ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS

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1 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE Abtract. We tudy the aymptotic behaviour of a cla of mall-noie diffuion driven by fractional Brownian motion, with random tarting point. Different caling allow for different aymptotic propertie of the proce mall-time and tail behaviour in particular. In order to do o, we extend ome reult on ample path large deviation for uch diffuion. A an application, we how how thee reult characterie the mall-time and tail etimate of the implied volatility for rough volatility model, recently propoed in mathematical finance. 1. Introduction Large deviation are ued extenively in Phyic thermodynamic, tatitical mechanic a well a in Mathematic information theory, tochatic analyi, mathematical finance to etimate the exponential decay of probability meaure of rare event. Varadhan [61], Schilder [58], Freidlin and Wentzell [32] proved, in different degree of generality, large deviation principle in R n and on path pace for olution of tochatic differential equation with mall noie, and the monograph by Dembo and Zeitouni [22] and Deuchel-Stroock [25] provide a precie account of thoe advance at leat up to the mid-199. In the pat decade, thi et of technique and reult ha been adopted by the mathematical finance community: finite-dimenional large deviation in the ene of Gärtner-Elli have been ued to prove mall-and large-time aymptotic of implied volatility in affine model [28, 41], ample-path LDP à la Freidlin-Wentzell [32] have proved efficient to determine importance ampling change of probability [38, 39, 57], and heat kernel expanion following Ben Arou [1] and Bimut [13], have led to a general undertanding of mall-time and tail behaviour of multi-dimenional diffuion [6, 7, 23, 24]. Thee aymptotic have overall provided a deeper undertanding of the behaviour of model, and, ultimately, allow for better calibration of real data; a general overview can be found in [55]. Motivated by financial application, we derive here aymptotic mall-time and tail behaviour of the olution to a generalied verion of the Stein-Stein tochatic volatility model, originally propoed in [59, 6]. We in particular conider two important in light of the recent trend in the literature propoed model extenion: i the SDE driving the intantaneou volatility proce i tarted from a random ditribution; thi o-called randomied type of model wa recently propoed in [46, 47, 51], in particular to undertand the behaviour of the o-called forward volatility ; ii the volatility proce i driven by a fractional Brownian motion. Fractional tochatic volatility model, originally propoed by Comte and Renault [18, 19] with H 1/2, 1, have recently been extended to the cae H, 1/2, and a recent flourihing activity in thi area [2, 8, 33, 37, 42] ha etablihed thee model a the go-to tandard for etimation and calibration. The original motivation behind randomiation of the initial tarting point i rooted in financial practice, where only the initial value of the tock price proce i oberved directly and the intantaneou value of volatility i Date: June 5, Mathematic Subject Claification. 41A6, 6F1, 6G15, 6G22. Key word and phrae. Rough volatility, large deviation, implied volatility aymptotic. BH acknowledge financial upport from the SNSF Early Potdoc.Mobility grant

2 2 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE ubject to calibration. The effect of randomiation of the initial volatility on the implied volatility urface wa explored by Jacquier and Roome [45] in a imple random environment etting, where the volatility component wa aumed to follow CEV dynamic. Their reult give an impetu both on the theoretical and the practical level: they olve a practical modelling problem in a imple tractable etting and at the ame time raie awarene for the potential prowe of applying random evolution equation for financial modelling. In thi paper we follow up on thi direction and blend more involved approache propoed by [5, 52] from the literature around random environment and random evolution equation into our financial model, where randomne alo appear in the drift and diffuion coefficient of the proce. On the practical level, independently from the reult of [45], Mechkov [51] goe a tep further in endoring the idea of randomiing the initial volatility and make a trong cae to move away from modelling hidden variable uch a tochatic volatility in the traditional way. He argue that tarting the volatility from a fixed tarting point heavily underetimate the effect of the hidden variable on the lope of the implied volatility mile, and therefore hot tart volatility model with random tarting point ignificantly outperform traditional one altogether. Indeed, both randomied model [45, 51] produce the deired exploion in the mile at hort maturitie. Jacquier and Shi [47] develop thi further by providing a precie link between the rate of exploion of implied volatilitie on the hort end and the tail ditribution of the initial ditribution of the volatility proce in a randomied Heton model. Thee output confirm that tochatic volatility model with random tarting point contitute a cla of counterexample to the long-tanding belief formulated by Gatheral [36, Chapter 5], that jump in the tock price proce are needed to produce teep hort-dated implied volatility kew. Another example of broadly different deign wa provided by Caravenna and Corbetta [15]. In their multicaling model, the tock price proce i continuou, while the volatility proce ha carefully deigned jump, and teepne of the mile i achieved with a heavy-tail ditribution of the mall-time ditribution of the volatility. Rough fractional volatility model with continuou volatility path have recently been propoed, and are able to capture the volatility kew [2, 8, 9, 26, 29, 33, 37, 4, 43]. In thi paper we analye the combined effect of a rough fractional Brownian driver with Hurt parameter H, 1 in the volatility and a random tarting point. We quantify how the tail behaviour parametried by a caling coefficient b > of the random tarting point modulate the rate of exploion in the implied volatility in the preence of rough fractional volatility. Finally, in a pecific implified etting we highlight how our model blend naturally into the etting of forward-tart option in tochatic volatility model, whoe aymptotic propertie have been tudied in [46]. In proving our reult, we improve the large deviation literature on both SDE with random tarting point and fractional SDE. In Section 2, we recall ome concept that will be ued in the paper and et the notation. We alo introduce the model 2.6, and the main aumption on it dynamic and on the initial random tarting point. Section 3 collect the main large deviation etimate in different regime: tail behaviour Section 3.1, and mall-time behaviour Section 3.2. In each cae, we preent two different cenario coniting of an appropriately recaled fractional model Theorem 3.1 and 3.7 and a implified diffuive model Theorem 3.6 and 3.13 with more retrictive condition on the random tarting point allowing for impler large deviation rate function. Section 4 diplay application to implied volatility aymptotic Corollarie 4.1 and 4.2, and preent an application to forward-tart option. Proof can be found in Appendix.

3 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 3 2. Set up and notation A outlined in the introduction, we prove pathwie large deviation for a two-dimenional ytem generaliing the Stein-Stein model [59, 6], with random initial datum. In particular, via uitable recaling, we determine the mall-time and the large-tail behaviour of the ytem. Before delving into the core of the paper, let u recall ome ueful fact about large deviation and Gauian procee, which hall alo erve a etting the notation for the ret of the paper. Unle otherwie tated, we alway work on a finite time horizon, ay [, 1] without lo of generality, which we denote by T, and we write T := T \ {. We let C := CT, R be the pace of continuou function from T to R and Cb 2 the pace of twice differentiable function on T with bounded partial derivative up to the econd order. We write X LDPh, I when the equence X > atifie a large deviation principle Definition 2.2 on C, a tend to zero with good rate function I and peed h, where h denote a function atifying lim h =. For a random variable X, we denote by uppx it upport Large deviation and fractional Brownian motion. We hall ue [22, Chapter 1.2] and [22, Chapter 1] a our guide through large deviation. Given a topological pace X and the completed Borel σ-field B X correponding to X, for any A B X, we denote by Å and A repectively it interior and cloure, and conider a equence X > on X, B X. Definition 2.1. A good rate function i a lower emi-continuou mapping I : X [, ] uch that the level et {x : Ix z are cloed compact ubet of X for any z. Definition 2.2. The equence X > atifie a large deviation principle LDP on X, B X a tend to zero, with peed h, and rate function I, if for any Borel ubet A X, the following inequalitie hold: 2.1 inf A o Iφ lim inf h log PX A lim up h log PX A inf Iφ. A A particularly convenient tool to prove large deviation i the o-called exponential equivalence, which we recall from [22, Definition 4.2.1] a follow: Definition 2.3. On a metric pace Y, d, two Y-valued equence X > and X > are called exponentially equivalent with peed h if there exit probability pace Ω, B, P > uch that for any >, P i the joint law and, for each δ >, the et {ω : X, X Γ δ i B -meaurable, and lim up h log P {ỹ, y : dỹ, y > δ =. Theorem 2.4. Let X > and X > be two exponentially equivalent equence with peed h on ome metric pace. If X LDPh, Λ X for ome good rate function Λ X, then X LDPh, Λ X. On a real, eparable Banach pace E,, we denote by B the aociated Borel igma field. Letting E denote the topological dual of E, we define a Gauian meaure a follow: Definition 2.5. A Gauian meaure µ on E, i uch that every ϑ E, when viewed a a random variable via the dual pairing ϑ ϑ, ϑ E E, i a real Gauian random variable on E, B, µ. We aociate a Gauian proce to a Gauian meaure in the uual way [16, Section 3.2]. Particular example of Gauian procee, crucial for the ret of the paper, include tandard Brownian motion on the time interval T, where E = C equipped with the upremum norm and with the topology of uniform convergence, and E i the

4 4 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE pace of igned meaure on T. In fact, thi contruction applie to all centered continuou Gauian procee, which are uniquely characteried by their covariance operator. A fractional Brownian motion W H with Hurt parameter H, 1 i uch a Gauian proce, tarting from zero, with covariance W H t, W H 1 = t 2H + 2H t 2H, for any, t T. 2 Of primary importance in undertanding mall-noie behaviour of Gauian ytem i the concept of reproducing kernel Hilbert pace RKHS, which we recall following [16, Definition 3.3]: Definition 2.6. Let µ be a Gauian meaure on E and define the map R : E E by Rx := E x, x xµdx. The RKHS H µ of µ i the completion of the image RE for the norm Rx Hµ := x, Rx 1/2, for all x E. To characterie the RKHS of fractional Brownian motion, the uual tool i it Volterra repreentation [54] 2.2 W H t = K H t, db, which hold almot urely for all t T, where B i a tandard Brownian motion generating the ame filtration a W H, and K H i the Volterra kernel defined, for any, t T with < < t, by [54, Theorem 5.2] κ t H u [tt H ] H H H u 1 H du, if H < 1 2, 2.3 K H t, = κ H H t u H du H u 1 H, if H > 1 2, 1, if H = 1 2, 1/2 2HΓ1 with H ± := H ± 1 2 and κ H H :=. For t T, the map K H t, i quare integrable ΓH + Γ2 2H around the origin, and the reproducing kernel Hilbert pace of the fractional Brownian motion i given by { H K H := K H t, fd, t T : f L 2 T with inner product K H, f 1 d, K H, f 2 d H K H := f 1, f 2 L2 T. The notation H K H, emphaiing the link with the underlying kernel, will be ueful later in Definition 2.7 for more general kernel. In particular, the RKHS aociated to tandard Brownian motion H = 1/2 i the Cameron-Martin pace, and correpond to the pace of abolutely continuou function tarting at zero, with quare integrable derivative. In other word, for any H, 1 \ {1/2, the identity H K H = K K H L 2 T hold, where K K H : L 2 T f KH t, fd. Thi characteriation motivate the following definition: Definition 2.7. For any trictly poitive function Φ : R 2 + R uch that Φt, L 2 T for any t T, the correponding RKHS i defined a { 2.4 H Φ := with inner product Φ, f 1 d, Φt, fd, t T : f L 2 T, Φ, f 2 d H Φ := f 1, f 2 L 2 T.

5 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 5 Reproducing Kernel Hilbert Space, together with their inner product, turn out to provide the right pace to characterie large deviation rate function. In particular, for a given Gauian Volterra proce of the form Φ, db, for ome Volterra kernel Φ, it follow from [25, Theorem 3.4.5] that the equence Φ, db > atifie a large deviation principle with peed 2 and rate function Λ Φ φ = 2 φ H Φ, if φ H Φ, +, otherwie. An obviouly pecial role i played by the tandard Brownian motion H = 1/2, and we hall adopt the implified notation H the claical Cameron-Martin pace and Λ in place of H K 1/2 and Λ K 1/ Setting and aumption. The particular ytem we are intereted in i dx t = σy t 2 dt + σy t ρdb t + ρdb t, X =, dy t = λ + βy t dt + ξdwt H, Y Θ, where W H i a fractional Brownian motion, with Hurt parameter H, 1, B, B i a two-dimenional tandard Brownian motion, β <, λ, ξ >, ρ 1, 1, ρ := 1 ρ 2, and Θ i a quare-integrable continuou random variable. In order to guarantee exitence and uniquene of a trong olution, we further aume [53, Theorem 3.1.3] that σ i Lipchitz continuou, atifie the growth condition σy C1 + y for y R, and i differentiable with locally Hölder continuou derivative. In order to prove our main reult below, we make the following technical aumption: Aumption A b : There exit a meaurable function σ : R R, with locally Hölder continuou derivative, uch that σ i Lipchitz continuou, atifying the growth condition σy C1 + y for y R, a well a a contant b > uch that the caling property lim b σy/ b = σy hold true uniformly on R. 3. Main reult Centrepiece of our analyi are large deviation etimate for uitably recaled verion of 2.6. The firt recaling preented in Section 3.1 i tailored to the analyi of the tail behaviour of 2.6, while the econd recaling Section 3.2 i bepoke to it hort-time aymptotic propertie. In addition to thee aymptotic reult in the general fractional cae Theorem 3.1 and 3.7, we preent two pecial implified diffuive cae, where particularly tractable rate function can be obtained Theorem 3.6 and 3.13 repectively. For thi we impoe tronger condition Aumption 3.4 on the random tarting point. Thi allow u to etablih, following [52] in Section 3.1.2, an exponential equivalence between 2.6 and an analogou proce with fixed tarting point. In Section we contruct a third recaling 3.9 inpired by Mellouk [5] in the hort-time diffuive cae under the aumption that the upport of the random tarting point i bounded. We hall work with recaled verion Y of the proce Y in 2.6 ee 3.1 and 3.7 for pecific example, together with a function h decribing the peed of the large deviation etimate, for which we introduce the following aumption: Aumption A b : There exit a family of continuou function σ n n on R uch that i σ n n converge uniformly to σ on R; ii for all δ >, lim lim up h log P σ n Y σy δ =. n Aumption A Θ b tail behaviour of Θ: The limit lim up h log P b Θ > 1 = hold. In Aumption A b and AΘ b above, the large deviation peed h take the value 2b in Section 3.1 and 4H+2b

6 6 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE in Section 3.2 repectively. The contant b which may vary below play an eential role in ubequent large deviation etimate, via exponential equivalence technique Definition 2.3. Aumption A b and A b are naturally atified in the fractional Stein-Stein cae where σy y, but impoing them allow u to tate our reult for more general σ, in particular when uing extended Contraction Principle [49, Propoition 2.3] Tail behaviour The general cae. For b, >, introduce the recaling X, Y := 2b X, b Y, o that 2.6 become dxt = 2b Y σ t Y b dt + 2b σ t b ρdb t + ρdb t, X =, dy t = b λ + βy t dt + b ξdwt H, Y b Θ. The particular recaling conidered here i perfectly uited for tail behaviour, a large deviation provide etimate for PX 1 = PX 2b. Our main reult i a follow, and i proved in Appendix A.1.3: Theorem 3.1. For any H, 1, the following hold: i for any b > uch that A Θ b hold, Y LDP 2b, Λ F H, with ΛF H in 2.5 and F H in Lemma 3.2; ii for any b 1 2 uch that Aumption A b, A b, AΘ b hold, X LDP 2b, Λ, with Λ in A.3. The proof of the theorem, developed later, require a precie analyi of the Reproducing Kernel Hilbert Space of the procee under conideration, and we firt tate the following two key ingredient proved in Appendice A.1.1 and A.1.2, which are alo of independent interet: Lemma 3.2. For any H, 1 and β >, there exit a tandard Brownian motion Z, uch that 3.2 ξ e βt dw H = F H t, dz, hold almot urely for t T, where F H : T T R i defined for < < t, with κ H in 2.3, a [ ξκ t { ] H H [tt ] H 1 2H + + β [uu ] H e βt u du, if H < 1 2u 2, F H t, := ξκ H H t u H e βt u du H u 1 H, if H > 1 2, ξe βt, if H = 1 2. The cae β = and ξ = 1 i excluded ince, in that cae, the lemma boil down to the claical Volterra repreentation of the fractional Brownian motion 2.2 and the function F H i nothing ele that K H given in 2.3. For H 1 2, 1, the expreion for F H i in agreement with [62, Definition 2.1], but, for H, 1 2, it correct the lightly erroneou expreion therein. Thi function F H allow u to fully characterie the following RKHS, with proof potponed to Section A.1.2: Propoition 3.3. For any H, 1, β >, the pace H F H i the RKHS of the Gauian proce ξ eβ dw H The Millet-Nualart-Sanz approach. In [52], Millet, Nualart and Sanz conider a perturbed tochatic differential equation of the form 3.3 dx t = b, X t dt + ax t dw t. Here, for any >, the function b, : R n R n and a : R n M n,d R are bounded Borel meaurable and uniformly Lipchitz, b, converge uniformly to a function b a tend to zero, W i a d-dimenional

7 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 7 Brownian motion, and X i an R n -valued quare-integrable random variable. Exitence and uniquene of a trong olution can be found in [31, Chapter 5, Theorem 2.1]. Following claical large deviation tep, conider, for any φ H, the controlled ordinary differential equation on T : 3.4 ψ t = ag t φ t + bψ t, the olution flow of which, tarting from x R n i denoted by S x φ. Millet, Nualart and Sanz [52] proved a large deviation principle [52, Theorem 4.1] for the equence X > under the following aumption: Aumption 3.4. Both a and b belong to C 2 b, and there exit x R n uch that, for any δ >, 3.5 lim up 2 log P X x > δ =. Theorem 3.5. Under Aumption 3.4, X > LDP 2, I with Iψ = inf{λφ : φ H, ψ = S x φ. Condition 3.5 i an exponential equivalence property between the initial random variable X and the contant x, and enure that large deviation are preerved under exponentially mall perturbation of the tarting point. Therefore, in the tandard diffuion cae H = 1 2, it i poible to obtain a imilar reult to Theorem 3.1 with a implified rate function albeit with lightly more retriction on the tarting point, by uing the approach conidered by Millet-Nualart-Sanz in [52]. For thi, we rewrite 3.3 to correpond to 3.1, albeit with tronger aumption on the coefficient, with W := W 1, W 2 a Brownian motion, X = X, Y, H = 1/2, b, X =, Θ, and ρ := 1 ρ 2, b, X t = 1 2 σy t 2 λ + βyt and ax ρ σy t = t ρ σy t. ξ The correlation between the two component of W i explicitly repreented in the diffuion matrix a. Theorem 3.6. Under Aumption 3.4, the olution X to 3.3 atifie X > LDP 2, I with Iχ = ρ ρ inf {Λφ : φ H, χ = S x Ψ ρ φ and Ψ ρ : R 2 z z,. 1 Proof. The proof of Theorem 3.5 relie firt on proving a large deviation principle for the flow S x Schilder theorem [22, Theorem 5.2.3], then on extending thi LDP to the original ytem. One can eaily extend it to include a correlation parameter ρ 1, 1, the main difference being the rate function. Indeed, ince W2 and W 2 are independent, Schilder theorem yield that W2, W 2 LDP 2, Λ. Since the map Ψ ρ i continuou on C, and W = Ψ ρ W2, W 2, the theorem follow from the Contraction Principle giving an LDP for W a tend to zero with peed 2 and good rate function 3.6 Λ ρ ψ := inf {Λφ : φ H, ψ = Ψ ρ φ Small-time behaviour. We now tackle the mall-time behaviour of the proce 2.6. Under the general et of aumption A b, A b, AΘ b, we need to introduce a particular recaling, both in time and in pace in order to oberve ome weak convergence. Thi i different from the claical Itô diffuion cae with fixed tarting point, where olution of uch SDE generally converge in mall time. In the Itô cae, though, if the ditribution of the tarting point ha compact upport, we how in Section that pace recaling i not required any longer. uing

8 8 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE The general cae. With the recaling Xt, Yt := 2H+2b 1 X 2 t, b Y 2 t, with b >, 2.6 become dxt = 2H+1+2b Y 2 σ t Y b dt + 2H+2b ρdbt σ t b + ρdb t, X =, dyt = b+2 λ + β 2 Yt dt + 2H+b ξdwt H, Y b Θ. Our main reult i a follow: Theorem 3.7. For any H, 1, i for any b > uch that A Θ b ii if b 1 2 2H uch that A b, A b, AΘ b hold Y LDP 4H+2b, Λ G H, with Λ G H a in 2.5; hold, X LDP 4H+2b, I, with I defined in A.6. The proof of i i imilar to that of Theorem 3.1i and relie on proving LDP for an auxiliary proce, defined in A.1, exponentially equivalent to the original recaled proce Y. The proof of ii i more involved and potponed to Appendix A.2. In order to tate the following key reult, define, for any >, G H a the function F H in Lemma 3.2, replacing β by β 2, and for, t T with < < t, it pointwie limit ξκ H H t u H 1 du, for H H 1 H u 2, 1, G H t, := lim G H t, = ξκ t H tt H H u H u H 1 du, for H, 1, H 2 ξ, for H = 1 2. The following propoition i imilar to Propoition 3.3, a G H t, L 2 T and for all < < t, G H t, >, and it proof i omitted. Propoition 3.8. For any H, 1, the pace H G H i the RKHS of the Gauian proce GH, dz Small-time aymptotic for bounded upport in the diffuion cae. In the tandard cae H = 1 2, the recaling in the previou ubection i not really natural, in the ene that mall-time weak convergence uually hold for Itô diffuion without pace recaling. In thi cae, uing an approach introduced by Bezuidenhout [11] and further developed by Mellouk [5], we can obtain impler large deviation etimate if the upport of the initial datum Θ i bounded. A implified verion of Mellouk conider, for any >, the ytem, on T, 3.8 dx t = bx t, Zdt + ax t, ZdW t, X = x R n, where b : R n R m R n and a : R n R m M n,d are bounded Borel meaurable, uniformly Lipchitz continuou, Z i a random variable with compact upport on R m and W a d-dimenional tandard Brownian motion, independent of Z. The main reult of the paper i a large deviation principle on C α T, R n, the pace of α-hölder continuou function, for α < 1 2, for the equence X >, under the following aumption: Aumption 3.9. H1 b, i jointly meaurable on R n R m and there exit C > uch that, for all x, x R n, z, z R m, bx, z C1 + x and bx, z bx, z C x x + z z. H2 a, i jointly meaurable on R n R m and there exit C > uch that, for all x, x R n, z, z R m, ax, z C and ax, z ax, z C x x + z z.

9 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 9 For f H, u uppz and x R n, let S x f, u denote the unique olution to the controlled ODE g t = x + bg, u d + ag, u f d, for t T. Let u now introduce the following definition: Definition 3.1. Let α [, 1 2 and B a be the ball of radiu a in the α-hölder norm. The lower emi-continuou regulariation Ĭ : Cα T, R m C α T, R m of a functional I : C α T, R m C α T, R m i defined a Ĭψ := lim a inf {Iφ : φ B a ψ. Theorem 3.11 Theorem 2.1 in [5]. Under Aumption 3.9, X > LDP 2, Ĭα, where with Λ in 2.5 I α ψ := inf {Λφ : φ H, S x φ, u = ψ, for ome u uppz. Coming back to our model, the recaling X t, Y t := X 2 t, Y 2 t, equivalent to that of Section with b = and H = 1 2, the mall-noie ytem 2.6, under Aumption A b, become, imilarly to Section 3.2.1, 3.9 dxt = 2 2 σy t 2 dt + σy t ρdb t + ρdb t, X =, dyt = 2 λ + β 2 Yt dt + ξdbt, Y Θ, with B a tandard Brownian motion. Subtracting the initial random datum X = X =, Θ, thi ytem can be expreed in the form 3.8 with X = X, Y, 3.1 b, X, X = σy t + Θ 2 and ax, X = λ + βyt + Θ ρ σy t + Θ ρ σy t + Θ, ξ and note that b,, converge to the null map a tend to zero. The aumption impoed in [5] on the drift and diffuion coefficient are clearly atified here. While Mellouk allow the drift and diffuion to depend explicitly on external random factor, we can write our etting dependence on a random tarting point into thi framework. The large deviation etimate for the equence X > = X, Y > thu obtained i tronger than that in the previou ection, a it hold on C α T, R n, for any α < 1 2. Note that the mild condition on the coefficient [5, H -H 2 ] are eaily atified in our cae, o the only additional aumption i the boundedne of the upport on Θ. We remark here that [5] i not directly applicable to the current etting but ha to be extended to include -dependence in the drift, and we do o following the Azencott [4] inpired approach developed by Peithmann [56, Subection 2.2.1]. In order to tate LDP proved in Section A.3, we impoe the following aumption: Aumption For u uppx and φ H, the ODE ψ t = aψ, u φ d ha a unique olution on T, denoted by S φ, u. Theorem Under Aumption 3.12, if Θ ha compact upport, then X > LDP 2, Ĭα, with I α ψ := inf {Λ ρ φ : φ H, uch that S φ, u = ψ, for ome u uppx, with Λ ρ defined in 3.6.

10 1 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE 4. Application to Implied volatility aymptotic A announced in the introduction, we unify here two branche of reearch, both aimed at reproducing the teepne of the implied volatility urface on the hort end via model with continuou path. While there are now numerou output [2, 8, 9, 29, 33, 37, 43] in the literature confirming that a fractional driving noie with Hurt exponent H < 1/2 in the volatility lead to the oberved teepne of the mile, recent reult [45, 47] reproduce thi effect by randomiing the initial volatility in claical diffuive model. In thi ection we demontrate how to modulate the two effect with repect to one another. In the Black-Schole-Merton model, the price of a European Call option i C BS t, e k, Σ, with aociated volatility Σ. Conidering a market with oberved Call option price C ob t, e k, with maturity t and trike e k, we denote by Σ t k the implied volatility, defined a the unique non-negative olution to C BS t, e k, Σ t k = C ob t, e k General fractional cae. From Theorem 3.1 and 3.7, we can deduce the aymptotic behaviour of the implied volatility for large trike and for mall maturitie. We tate thoe below, and potpone the proof to Appendice A.4 and A.5. Corollary 4.1. Large-trike implied volatility aymptotic For any H, 1 and any b 1/2 uch that Theorem 3.1 hold, we have the following large-trike aymptotic etimate of the implied volatility: Σ 2 t kt lim = 1 1 inf Λϕ k k 2 y 1 ϕt=y with Λ a in A.3, and for any t T. Similarly, from Theorem 3.7 we can deduce the aymptotic behaviour of the implied volatility when time become mall. The following Corollary generalie [29, Corollary 4.1]. Corollary 4.2 Small-time Implied volatility aymptotic. For any H, 1 and any b 1/2 2H uch that Theorem 3.7 hold, the following mall-time etimate i true for any k : lim t b Σ 2 t t 1/2 H b k = k2 inf t 2 Iϕ ϕ 1 =y, with I a in A.6. y k Thi implie that the implied volatility explode with rate t b. For b =, it i identical to [29, Formula 26] Refined aymptotic reult in the pecial diffuive cae from Section and Large-trike aymptotic. We conider here a pecific cae of a multidimenional diffuion, a we are only intereted in tudying it tail aymptotic. Let X ζ := X ζ,1,, X ζ,n be the unique trong olution in R n to dx ζ t = bx ζ t dt + ax ζ t dw t, X ζ = ζ, for ome d-dimenional tandard Brownian motion and ome quare integrable random variable ζ, and with b : R n R n and a : R n M n,d R. Conider the following caling aumption: Aumption 4.3. There exit b 1,, b n > with b 1 = 2 uch that X,ζ := b 1 X ζ,1,, b n X ζ,n atifie 4.2 dx,ζ t = b, X,ζ t dt + ax,ζ t dw t, X ζ = b 1 ζ 1,, b n ζ n. Furthermore, b, / converge uniformly to ome function b a tend to zero. We can then tate the main reult about tail aymptotic:

11 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 11 Propoition 4.4. Under Aumption 4.3, if there exit x R n uch that for any δ >, lim up 2 log P ζ x > δ =, and if the triple x, b, a atifie Aumption 3.4, then lim 2 log PX ζ,1 t 1 = lim 2 log PX x,1 t 1, for any t T. The caling from Aumption 4.3 may be odd at firt, but reflect the fact that component of tochatic model may each act on different cale. Conider for example the Orntein-Uhlenbeck proce, olution to dx t = 1 2 Y t 2 dt + Y t dw t, X = ζ, dy t = λ + βy t dt + ξdb t, Y = y >, where W and B are two correlated Brownian motion. The recaling X, Y := 2 X, Y correponding to b 1 = 2 and b 2 = 1 yield dxt dyt namely 4.2, and the aumption are atified. = 1 2 Y t 2 dt + Yt dw t, X = 2 ζ, = λ + βy t dt + ξdb t, Y = y >, The proof of Propoition 4.4 i potponed to Appendix A.6. The aumption of the random initial condition X ζ = ζ being F -meaurable ditribution can be relaxed. Indeed, the filtration F i the filtration generated by the d-dimenional Brownian motion W. Intead, one could work on a filtration F := F t t T generated by F t := σ{w u, u t {ζ, for all t T. Then the random initial point X ζ ha a F -meaurable ditribution and the reult above till hold, in particular Theorem 3.5 and Propoition 4.4, on the new filtered probability pace Ω, F, F t T, P. In the context of implied volatility aymptotic, thi reult ha the following meaning: Corollary 4.5. The wing of the mile are independent of the tarting point ζ or x. Proof. Gao and Lee [34] how that aymptotic behaviour of the implied volatility can be directly inferred from comparing tail probabilitie to thoe of the Black-Schole model. It i traightforward to ee that the caling of Propoition 4.4 i the ame in Black-Schole, and the corollary follow immediately Small-time aymptotic for the forward Stein-Stein model. We are intereted in a forward proce, a defined by Jacquier and Roome [45] in the context of forward-tart European option: + St+τ E e k = E e X t+τ X t e k + =: E e Xt τ e k +, S t with X τ t τ the o-called forward proce, defined path-wie by X τ t := X t+τ X t, for ome fixed t >, and for all τ. The forward proce X t τ 4.3 dx t τ dy t τ The tochatic differential equation for X t τ τ then atifie the following tochatic differential equation: = 1 t Y τ 2 dτ + Y τ t dw 1,τ, X t =, 2 = λ + βy τ t dτ + ξdw 2,τ, Y t σ t., Y t τ i the ame a that for X t, Y t t, albeit with an initial τ random ditribution δ, σ t, where σ t i Gauian with mean e βt σ + λ β λ ξ2 β and variance 2β e2βt 1. We now apply the reult of Section to obtain mall-time aymptotic for a verion of the Stein-Stein forward model with a generalied random tarting point. Propoition 4.6. With the caling Xτ, Yτ := X t 2 τ, Y t 2 τ for, t >, the randomied Stein-Stein recaled model 4.3 i the ame a 3.9 with coefficient given in 3.1, with σy y, and Theorem 3.13 applie.

12 12 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE We can tranlate thi reult into forward implied volatility aymptotic directly uing [27, Theorem 2.4], and refer to thi very paper for a precie definition of the forward implied volatility Σ t,τ : 1 Corollary 4.7. The mall-time forward mile read lim Σ 2 t,τ k = k2 inf Ĭ α ϕ ϕ1 =y, with τ 2 y k Ĭα in Theorem A.1. Proof of Theorem 3.1. Appendix A. Proof A.1.1. Proof of Lemma 3.2. From [62, Definition 2.1], for any mooth function f on T with bounded derivative uch that f := T Γ H,t ft2 dt <, define, for t T, fudw u H := Γ H,t fudz u, with Γ H,tf := κ H d H d u H u H fudu, Applying thi to f := ξe βt, one obtain the following: 1 for H < 1 2, uing integration by part and Leibniz integration rule, d d = d d u H u H fudu { ξt H + t H H + = ξt H t H + ξh Hence, Γ H,tf := κ H H = ξκ H H + d { ξ H d H + d d t H t H + β u H u 1 H eβt u du ξβ u H u H fudu 2 for H > 1 2, uing Leibniz integration rule, u H+ u eβt u du + ξ β 1 H for all, t]. H + u H u H e βt u du. u H u H e βt u du H u H + u H e βt u du u H u eβt u du. 1 H d d u H u H fudu = and hence Γ H,tf := κ H d H d ξ d d uh u H e βt u du = ξh u H u H 1 e βt u du, u H u H fudu = ξκ HH H u H u H 1 e βt u du A.1.2. Proof of Propoition 3.3. In order to prove that H F H i the RKHS of the Gauian proce ξ eβ dw H, one need to how that the operator K F H i a bijection from L 2 T to H F H and that H F H i dene in C. The operator K F H acting on L 2 T defined by K F H ft := F H t, fd atifie K F H L 2 T =: H F H and i urjective. It i alo injective on H F H : let f L 2 T a non-zero function uch that K F H f = on T. Similar to [29], there i an interval [t 1, t 2 ] T where f ha contant ign. Uing previou notation, F H i defined, for t T and, t], a F H t, := ξγ H,t eβt. The function g := ξ t uh u H e βt u du i increaing, hence F H t, i a poitive function on [, t] and 2 t 1 F H t, fd >, leading to a contradiction. Thu K F H i injective on H F H, hence bijective from L 2 T to H F H. Since K F H i a linear operator, one can then define K F H f 1, K F H f 2 HF H := f 1, f 2 L2 T a an inner product.

13 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 13 The econd part of the proof conit in howing that H F H := K F H L 2 T i dene in C. We claim that F H t, 1 L 2 T. Indeed, a hown in the proof of [62, Theorem 3.1], the following hold for all < < t: if < H < 1 2, F H t, ξκ H 1 2 H τ H t H >, leading d F H t, 2 1 ξ 2 κ 2 H t2h 2H d t if 1 2 < H < 1, F H t, ξκ H e β t t H, o that 1 ξ 2 κ 2 H t4h t2 2H 2H d = 2Hξ 2 κ 2 H < ; d F H t, 2 1 ξ 2 κ 2 H e2βt Given a >, take for [, t], f a := e 2β d t 2H 1 ξ 2 κ 2 H e2βt t 2H a F H t, e 2β d = 1 e 2βt 2βξ 2 κ 2 H t2h <. L2 T. Then, F H t, f a d = ta+1 a+1. Hence, H F H contain all polynomial null at the origin, and by the Stone-Weiertra Theorem, i dene in C. A.1.3. Proof of Theorem 3.1i. Let Y denote the olution to the econd SDE 3.1, tarting from zero. The product rule for fractional Brownian motion [53, Theorem 3.1.4] yield, for any t, A.1 Y t = λb β 1 e βt + ξ b e βt u dwu H, Since ξ eβ u dw H u i Gauian, Lemma 3.2 combined with [25, Theorem 3.4.5] yield a large deviation principle on C for the equence ξ b eβ dw H, with peed > 2b and good rate function Λ F H a in 2.5. Since the two equence Y > and ξ b eβ dw H > only differ by ome determinitic quantity, they are clearly exponentially equivalent: for any δ > and t T, lim up 2b log P Y ξ b e β dw H > δ = lim up 2b λ b log P eβ 1 β > δ =, ince β <, o that Y LDP 2b, Λ F H. Finally, the LDP for Y follow again by exponentially equivalence: Y P Y > δ = P b Θe β > δ P Θ > δ b, for t,, δ >, and the theorem follow from Aumption A Θ b. A.1.4. Proof of Theorem 3.1ii. We firt prove large deviation for the Gauian driver of the proce, which we then, by mean of iterated Contraction Principle, tranlate to large deviation for the whole caled proce. 1 When H 1 2, Lemma B.1 in [29] implie that ρb, W H i Gauian, o that b ρb, W H atifie a large deviation principle on CT, R 2, with peed 2b and good rate function 1 f 2 d, φ, ψ H H, A.2 I 1 φ, ψ := 2 T +, φ, ψ / H H, where H H i the RKHS for ρb, W H defined a { H H := φ, ψ C 2 : φ = ρ fd and ψ = K H, fd, for ome f L 2 T. By independence of B with repect to B and W H and uing the Contraction Principle, b ρb + ρb, W H atifie a large deviation principle on CT, R 2, with peed 2b and good rate function I 3 φ, ψ := inf φ=u+w {I 1u, ψ + I 2 w = inf u H {I 1u, ψ + I 2 φ u,

14 14 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE where, with H ρ a in 2.4, ρ 1 f 2 d, w H ρ, I 2 w := 2 +, w / H ρ. 2 For H = 1 2, uing the proof of Theorem 3.6, b ρb + ρb, B atifie a large deviation principle on CT, R 2, when tend to zero with peed 2b and good rate function Λ ρ defined in 3.6. We now introduce the proce X atifying the following SDE d X t = 1 2 σ2 Yt dt+ b σy t ρdb t + ρdb t, and tranlate thi large deviation into one for the equence X. Since the proof for the H 1 2 cae i the ame a the one for H = 1 2, albeit with a different rate function, we concentrate on the former cae. Since Yt = b e βt Θ λb β 1 eβt + b ξ eβt u dwu H, one can define a continuou map G, 1 C C, uch that b ρb + ρb, Y = b ρb + ρb, G, b W H t. Moreover, uing A Θ b, G, b W H i exponentially equivalent to G b W H, with Gφt := ξ eβt u dφ u. The Contraction Principle thu yield b ρb + ρb, Y LDP 2b, I 4, with I 4 : φ, ψ inf{i 3 φ, v ψ C : ψ = Gv. Under A b, one can apply the extended Contraction Principle proved in [49, Propoition 2.3], o that b ρb + ρb, σy LDP 2b, I 5 with I 5 φ, ψ := inf {I 4 φ, v ψ = σv = inf { I 3 φ, v ψ = σgv. Finally, etting b 1 2, the equence of emi-martingale b W i uniformly exponentially tight UET in the ene of [35, Definition 1.1], and the equence σy i càdlàg Aumption A b, and adapted to the filtration F. Denoting X Y := XdY the tochatic integral with repect to a emi-martingale, Theorem 1.2 in [35] yield a large deviation principle on CT, R 3 for b ρb + ρb, σy, b σy ρdb + ρdb, with peed 2b and good rate function I 6 defined a I 6 ϕ := I 5 φ, ψ if ϕ = φ ψ and ψ BV the pace of function of finite variation, and infinity otherwie. Applying another Contraction Principle, ince, for t T, X t = 1 2 Hence X LDP 2b, Λ with σy 2 d + b σy dw =: I σy, σy b ρb + ρb, t. A.3 Λϕ := inf{i6 χ ϕ = Iφ, φ ψ, χ = φ, ψ, ψ BV. The lat tep i proving that the procee X and X are exponentially equivalent. Indeed, for t T, X X 1 Y 2 2b σ 2 b σy d + b Y b σ b σy dρb + ρb. Uing the linear growth aumption a well a Aumption A b, we have for [, t], Y 2b σ 2 b σy 22b C Y 2 b + 2C Y 2 2C b + 2 Y 2, hence Y 2b σ 2 σy b d 2C b + 2 Y 2 d 2C b + 4C 2 Y 2 d. Since Y = b e β Θ + b λ β eβ 1 + b ξ [ W H + β W H u e βu du ], we obtain for T, recalling that β <, Y b e β Θ + b λ e β 1 + ξ b W H, β

15 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 15 and A.4 Y 2 2 2b e 2β Θ b λ β 2 e 2β ξ 2 2b W H 2, o that t Y 2 d 2 2b tθ 2 + 2b λ β 2 2t 1 β + ξ 2 2b t W H 2 d. Introducing J := Y 2 d, we obtain for any δ > and a >, X P X > δ P 2C 2 J + b P 2C 2 J + b + P 2C 2 J + b P b σ Y b b σ b σ b σ Y b Y b Y b σy σy dρb + ρb > δ σy dρb + ρb > δ 2C 2 J < a 1 P σy dρb + ρb > δ 2C 2 J a P dρb + ρb > δ a 2C2 J < a + P, with P := P 2C 2 J a and δ := δ C b. Uing A.4 and introducing J t := W H 2 d, we obtain P J > δ 2 λ P 2b Θ 2 + 2b ξ 2 2b J 1 > δ β β 2 2 λ P 2b Θ 2 + 2b 2 1 β β + ξ2 2b J 1 > δ 2 ξ2 2b J 1 < a P ξ 2 2b J 1 < a 2 λ + P 2b Θ 2 + 2b 2 1 β β + ξ2 2b J 1 > δ 2 ξ2 2b J 1 a P ξ 2 2b J 1 a P Θ 2 > 1 2 δ λ 2b 2 2b 2 1 a β β ξ2 2b J 1 < a + P ξ 2 2b J 1 a, P Θ > 1 2 δ λ b 2 2b 2 1 β β a ξ2 2b J 1 < a + P ξ 2 2b J 1 a, b for any δ > and a >. One can then ue Markov inequality: P ξ 2 2b J 1 a E [ ξ 2 2b J 1 ] a = ξ2 2b 1 a E W H 2 d. Since VW H = 2H, one obtain P ξ 2 2b J 1 a ξ2 2b that for all, Pξ 2 2b J 1 < a 1 η. Aumption A Θ b a 1 2H+1. Hence there exit η [, 1 and > uch then implie that there exit M, > uch that P Θ > 1 2 δ λ b 2 2b 2 1 a β β exp M 2b, for all.

16 16 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE Thu, for all min{,, Baye Theorem yield P Θ > 1 2 δ λ b 2 2b 2 1 β β a ξ2 2b J 1 < a 2 P Θ > 1 δ b 2 2b λ 2 β 1 β a Pξ 2 2b exp M 2b. J 1 < a 1 η Finally, for all min{,, A.5 P J > δ P Θ > 1 b δ 2 2b e M/2b 1 η + ξ2 2b 1, a 2H + 2 λ 2 1 β β a ξ2 2b J 1 < a + P ξ 2 2b J 1 a, o that for all min{,, P 2C 2 J a e M/2b 1 1 η 2H +. Hence, there exit η [, 1 and < min{, uch that for all, P2C 2 J < a 1 η. Moreover, + ξ2 2b a uing the uniform convergence aumed in A b, for all y R there exit N > and > uch that, for all <, we have that b Y σ σy N. Hence one can then apply the Borell-TIS inequality in b particular [56][Propoition A.1], and obtain, Y P b σ b σy dρb + ρb > 1 δ C 2 b 1 + 2b { δ C b 2 a a 2 exp 2N 2 2b. Thu, for all min{,, Baye Theorem yield Y P b σ b σy dρb + ρb > 1 δ C 2 b 1 + 2b a 2C 2 J < a P b Y σ σy b dρb + ρb > 1 δ C b a b P2C 2, J < a 2 exp { δ C2 1+ 2b a 2 2N 2 2b. 1 η Finally, for all min{,, X P X 2 exp { δ C2 1+ 2b a 2 > δ 1 η 2N 2 2b o that X and X are exponentially equivalent and X LDP 2b, Λ. + e M/2b 1 η + ξ2 2b 1, a 2H + Remark A.1. For computational purpoe, uch infinite-dimenional optimiation problem are dicretied over a et of bai function, uch a orthonormal polynomial. If we then conider φ continuouly differentiable and I 1 finite, we can implify I 1 φ = I 1 φ, ψ = 1 2ρ T [ 2 φt]2 dt. Further, if u C 1, then { I 3 φ 1 2 inf t 2 u u H C 1 ρ d + 1 t ρ φ u 2 d, and I 4 φ = I 3 φ. A.2. Proof of Theorem 3.7.

17 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 17 A.2.1. Proof of Theorem 3.7i. Let u introduce the proce Y defined, for t T, a the unique olution of dy t = b+2 λ + β 2 Y t dτ + 2H+b ξdw H t Y t = λb β, with Y =, i.e. 1 e β2 t + ξ 2H+b e β2 t u dwu H. From Lemma 3.2, Propoition 3.8 and Theorem in [25], the equence 2H+b GH, dz > atifie a large deviation principle on C, with peed 4H+2b and good rate function Λ G H a in 2.5. Moreover, 1 When H 1 2, 2H+b t GH t, dz > and 2H+b t GH t, dz > are exponentially equivalent: indeed the aymptotic expanion of the Gauian denity [3, Formula ] yield, for any δ >, P 2H+b G H, dz 2H+b G H, dz > δ = P G H, G H, dz > δ 2H+b = P 2 V = 2H+b δ 2 1 exp + O 4H+2b π δ 2 4H+2b V 2, N, 1 > δ V 2H+b, with V 2 := G H t, G H t, 2 d and note that lim V 2 =. Now, for > and < < t, i if H < 1 2, [ G H t, G H t, ] 2 2κ 2 H ξ2 2H { H 2 [ H 2κ 2 Hξ 2 2 2H 2 u H t 2 2 u eβ t u 1du + β 2 4 u H u H e β2 t u du 1 H 2 u H u du + β2 4 2 ] uu H du. 1 H 2H Uing [62, Lemma A.3], we further obtain, for > and < < t, V 2 2κ 2 H ξ2 [ C H t 2H + β 2 4 CH t 2H+2 ], with C H, C H > two contant depending on H. Hence, ii if H > 1 2, < 4H+2b V 2 2κ 2 Hξ 2 C H 4H+2b t 2H + β 2 4H+2b+4 CH t 2H+2 ; [ G H t, G H t, ] 2 κ 2 = H ξ2 u H 2 2 2H H2 u 1 H eβ t u 1du κ2 H ξ2 2H H2 u H du u 1 H 2 κ2 H ξ2 H2 t 2H 2H 2 u H 1 du = κ2 H ξ2 [tt. 2H ]2H Hence, for H 1 2, a b >, lim 4H+2b V 2 = a well a lim 2H+b V =. 2 When H = 1 2, ξ1+b t t db eβ2 > and ξ 1+b B t > are exponentially equivalent: the aymptotic expanion of the Gauian denity near infinity [3, Formula ] yield, for any δ >, P ξ e β2 db ξ db > δ 1+b = P ξe β2 1dB > δ 1+b δ = P N, 1 > V 1+b ξ 2 V = 1+b ξ δ 2 1 exp + O 2+2b π δ 2 2+2b V 2, ξ2

18 18 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE with V 2 := [ t 2d e β2 t 1] = t + e 2β2t 1 2β 2 2 t + 21 eβ β. Hence < 2+2b V b t + 2 2b t eβ 1 2β, and, a b >, lim 2+2b V 2 =. Beide, note that lim V 2 =. The required exponential equivalence then follow, and hence for all H, 1, 2H+b GH, dz > LDP 4H+2b, Λ G H. The final tep i exponential equivalence between Y and 2H+b P Y 2H+b G H, dz > δ = P Y 2H+b = P λb 1 eβ β GH, dz >. For any δ, t >, e β2 dw H > δ 2 > δ. Hence lim up 4H+2b log P Y 2H+b G, dz > δ =, yielding exponential equivalence and Y LDP 4H+2b, Λ G H. Finally, ince for any δ >, Y P Y > δ P b Θ > δ, a β <, and Y and Y are exponentially equivalent, and Y LDP 4H+2b, Λ G H by Aumption A Θ b. A.2.2. Proof of Theorem 3.7 ii. The idea of the proof i the ame a the proof of Theorem 3.1ii. 1 H 1 2 : we already etablihed that 2H+b ρb + ρb, W H atifie a large deviation principle on CT, R 2, with peed 4H+2b and good rate function I 3 defined earlier. 2 H = 1 2 : uing the proof of Theorem 3.6, b+1 ρb + ρb, B atifie a large deviation principle on CT, R 2, with peed 2+2b and good rate function Λ ρ defined in 3.6. Similar to above, we only prove the cae H 1 2, the other one being analogou. We introduce the proce X atifying the following SDE d X t = 2H+1 2 σ 2 Yt dt + 2H+b σy t ρdb t + ρdb t, and tranlate thi large deviation into one for the equence X. Uing that for > and t T, Yt = b e β2t Θ λb β 1 eβ2t + 2H+b ξ t u dw H eβ2 u, one can define a continuou map G, 1 C C, uch that 2H+b ρb+ ρb, Y = 2H+b ρb+ ρb, G, 2H+b W H t. Uing A Θ b and the aymptotic expanion of the Gauian denity near infinity [3, Formula ], G, 2H+b W H i exponentially equivalent to Ĝ 2H+b W H, defined by Ĝφt := ξ dφ u. Hence, the Contraction Principle yield an LDP on CT, R 2 for 2H+b ρb + ρb, Y, with peed 4H+2b and good rate function Ĩ4 : φ, ψ inf{i 3 φ, v ψ C : ψ = Ĝv. Under A b, the extended Contraction Principle [49, Propoition 2.3] yield an LDP on CT, R 2 for 2H+b { ρb + ρb, σy, with peed 4H+2b and good rate function Ĩ5 : φ, ψ inf {Ĩ4 φ, v ψ = σv = inf I 3 φ, v ψ = σĝv. Finally, etting b 1 2 2H, the equence of emi-martingale 2H+b W i UET and the equence σy i càdlàg Aumption A b, and adapted to the filtration F. Theorem 1.2 in [35] thu give an LDP on C for the equence of tochatic integral 2H+b σy ρdb + ρdb, with peed 4H+2b and good rate function A.6 Iχ := inf {Ĩ5 φ, ψ : φ ψ = χ, ψ BV. We now prove that X and 2H+b σy ρdb + ρdb are exponentially equivalent. P X 2H+b σy dw > δ = P 1 2 P Y 2 d H+ 1 2 σy 2 d > δ > δ C 2 2H+1 1, = P σy 2 d > 2δ 2H+1,

19 ASYMPTOTIC BEHAVIOUR OF RANDOMISED FRACTIONAL VOLATILITY MODELS 19 uing the growth condition aumption σx C1 + x for x R. Since Y = b e β2 Θ + b λ β eβ2 1 + [ 2H+b ξ W H + β ] 2 W u H e β2u du and β <, we obtain for T, Y b e β2 Θ + b λ e β2 1 + ξ 2H+b W H, β o that 2 λ Y 2 2 2b e 2β2 Θ b e 2β ξ 2 4H+2b W H 2, β and hence t 2 Y 2 d 2b tθ 2 + 2b λ β 2t 1 β +ξ 2 4H+2b t 2 W H 2 d. Uing a imilar argument to A.5, ued in the proof of Theorem 3.1, we have, for all min{,, P Y 2 d > δ C 2 2H 1 e M/4H+2b + ξ2 4H+2b 1, + 1 η a 2H + o that X and 2H+b σy ρdb + ρdb are exponentially equivalent. Finally, one can ue the ame argument that in Appendix A.1.3, replace β by β 2 and the peed 2b by 4H+2b and prove that X and X are exponentially equivalent. Hence X LDP 4H+2b, I. A.3. Proof of Theorem The proof i imilar to that of [56, Theorem 2.9], with ome minor modification. One key ingredient in the proof of the theorem i an exponential equivalence between X,x > and it limit ytem a tend to zero. In thi proof, we denote X by X,x to tre the dependence on the tarting point. Lemma A.2. Under Aumption 3.12, for each R, δ, β >, there exit γ, ρ, > uch that 2 log P X,x S φ, u > δ, W φ γ R hold when for all u uppx and all φ H atifying Λ ρ φ β, x B ρ,. { Proof. Firt, introduce the Radon-Nikodym derivative D 1 φ := exp T φ dw 1 2 T φ 2 d. Following [56], conider the family Y > 2 of unique trong olution of dy t = c, t, Y t, X dt + ay t, X db t, for t >, with initial condition Y =, R 2. We denote c, t, x, y := b, x, y + ax, y f t, for x, y R 2, t T and >, and B t := W t 1 f t, for t T. Peithmann aumption are here updated a: i c :, T R 2 R 2 R 2 converge to c : T R 2 R 2 R 2 in the ene that lim up c, t, x, y c t, x, y dt =, for all y R 2. x R 2 T In particular, note that c t, x, y = ax, y f t ; ii there exit ϱ L 2 T uch that for all x, y R 2, c, t, x, y + c t, x, y ϱt, for t T ; iii there exit κ L 1 T uch that for all y, x 1, x 2 R 2, c t, x 1, y c t, x 2, y κt x 1 x 2 on T. Partitioning T into {t k := kt n k=,...,n, et Y,n t := Y t k for t k t < t k+1, and Peithmann argument yield i for any δ >, lim n lim up 2 log P Y Y,n > δ = uniformly with repect to y R 2 ; ii given M t := T ay, X dw and M,n t := T ay,n, X dw, for all δ >, lim lim up 2 log P M M,n > δ, Y Y,n β =, β uniformly with repect to n N, y R 2 ; iii for all δ >, lim γ lim up 2 log P M > δ, W γ = ;

20 2 BLANKA HORVATH, ANTOINE JACQUIER, AND CHLOÉ LACOMBE iv let ζ denote the olution of the ordinary differential equation ζ t = c t, ζ t, Y t tarting from ζ = y. For all R >, δ >, there exit γ, ρ, > uch that P{ Y, ζ > δ { W γ exp R/ 2, for all y R 2, y B ρ y and. The theorem follow from Giranov theorem with Radon-Nikodym derivative D f. We tart with the lower bound. For any open ubet G of CT, R 2, let η > and chooe ψ G uch that I α ψ inf ψ G I α ψ + η. Then, let u uppx and φ H uch that S φ, u = ψ and Λ ρ φ = I α ψ. Let δ > uch that B δ ψ G. Then, for each γ >, x R 2, PX,x G P X,x ψ δ P W φ γ P X,x ψ > δ, W φ γ. Schilder Theorem [22, Theorem 5.2.3], then yield the lower bound lim inf 2 P W φ γ Λ ρ φ = I α ψ inf ψ G Iα ψ η. Then, we bound the econd probability from above uing Lemma A.2: fix β Λ ρ φ and R > inf ψ G I α ψ+η, and find γ, ρ, > uch that, for x B ρ,,, the bound 2 log P X,x ψ > δ, W φ γ R hold. Thee two bound then imply the required lower bound: lim inf ρ 2 log { inf P x B ρ, X,x G min R, inf ψ G Iα ψ η = inf ψ G Iα ψ η. We now prove the upper bound. For any cloed et F of CT, R 2, take β, inf ψ G I α ψ and R > β. Let u uppx and ψ H with I α ψ β. We find δ > uch that B δ ψ F = and φ {Λ ρ β uch that S φ, u = ψ. Uing Lemma A.2, there exit γ, ρ, > uch that for x B ρ,,, 2 log P X,x ψ > δ, W φ γ R. The et {B γ φ : ψ H, I α ψ β form a cover of the compact et {Λ ρ φ β, o that we can extract a finite ub-cover {B γi φ i i=1,...,k and et A := k i=1 B γ i f i and ψ i := S φ i, r. For any i = 1,..., k, there exit δ i, ρ i, i > uch that, for any x B ρi, and i, 2 log P X,x ψ i > δ, W φ i γ R. Set := min { 1,..., k, ρ := min {ρ 1,..., ρ k, take and x B ρ,. Since F B δi ψ i = for every i = 1,..., k, we obtain PX,x F PX,x F, W A + PW A c k P X,x g i > δ i, W φ i γ i + PW A c k exp R 2 + exp β 2, i=1 ince I α ψ i β by definition of β. Finally, ince R > β, the theorem follow from the upper bound lim up ρ 2 log up P X,x F inf x B ρx ψ F Iα ψ. A.4. Proof of Corollary 4.1. Thi i a traightforward application of [34, Corollary 7.1]. Taking 2b to be k we have, from Theorem 3.1, that 2b X t LDP 2b, Λ a goe to zero. Then, 1 lim k k log PX t k = inf Λϕ y 1 ϕt=y. Σ 2 t k Similarly, in Black-Schole, lim k k 2 log PX t k = 1, and the proof follow from [34, Corollary 7.1]. 2t