A comparison principle between rough and non-rough Heston models - with applications to the volatility surface
AA COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGHHESTON MODELS – WITH APPLICATIONS TO THE VOLATILITYSURFACE
MARTIN KELLER-RESSEL AND ASSAD MAJIDA bstract . We present a number of related comparison results, which allow tocompare moment explosion times, moment generating functions and critical mo-ments between rough and non-rough Heston models of stochastic volatility. Allresults are based on a comparison principle for certain non-linear Volterra inte-gral equations. Our upper bound for the moment explosion time is different fromthe bound introduced by Gerhold, Gerstenecker and Pinter (2018) and tighter fortypical parameter values. The results can be directly transferred to a compari-son principle for the asymptotic slope of implied volatility between rough andnon-rough Heston models. This principle shows that the ratio of implied volatil-ity slopes in the rough vs. the non-rough Heston model increases at least withpower-law behavior for small maturities.
1. I ntroduction
It is well-known that classic stochastic volatility models are not able to accuratelyreproduce all features of the observed implied volatility surface. In particular forshort maturities, it is frequently seen that Markovian diffusion-driven models,such as the Heston model [Hes93], produce a smile which is flatter and lessskewed than the implied smile of observed market data [BCC97]. While addingjumps to the stock price dynamics can mitigate some of these deficiencies (cf.[Bat96, JKRM13]) a recently emerging alternative is given by rough volatility models [GJR18]. In such models, volatility is modeled by a non-Markovian stochasticprocess comparable to fractional Brownian motion with low Hurst index (e.g. H « `
19] (short-time asymptotics of at-the-money skew).Here, we focus on wing asymptotics (small- and large-strike) of implied volatilityin the rough Heston model of [ER19] (see also [ER18]), which is becoming in-creasingly popular due to its tractability and its connections with affine processes(see [AJLP17, GKR19, KRLP18]).Starting with the results of [Lee04] (see also [BFL09]) it has become well-understoodthat wing asymptotics of implied volatility are intimately connected to momentexplosions in the underlying stochastic model (see also [FKR10]). Consequently,a first study of moment explosions in the rough Heston model has been under-taken by Gerhold, Gerstenecker and Pinter [GGP18]. The authors derive a lower a r X i v : . [ q -f i n . M F ] J un M. KELLER-RESSEL AND A. MAJID and upper bound for moment explosion times and a method for their numeri-cal approximation (valid in a certain parameter range). We build on the resultsof [GGP18] and derive a new upper bound for the moment explosion time inthe rough Heston model (Thm. 4.1). Our new bound is usually tighter than theupper bound of [GGP18] and, more importantly, given by a transformation ofthe classic Heston explosion time, thus allowing for direct comparison betweenrough and non-rough Heston models. The result rests on a comparison principlefor non-linear Volterra integral equations and leads to a number of further com-parison results: For the moment generating functions of rough and non-roughHeston model (Thm. 5.1), for their critical moments (Thm. 6.3) and finally forthe implied volatility slope in the wings of the smile. We highlight Theorem 7.2,which concerns the asymptotic slope of left-wing implied volatility (
AIVS ´ α p T q )in a (negative-leverage) rough Heston model in dependency on maturity T andthe roughness parameter α “ H ` . This slope can be lower bounded by the time-changed and rescaled slope of a non-rough Heston model ( AIVS ´ p T q ) as AIVS ´ α p T q ě T α ´ α Γ p α q AIVS ´ ˆ T α α Γ p α q ˙ for all T smaller than a certain threshold T α . Slightly weaker results that alsoapply to the right wing are finally given in Theorem 7.4.2. P reliminaries The rough Heston model.
We consider the rough Heston model [ER19,ER18] for a risk-neutral asset-price S with spot variance V , given by dS t “ S t a V t dW t (2.1a) V t “ V ` ż t κ α p t ´ s q λ p θ p s q ´ V s q ds ` η ż t κ α p t ´ s q a V s dB s (2.1b)where V , λ and η are positive, θ P L p R ě , R ě q , p B , W q are Brownian motionswith constant correlation ρ P p´
1, 1 q and κ α is the power-law kernel κ α p t q “ Γ p α q t α ´ , α P p , 1 s .It was shown in [ER19, Thm. 2.1] that V is H ¨older-continuous with exponent in r α ´ q and therefore that α controls the ‘roughness’ of the variance process V .Other important properties of the rough Heston model, such as the decay of at-the-money implied volatility slope are also linked to the parameter α , cf. [ER19,Sec. 5.2].In the particular case α “
1, the kernel becomes constant, i.e., κ ” V canbe written in the familiar SDE form(2.2) dV t “ λ p θ p t q ´ V t q dt ` η a V t dB t .In this case p S , V q becomes an extended Heston model with time-varying meanreversion level, as considered by [B ¨uh06, Ex. 3.4] in the context of variance curvemodels. If also θ is constant, the Heston model of [Hes93] (‘classic Heston model’)is recovered. COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 3
The moment generating function of the rough Heston model.
Our com-parison principle rests on the moment generating function of X “ log S , whichhas been studied in [ER18] and [GGP18]. Define the moment explosion time of the α -rough Heston model(2.3) T ˚ α p u q : “ sup ! t ě E ” e uX t ı ă 8 ) , u P R and the quadratic polynomial(2.4) R p u , w q : “ u p u ´ q ` w p ρη u ´ λ q ` η w ,which can be considered the ‘symbol’ of the Heston model in the sense of [Hoh98].Moreover denote the Riemann-Liouville left-sided fractional integral and deriva-tive operators by I α f p t q : “ Γ p α q ż t p t ´ s q α ´ f p s q ds and D α f p t q : “ ddt I ´ α t f p t q The moment generating function of the rough Heston model is given by the fol-lowing result:
Theorem 2.1 ([ER18, GGP18]) . In the rough Heston model, the log-price X “ log Ssatisfies (2.5) E ” e uX t ı “ exp ˆ λ ż t θ p t ´ s q ψ α p s , u q ds ` V I ´ α t ψ α p t , u q ˙ for all u P R , t P r T ˚ α p u qq and ψ α p¨ , u q solves the fractional Riccati equation (2.6) D α ψ α p t , u q “ R p u , ψ α p t , u qq , I ´ α ψ α p u q “ α “
1) the operator I ´ α vanishes, D α becomes an ordinary derivative and the fractional Riccati equation (2.6) turnsinto the familiar Riccati ODE. Moreover, the solution ψ and the moment explo-sion time T ˚ p u q are explicitly known (cf. [AP07, KR11]) in the classic Heston case.The above theorem is complemented by the following result: Theorem 2.2 ([GGP18]) . The fractional Riccati equation (2.6) is equivalent to the Riccati-Volterra integral equation (2.7) ψ α p t , u q “ ż t κ α p t ´ s q R p u , ψ α p s , u qq dsand T ˚ α p u q “ ˆ T α p u q , where (2.8) ˆ T α p u q : “ sup t t ě ψ α p t , u q ă 8u .For the equivalence of (2.6) and (2.7) see also [KST06, Thm. 3.10]. The second partof the theorem states that the functions t ÞÑ E “ e uX t ‰ and t ÞÑ ψ α p t , u q blow up atexactly the same time. Therefore, the solution ψ α of (2.7) contains all relevantinformation needed for the analysis of both moments and moment explosions inthe rough Heston model. Related results for more general kernels κ and multivariate generalizations (‘affine Volterraprocesses’) can be found in [AJLP17, GKR19] M. KELLER-RESSEL AND A. MAJID
Calibration to the forward variance curve.
Given a stochastic volatility modelwith spot variance process V , the associated forward variance curve is given by(2.9) ξ p T q : “ E r V T s ,and represents the market expectation of future variance. It is well-understoodthat forward variance is closely linked to the prices of variance swaps and othervolatility-dependent products and that for these products the forward variancecurve has a similar role as the forward curve for interest rates. In the roughHeston model (2.1), it is known from [ER18, Prop. 3.1] (see also [KRLP18]) thatthe forward variance curve is given by(2.10) ξ p T q “ V ˜ ´ ż T r α , λ p s q ds ¸ ` ż T θ p T ´ s q r α , λ p s q ds ,where r λ , α is the so-called resolvent of λκ α , given by(2.11) r λ , α p t q “ λ t α ´ E α , α p´ λ t α q , α P p
0, 1 q λ e ´ λ t α “ E α , α denoting the Mittag-Leffler function, cf. [HMS11]. Given a variancecurve ξ of suitable regularity, equation (2.10) can be inverted and solved for θ ,with solution(2.12) θ p t q “ λ D α p ξ p t q ´ V q ` ξ p t q ,see also [ER18, Rem. 3.2]. We refer to a model with this choice of θ p . q as calibrated to a given forward variance curve. For a calibrated model, the moment generatingfunction (2.5) can be expressed in terms of the variance curve ξ p . q instead of θ p . q and written as(2.13) E ” e uX t ı “ exp ˆż t ξ p t ´ s q p R p u , ψ α p s , u qq ` λψ α p s , u qq ds ˙ ,for all u P R , t P r T ˚ p u qq , see [KRLP18].3. A comparison principle for R iccati -V olterra equations Our comparison results for moments, moment explosion times and implied volatil-ities in the rough Heston model will all be derived from comparison results forthe Volterra-Riccati integral equation (2.7). In fact, the comparison results in thissection are obtained for the more general Volterra integral equation(3.1) ψ κ p t , u q “ ż t κ p t ´ s q R p u , ψ κ p s , u qq ds ,where only the following assumptions on the kernel κ are imposed: Assumption 3.1.
The kernel κ ‚ is non-negative and decreasing, and ‚ satisifies ş T κ p s q ds ă 8 for all T ą . COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 5
Clearly, this assumption includes the power-law kernels κ α for all α P p
0, 1 s . Theslight abuse of notation that we have introduced should not cause any confusions: ψ κ denotes the solution of (3.1) for a general kernel κ ; ψ α for the power-law kernel κ α ; and ψ for the plain Heston case κ ” ψ κ , in particular its maximal life-time, cruciallydepend on the nature of R p u , w q . As in [GGP18], we distinguish between thefollowing cases, illustrated in Figure 1(A) R p u , 0 q ą B w R p u , 0 q ě R p u , 0 q ą B w R p u , 0 q ă R p u , ¨q has no roots,(C) R p u , 0 q ą B w R p u , 0 q ă R p u , ¨q has positive roots,(D) R p u , 0 q ď wR p u , w q case (A) wR p u , w q case (B) wR p u , w q case (C) wR p u , w q case (D) F igure
1. Schematic plot of R p u , ¨q , with u P R satisfying case (A),(B), (C), or (D).These cases can be analyzed by applying the familiar theory of quadratic equa-tions to the polynomial R p u , w q . Following the notation from [GGP18], we rewrite R p u , w q as(3.2) R p u , w q “ c p u q ` c p u q w ` η w ,with coefficients c p u q “ u p u ´ q , c p u q “ ρη u ´ λ .The discriminant of w ÞÑ R p u , w q is given by(3.3) ∆ p u q “ ´ p ρη u ´ λ q ´ η p u ´ u q ¯ . M. KELLER-RESSEL AND A. MAJID
If and only if ∆ p u q is positive, R p u , ¨q has two real roots located at η p´ c p u q ˘ a ∆ p u qq . In the case ρ ă Lemma 3.2.
Suppose that ρ ă and denote the roots of ∆ p u q by (3.4) d ˘ : “ η ´ ρ ˘ a p η ´ ρ q ` λ p ´ ρ q η p ´ ρ q . Then λρη ă d ´ ă ; ă d ` and ‚ u satisfies case (A) ðñ u ď λρη , ‚ u satisfies case (B) ðñ u P p λρη , d ´ q Y p d ` , , ‚ u satisfies case (C) ðñ u P r d ´ , 0 q Y p d ` s , ‚ u satisfies case (D) ðñ u P r
0, 1 s .Proof. The mapping of the cases (A-D) to the corresponding intervals is basedon the following observations: R p u , 0 q “ c p u q is negative on r
0, 1 s and strictlypositive outside; B w R p u , 0 q “ c p u q is positive for u ď λρη and strictly negative else-where. Finally, the discriminant ∆ p u q of w ÞÑ R p u , w q is positive within r d ´ , d ` s and strictly negative outside. It remains to show the stated inequalities for d ˘ .Directly from (3.4) it can be seen that d ´ ă d ` ě p η ´ ρ q η p ´ ρ q ě ´ ρ ą e ˆ λρη ˙ “ ´ ρ p λ ´ λρη q ă λρη ă d ´ . (cid:3) If ş κ p s q ds “ 8 , then the four cases introduced above have the following connec-tion to the properties of ψ κ : ‚ In cases (A) and (B), the solution ψ κ explodes in finite time. ‚ In cases (C) and (D), the solution ψ κ exists globally.In the power-law case κ “ κ α this has already been shown in [GGP18]. How-ever, our goal is not just to characterize the domains where ψ κ exists globally, butrather to give a more refined comparison principle between ψ κ and the non-roughHeston solution ψ . Such comparison results have been shown in [GKR19, Ap-pendix A] in the non-exploding case (C) and we will extend those arguments tocover all situations (A-D). To formulate these results let w p u q : “ ´ c p u q η “ ´ η p ρη u ´ λ q COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 7 denote the location of the global minimum of w ÞÑ R p u , w q , and w ˚ p u q : “ η ´ ´ c p u q ´ a ∆ p u q ¯ the location of its first root, whenever ∆ p u q ě
0. The next Lemma is closely relatedto [GKR19, Lem. A.3].
Lemma 3.3.
Let u P R and Q p u , ¨q defined by (3.5) Q p u , w q : “ ż w d ζ R p u , ζ q , w P R . Furthermore, we definev p u q : “ a ´ ∆ p u q ˜ π ´ arctan ˜ c p u q a ´ ∆ p u q ¸¸ , v p u q : “ a ∆ p u q log ˜ c p u q ` a ∆ p u q c p u q ´ a ∆ p u q ¸ .(a) If u satisfies case (A) and ∆ p u q ă , the function Q p u , ¨q maps r onto r v p u qq , is strictly increasing, and has an inverse Q ´ p u , ¨q , which maps r v p u qq onto r . If ∆ p u q ą , v p u q has to be replaced by v p u q . (b) If u satisfies case (B), the same assertion as in (a) holds (with the restriction thatonly v p u q is needed). (c) If u satisfies case (C), it holds that w ˚ p u q ą and the function Q p u , ¨q maps r w ˚ p u qq onto r , is strictly increasing, and has an inverse Q ´ p u , ¨q , whichmaps r onto r w ˚ p u qq . (d) If u satisfies case (D), it holds that w ˚ p u q ă and the function Q p u , ¨q maps p w ˚ p u q , 0 s onto r , is strictly decreasing, and has an inverse Q ´ p u , ¨q , whichmaps r onto p w ˚ p u q , 0 s . Remark 3.4.
While the lemma is mainly a technical result on the properties of the func-tion Q p u , w q , the connection to moment explosions in the Heston model should becomeapparent from the fact that T ˚ p u q can be written asT ˚ p u q “ lim w Ñ8 Q p u , w q “ ż d ζ R p u , ζ q in cases (A) and (B), cf. [KR11, Sec. 6.1] .Proof. (a) Due to the fact that the integrand 1 { R p u , ζ q is positive on r if u satisfies case (A), we can conclude that Q p u , ¨q is strictly increasing. It just remainsto show that the integral attains the limit v p u q resp. v p u q . If ∆ p u q ă
0, we getlim w Ñ8 Q p u , w q “ ż d ζ R p u , ζ q“ a ´ ∆ p u q arctan ˜ η w ` c p u q a ´ ∆ p u q ¸ˇˇˇˇˇ “ v p u q , M. KELLER-RESSEL AND A. MAJID and if ∆ p u q ą
0, we obtain ż d ζ R p u , ζ q “ a ∆ p u q log ˆ η w ` c ´ a ∆ p u q η w ` c ` a ∆ p u q ˙ˇˇˇˇˇ “ v p u q .(b) Restricted to ∆ p u q ă
0, the proof of case (B) is analogue to (a).(c) In case (C) we can argue similar, since the integrand 1 { R p u , ζ q is positive on r w ˚ p u qq . The assertion follows if we replace the upper limit in the above inte-grals by w ˚ p u q .(d) The proof of case (D) is analogue to (c), only the different sign of R p u , ¨q on p w ˚ p u q , 0 s has to be taken into account. (cid:3) Now, we are ready to adapt the results of [GKR19, Appendix A] to our frame-work.
Theorem 3.5.
Let u P R . (a) If u satisfies case (A), then ψ κ p¨ , u q satisfies (3.6) 0 ď ψ ˆż t κ p s q ds , u ˙ ď ψ κ p t , u q , t ě If u satisfies case (B), then ψ κ p¨ , u q satisfies (3.7) 0 ď ψ ˆż t κ p s q ds , u ˙ ď ψ κ p t , u q , t ě where ψ is the solution of ψ p t , u q “ ż t R ` u , ψ p s , u q ˘ ds , t ě with R given by (3.8) R p u , w q : “ R p u , w p u qq w ď w p u q R p u , w q w ą w p u q .(c) If u satisfies case (C), then ψ κ p¨ , u q exists globally and satisfies (3.9) 0 ď ψ κ p t , u q ď ψ ˆż t κ p s q ds , u ˙ ď w ˚ p u q , t ě If u satisfies case (D), then ψ κ p¨ , u q exists globally and satisfies (3.10) w ˚ p u q ă ψ ˆż t κ p s q ds , u ˙ ď ψ κ p t , u q ă t ě Remark 3.6.
The function w ÞÑ R p u , w q introduced in (3.8) should be interpreted as in-creasing lower envelope of w ÞÑ R p u , w q , i.e., the largest increasing function boundingit from below. COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 9
Proof.
By [GLS90, Thm. 12.11] equation (3.1) has a continuous local solution ψ κ p¨ , u q on some non-empty time interval r T κ p u qq . In addition, ψ κ p¨ , u q can be continuedup to (but not beyond) a maximal interval of existence r
0, ˆ T κ p u qq , which is opento the right, and for ˆ T κ p u q ă 8 it holds that(3.11) lim sup t Ñ ˆ T κ p u q ψ κ p t , u q “ 8 .In particular, this means that ˆ T κ p u q can be written asˆ T κ p u q “ sup t t ą ψ κ p t , u q ă 8u ,consistent with (2.8).(a) Let u satisfy case (A). Recall the Riccati equation in the non-rough Hestonmodel:(3.12) B t ψ p t , u q “ R p u , ψ p t , u qq .We claim that its solution satisfies(3.13) Q p u , ψ p t , u qq “ t , @ t P r
0, ˆ T p u qq ,where Q is given by (3.5). Dividing by R p u , ψ p t , u qq and integrating both sides of(3.12) yields ż t B s ψ p s , u q R p u , ψ p s , u qq ds “ t .Now we substitute η “ ψ p s , u q , d η “ B s ψ p s , u q ds , and get(3.14) ż ψ p t , u q d η R p u , η q “ t ,which verifies (3.13).We remember that for u satisfying case (A), the function R p u , ¨q is positive andincreasing on r . Since the kernel κ is decreasing, we can deduce the followinginequality ψ κ p t , u q “ ż t κ p t ´ s q R p u , ψ κ p s , u qq ds ě ż t κ p T ´ s q R p u , ψ κ p s , u qq ds “ : v p t , T q ,(3.15)for 0 ď t ď T ă ˆ T κ p u q . It is easily seen that the above defined function v p t , T q hasthe boundary values v p T q “ v p t , t q “ ψ κ p t , u q ,(3.16)and, since w ÞÑ R p u , w q is increasing on r , satisfies the differential inequality(3.17) B t v p t , T q “ κ p T ´ t q R p u , ψ κ p t , u qq ě κ p T ´ t q R p u , v p t , T qq .Now, we can use a standard comparison principle for differential equations (seee.g. Chapter II, § v p t , T q ě r p t , T q , with r p t , T q , being the solution of(3.19) B t r p t , T q “ κ p T ´ t q R p u , r p t , T qq .Note that this differential equation differs from (3.12) only by the factor κ p T ´ t q .Thus, if we divide by R p u , r p t , T qq , integrate both sides up to T and substituteanalogue to the Heston case in (3.14) with η “ r p t , T q , d η “ B t r p t , T q dt , we get(3.20) Q p u , r p T , T qq “ ż r p T , T q d η R p u , η q “ ż T κ p T ´ t q dt “ ż T κ p t q dt ,with T ă T ˚ κ p u q . Applying Q ´ p u , ¨q to (3.13) and (3.20), it holds that ψ p t , u q “ Q ´ p t , u q , r p t , t q “ Q ´ ˆ u , ż t κ p s q ds ˙ ,(3.21)and we can deduce(3.22) r p t , t q “ ψ ˆ u , ż t κ p s q ds ˙ , t P “
0, ˆ T κ p u q ˘ .The inequalities (3.15) and (3.18) finally yield(3.23) ψ κ p t , u q “ lim T Ó t v p t , T q ě lim T Ó t r p t , T q “ ψ ˆ u , ż t κ p s q ds ˙ , t P “
0, ˆ T κ p u q ˘ .(b) If u satisfies case (B), the inequality (3.15) still holds. However, we cannotargue as in (3.17), because the function R p u , ¨q is decreasing on r w p u qq . Tocircumvent this obstacle, we use the adjusted function R p u , w q from (3.8) andconclude the inequalities B t v p t , T q “ κ p T ´ t q R p u , ψ κ p t , u qqě κ p T ´ t q R p u , ψ κ p t , u qqě κ p T ´ t q R p u , v p t , T qq ,for all 0 ď t ď T ă ˆ T κ p u q . From this point we can proceed as in (a) with thefunction r p t , T q , being the solution of B t r p t , T q “ κ p T ´ t q R p u , r p t , T qq .(c) Let u be satisfying case (C) and set(3.24) r T κ p u q : “ inf (cid:32) t P `
0, ˆ T κ p u q ˘ : ψ κ p t , u q “ w ˚ p u q or ψ κ p t , u q “ ( .Due to the behavior of the function R p u , ¨q in case (C), and because of (3.1) we canconclude(3.25) ψ κ p t , u q ą @ t P ´ r T κ p u q ¯ .This clearly indicates that ψ κ p¨ , u q is increasing for t P ´ r T κ p u q ¯ , and thereforethe upper bound in (3.24) is always hit before the lower bound.Now we can continue similarly to (a):Considering 0 ď t ď T ď r T κ p u q , it can be seen that the inequality (3.15) is satisfied. COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 11
In (3.17), however, the inequality sign has to be reversed, since R p u , ¨q is decreasingon r w ˚ p u qq . Therefore, the solution r p t , T q of (3.19) satisfies(3.26) r p t , T q ě v p t , T q , 0 ď t ď T ď r T κ p u q .Using (3.15),(3.16), (3.22), and (3.26), we obtain(3.27) ψ ˆ u , ż t κ p s q ds ˙ “ lim T Ó t r p t , T q ě lim T Ó t v p t , T q “ ψ κ p t , u q ,for all t P r r T κ p u qq . By means of (3.21), (3.22), and Lemma 3.3 (c), this impliesthat(3.28) lim t Ñ r T κ p u q ψ κ p t , u q ď ψ ˜ u , ż r T κ p u q κ p s q ds ¸ ă w ˚ p u q .Considering (3.24), we now obtain r T κ p u q “ ˆ T κ p u q , i.e. the bounds (3.9) hold for all t P r
0, ˆ T κ p u qq and we have lim t Ñ ˆ T κ p u qq ψ κ p t , u q P r w ˚ p u qs .If ˆ T κ p u qq ă 8 , this is a contradiction to (3.11), and we conclude that ˆ T κ p u q “ 8 .(d) The proof of the bounds in case (D) is analogous to (c) with the followingadaptations: The inequality sign in (3.15) has to be reversed, since the function R p u , ¨q is negative on p w ˚ p u q , 0 s . Thus, in contrast to (c), the inequality sign of(3.17) remains. It follows that the inequality signs of (3.25), (3.26), (3.27), and(3.28) have to be reversed, and the proof is complete. (cid:3) First consequences.
We state two immediate corollaries from Theorem 3.5.The first generalizes [GGP18, Thm. 2.4] from power-law kernels to a large classof other kernels.
Corollary 3.7.
Let κ be a kernel satisfying Assumption 3.1 and with ş κ p s q ds “ 8 .Then ˆ T κ p u q is finite if and only if u satisfies case (A) or (B), and it is infinite if and onlyif u satisfied case (C) or (D). In particular, the set (cid:32) u P R : ˆ T κ p u q ă 8 ( is independentof κ .Proof. In case (A) it is known that ψ p t , u q blows up in finite time, cf. [AP07, KR11].Since ş κ p s q ds “ 8 and, by Theorem 3.5, ψ ˆż t κ p s q ds , u ˙ ď ψ κ p t , u q for all t ě
0, also ψ κ p t , u q must blow up in finite time.In case (B), ψ p t , u q has to be used instead of ψ p t , u q . It can be seen by directcalculation that also ψ p t , u q blows up in finite time, see also Lemma 4.2 below.In cases (C) and (D) Theorem 3.5 shows global existence of ψ κ p t , u q , i.e., no finite-time blow-up can take place. (cid:3) In many cases of interest, the time-change T ÞÑ ş T κ p s q ds contracts time for small T up to a time T κ ; see Figure 2. This allows to reformulate Theorem 3.5 withouttime-change, at the expense weakening the inequalities. Corollary 3.8.
Suppose that κ is strictly decreasing and there exists t ˚ P p with κ p t ˚ q “ . Then, there is a unique solution T κ P p of (3.29) T “ ż T κ p s q dsand the following holds: (a) If u satisfies case (A), it holds that ψ p t , u q ď ψ κ p t , u q , @ t ď T κ .(b) If u satisfies case (B), it holds that ψ p t , u q ď ψ κ p t , u q , @ t ď T κ . Proof.
Under the given assumptions, the function t ÞÑ ş t κ p s q ds starts at t “
0, isincreasing, strictly concave, and has derivative one at t ˚ P p . It is obviousthat this implies the existence of a unique fixed point T κ P p , i.e., of a uniquesolution of (3.29). Moreover, t ď ż t κ p s q ds must hold for all t ď T κ . Since ψ p¨ , u q is strictly increasing in cases (A-C) (seeChapter 2 in [Gat06]), we obtain from Theorem 3.5 that ψ p t , u q ď ψ ˆż t κ p s q ds , u ˙ ď ψ α p t , u q for t ď T κ , completing case (a). The proof of (b) is analogue. (cid:3) t t α α Γ p α q α “ α “ α “ F igure
2. The graph of the time-change ş t κ α p s q ds “ t α α Γ p α q for differ-ent α .In the rough Heston model with power-law kernel κ α p t q “ Γ p α q t α ´ the relevanttime-change can be easily computed and is given by ş t κ α p s q ds “ t α α Γ p α q . The kernel κ α also satisfies the requirements of Corollary 3.8 and the solution of (3.29) isgiven by T α “ p α Γ p α qq {p α ´ q . An illustration is given in Figure 2 COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 13
4. C omparison of moment explosion times
In this section we study the temporal evolution of moments E r S ut s of the priceprocess in the rough Heston model (2.1). Whereas in the Black-Scholes model mo-ments of all orders exist for all maturities, it is well-known that moments in sto-chastic volatility models can become infinite at a certain time (see e.g. [FKR10]).Recall from (2.3) the definition of the time of moment explosion T ˚ α p u q “ sup t t ě E r S ut s ă 8u for the moment of order u in the rough Heston model with index α P p , 1 s . In the Heston case, T ˚ p u q is known explicitly and given by(4.1) T ˚ p u q “ $’’’’&’’’’% ? ´ ∆ p u q ˆ π ´ arctan ˆ c p u q ? ´ ∆ p u q ˙˙ , ∆ p u q ă
0, (A) or (B) ? ∆ p u q log ˆ c p u q` ? ∆ p u q c p u q´ ? ∆ p u q ˙ , ∆ p u q ą c p u q ą
0, (A) , ∆ p u q ě c p u q ă
0, (C) or (D),see [AP07, KR11]. For α ă ψ α p t , u q is not known explicitly andtherefore also no explicit expression for T ˚ α p u q can be derived. As discussed inthe introduction, an upper bound, a lower bound an an approximation method(valid in case (A)) for T ˚ α p u q have been derived in [GGP18]. Here, we obtain analternative upper bound of T ˚ α p u q in terms of T ˚ p u q as a direct consequence ofTheorem 3.5: Theorem 4.1.
Let u P R , such that case (A) holds. Then the blow-up time T ˚ α p u q satisfies (4.2) T ˚ α p u q ď p α Γ p α q T ˚ p u qq { α . If, in addition, T ˚ p u q ď T α , where T α “ p α Γ p α qq {p α ´ q (as in Corollary 3.8), then (4.3) T ˚ α p u q ď T ˚ p u q . The two inequalities also hold in case (B) when T ˚ p u q is replaced by T ˚ p u q . In cases (C)and (D) it holds that T ˚ α p u q “ 8 .Proof. By Theorem 3.5, we know that ψ ˆż t κ α p s q ds , u ˙ ď ψ α p t , u q for all t P r
0, ˆ T α p u qq . Clearly, the right hand side must blow-up before the lefthand side, and therefore the blow-up time of ψ p ş t κ α p s q ds , u q represents an upperbound T ` α p u q of ˆ T α p u q . Since the blow-up time T ˚ p u q of ψ p t , u q is known, we candetermine T ` α p u q by solving the equation ż T ` α p u q κ α p s q ds “ T ˚ p u q ,which leads us to T ` α p u q “ p α Γ p α q T ˚ p u qq { α . By Theorem 2.2 T ˚ α p u q “ ˆ T α p u q , which proves (4.2). Using the same argument asin the proof of Corollary 3.8, we obtain that T ` α p u q ď ż T ` α p u q κ α p s q ds “ T ˚ p u q ,as long as T ˚ p u q ď T α , and (4.3) follows. The proof of case (B) is analogue. (cid:3) The explicit form of T ˚ p u q has been given in (4.1). The bound T ˚ p u q , relevant incase B, can also be computed explicitly: Lemma 4.2.
For u in in case (B), the explosion time T ˚ p u q of ψ p t , u q is given by (4.4) T ˚ p u q “ a ´ ∆ p u q ˜ π ´ c p u q a ´ ∆ p u q ¸ . Remark 4.3.
Direct comparison of (4.1) and (4.4) shows that the difference betweenT ˚ p u q and T ˚ p u q can be reduced to the linearization arctan p x q „ x of the arctangentaround zero. This observation can be used to show that for ρ ă , the piecewise definedfunction r T ˚ p u q : “ T ˚ p u q , u ď λρη T ˚ p u q , u P p λ {p ρη q , d ´ q Y p d ` , is twice continuously differentiable at the cut-point u “ λ {p ρη q , i.e. T ˚ p u q transitionssmoothly into T ˚ p u q at the boundary between case (A) and (B). See Figure 3 for anillustration. T crit λρη d ´ d ` « case (A) (B) (C) (D) (C) (B) F igure
3. Illustration of the boundaries between cases (A-D) in thenegative-leverage case ρ ă T ˚ p u q (blue solid), and of the auxiliary explosiontime T ˚ p u q (grey dashed). The time T crit introduced in (6.7) is alsoindicated. COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 15
Proof.
From the differential equation BB t ψ p t , u q “ R p u , ψ p t , u qq with initial condition ψ p u q “
0, we derive that t “ ż ψ p t , u q d η R p u , η q Sending t Ñ T ˚ p u q and taking into account the definition of R p u , w q in (3.8) yields T ˚ p u q “ ż d η R p u , η q “ ż w p u q d η R p u , w p u qq ` ż w p u q d η R p u , η q .A primitive of η ÞÑ { R p u , η q is given by F p w q : “ a ´ ∆ p u q arctan ˜ η w ` c p u q a ´ ∆ p u q ¸ .Note that F p w p u qq “ w p u q{ R p u , w p u qq “ c p u q{p ∆ p u qq . Hence, T ˚ p u q “ c p u q ∆ p u q ` F p8q ´ F p w p u qq “ a ´ ∆ p u q ˜ π ´ c p u q a ´ ∆ p u q ¸ as claimed. (cid:3)
5. C omparison of moments
For the comparison of moments, we fix the parameters ρ , λ and η of both therough and the non-rough Heston model, but not θ p . q . Instead we assume that θ p . q is determined by calibrating each model to a fixed forward variance curve; seeSection 2.3. We write Φ α p t , u q “ E r S ut s , S is α -rough Hestonfor the moment generating function in dependency on α and p t , u q . In addition,we set K α p t q : “ ż t κ α p s q ds “ t α {p α Γ p α qq L α , λ p t q “ λ ż t r α , λ p s q ds “ ż t s α ´ E α , α p´ λ s α q ds ,(5.1)where r α , λ is the λ -resolvent kernel from (2.11). Note that both are continuous,positive, strictly increasing functions (‘time-changes’) with infinite derivative at t “
0. However, K α p t q Ñ 8 as t Ñ 8 , while L α , λ p t q Ñ λ . For both time-changes,there exists a unique solution in p of K α p t q “ t , and L α , λ p t q “ t ,which we denote by T α and T α , λ respectively; see also Cor. 3.8 where thesetimes are introduced for a generic kernel κ . It is easy to calculate that T α “p α Γ p α qq {p α ´ q , while T α , λ cannot be given in explicit form. Theorem 5.1.
Let α P p , 1 q , ρ ă and let Φ p t , u q and Φ α p t , u q be the moment gen-erating functions of a non-rough and a rough Heston model, which are calibrated to thesame forward variance curve ξ . Then Φ p t , u q ď Φ α p t , u q holds (a) for all u ď λ {p ρη q and t ď T α , and (b) for all u P p λ {p ρη q , 0 s and t ď T α , λ . This theorem allows the direct comparison of the moment generating functionsof rough and non-rough Heston models for small enough times t and negative u .To extend the result to all t , we have to make a monotonicity assumption on theforward variance curve and use the time-changes introduced in (5.1). Corollary 5.2.
Let the assumptions of Theorem 5.1 hold. In addition, assume that theforward variance curve is flat or increasing. Then (a) for all u ď λ {p ρη q it holds that Φ ´ t ^ K α p t q , u ¯ ď Φ α p t , u q (b) for all u P p λ {p ρη q , 0 s it holds that Φ ´ t ^ L α , λ p t q , u ¯ ď Φ α p t , u q , with K α and L α , λ as in (5.1) .Proof of Theorem 5.1. Our starting point is the representation (2.13) of the momentgenerating function in a (rough or non-rough) Heston model calibrated to a for-ward variance curve ξ . From this representation, it is clear that the statement(5.2) Φ p t , u q ď Φ α p t , u q for some t , t P R ě is equivalent to(5.3) ż t ξ p t ´ s q R p u , ψ p s , u qq ds ď ż t ξ p t ´ s q R p u , ψ α p s , u qq ds ,where we set(5.4) R p u , w q “ R p u , w q ` λ w “ p u ´ u q ` ρη uw ` η w .From Corollary 3.8a we obtain that ψ p s , u q ď ψ α p s , u q for all s ď T α and u ď λ {p ρη q . Since w ÞÑ R p u , w q is increasing for positive arguments, (5.3) followswith t “ t and part (a) of the Theorem is shown.For u P p λ {p ρη q , 0 s , we are in the domain of case (B) or (C). Instead of usingCorollary 3.8b (which does not allow direct comparison with the non-roughHeston model) we transform the Volterra-Riccati integral equation (2.7) usingthe resolvent kernel r α , λ from (2.11). Using the convolution notation f ‹ g “ ş t f p t ´ s q g p s q ds , the resolvent kernel is characterized by the property λκ α ´ r α , λ “ λ r α , λ ‹ κ α ,see e.g. [GLS90, Ch. 2]. Convolving ψ α (and suppressing its dependency on u )with r α , λ , we obtain r α , λ ‹ ψ α “ r α , λ ‹ κ α ‹ R p u , ψ α q “ κ α ‹ R p u , ψ α q ´ λ r α , λ ‹ R p u , ψ α q . COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 17
Subtracting this from the Volterra integral equation ψ α “ κ α ‹ R p u , ψ α q we obtain(5.5) ψ α p t , u q “ λ ż t r α , λ p t ´ s q R p u , ψ α p s , u qq ds ,another Volterra integral equation for ψ α , now involving the kernel λ r λ , α . Thiskernel satisfies Assumption 3.1 and an application of Corollary 3.8 yields that ψ p s , u q ď ψ α p s , u q for all s ď T α , λ . Note that the domain of case (A) has to bedetermined relative to R p u , w q , which now includes all u ď
0. The remainingproof of part (b) follows by repeating the arguments of part (a). (cid:3)
Proof of Corollary 5.2.
Assume u ď λ {p ρη q and observe that t ^ K α p t q “ t , t ă T α K α p t q t ě T α .Thus, for t ă T α the claim of the Corollary is already covered by Theorem 5.1 andit remains to treat the case t ě T α . From the concavity of K α , it follows that(5.6) K α p t q ´ K α p r q ď κ α p r qp t ´ r q ď t ´ r for all T α ď r ď t ; note that κ α p r q ď r . Moreover, from Theorem 3.5we know that ψ p K α p t q , u q ď ψ α p t , u q for all t ě
0. Thus, ż K α p t q T α ξ p K α p t q ´ s q R p u , ψ p s , u qq ds “ ż t T α ξ p K α p t q ´ K α p s qq R p u , ψ p K α p s q , u qq κ α p s q ds ď ż t T α ξ p t ´ s q R p u , ψ α p s , u qq ds where we have used (5.6) and the assumption that ξ is increasing in the lastinequality. Combining this estimate with ż T α ξ p K α p t q ´ s q R p u , ψ p s , u qq ď ż T α ξ p t ´ s q R p u , ψ α p s , u qq part (a) follows. The proof of part (b) is analogous, replacing K α by L α , λ and using(5.5) as in the proof of Theorem 5.1. (cid:3)
6. C omparison of critical moments
For the rough Heston model S with kernel κ α , α P p {
2, 1 s , the lower resp. uppercritical moments are defined by u ´ α p t q : “ inf t u ă E r S ut s ă 8u , t ą u ` α p t q : “ sup t u ą E r S ut s ă 8u , t ą S . Moreover, the critical moments can be written in terms of moment explosion times as u ´ α p t q : “ inf t u ă t ă T ˚ α p u qu , t ą u ` α p t q : “ sup t u ą t ă T ˚ α p u qu , t ą u ÞÑ T ˚ α p u q the mappings t ÞÑ u ˘ α p t q are its piecewise inverse functions. In the case α “ T ˚ p u q : Lemma 6.1.
Let ρ ă and let d ˘ be defined as in (3.4) . The function u ÞÑ T ˚ p u q is a strictly increasing continuous function from p´8 , d ´ q onto p and a strictlydecreasing continuous function from p d ` , onto p . Its inverse functions are givenby t ÞÑ u ´ p t q and t ÞÑ u ` p t q on the respective domains, and hence (6.3) T ˚ p u ˘ p t qq “ t , @ t ą d ˘ are precisely the boundaries between case (B) and (C) andthat T ˚ p u q “ 8 for all u P r d ´ , d ` s . In the rough Heston model ( α ă
1) it iscurrently only known (from [GGP18]) that u ÞÑ T ˚ α p u q are monotone functions(not necessarily in the strict sense) on the same domains as T ˚ p u q . For our pur-poses, however, the following property will be good enough: Directly from (6.2),it follows that u ă u ` α p t q implies t ă T ˚ α p u q and u ą u ´ α implies t ă T ˚ α p u q . Bycontraposition, we obtain(6.4) t ě T ˚ α p u q ùñ u ď u ´ α p t q if u ă d ´ u ě u ` α p t q if u ą d ` .In case (B), Theorem 4.1, the key comparison principle for the moment explosiontime T ˚ α p u q , is based on T ˚ p u q rather than on T ˚ p u q . Therefore, we also define u ` p t q : “ sup t u ą t ă T ˚ p u qu , t ą u ´ p t q : “ inf t u ă t ă T ˚ p u qu , t ą u ˘ p t q represent the critical mo-ments and therefore we refer to them as critical pseudo-moments . In analogy toLemma 6.1, the following can be derived by elementary calculus from (4.4): Lemma 6.2.
Let ρ ă , let d ˘ be defined as in (3.4) and set (6.7) T crit : “ T ˚ ˆ λρη ˙ “ | ρ | π a λ p λ ´ ρη q . The function u ÞÑ T ˚ p u q is a strictly increasing continuous function from p λ {p ρη q , d ´ q onto p T crit , and a strictly decreasing continuous function from p d ` , onto p .Its inverse functions are given by t ÞÑ u ´ p t q and t ÞÑ u ` p t q on the respective domains,and hence T ˚ p u ´ p t qq “ t , @ t ą T crit , T ˚ p u ` p t qq “ t , @ t ą COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 19
Theorem 6.3.
Let ρ ă and set (6.9) T crit : “ p α Γ p α q T crit q { α “ ˜ α Γ p α q| ρ | π a λ p λ ´ ρη q ¸ { α . Then the critical moments of the rough Heston model satisfyu ´ α p t q ě u ´ ˆ t α α Γ p α q ˙ @ t P p T crit s (6.10a) u ´ α p t q ě u ´ ˆ t α α Γ p α q ˙ @ t P p T crit , (6.10b) u ` α p t q ď u ` ˆ t α α Γ p α q ˙ @ t P p .(6.10c) For any t ď T α the inequalities also remain valid with t α α Γ p α q replaced by t.Proof. First, observe that u ´ p t α {p α Γ p α qqq is in the domain of case (A) if and only if t α α Γ p α q ď T ˚ ˆ λρη ˙ “ T crit ,which is easily transformed into t ď T crit “ ˜ α Γ p α q| ρ | π a λ p λ ´ ρη q ¸ { α .For any such t we obtain, using Theorem 4.1 and (6.3), that T ˚ α ˆ u ´ ˆ t α α Γ p α q ˙˙ Thm . 4.1 ď ˆ α Γ p α q T ˚ ˆ u ´ ˆ t α α Γ p α q ˙˙˙ { α (6.3) “ ˆ α Γ p α q ˆ t α α Γ p α q ˙˙ { α “ t .By (6.4), this implies u ď u ´ α p t q , showing the first inequality of (6.10). The othertwo inequalities are shown analogously, but – owing to the fact that case (B)applies – the critical pseudo-moments u ˘ p t q have to be used instead of u ˘ p t q .The last claim follows from the fact that t ď t α {p α Γ p α qq for all t ď T α and themonotonicity of u ˘ p . q and u ˘ p . q . (cid:3)
7. A pplications to I mplied V olatility As known from the work of Roger Lee [Lee04], moment explosions and criticalmoments are closely related to the shape of the implied volatility smile for deepin-the-money or out-of-the-money options. In this section we will apply Lee’smoment formula to our results and compare the smile’s asymptotic steepness inthe rough and classic Heston model.
For any given strike K of a European option with maturity T , let x “ log ´ KS ¯ denote the log-moneyness. Let σ iv p T , x q be the associated implied Black-Scholesvolatility and define the asymptotic implied volatility slope as(7.1) AIVS ˘ p T q “ lim sup x Ñ˘8 σ p T , x q{| x | .Note that the superscript ˘ refers to the left ( ´ ) and right ( ` ) wing of the smilerespectively. We also remark that in most models of practical interest, such as theHeston model, the ’lim sup’ can be replaced by a genuine limit, e.g. by apply-ing the theory of regularly varying functions; see [BF09]. For the rough Hestonmodel, however, it is currently an open question whether the lim sup in (7.1) canbe replaced by a genuine limit.The connection between critical moments and the asymptotic implied volatilityslope is given by Lee’s moment formula: Proposition 7.1 ([Lee04]) . For all T ą it holds thatAIVS ´ p T q “ ς p´ u ´ p T qq T , and AIVS ` p T q “ ς p u ` p T q ´ q T , where ς p y q “ ´ p a y ` y ´ y q and u ˘ p T q are the critical moments. Applying our comparison results for critical moments to Proposition 7.1 we ob-tain the following result.
Theorem 7.2.
Let ρ ă and let AIVS ˘ α p T q and AIVS ˘ p T q be the asymptotic impliedvolatility slope in the rough resp. classic Heston model for maturity T ą . ThenAIVS ´ α p T q ě T α ´ α Γ p α q AIVS ´ ˆ T α α Γ p α q ˙ for all T ď T crit , with T crit as in (6.9) . Remark 7.3.
This results shows that for small maturities the slope of left-wing impliedvolatility in the rough Heston model is dramatically steeper than in the non-rough Hestonmodel. This complements known results on small-time behavior of the at-the-money skewin rough models, which explodes at the same rate, cf. [Fuk17] .Proof.
Since T ď T crit , we can apply (6.10a) from Theorem 6.3 to estimate thelower critical moment. Since the function ς is strictly decreasing on R ě , a COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 21 straightforward application of Proposition 7.1 yields
AIVS ´ α p T q “ ς ` ´ u ´ α p T q ˘ T ě ς ´ ´ u ´ p T α α Γ p α q q ¯ T “ T α ´ α Γ p α q ς ´ ´ u ´ p T α α Γ p α q q ¯ T α α Γ p α q “ T α ´ α Γ p α q AIVS ´ ˆ T α α Γ p α q ˙ . (cid:3) Theorem 7.2, which is non-asymptotic in T , can be complemented by anotherresult, which is asymptotic in T , but also contains information on the right-wingimplied volatility slope. Here and below, we use the notation f p t q „ g p t q ðñ lim t Ñ f p t q g p t q “ f p t q Á g p t q ðñ lim t Ñ f p t q g p t q ě t Ñ 8 ) when indicated.
Theorem 7.4.
Let ρ ă and setC ˘ “ π ´ ˜ ˘ ρ a ´ ρ ¸ , D “ π ´ ρ a ´ ρ . In the classic Heston model, the limits AIVS ˘ p q : “ lim T Ó AIVS ˘ p T q exist and aregiven by (7.2) AIVS ˘ p q “ η a ´ ρ C ˘ . In the rough Heston model, it holds thatAIVS ´ α p T q Á T α ´ α Γ p α q AIVS ´ p q p as T Ñ q (7.3a) AIVS ` α p T q Á C ` D T α ´ α Γ p α q AIVS ` p q p as T Ñ q .(7.3b) Remark 7.5.
This result shows that as T Ñ the right-wing asymptotic implied volatil-ity slope of the rough Heston model explodes at the same power-law rate as the left-wingasymptotic implied volatility slope. We remark that the constant C ` { D which distin-guishes the estimates at the left and the right wing, is always within p
0, 1 q , given that ρ ă .Proof. We first analyze the behavior of T ˚ p u q and T ˚ p u q as | u | Ñ 8 . To this endnote that it follows from (3.3) that ∆ p u q ă | u | large enough and thatlim u Ñ˘8 c p u q a ´ ∆ p u q “ ˘ ρ a ´ ρ . Inserting into (4.1), we obtain T ˚ p u q „ | u | ´ C ˘ η a ´ ρ p as u Ñ ˘8q ,and from (4.4), we obtain T ˚ p u q „ u ´ D η a ´ ρ p as u Ñ 8q .The critical (pseudo-)moments are the piecewise inverse functions of the momentexplosion times, and hence u ˘ p t q „ ´ t ´ C ˘ η a ´ ρ p as t Ñ q u ` p t q „ t ´ D η a ´ ρ p as t Ñ q .To obtain the small-time behaviour of the asymptotic implied volatility slope, itremains to insert these relations into Lee’s moment formula. First, note thatlim ε Ñ ς p { ε q ε “
12 .Hence, with focus on the right wing, we conclude that for the classic Hestonmodel(7.4) lim T Ñ AIVS ` p T q “ lim T Ñ ς p u ` p T q ´ q T “ η a ´ ρ C ` ,and similarly at the left wing. For the rough Heston model we estimate withTheorem 6.3 (again at the right wing)lim T Ñ α Γ p α q T α ´ AIVS ` α p T q “ lim T Ñ α Γ p α q T α ς p u ` α p T q ´ qě lim T Ñ α Γ p α q T α ς ˆ u ` ˆ T α α Γ p α q ˙ ´ ˙ “ lim T Ñ α Γ p α q " T α ˆ u ` ˆ T α α Γ p α q ˙ ´ ˙* ´ ¨ “ η a ´ ρ D .Comparison with (7.4) yields (7.3b). The calculation on the left wing uses ´ u ´ instead of u ` ´ T Ñ α Γ p α q T α ´ AIVS ´ α p T q ě η a ´ ρ C ´ ,completing the proof. (cid:3) COMPARISON PRINCIPLE BETWEEN ROUGH AND NON-ROUGH HESTON MODELS 23
8. N umerical I llustration In this section we graphically illustrate and compare the bounds of moment ex-plosions (Thm. 4.1) and of the asymptotic implied volatility slope (Thms. 7.2 and7.4) for a concrete choice of the rough Heston model’s parameters. We set ρ “ ´ λ “ η “ α “ H “ Moment explosion times.
To provide a better readability, we use the follow-ing notations: ‚ T ` KM p u q “ $&%` α Γ p α q T ˚ p u q ˘ { α u sat. case (A) ´ α Γ p α q T ˚ p u q ¯ { α u sat. case (B) ,denotes the combined upper bounds of T ˚ α p u q , introduced in Theorem 4.1, ‚ T ` GGP p u q and T ´ GGP p u q denote the upper and lower bound of T ˚ α p u q , intro-duced in Theorem 4.1 and 4.2 in [GGP18], ‚ T ˚ α ,aprx p u q denotes the approximation of the explosion time T ˚ α p u q , com-puted by Algorithm 7.5 in [GGP18], which is valid for u in case (A).Figure 4 shows a comparison of the moment explosion bounds and the approx-imation T ˚ α ,aprx p u q in the given setting. It can be seen that the bound T ` KM p u q istighter than T ` GGP p u q on both sides of u “
0. Numerical experiments confirm thatthis relation persists in a large range of parameters, except in very close proximityto the boundary case α “ Implied volatility asymptotics.
In Figures 5 and 6 we illustrate the boundsfor the asymptotic implied volatility slope from Theorems 7.2 and 7.4. The boundsshown in the plots are generated as follows: First, we use T ˚ p u q as function of u to compute the critical moments u ˘ p t q of the classic Heston model by numericalroot finding. Afterwards we use Lee’s moment formula to determine AIVS ˘ p T q ,the asymptotic implied volatility slope in the classic Heston model. The boundsof the rough Heston implied volatility slope AIVS ˘ α from Theorems 7.2 and 7.4are then computed from AIVS ˘ . On the left wing of the smile (where case (A)applies for small T ), we additionally compute an approximation of AIVS ´ α p T q , byapplying the same procedure to the approximate explosion time T ˚ α ,aprx p u q .R eferences [AJLP17] Eduardo Abi Jaber, Martin Larsson, and Sergio Pulido. Affine Volterra processes. arXiv:1708.08796 , 2017. ´ ´
20 12 c a s e ( A ) c a s e ( B ) c a s e ( C ) ` ( D )
50 100 150 c a s e ( B ) « F igure
4. Bounds of the moment explosion time in the rough Hes-ton model for u P r´
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