iii)Suppose that
*S*is a martingale. If \(T^*(u)\) decreases strictly on the intervalthen \(T^*=(u^+)^{-1}\) on \(\mathcal {D}^+\). Analogously, if \(T^*(u)\) increases strictly on the interval$$\begin{aligned} \mathcal {D}^+:= \{ u\ge 1 : T^*(u)<\infty \}, \end{aligned}$$then \(T^*=(u^-)^{-1}\) on \(\mathcal {D}^-\).$$\begin{aligned} \mathcal {D}^-:= \{ u\le 0 : T^*(u)<\infty \}, \end{aligned}$$

### Proof

- (i)
As \(T^*(0)=\infty \), we may assume that \(u>0\) (\(u<0\) is analogous). The assertion follows from Jensen’s inequality, since \(x\mapsto x^{v/u}\) is convex for \(0<u\le v\).

- (ii)We just consider \(u^+\). Since
*S*is a martingale, we have \(u^+\ge 1\). For any number \(u\ge 1\) and \(0<T\le T'\), we haveby the conditional Jensen inequality. This shows the assertion.$$\begin{aligned} \mathbb {E}[S_{T'}^u]=\mathbb {E}\big [\mathbb {E}[S_{T'}^u|\mathcal {F}_T]\big ] \ge \mathbb {E}[S_{T}^u] \end{aligned}$$ - (iii)We just prove the first statement. Note that (i) implies that \(\mathcal D^+\) is an interval. Now suppose for contradiction that \(u\in \mathcal D^+\) satisfiesThis means that there is \(v>u\) satisfying \(\mathbb {E}[S_{T^*(u)}^v]<\infty \). Hence, \(T^*(v)= T^*(u)\), contradicting the strict decrease of \(T^*\). Finally, suppose for contradiction that \(u\ge 1\) satisfies \(u > u^+(T^*(u))\). Then, there is \(1\le v<u\) such that \(\mathbb {E}[S_{T^*(u)}^v]=\infty \). For arbitrary \(t\ge T^*(u)\), we get$$\begin{aligned} u< u^+(T^*(u)) =\sup \{ v: \mathbb {E}[S_{T^*(u)}^v]<\infty \}. \end{aligned}$$This implies \(T^*(v)=T^*(u)\), again contradicting the strict decrease of \(T^*\).$$\begin{aligned} \mathbb {E}[S_t^v]=\mathbb {E}\big [\mathbb {E}[S_t^v|\mathcal {F}_{T^*(u)}]\big ] \ge \mathbb {E}[S_{T^*(u)}^v]=\infty . \end{aligned}$$

If the assumptions of part (iii) hold, then we can compute the critical moments from the explosion time, by numerically solving the equations \(T^*(u^+(T)) = T\) resp. \(T^*(u^-(T)) = T\). Note, however, that *strict* monotonicity may fail for reasonable stochastic volatility models. In the 3 / 2-model (Lewis 2000), the explicit characteristic function shows that the critical moments do not depend on maturity (for positive maturity), and the explosion time assumes only the values zero and infinity.

*u*: On the set where it is finite, \(T_1^*\) strictly increases for negative

*u*and strictly decreases for positive

*u*. By part (iii) of Lemma 13, this implies that we have

*rough*Heston model, this seems not easy to verify. If we accept it as given, then the lower critical moment can be computed for \(\rho <0\) as the unique solution of

*lower*critical moment, because then we can apply Algorithm 1 to compute \(T_\alpha ^*\) for \(\rho <0\). Recall that this algorithm works only for \(u\le \lambda /(\rho \xi )\), which amounts to case (A). Thus,

*T*must not be too large in (9), namely such that \(u^-(T)\) satisfies case (A). (Usually, this requirement is not too prohibitive.)

To provide some indication for the strict monotonicity of \(T_\alpha ^*\), recall that according to Lemma 6, \( f(u, t) = c_3 \psi (u, t) \) (see Sect. 2) strictly decreases w.r.t. *u*, if \(u<0\) satisfies case (A). It is then plausible (although not proven) that the strictly smaller function \( f(u_2,\cdot )\) explodes at a larger time than \( f(u_1,\cdot )\), where \(u_1<u_2<0\). As another indication, the bounds in Theorems 4 and are strictly monotonous, as seen by differentiating them w.r.t. *u*.

*T*). The validity of (61) and (62) is clear from (40): If \(T^*_\alpha (\cdot )\) 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.

## 10 Option pricing

*K*and maturity

*T*can be computed by the Fourier integral

Even for Markovian stochastic volatility models, it is well known that the choice of the parameter \(\beta \) greatly affects the performance of numerical quadrature algorithms when computing call prices in this way. See Lord and Kahl (2007) for a detailed study. For the rough Heston model, this is even more important, because evaluating the mgf is far more costly than in the classical case. A bad choice of \(\beta \) can lead to unnecessary oscillations of the integrand, which require a large number of grid points. Moreover, \(\beta <u^-(T)\) might lead to a numerical result that is not the correct option price, because then the Fourier representation of the call is no longer valid. Our results allow to compute the range of possible values of \(\beta \) (with the caveats described in the preceding section). For \(\rho <0\), we recommend to use (63) to compute the call price, because our Algorithm 1 allows to compute \(u^-(T)\). Figures 4 and 5 show how the amount of oscillations is affected by the choice of \(\beta ={ Re}(u)\).

## 11 Application to asymptotics

*lower*critical moment, which Algorithm 1 computes in the important case \(\rho <0,\) if

*T*is not too large.

*T*is not too large.

*T*and \(\alpha \). In the classical Heston model, the factor \(\hbox {e}^{c_2 (\log x)^{1-1/(2\alpha )}}\) becomes \(\hbox {e}^{c_2 \sqrt{\log x}}\), in line with Friz et al. (2011). From previous studies on simpler models, it is well known that Lee’s formula (64) only becomes accurate for very large values of |

*k*|. Therefore, a refined analysis of the wings is necessary to obtain an asymptotic formula of practical use. Extending the analysis of Friz et al. (2011) to \(\tfrac{1}{2}<\alpha <1\) will require a detailed study of the blowup behavior of the Volterra integral equation (9). Among other things, (a special case of) the heuristic analysis in Roberts and Olmstead (1996), which we already mentioned in Sect. 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 (57) might be useful in this context.

## Notes

### Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). We thank Omar El Euch, Paul Gassiat, Antoine Jacquier and Martin Keller-Ressel for helpful comments.

## References

- Abi Jaber, E., Larsson, M., Pulido, S.: Affine Volterra processes. Preprint arXiv:1708.08796 (2017)
- Alòs, E., León, J.A., Vives, J.: On the short-time behavior of the implied volatility for jump-diffusion models with stochastic volatility. Finance Stoch.
**11**(4), 571–589 (2007)CrossRefGoogle Scholar - Alòs, E., Gatheral, J., Radoicic, R.: Exponentiation of conditional expectations under stochastic volatility. Preprint (2017). https://ssrn.com/abstract=2983180
- Andersen, L.B.G., Piterbarg, V.V.: Moment explosions in stochastic volatility models. Finance Stoch.
**11**(1), 29–50 (2007)CrossRefGoogle Scholar - Banaś, J., Martinon, A.: Monotonic solutions of a quadratic integral equation of Volterra type. Comput. Math. Appl.
**47**(2–3), 271–279 (2004)CrossRefGoogle Scholar - Bayer, C., Friz, P.K., Gassiat, P., Martin, J., Stemper, B.: A regularity structure for rough volatility. Preprint arXiv:1710.07481 (2017)
- Brunner, H.: Collocation methods for Volterra integral and related functional differential equations. In: Ciarlet, P.G., et al. (eds.) Cambridge Monographs on Applied and Computational Mathematics, vol. 15. Cambridge University Press, Cambridge (2004)Google Scholar
- Brunner, H.: Volterra Integral Equations, Cambridge Monographs on Applied and Computational Mathematics, vol. 30. Cambridge University Press, Cambridge (2017)CrossRefGoogle Scholar
- Brunner, H., Yang, Z.W.: Blow-up behavior of Hammerstein-type Volterra integral equations. J. Integral Equ. Appl.
**24**(4), 487–512 (2012)CrossRefGoogle Scholar - Callegaro, G., Grasselli, M., Pagès, G.: Fast hybrid schemes for fractional Riccati equations (rough is not so tough). Preprint arXiv:1805.12587 (2019)
- Comte, F., Renault, E.: Long memory in continuous-time stochastic volatility models. Math. Finance
**8**(4), 291–323 (1998)CrossRefGoogle Scholar - Comte, F., Coutin, L., Renault, E.: Affine fractional stochastic volatility models. Ann. Finance
**8**(2–3), 337–378 (2012)CrossRefGoogle Scholar - Cont, R.: Encyclopedia of Quantitative Finance. Wiley, Berlin (2014)Google Scholar
- El Euch, O., Rosenbaum, M.: Perfect hedging in rough Heston models. Ann. Appl. Probab.
**28**(6), 3813–3856 (2018)CrossRefGoogle Scholar - El Euch, O., Rosenbaum, M.: The characteristic function of rough Heston models. Math. Finance
**29**(1), 3–38 (2019)CrossRefGoogle Scholar - Elstrodt, J.: Maß- und Integrationstheorie. Springer-Lehrbuch, 6th edn. Springer, Berlin (2009)Google Scholar
- Flajolet, P., Sedgewick, R.: Analytic Combinatorics. Cambridge University Press, Cambridge (2009)CrossRefGoogle Scholar
- Friz, P., Gerhold, S., Gulisashvili, A., Sturm, S.: On refined volatility smile expansion in the Heston model. Quant. Finance
**11**(8), 1151–1164 (2011)CrossRefGoogle Scholar - Fukasawa, M.: Asymptotic analysis for stochastic volatility: martingale expansion. Finance Stoch.
**15**(4), 635–654 (2011)CrossRefGoogle Scholar - Gatheral, J.: The Volatility Surface, a Practitioner’s Guide. Wiley, Berlin (2006)Google Scholar
- Gatheral, J., Keller-Ressel, M.: Affine forward variance models. Finance Stoch
**23**(3), 501–533 (2019)CrossRefGoogle Scholar - Gatheral, J., Jaisson, T., Rosenbaum, M.: Volatility is rough. Quant. Finance
**18**(6), 933–949 (2018)CrossRefGoogle Scholar - Gerstenecker, C.: Moment explosion time in the rough Heston model. Master’s Thesis, TU Wien (2018)Google Scholar
- Greene, R.E., Krantz, S.G.: Function theory of one complex variable. In: In: Krantz, S.G., et al. (eds.) Graduate Studies in Mathematics, vol. 40, 3rd edn. American Mathematical Society, Providence (2006)Google Scholar
- Gripenberg, G., Londen, S.O., Staffans, O.: Volterra Integral and Functional Equations, Encyclopedia of Mathematics and its Applications, vol. 34. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
- Keller-Ressel, M.: Moment explosions and long-term behavior of affine stochastic volatility models. Math. Finance
**21**(1), 73–98 (2011)CrossRefGoogle Scholar - Keller-Ressel, M., Majid, A.: A comparison principle between rough and non-rough Heston models—with applications to the volatility surface. Preprint arXiv:1906.03119 (2019)
- Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)Google Scholar
- Krantz, S.G.: Function Theory of Several Complex Variables, 2nd edn. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove (1992)Google Scholar
- Lee, R.W.: The moment formula for implied volatility at extreme strikes. Math. Finance
**14**(3), 469–480 (2004a)CrossRefGoogle Scholar - Lee, R.W.: Option pricing by transform methods: extensions, unification, and error control. J. Comput. Finance
**7**(3), 51–86 (2004b)CrossRefGoogle Scholar - Lewis, A.L.: Option Valuation Under Stochastic Volatility. Finance Press, Newport Beach (2000)Google Scholar
- Lord, R., Kahl, C.: Optimal Fourier inversion in semi-analytical option pricing. Tinbergen Institute Discussion Papers 06-066/2, Tinbergen Institute (2007)Google Scholar
- Lubich, C.: Runge–Kutta theory for Volterra and Abel integral equations of the second kind. Math. Comput.
**41**(163), 87–102 (1983)CrossRefGoogle Scholar - Miller, R.K., Feldstein, A.: Smoothness of solutions of Volterra integral equations with weakly singular kernels. SIAM J. Math. Anal.
**2**, 242–258 (1971)CrossRefGoogle Scholar - Remmert, R.: Theory of Complex Functions, Graduate Texts in Mathematics, vol. 122. Springer, New York (1991)CrossRefGoogle Scholar
- Roberts, C.A.: Analysis of explosion for nonlinear Volterra equations. J. Comput. Appl. Math.
**97**(1–2), 153–166 (1998)CrossRefGoogle Scholar - Roberts, C.A., Olmstead, W.E.: Growth rates for blow-up solutions of nonlinear Volterra equations. Quart. Appl. Math.
**54**(1), 153–159 (1996)CrossRefGoogle Scholar - Schilling, R.L.: Measures, Integrals and Martingales. Cambridge University Press, New York (2005)CrossRefGoogle Scholar
- Whittaker, E.T., Watson, G.N.: A course of modern analysis. Cambridge Mathematical Library, Cambridge University Press, Cambridge (1996). Reprint of the fourth (1927) editionGoogle Scholar
- Widder, D.: The Laplace Transform. Princeton Mathematical Series. Princeton University Press, Princeton (1941)Google Scholar

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