Gaussian stochastic volatility models: Scaling regimes, large deviations, and moment explosions
aa r X i v : . [ q -f i n . M F ] J un GAUSSIAN STOCHASTIC VOLATILITY MODELS: SCALING REGIMES, LARGEDEVIATIONS, AND MOMENT EXPLOSIONS
ARCHIL GULISASHVILIA
BSTRACT . In this paper, we establish sample path large and moderate deviation prin-ciples for log-price processes in Gaussian stochastic volatility models, and study the as-ymptotic behavior of exit probabilities, call pricing functions, and the implied volatility. Inaddition, we prove that if the volatility function in an uncorrelated Gaussian model growsfaster than linearly, then, for the asset price process, all the moments of order greater thanone are infinite. Similar moment explosion results are obtained for correlated models.
AMS 2010 Classification : 60F10, 60G15, 60G18, 60G22, 41A60, 91G20.
Keywords : Gaussian stochastic volatility models, Volterra type models, sample pathlarge and moderate deviations, central limit regime, moment explosions, implied volatil-ity asymptotics. 1. I
NTRODUCTION
This paper deals with Gaussian stochastic volatility models. In such a model, thevolatility process is a positive function σ of a Gaussian process b B . The main results ob-tained in the paper are the following (see the end of the introduction for a more detailedoverview): • A sample path large deviation principle for the log-price process in a Volterra typeGaussian stochastic volatility model (see Theorem 2.9) with application to the exittime probability function asymptotics (see Theorem 2.16). • A sample path moderate deviation principle for the log-price process in a Gaussianstochastic volatility model (see Theorem 3.1). • The results in Section 6 concerning moment explosions for asset price processes inGaussian stochastic volatility models, especially Theorem 6.11.In the present paper, we also suggest a unified approach to various scaling regimes asso-ciated with Gaussian stochastic volatility models. More precisely, large deviation, mod-erate deviation, and central limit scalings are considered. Sample path large and moder-ate deviation principles are established in this paper under very mild restrictions on thevolatility function and the volatility process. We also find leading terms in asymptoticexpansions of call pricing functions and the implied volatility in mixed scaling regimes.To find more terms in such expansions, additional smoothness restrictions have to be im-posed on the volatility function σ (see, e.g., [4]). Higher order expansions of call pricingfunctions and the implied volatility are not discussed in the present paper. We refer theinterested reader to an important paper [19], where such expansions are studied. Department of Mathematics, Ohio University, Athens OH 45701; e-mail: [email protected] he asset price process S in a Gaussian stochastic volatility model satisfies the followingstochastic differential equation: dS t = S t σ ( b B t ) dZ t , S = s >
0, 0 ≤ t ≤ T , (1.1)where s is the initial price, and T > Z in (1.1) is stan-dard Brownian motion. The equation in (1.1) is considered on a filtered probability space ( Ω , F , {F t } ≤ t ≤ T , P ) , where {F t } ≤ t ≤ T is the augmentation of the filtration generated bythe process Z (see [34], Definition 7.2). The filtration {F t } is right-continuous ([34], Corol-lary 7.8). It is assumed in (1.1) that σ is a nonnegative continuous function on R , and b B is a nondegenerate continuous Gaussian process adapted to the filtration {F t } ≤ t ≤ T . InSection 2 devoted to large deviation principles, the process b B is a continuous Gaussianprocess possessing a Volterra type representation with respect to the process B . It followsfrom (1.1) that the evolution of volatility in a Gaussian stochastic volatility model is de-scribed by the stochastic process σ ( b B ) . The function σ and the process b B will be called thevolatility function and the volatility process, respectively.We will often need to take into account the correlation structure between the asset priceand the volatility. It will be assumed in such a case that standard Brownian motion Z , ap-pearing in (1.1), has the following form: Z t = ¯ ρ W t + ρ B t , where W and B are independentstandard Brownian motions, ρ ∈ [ −
1, 1 ] is the correlation coefficient, and ¯ ρ = p − ρ .Then, the model for the asset price takes the following form: dS t = S t σ ( b B t )( ¯ ρ dW t + ρ dB t ) , S = s >
0, 0 ≤ t ≤ T . (1.2)In the special case, where the correlation coefficient ρ equals zero, the model in (1.2) iscalled uncorrelated. The asset price process in such a model satisfies the stochastic differ-ential equation dS t = S t σ ( b B t ) dW t , S = s , 0 ≤ t ≤ T .Let us denote by { e F t } ≤ t ≤ T the augmentation of the filtration generated by the process B . If the volatility process b B is a Volterra type continuous Gaussian process (see Defini-tions 2.1 and 2.2 in Section 2), then it is adapted to the filtration { e F t } ≤ t ≤ T , and the modelin (1.2) looks like a classical correlated stochastic volatility model. We call such a model aVolterra type Gaussian stochastic volatility model. Note that Definition 2.2 of a Volterratype process with H ¨older kernel includes an r -H ¨older-type condition in L for the kernelof the volatility process.The unique solution to the equation in (1.1) is the Dol´eans-Dade exponential S t = s exp (cid:26) − Z t σ ( b B s ) ds + Z t σ ( b B s ) dZ s (cid:27) , 0 ≤ t ≤ T .Therefore, the log-price process X t = log S t satisfies X t = x − Z t σ ( b B s ) ds + Z t σ ( b B s ) dZ s , (1.3)where x = log s .Suppose H > β ∈ [ H ] , and let ε ∈ (
0, 1 ] be a small-noise parameter. We will workwith the following scaled version of the model in (1.1): dS ε , β , Ht = ε H − β S ε , β , Ht σ (cid:16) ε H b B t (cid:17) dZ t , here 0 ≤ t ≤ T . For the sake of simplicity, we often assume that the initial condition s for the asset price satisfies s =
1. The asset price process in the scaled model is given by S ε , β , Ht = exp (cid:26) − ε H − β Z t σ ( ε H b B s ) ds + ε H − β Z t σ ( ε H b B s ) dZ s (cid:27) , 0 ≤ t ≤ T , (1.4)while the log-price process is as follows: X ε , β , Ht = − ε H − β Z t σ ( ε H b B s ) ds + ε H − β Z t σ ( ε H b B s ) dZ s , 0 ≤ t ≤ T . (1.5)It is easy to understand how the results obtained in the present paper transform if s = X ε , β , H by the process X ε , β , H − x .We will next provide a brief overview of the results obtained in the paper. Sections 2and 3 are devoted to sample path large and moderate deviation principles for log-priceprocesses. The theory of sample path LDPs for solutions of stochastic differential equa-tions goes back to a celebrated work of Freidlin and Wentzell (see [18]; for more informa-tion consult [11, 12, 50]). We also refer the reader to [3, 9, 42, 46] for applications of samplepath large deviation principles in financial mathematics.In the case where β =
0, the model is in the large deviation scaling regime. In Sec-tion 2, we prove a sample path large deviation principle (LDP) for the log-price process ε X ε ,0, H (see Theorem 2.9). A similar sample path LDP was obtained in a recent pre-print [7] of Cellupica and Pacchiarotti under more restrictive assumptions on the volatilityfunction σ . It is common to apply small-noise sample path large deviation principles inthe study of the asymptotic behavior of the exit probabilities. Such results go back to thework of Freidlin and Wentzell (see [51, 52, 18]). Different proofs of these results, usingstochastic control theory, were given by Fleming in [15]. In Section 2, we characterizethe leading term in the asymptotic expansion of the exit time probability function, usingthe large deviation principle obtained in Theorem 2.9 (see Theorem 2.16). A similar re-sult was obtained in [7] under more restrictions on the volatility function σ . Note thata large deviation principle for the process ε X ε ,0, HT with state space R was earlier es-tablished in Forde and Zhang [17] in the case, where the function σ satisfies the globalH ¨older condition, while the process b B is fractional Brownian motion. In [25], we provedthe Forde-Zhang LDP under very mild restrictions on σ and b B . We formulate the latterresult in Section 2.If 0 < β < H , then the model is in the moderate deviation scaling regime (see, e.g.,[4, 13, 20], and the references therein for more information on moderate deviations). InSection 3, we prove a sample path moderate deviation principle (MDP) for the process ε X ε , β , H (see Theorem 3.1), and derive a corresponding MDP for the process ε X ε , β , HT (see Corollary 3.5). As it often happens in the theory of moderate deviations, the ratefunction in Corollary 3.5 is quadratic. At the end of Section 3, we explain how to passfrom small-noise large and moderate deviation principles to small-time ones under thecondition that the volatility process is self-similar.The case where β = H corresponds to the central limit (CL) scaling regime. In Section4, we characterize the limiting behavior on the path space of the distribution function ofthe process ε X ε , H , H (see Theorem 4.1), and also that of the process ε X ε , H , HT in thespace R (see Theorem 4.3). The results in the CL regime can be considered as degenerateMDPs with the rate function equal to a constant (see Remark 4.4 in Section 4). An example f a CL scaling can be found in [22]. The volatility of an asset in [22] is modeled by theprocess δ σ ( δ e U ) , where σ is a smooth function, while e U is the stationary fractionalOrnstein-Uhlenbeck process (our notation is different from that used in Section 3 of [22]).The CL scaling in [22] corresponds to the following values of the parameters: H = β = H > , while thevolatility function satisfies an additional condition (condition (G) formulated before The-orem 6.11), was obtained independently, but a little later, by Gassiat (see Theorem 2 inthe arXiv:1811.10935v1 (November 27, 2018) version of [24]). We also show that in a cor-related Volterra type Gaussian model ( ρ = γ > − ρ explode (see part (ii) of Theorem 6.11). Such a result was first established by Jourdainfor the Scott model (see [31]). In the Scott model, the volatility process is the Ornstein-Uhlenbeck process, while the volatility function is σ ( x ) = e x . Gassiat obtained a similarresult for a model with ρ <
0, the volatility function satisfying condition (G), and theRiemann-Liouville fractional Brownian motion with H > as the volatility process (seeTheorem 2 in [24]). We do not assume in part (ii) of Theorem 6.11 that the model is nega-tively correlated and the volatility function satisfies condition (G). Moreover, the volatilityprocess in Theorem 6.11 may be any Volterra type continuous Gaussian process. In thepresent paper, we also obtain partial results concerning the explosion of the moment of rder γ = − ρ in a correlated Gaussian stochastic volatility model (see Theorem 6.13).More information can be found in Remarks 5.3 and 5.7 below. We would also like to bringthe attention of the reader to the paper [41], where the author explains how the exponen-tial integrability of the maximal function of a continuous local martingale depends on thegrowth of the moments of its quadratic variation.The last section (Section 7) of the present paper is devoted to the study of small-noiseasymptotic behavior of the implied volatility in Gaussian stochastic volatility models un-der various scaling regimes. 2. L ARGE DEVIATIONS : β = K be a square integrable kernel on [ T ] such that sup t ∈ [ T ] R T | K ( t , s ) | ds < ∞ . Let K : L [ T ] L [ T ] be the linear operatordefined by K h ( t ) = R T K ( t , s ) h ( s ) ds , and let b B be a centered Gaussian process having thefollowing representation in law: b B t = Z T K ( t , s ) dB s , 0 ≤ t ≤ T , (2.1)where B is the Brownian motion appearing in (1.2). Such representations of Gaussianprocesses are called Fredholm representations. Actually, for every centered continuousGaussian process there exists a Fredholm representation with Brownian motion depend-ing on the process (see [48], Theorem 3.1). In this paper, we assume that the process b B hasa representation in (2.1) with the process B appearing in (1.2).The modulus of continuity of the kernel K in the space L [ T ] is defined as follows: M ( h ) = sup { t , t ∈ [ ] : | t − t |≤ h } Z T | K ( t , s ) − K ( t , s ) | ds , 0 ≤ h ≤ T .We will next define Volterra type processes and Volterra type processes with H ¨olderkernels. Definition 2.1.
The process in (2.1) is called a Volterra type Gaussian process if the followingcondition holds for the kernel K:(a) K ( t , s ) = for all ≤ t < s ≤ T. Definition 2.2.
The process in (2.1) will be called a Volterra type Gaussian process with a H¨olderkernel, if condition (a) is satisfied, and the following additional condition holds:(b) There exist constants c > and r > such that M ( h ) ≤ ch r for all h ∈ [ T ] . emark 2.3. Condition (a) is a typical Volterra type condition for the kernel. The smoothnesscondition (b) was included in the definitions of a Volterra type Gaussian process in [28, 29] . Itwas also used in [25] . We will next introduce classical fractional processes. For 0 < H <
1, fractional Brown-ian motion B Ht , t ≥
0, is a centered Gaussian process with the covariance function givenby C H ( t , s ) = (cid:16) t H + s H − | t − s | H (cid:17) , t , s ≥ B H was first implicitly considered by Kolmogorov in [35], and was studiedby Mandelbrot and van Ness in [39]. The constant H is called the Hurst parameter. TheRiemann-Liouville fractional Brownian motion is defined as follows: R Ht = Γ ( H + ) Z t ( t − s ) H − dB s , t ≥ < H <
1. This stochastic process was introduced by L´evy in [36]. More infor-mation about the process R H can be found in [38, 43]. The fractional Ornstein-Uhlenbeckprocess is defined for 0 < H < a >
0, by the following formula: U Ht = Z t e − a ( t − s ) dB Hs , t ≥ r = H (see Lemma 2 in [25]). For fractional Brownian motion, theprevious statement was established in [53]. We refer the reader to [10, 14, 28, 29, 30, 40]for more information on Volterra type processes. Remark 2.4.
We will assume throughout the paper that the Gaussian process b B is non-degenerated.This means that the variance function v of b B satisfies the condition v ( s ) > for all s ∈ ( T ] . Remark 2.5.
Volatility processes satisfying the conditions in Definition 2.2 are used in the paperonly in the results associated with the large deviation regime. In all the other regimes, Definition2.1 is used.
Definition 2.6.
Let ω be an increasing modulus of continuity on [ ∞ ) , that is, ω : R + R + is an increasing function such that ω ( ) = and lim s → ω ( s ) = . A function σ defined on R iscalled locally ω -continuous, if for every δ > there exists a number L ( δ ) > such that for allx , y ∈ [ − δ , δ ] , the following inequality holds: | σ ( x ) − σ ( y ) | ≤ L ( δ ) ω ( | x − y | ) .A special example of a modulus of continuity is ω ( s ) = s γ with γ ∈ (
0, 1 ) . In this case,the condition in Definition 2.6 is a local γ -H ¨older condition. If γ =
1, then the conditionin Definition 2.6 is a local Lipschitz condition.Denote by C [ T ] the space of continuous functions on the interval [ T ] . For a func-tion f ∈ C [ T ] , its norm is defined by || f || C [ T ] = sup t ∈ [ T ] | f ( t ) | . In the sequel, thesymbol H [ T ] will stand for the Cameron-Martin space, consisting of absolutely con-tinuous functions f on [ T ] such that f ( ) = f ∈ L [ T ] , where ˙ f is the derivative f f . For a function f ∈ H [ T ] , its norm in H [ T ] is defined by || f || H [ T ] = (cid:26) Z T ˙ f ( t ) dt (cid:27) .The following notation will be used below: b f ( s ) = Z s K ( s , u ) ˙ f ( u ) du .We will next formulate the large deviation principle for Volterra type Gaussian stochas-tic volatility models established in [25]. We adapt the formulation to the notation used inthe present paper. Theorem 2.7.
Suppose σ is a positive function on R that is locally ω -continuous for some mod-ulus of continuity ω . Let H > , and let b B be a Volterra type Gaussian process with a H¨olderkernel. Set I T ( x ) = inf f ∈ H [ T ] (cid:16) x − ρ R T σ ( b f ( s )) ˙ f ( s ) ds (cid:17) ( − ρ ) R T σ ( b f ( s )) ds + Z T ˙ f ( s ) ds . (2.2) Then the function I T is a good rate function. Moreover, a small-noise large deviation principle withspeed ε − H and rate function I T given by (2.2) holds for the process ε X ε ,0, HT , where X ε ,0, HT isdefined by (1.5). More precisely, for every Borel measurable subset A of R , the following estimateshold: − inf x ∈ A ◦ I T ( x ) ≤ lim inf ε ↓ ε H log P (cid:16) X ε ,0, HT ∈ A (cid:17) ≤ lim sup ε ↓ ε H log P (cid:16) X ε ,0, HT ∈ A (cid:17) ≤ − inf x ∈ ¯ A I T ( x ) . The symbols A ◦ and ¯ A in the previous estimates stand for the interior and the closure of the set A,respectively.
Remark 2.8.
Recall that a rate function on a topological space X is a lower semi-continuousmapping I : X 7→ [ ∞ ] , that is, for all y ∈ [ ∞ ) , the level set L y = { x ∈ X : I ( x ) ≤ y } is aclosed subset of X . A rate function I is called a good rate function if for every y ∈ [ ∞ ) , the setL y is a compact subset of X . We refer the reader to [4, 17, 25] for more information on large deviation principles forVolterra type Gaussian stochastic volatility models.Let us define a measurable functional Φ from the space M = C [ T ] into the space C [ T ] as follows: For l ∈ H [ T ] and ( f , g ) ∈ C [ T ] such that f ∈ H [ T ] and g = b f , Φ ( l , f , g )( t ) = ¯ ρ Z t σ ( b f ( s )) ˙ l ( s ) ds + ρ Z t σ ( b f ( s )) ˙ f ( s ) ds , 0 ≤ t ≤ T .In addition, for all the remaining triples ( l , f , g ) , we set Φ ( l , f , g )( t ) = t ∈ [ T ] .The next statement is a sample path large deviation principle for the process ε X ε ,0, H with state space C [ T ] . heorem 2.9. Suppose the conditions in Theorem 2.7 hold. Then the process ε X ε ,0, H definedby (1.5) satisfies the small-noise large deviation principle with speed ε − H and good rate functionQ T given by Q T ( g ) = ∞ , for all g ∈ C [ T ] \ H [ T ] , andQ T ( g ) = inf f ∈ H [ T ] Z T " ˙ g ( s ) − ρσ ( b f ( s )) ˙ f ( s ) ¯ ρσ ( b f ( s )) ds + Z T ˙ f ( s ) ds , (2.3) for all g ∈ H [ T ] . The validity of the large deviation principle means that for every Borelmeasurable subset A of C [ T ] , the following estimates hold: − inf g ∈A ◦ Q T ( g ) ≤ lim inf ε ↓ ε H log P (cid:16) X ε ,0, H ∈ A (cid:17) ≤ lim sup ε ↓ ε H log P (cid:16) X ε ,0, H ∈ A (cid:17) ≤ − inf g ∈ ¯ A Q T ( g ) . Remark 2.10.
A similar LDP was recently obtained in [7] under more restrictive assumptions.For instance, it is supposed in [7] that the volatility function σ is α -H¨older continuous andbounded from above on R , while we assume in Theorem 2.9 that σ is locally ω -continuous forsome modulus of continuity ω . Remark 2.11.
To make a sanity check, let us consider a special case of Theorem 2.9 where ρ = and σ ( u ) = for all u ∈ [ T ] . ThenQ T ( g ) = inf f ∈ H [ T ] (cid:20) Z T ˙ g ( s ) ds + Z T ˙ f ( s ) ds (cid:21) = Z T ˙ g ( s ) ds , and we recover Schilder’s theorem. Remark 2.12.
Recall that a set
A ⊂ C [ T ] is called a set of continuity for the rate function Q T if inf g ∈A ◦ Q T ( g ) = inf g ∈ ¯ A Q T ( g ) . (2.4) For such a set, Theorem 2.9 implies that lim ε ↓ ε H log P (cid:16) X ε ,0, H ∈ A (cid:17) = − inf g ∈A Q T ( g ) . (2.5) Proof of Theorem 2.9.
For every g ∈ C [ T ] , set Q T ( g )= inf l , f ∈ H [ T ] (cid:20) (cid:18) Z T ˙ l ( s ) ds + Z T ˙ f ( s ) ds (cid:19) : Φ ( l , f , b f )( t ) = g ( t ) , t ∈ [ T ] (cid:21) , (2.6)if g is such that the set on the right-hand side of (2.6) is not empty, and Q T ( g ) = ∞ ,otherwise. For the sake of simplicity, we assume that T = s = ε ε H ( W , B , b B ) withstate space R × C [
0, 1 ] satisfies the large deviation principle with speed ε − H and goodrate function given by e I ( y , f , g ) = y + I ( f , g ) , y ∈ R , ( f , g ) ∈ C [
0, 1 ] . n the previous definition, the function I is defined as follows: If f ∈ H [
0, 1 ] and g = b f ,then I ( f , g ) = R ˙ f ( s ) ds , and in all the remaining cases, I ( f , g ) = ∞ . Similarly, wecan prove that the process ε ε H ( W , B , b B ) with state space C [
0, 1 ] satisfies the largedeviation principle with speed ε − H and good rate function given by I ( v , f , g ) = Z ˙ v ( s ) ds + I ( f , g ) , y ∈ R , ( v , f , g ) ∈ C [
0, 1 ] .Here we take into account Schilder’s theorem and the fact that Brownian motions W and B are independent.Using the same ideas as in Section 5 of [25], we can show that if we remove the driftterm, then the LDP in Theorem 2.9 is not affected. More precisely, this means that itsuffices to prove the LDP in Theorem 2.9 for the process ε b X ε ,0, H , where b X ε ,0, Ht = ε H Z t σ ( ε H b B s )( ¯ ρ dW s + ρ dB s ) , 0 ≤ t ≤
1. (2.7)Our next goal is to show how to apply the extended contraction principle in our envi-ronment (see Theorem 4.2.23 in [11] for more details concerning the extended contractionprinciple). Let us define a sequence of functionals Φ m : M C [
0, 1 ] , m ≥ ( r , h , l ) ∈ C [
0, 1 ] , and t ∈ [
0, 1 ] , Φ m ( r , h , l )( t )= ¯ ρ [ mt − ] ∑ k = σ (cid:18) l (cid:18) km (cid:19)(cid:19) (cid:20) r (cid:18) k + m (cid:19) − r (cid:18) km (cid:19)(cid:21) + σ (cid:18) l (cid:18) k + m (cid:19)(cid:19) (cid:20) r ( t ) − r (cid:18) [ mt ] m (cid:19)(cid:21) + ρ [ mt − ] ∑ k = σ (cid:18) l (cid:18) km (cid:19)(cid:19) (cid:20) h (cid:18) k + m (cid:19) − h (cid:18) km (cid:19)(cid:21) + σ (cid:18) l (cid:18) k + m (cid:19)(cid:19) (cid:20) h ( t ) − h (cid:18) [ mt ] m (cid:19)(cid:21) .It is not hard to see that for every m ≥
1, the mapping Φ m is continuous.We will next establish that formula (4.2.24) in [11] holds in our setting. This formula isused in the formulation of the extended contraction principle (see [11], Theorem 4.2.23). Lemma 2.13.
For every ζ > and y > , lim sup m → ∞ sup { ( r , f ) ∈ H [ ] : R ˙ r ( s ) ds + R ˙ f ( s ) ds ≤ ζ } || Φ ( r , f , b f ) − Φ m ( r , f , ˆ f ) || C [ ] = Proof.
The proof of Lemma 2.13 is similar to that of Lemma 21 in [25]. It is not hard tosee that for all ( r , f ) ∈ H [
0, 1 ] and m ≥ Φ m ( r , f , ˆ f ) = ¯ ρ Z t h m ( s , f ) ˙ r ( s ) ds + ρ Z t h m ( s , f ) ˙ f ( s ) ds ,where h m ( s , f ) = m − ∑ k = σ (cid:18) b f (cid:18) km (cid:19)(cid:19) { km ≤ s ≤ k + m } , 0 ≤ s ≤ herefore, Φ ( r , f , b f ) − Φ m ( r , f , ˆ f ) = ¯ ρ Z t [ σ ( b f ( s )) − h m ( s , f )] ˙ r ( s ) ds + ρ Z t [ σ ( b f ( s )) − h m ( s , f )] ˙ f ( s ) ds . (2.8)For every η >
0, denote D η = { w ∈ H [
0, 1 ] : R ˙ w ( s ) ds ≤ η } . It is not hard to see thatto prove Lemma 2.13, it suffices to show that for all η > m → ∞ " sup f ∈ D η , w ∈ D η sup t ∈ [ ] (cid:12)(cid:12)(cid:12)(cid:12) Z t [ σ ( ˆ f ( s )) − h m ( s , f )] ˙ w ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) =
0, (2.9)We havesup f ∈ D η , w ∈ D η sup t ∈ [ ] (cid:12)(cid:12)(cid:12)(cid:12) Z t [ σ ( ˆ f ( s )) − h m ( s , f )] ˙ w ( s ) ds (cid:12)(cid:12)(cid:12)(cid:12) ≤ sup f ∈ D η , w ∈ D η Z (cid:12)(cid:12)(cid:12) σ ( ˆ f ( s )) − h m ( s , f ) (cid:12)(cid:12)(cid:12) | ˙ f ( s ) | ds ≤ √ η sup f ∈ D η sup s ∈ [ ] (cid:12)(cid:12)(cid:12) σ ( ˆ f ( s )) − h m ( s , f ) (cid:12)(cid:12)(cid:12) .(2.10)It was established in the proof of Lemma 21 in [25] thatsup f ∈ D η sup s ∈ [ ] (cid:12)(cid:12)(cid:12) σ ( ˆ f ( s )) − h m ( s , f ) (cid:12)(cid:12)(cid:12) → m → ∞ (the previous statement follows from (49) in [25]). Now, it is clear that (2.10)and (2.11) imply (2.9).This completes the proof of Lemma 2.13.It remains to prove that the sequence of processes ε Φ m (cid:16) ε H W , ε H B , ε H b B (cid:17) with statespace C [
0, 1 ] is an exponentially good approximation to the process ε b X ε ,0, H . Theprevious statement means that for every δ > m → ∞ lim sup ε ↓ ε H log P (cid:16) || b X ε ,0, H − Φ m (cid:16) ε H W , ε H B , ε H b B (cid:17) || C [ ] > δ (cid:17) = − ∞ . (2.12)Using the definitions of b X ε ,0, H and Φ m , we see that in order to prove the equality in (2.12),it suffices to show that for every τ > m → ∞ lim sup ε ↓ ε H log P ε H sup t ∈ [ ] (cid:12)(cid:12)(cid:12)(cid:12) Z t σ ( m ) s dB s (cid:12)(cid:12)(cid:12)(cid:12) > δ ! = − ∞ (2.13)and lim m → ∞ lim sup ε ↓ ε H log P ε H sup t ∈ [ ] (cid:12)(cid:12)(cid:12)(cid:12) Z t σ ( m ) s dW s (cid:12)(cid:12)(cid:12)(cid:12) > δ ! = − ∞ , (2.14)where σ ( m ) s = σ (cid:16) ε H b B s (cid:17) − σ (cid:18) ε H b B [ mt ] m (cid:19) , 0 ≤ s ≤ m ≥ inally, by taking into account (2.12), Lemma 2.13, and applying the extended contrac-tion principle (Theorem 4.2.23 in [11]), we show that the process ε b X ε ,0, H satisfies thelarge deviation principle with speed ε − H and good rate function Q (see the definitionin (2.6)). For any T >
0, the process ε b X ε ,0, H satisfies the large deviation principle withspeed ε − H and good rate function Q T defined in (2.6). The previous statement can beestablished using the methods employed in the reasoning before Definition 17 in [25].We will next prove that the function Q T satisfies the formula in (2.3). It is not hard tosee that if g ∈ C [ T ] , l , f ∈ H [ T ] , and Φ ( l , f , b f )( t ) = g ( t ) for all t ∈ [ T ] , then g ∈ H [
0, 1 ] . Moreover, if for g ∈ H [ T ] the previous equality holds, then it is not hardto see that ˙ l ( t ) = ˙ g ( s ) − ρσ ( b f ( s ))) ˙ f ( s ) ¯ ρσ ( b f ( s )) .Now it is clear that formula (2.3) holds for the function Q T defined by (2.6).This completes the proof of Theorem 2.9.Our next goal in the present section is to prove that for every g ∈ H [ T ] , there existsat least one minimizer f g in the minimization problem on the right-hand side of (2.3). Lemma 2.14.
For every function g ∈ H [ T ] there exists a function f g ∈ H [ T ] such thatQ T ( g ) = Z T " ˙ g ( s ) − ρσ ( b f g ( s )) ˙ f g ( s ) ¯ ρσ ( b f g ( s )) ds + Z T ˙ f g ( s ) ds . (2.15) Proof.
It is not hard to see that it suffices to prove that for every j ∈ L [ T ] , the follow-ing minimization problem on the space L [ T ] has a solution: H ( j ) = inf h ∈ L [ T ] e H j ( h ) ,where e H j ( h ) = Z T (cid:20) j ( s ) ¯ ρσ ( K h ( s )) − ρ ¯ ρ h ( s ) (cid:21) ds + Z T h ( s ) ds . (2.16)In (2.16), K h ( s ) = R T K ( s , u ) h ( u ) du , and K is a Volterra type kernel satisfying condition(b) in Definition 2.2. It is known that a solution to the minimization problem formulatedabove exists if the functional e H j : L [ T ] R is coercive and weakly sequentially lowersemi-continuous (see, e.g., [49], Ch. 1, Theorem 1.2). If the latter property holds for thefunctional F , then, for the sake of shortness, we will write F ∈ W LS . The coercivity ofthe functional in (2.16) is clear. It is also a known fact that the functional h R T h ( s ) ds belongs to the class W LS . Since the sum of two functionals from
W LS is also from
W LS ,and the square of a nonnegative functional from
W LS is in
W LS , it suffices to prove thatthe functional G j ( h ) = ( Z T (cid:20) j ( s ) ¯ ρσ ( K h ( s )) − ρ ¯ ρ h ( s ) (cid:21) ds ) belongs to the class W LS . We have G j ( h ) = sup || q || ≤ Z T (cid:20) j ( s ) ¯ ρσ ( K h ( s )) − ρ ¯ ρ h ( s ) (cid:21) q ( s ) ds . he supremum of any family of functionals from W LS is in
W LS . Therefore, to finishthe proof of Lemma 2.14 it is enough to show that for any j , q ∈ L [ T ] , the functional h Z T (cid:20) j ( s ) ¯ ρσ ( K h ( s )) − ρ ¯ ρ h ( s ) (cid:21) q ( s ) ds belongs to the class W LS . It is clear that the functional h
7→ − ρ ¯ ρ R T h ( s ) q ( s ) ds is weaklycontinuous, hence it is in W LS . It remains to analyze the functional D j , q ( h ) Z T j ( s ) q ( s ) ¯ ρσ ( K h ( s )) ds . (2.17)We will prove that this functional is weakly sequentially continuous. Suppose h k → h weakly in L [ T ] . Then sup k || h k || < ∞ . It follows from the restrictions on the kernel K that the operator K is continuous from L [ T ] into C [ T ] . Therefore, it is weaklycontinuous, and hence K h k ( s )
7→ K h ( s ) for all s ∈ [ T ] . Moreover, we havesup k ≥ s ∈ [ T ] |K h k ( s ) | ≤ ||K|| sup k || h k || < ∞ .It is assumed in Theorem 2.9 that the volatility function σ is strictly positive on R . There-fore, it is bounded away from zero on any finite interval. Next, using (2.17) and thedominated convergence theorem, we see that D j , q ( h k ) D j , q ( h ) as k → ∞ . Hence thefunctional D j , q is weakly sequentially continuous. Finally, by taking into account the factsestablished above, it is easy to complete the proof of Lemma 2.14.In the next lemma, we prove the continuity of the rate function Q T on the space H [ T ] . Lemma 2.15.
The functional Q T : H [ T ] R is continuous.Proof. The lower semi-continuity of the functional Q T in Lemma 2.15 follows from thefact that Q T is a rate function on C [ T ] and the Sobolev embedding H [ T ] ⊂ C [ T ] .We will next prove the upper semi-continuity of Q T on H [ T ] . For every f ∈ H [ T ] ,define the functional D f : H [ T ] R by D f ( g ) = Z T " ˙ g ( s ) − ρσ ( b f ( s )) ˙ f ( s ) ¯ ρσ ( b f ( s )) ds + Z T ˙ f ( s ) ds .It is not hard to see that in order to complete the proof of Lemma 2.15, it suffices toestablish that for every f ∈ H [ T ] , the functional D f is continuous. It is clear that b f is a bounded continuous function on [ T ] . Therefore there exist δ > M > M > σ ( b f ( s )) > δ for all s ∈ [ T ] . Suppose g k → g in H [ T ] . Then we have |D f ( g ) − D f ( g k ) | ≤ δ ( − ρ ) Z T | ˙ g ( s ) − ˙ g k ( s ) || ˙ g ( s ) + ˙ g k ( s ) − ρσ ( b f ( s )) ˙ f ( s ) | ds ≤ δ ( − ρ ) || g − g k || H [ T ] || g || H [ T ] + sup k || g || H [ T ] + (cid:26) Z T σ ( b f ( s )) ˙ f ( s ) ds (cid:27) ! ≤≤ δ ( − ρ ) || g − g k || H [ T ] || g || H [ T ] + sup k || g || H [ T ] + M || f || H [ T ] ! . ow, it is clear that D f ( g k ) → D f ( g ) as k → ∞ , and hence the functional D f is continuouson the space H [ T ] . It follows that the functional Q T is upper semi-continuous since itcan be represented as the infimum of a family of continuous on H [ T ] functionals.This completes the proof of Lemma 2.15.Our final goal in the present section is to apply Theorem 2.9 to characterize the leadingterm in the asymptotic expansion of the exit probability from an open interval. Recallthat we assumed S ε ,0, H =
1. Therefore, X ε ,0, H =
0. Let U = ( a , b ) be an interval such that0 ∈ ( a , b ) , and define the exit time from U by τ ε = inf n s ∈ ( T ] : X ε ,0, Hs / ∈ U o . For everyfixed t ∈ ( T ] , the exit time probability function v ε ( t ) is defined by v ε ( t ) = P ( τ ε ≤ t ) .Set A t = { f ∈ C [ T ] : f ( s ) / ∈ U for some s ∈ ( t ] } . Theorem 2.16.
Under the conditions in Theorem 2.9, for every t ∈ [ T ] , lim ε → ε H log v ε ( t ) = − inf f ∈A t Q T ( f ) . Proof.
It is not hard to see that { τ ε ≤ t } = (cid:8) X ε ,0, H ∈ A t (cid:9) . We will next show that A t is aset of continuity for Q T . Indeed, the interior A ◦ t of A t consists of all the paths f ∈ C [ T ] ,for which there exists s < t such that f ( s ) / ∈ ¯ U . In addition, the boundary of A t coincideswith the set of all paths f ∈ C [ T ] , which hit the boundary of U before t or at s = t ,but never exit ¯ U before t . It is not hard to see that the set A ◦ t ∪ H [ T ] is dense in the set¯ A t ∪ H [ T ] in the topology of the space H [ T ] . Now, using Lemma 2.15, it is easy toprove that the equality in (2.4) holds for the set A t , and hence (2.5) is valid for A t .The proof of Theorem 2.16 is thus completed.3. M ODERATE DEVIATIONS : 0 < β < H In this section, we assume that 0 < β < H , and prove a sample path large deviationprinciple for the process ε X ε , β , H . We also obtain a similar result for the process ε X ε , β , HT . It is not assumed in the present section that the Gaussian stochastic volatilitymodel is of Volterra type.The next statement is the main result of the present section. Theorem 3.1.
Let < β < H, σ ( ) > , and suppose the function σ is locally ω -continuous on R for some modulus of continuity ω . Suppose also that b B is a nondegenerate continuous centeredGaussian process that is adapted to the filtration {F t } ≤ t ≤ T . Then the process ε X ε , β , H withstate space C [ T ] satisfies the LDP with speed ε β − H and good rate function defined by e I T ( f ) = ( σ ( ) R T ˙ f ( t ) dt , f ∈ H [ T ] ∞ , f ∈ C [ T ] \ H [ T ] . Proof.
Set b X ε , β , Ht = ε H − β Z t σ ( ε H b B s ) dZ s , 0 ≤ ε ≤
1. (3.1)We will first prove that in the environment of Theorem 3.1, the removal of the drift termdoes not affect the validity of the LDP. emma 3.2. Under the conditions in Theorem 3.1, the processes ε → b X ε , β , H and ε → X ε , β , H with state space C [ T ] are exponentially equivalent. Remark 3.3.
The definition of the exponential equivalence can be found in [11] . In our case, theexponential equivalence means that for every y > , lim ε → ε H − β log P (cid:16) || b X ε , β , H − X ε , β , H || C [ T ] ≥ y (cid:17) = − ∞ . Proof of Lemma 3.2.
A statement similar to that in Lemma 3.2 was obtained in a littledifferent setting in Section 5 of [25]. In our case, P (cid:16) || b X ε , β , H − X ε , β , H || C [ T ] ≥ y (cid:17) = P (cid:18) ε H − β Z T σ ( ε H b B s ) ds ≥ y (cid:19) ,and we can finish the proof of Lemma 3.2, using the same tools as in the proof in Section5 of [25].It follows from Lemma 3.2 that the processes ε → b X ε , β , H and ε → X ε , β , H satisfy the samelarge deviation principle (see [11] for the proof of the fact that the exponential equivalenceof two processes implies that they satisfy the same LDP). Hence, it suffices to prove The-orem 3.1 for the former process. Lemma 3.4.
Under the conditions in Theorem 3.1, the process ε b X ε , β , H is exponentially equiv-alent to the process ε e G ε , β , H : = ε H − β σ ( ) Z.Proof of Lemma 3.4.
Let δ > < η <
1. For every ε ∈ [
0, 1 ] , set M ( ε ) t = Z t h σ ( ε H b B s ) − σ ( ) i dZ s , 0 ≤ t ≤ T ,and define a stopping time by ξ ( ε ) η = inf n s ∈ [ T ] : ε H | b B s | > η o . Then we have P (cid:16) || b X ε , β , H − e G ε , β , H || C [ T ] > δ (cid:17) = P ε H − β sup t ∈ [ T ] (cid:12)(cid:12)(cid:12) M ( ε ) t (cid:12)(cid:12)(cid:12) > δ ! ≤ P ε H − β sup t ∈ [ ξ ( ε ) η ] (cid:12)(cid:12)(cid:12) M ( ε ) t (cid:12)(cid:12)(cid:12) > δ + P (cid:16) ξ ( ε ) η < T (cid:17) = J ( ε , δ , η ) + J ( ε , δ , η ) . (3.2)We will first estimate J . Set σ ( ε ) s = σ ( ε H b B s ) − σ ( ) . Since the function σ is locally ω -continuous (see Definition 2.6), | σ ( ε ) s | ≤ L ( ) ω ( η ) for all s ∈ h ξ ( ε ) η i . (3.3)It is clear that since the process t M ( t ∧ ξ ( ε ) η ) , t ∈ [ T ] . (3.4)can be represented as a stochastic integral with a bounded integrand, it is a martingale.Moreover, for 0 < ε < ε and a fixed λ >
0, the stochastic exponential E ( ε ) t = exp ( λε H − β Z t ∧ ξ ( ε ) η σ ( ε ) s dZ s − λ ε H − β Z t ∧ ξ ( ε ) η (cid:16) σ ( ε ) s (cid:17) ds ) s a martingale (use Novikov’s condition). We will assume in the rest of the proof that0 < ε < ε . It follows from (3.3) and the martingality condition formulated above that E " exp ( λε H − β Z t ∧ ξ ( ε ) η σ ( ε ) s dZ s ) = E " E ( ε ) t exp ( λ ε H − β Z t ∧ ξ ( ε ) η (cid:16) σ ( ε ) s (cid:17) ds ) ≤ exp (cid:26) T λ ε H − β L ( ) ω ( η ) (cid:27) < ∞ , (3.5)for all t ∈ [ T ] . Plugging t = T into (3.5), we get E " exp ( λε H − β Z ξ ( ε ) η σ ( ε ) s dZ s ) ≤ exp (cid:26) T λ ε H − β L ( ) ω ( η ) (cid:27) . (3.6)Since the process in (3.4) is a martingale, the integrability condition in (3.5) implies thatthe process t exp ( λε H − β Z t ∧ ξ ( ε ) η σ ( ε ) s dZ s ) is a positive submartingale (see Proposition 3.6 in [34]). Next, using (3.6) and the firstsubmartingale inequality in [34], Theorem 3.8, we obtain P sup t ∈ [ ξ ( ε ) η ] exp (cid:26) ε H − β λ Z t σ ( ε ) s dZ s (cid:27) > e λδ ≤ exp (cid:26) T ε H − β λ L ( ) ω ( η ) − λδ (cid:27) .Setting λ = δ T ε H − β L ( ) ω ( η ) , we get from the previous inequality that P sup t ∈ [ ξ ( ε ) η ] ε H − β Z t σ ( ε ) s dZ s > δ ≤ exp (cid:26) − δ T ε H − β L ( ) ω ( η ) (cid:27) . (3.7)It is possible to replace the process M by the process − M in the reasoning above. Thisgives the following inequality that is similar to (3.7): P sup t ∈ [ ξ ( ε ) η ] (cid:20) − ε H − β Z t σ ( ε ) s dZ s (cid:21) > δ ≤ exp (cid:26) − δ T ε H − β L ( ) ω ( η ) (cid:27) . (3.8)It follows from (3.7) and (3.8) that P sup t ∈ [ ξ ( ε ) η ] ε H − β (cid:12)(cid:12)(cid:12)(cid:12) Z t σ ( ε ) s dZ s (cid:12)(cid:12)(cid:12)(cid:12) > δ ≤ (cid:26) − δ T ε H − β L ( ) ω ( η ) (cid:27) , (3.9)for all δ > < η <
1. Then we have the following estimate for J introduced in(3.2) J ( ε , δ , η ) ≤ (cid:26) − δ T ε H − β L ( ) ω ( η ) (cid:27) nd lim sup ε → ε H − β log J ( ε , δ , η ) ≤ − δ TL ( ) ω ( η ) . (3.10)Our next goal is to estimate J defined in (3.2). We have J ( ε , δ , η ) ≤ P ε H sup s ∈ [ T ] | b B s | > η ! , (3.11)for all ε ∈ ( T ] , δ >
0, and η ∈ (
0, 1 ) . Using the large deviation principle for themaximum of a Gaussian process (see, e.g., (8.5) in [37]), we can show that there existconstants C > y > P sup t ∈ [ T ] | b B t | > y ! ≤ e − C y (3.12)for all y > y . Next, taking into account (3.11) and (3.12), we obtainlim sup ε → ε H − β log J ( ε , δ , η ) = − ∞ . (3.13)Finally, combining (3.2), (3.10), and (3.13), and using the inequalitylog ( a + b ) ≤ max { log ( a ) , log ( b ) } , a > b > ε → ε H − β log P (cid:16) || b X ε , β , H − e G ε , β , H || C [ T ] > δ (cid:17) = − ∞ ,for all δ > e G ε , β , H satisfies the LDP in the formulation of Theorem 3.1. Next, using theexponential equivalence in Lemmas 3.2 and 3.4, we see that the same LDP holds for theprocess b X ε , β , H .This completes the proof of Theorem 3.1. Corollary 3.5.
Under the restrictions in Theorem 3.1, the process ε X ε , β , HT with state space R satisfies the LDP with speed ε β − H and good rate function defined by b I T ( x ) = x T σ ( ) , x ∈ R .Corollary 3.5 can be derived from Theorem 3.1. Indeed, let A be a Borel subset of R ,and consider the Borel subset e A of C consisting of f ∈ C [ T ] such that f ( T ) ∈ A . Then,it is not hard to prove the LDP-estimates in Corollary 3.5 for the set A , by applying theLDP-estimates in Theorem 3.1 to the set e A . Remark 3.6.
The large deviation and moderate deviation results obtained in Theorem 2.7 andCorollary 3.5, and also the fact that the rate function I T is nondecreasing on [ ∞ ) (see [25] ), mply the following tail estimates: lim ε ↓ ε H − β log P (cid:16) X ε , β , HT ≥ x (cid:17) = ( − I T ( x ) , if β = − x T σ ( ) , if < β < H . (3.14)Our next goal is to discuss relations between small-time and small-noise LDPs for self-similar volatility processes. Definition 3.7.
Let < H < . The process b B t , ≤ t ≤ T, is called H-self-similar if for every ε ∈ (
0, 1 ] , b B ε t = ε H b B t , ≤ t ≤ T, in law.
Fractional Brownian motion B H and the Riemann-Liouville fractional Brownian motion R H are H -self-similar, while the fractional Ornstein-Uhlenbeck process U H is not.Suppose the volatility process b B is H -self-similar. Then, we can pass from small-noiseLDP and MDP in Theorem 2.7 and Corollary 3.5 to small-time LDP and MDP, by makingthe following observation (such methods are well-known). Set Z ε , Ht = − ε Z t σ ( ε H b B s ) ds + √ ε Z t σ ( ε H b B s ) dZ s .It is not hard to see that under the self-similarity condition for b B , for every ε ∈ (
0, 1 ] theequality X ε t = Z ε , Ht , 0 ≤ t ≤ T , holds in law. Here the process X is defined by (1.3). Then ε H − β − X ε t = − ε H − β + Z t σ ( ε H b B s ) ds + b X ε , β , Ht ,for all t ∈ [ T ] , ε ∈ (
0, 1 ] , H ∈ (
0, 1 ) , and β ∈ [ H ) . The process b X ε , β , Ht in the previousequality is defined in (2.7) and (3.1). Next, replacing t by T and ε by t , we obtain t H − β − X tT = − t H − β + Z T σ ( t H b B s ) ds + b X t , β , HT , (3.15)for all t ∈ [
0, 1 ] . For β = t X t , β , HT satisfies the LDP in Theorem 2.7,while for β ∈ ( H ) it satisfies the MDP in Corollary 3.5. Now, replacing the drift term − t H − β R T σ ( t H b B s ) ds in the process t X t , β , HT by a new drift term − t H − β + Z T σ ( t H b B s ) ds ,and using (3.15), we see that the process on the left-hand side of (3.15) satisfies the LDP inTheorem 2.7 for β =
0, and the MDP in Corollary 3.5 for β ∈ ( H ) . The possibility of driftreplacement can be justified using the ideas employed in Section 5 of [25]. The previousreasoning shows how to obtain small-time large and moderate deviation principles fromthe small-noise ones. For β =
0, a small-time analogue of the LDP in Theorem 2.7 wasobtained in [25], Theorem 18.4. C
ENTRAL LIMIT REGIME : β = H We will next describe what happens if β = H . Recall that in LDP and MDP regimes,we can ignore drift terms. For β = H , this is no more the case, and drift terms have to e taken into account. In the rest of the paper, the symbol ¯ N will stand for the standardnormal complementary cumulative distribution function defined by¯ N ( z ) = √ π Z ∞ z exp (cid:26) − u (cid:27) du , z ∈ R .Let us assume that the restrictions on the function σ imposed in Theorem 3.1 hold. Wehave X ε , H , Ht = − Z t σ ( ε H b B s ) ds + Z t σ ( ε H b B s ) dZ s , 0 ≤ t ≤ T .If β = H , then the expression on the left-hand side of (3.14) has the following form: L ( x ) = lim ε ↓ log P (cid:16) X ε , H , HT ≥ x (cid:17) , x >
0. (4.1)It will be shown below that the limit in (4.1) exists for every x >
0, and its value will becomputed.We will first study the behavior of the process ε X ε , H , H on the path space. Set U t = − t σ ( ) + σ ( ) Z t , t ∈ [ T ] . (4.2) Theorem 4.1.
Under the restrictions on the function σ and the process b B imposed in Theorem3.1, the following formula holds for all y > : lim ε → P (cid:16) || X ε , H , H − U || C [ T ] ≥ y (cid:17) = Proof.
For every y > P (cid:16) || X ε , H , H − U || C [ T ] ≥ y (cid:17) ≤ P sup t ∈ [ T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t h σ ( ) − σ ( ε H b B s ) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≥ y ! + P sup t ∈ [ T ] (cid:12)(cid:12)(cid:12)(cid:12) Z t h σ ( ε H b B s ) − σ ( ) i dZ s (cid:12)(cid:12)(cid:12)(cid:12) ≥ y ! = L ( ε , y ) + L ( ε , y ) . (4.3)We will first show that lim ε → L ( ε , y ) =
0. (4.4)To prove the equality in (4.4), we employ the methods used in the proof of Lemma 3.4.Analyzing the proof preceding (3.9), we see that the estimate in (3.9) holds for β = H too.This gives P sup t ∈ [ ξ ( ε ) η ] (cid:12)(cid:12)(cid:12)(cid:12) Z t σ ( ε ) s dZ s (cid:12)(cid:12)(cid:12)(cid:12) > δ ≤ (cid:26) − δ L ( ) ω ( η ) (cid:27) .Now, it is not hard to see how to prove (4.4) using (3.11) and (3.12).Our next goal is to show that lim ε → L ( ε , y ) =
0. (4.5) or all η ∈ (
0, 1 ) , we have L ( ε , y ) ≤ P sup t ∈ [ ξ ( ε ) η ] (cid:12)(cid:12)(cid:12)(cid:12) Z t h σ ( ) − σ ( ε H b B s ) i ds (cid:12)(cid:12)(cid:12)(cid:12) ≥ y + P (cid:16) ξ ( ε ) η < T (cid:17) ≤ P sup t ∈ [ ξ ( ε ) η ] Z t (cid:12)(cid:12)(cid:12) σ ( ) − σ ( ε H b B s ) (cid:12)(cid:12)(cid:12) (cid:16) σ ( ) + σ ( ε H b B s ) (cid:17) ds ≥ y + P ε H sup s ∈ [ T ] | b B s | > η ! ≤ P TL ( ) ω ( η ) sup ≤ u ≤ [ σ ( u )] ≥ y ! + P ε H sup s ∈ [ T ] | b B s | > η ! . (4.6)For a fixed y > η small enough, the first term on the last line in (4.6) is equal tozero, since ω ( η ) → η →
0. Moreover, for a fixed η ∈ (
0, 1 ) , we havelim ε → P ε H sup s ∈ [ T ] | b B s | > η ! = D H , T ( x ) = lim ε → P sup t ∈ [ T ] X ε , H , Ht > x ! can one derive using Theorem 4.1? We will only deal with the right tail. Theorem 4.2.
Let H > and x > . ThenD H , T ( x ) = Φ x σ ( ) √ T + σ ( ) √ T ! + exp {− x } " − Φ x σ ( ) √ T + σ ( ) √ T ! . (4.7) where the symbol Φ stands for the standard normal cumulative distribution function.Proof. Fix δ with 0 < δ <
1. Then we have P sup t ∈ [ T ] X ε , H , Ht > x ! ≤ P (cid:16) || X ε , H , H − U || C [ T ] > ( − δ ) x (cid:17) + P sup t ∈ [ T ] U t > δ x ! , here U is defined by (4.2). It follows from Theorem 4.1 thatlim sup ε → P sup t ∈ [ T ] X ε , H , Ht > x ! ≤ lim sup ε → P (cid:16) || X ε , H , H − U || C [ T ] > ( − δ ) x (cid:17) + P sup t ∈ [ T ] U t > δ x ! = P sup t ∈ [ T ] U t > δ x ! = P sup t ∈ [ T ] (cid:18) − t σ ( ) + Z t (cid:19) > δ x σ ( ) ! . (4.8)The distribution of the maximum of Brownian motion with drift is known. The follow-ing formula holds for every y > µ ∈ R : P sup t ∈ [ T ] ( µ t + Z t ) > y ! = Φ (cid:18) y − µ T √ T (cid:19) + exp { µ y } (cid:20) − Φ (cid:18) y − µ T √ T (cid:19)(cid:21) (4.9)(see, e.g., [2]). Therefore, (4.8) implies thatlim sup ε → P sup t ∈ [ T ] X ε , H , Ht > x ! ≤ Φ δ x σ ( ) √ T + σ ( ) √ T ! + exp {− δ x } " − Φ δ x σ ( ) √ T + σ ( ) √ T ! .Next, by letting δ →
1, we obtainlim sup ε → P sup t ∈ [ T ] X ε , H , Ht > x ! ≤ Φ x σ ( ) √ T + σ ( ) √ T ! + exp {− x } " − Φ x σ ( ) √ T + σ ( ) √ T ! . (4.10)Our next goal is to obtain a lower estimate. We have P sup t ∈ [ T ] X ε , H , Ht > x ! ≥ P sup t ∈ [ T ] U t > ( + δ ) x ! − P (cid:16) || X ε , H , H − U || C [ T ] > δ x (cid:17) and lim inf ε → P sup t ∈ [ T ] X ε , H , Ht > x ! ≥ P sup t ∈ [ T ] U t > ( + δ ) x ! − lim sup ε → P (cid:16) || X ε , H , H − U || C [ T ] > δ x (cid:17) = P sup t ∈ [ T ] U t > ( + δ ) x ! = P sup t ∈ [ T ] (cid:18) − t σ ( ) + Z t (cid:19) > ( + δ ) x σ ( ) ! . (4.11) ow, using (4.11), (4.9), and letting δ →
0, we obtainlim inf ε → P sup t ∈ [ T ] X ε , H , Ht > x ! ≥ Φ x σ ( ) √ T + σ ( ) √ T ! + exp {− x } " − Φ x σ ( ) √ T + σ ( ) √ T ! . (4.12)Finally, it is clear that (4.7) follows from (4.10) and (4.12).This completes the proof of Theorem 4.2.The next statement is a corollary of Theorem 4.1. Theorem 4.3.
Under the restrictions on the function σ imposed in Theorem 3.1, the followingformula is valid: lim ε ↓ P (cid:16) X ε , H , HT ≥ x (cid:17) = ¯ N x √ T σ ( ) + √ T σ ( ) ! . Therefore the limit in (4.1) exists for every x > , and moreoverL ( x ) = log ¯ N x √ T σ ( ) + √ T σ ( ) ! . (4.13) Proof.
By Theorem 4.1, the process X ε , H , HT converges in probability as ε ↓ − T σ ( ) + σ ( ) Z T . It is known that convergence in probability impliesconvergence in distribution. Since for every x >
0, the set [ x , ∞ ) is a set of continuity ofthe distribution of Z T , we havelim ε ↓ P (cid:16) X ε , H , HT ≥ x (cid:17) = √ π √ T σ ( ) Z ∞ x exp ( − T σ ( ) (cid:18) r + T σ ( ) (cid:19) ) dr = ¯ N x √ T σ ( ) + √ T σ ( ) ! . (4.14)Now it is clear that (4.13) follows from (4.14).This completes the proof of Theorem 4.3. Remark 4.4.
In the case where β = H, one can consider the function − L T ( x ) = − log ¯ N x √ T σ ( ) + √ T σ ( ) ! , x > as a replacement for the rate function I T in the large deviation principle in Theorem 2.7, or the ratefunction b I T ( x ) = √ T σ ( ) x in the moderate deviation principle in Corollary 3.5. However, for β = H, the corresponding moderate deviation principle is degenerated since in this case the speed ε H − β is identically equal to one. Remark 4.5. If β → , then the rate function in the MDP regime in Corollary 3.5 does nottend to the rate function in the LDP regime in Theorem 2.7. This discontinuity disappears for mall x > , if we tolerate an O ( x ) -approximation. Indeed for β = , the following asymptoticexpansion was established in [4] under a stronger smoothness restriction on the volatility function:I T ( x ) = x T σ ( ) + O ( x ) as x → (actually, more terms in the Taylor expansion above were found in [4] ). Note that thereis also a discontinuity in the asymptotic formulas at β = H. One of the reasons for the above-mentioned discontinuities is that it is in general not possible to pass to the limit with respect to anextra parameter in asymptotic formulas.
5. A
SYMPTOTIC BEHAVIOR OF SMALL - NOISE CALL PRICING FUNCTIONS IN MIXEDREGIMES
In this section, we study the asymptotic behavior of small-noise call pricing functionsin mixed regimes. The following small-noise call pricing functions will be considered: C β , H , T ( ε , x ε α ) = E (cid:20)(cid:16) S ε , β , HT − exp { x ε α } (cid:17) + (cid:21) .The parameters appearing in the previous definition satisfy x > H > β ≤ H , α ≥ ≤ α + β ≤ H . Note that the parameter β may take negative values. In the previousformula, the maturity is parametrized by ε , while the log-strike follows the path ε x ε α (see [21] for the discussion of various parametrizations of the call).In the present section, we deal with the case where the volatility function satisfies thelinear growth condition. What happens when the volatility function grows at infinityfaster than linearly is discussed in the next section. Definition 5.1.
It is said that the linear growth condition holds for the function σ if there existconstants c > and c > such that σ ( x ) ≤ c + c x for all x ≥ . It is known that if the linear growth condition holds for the function σ , then the assetprice process S in the model described by (1.1) is a martingale, and hence P is a risk-neutral measure (see, e.g., [17, 25]). The process S can be a martingale even for morerapidly growing functions σ . For example, it was established in [31] that for the Scottmodel (see [47]), where σ ( x ) = e x and b B is the classical Ornstein-Uhlenbeck process, theasset price process S is a martingale if and only if − < ρ ≤
0. A similar result for generalVolterra type Gaussian models was obtained in a recent preprint [24].In the next assertion, asymptotic formulas for call pricing functions are derived fromlarge deviation principles.
Theorem 5.2.
Suppose the volatility function σ satisfies the linear growth condition. Then thefollowing are true:(i) Assume that the conditions in Theorem 2.7 hold, and let α + β = . Then lim ε ↓ ε H log C β , H , T ( ε , x ε α ) = − I T ( x ) . (ii) Assume that the conditions in Corollary 3.5 hold, and let < α + β < H. Then lim ε ↓ ε H − α − β log C β , H , T ( ε , x ε α ) = − x T σ ( ) . roof. The methods allowing to derive estimates for call pricing functions from largedeviation principles under the linear growth condition are well-known (see, e.g., [16, 17,25, 42]). We will only sketch the proof of the upper estimate in part (i) of Theorem 5.2. Letus start with tail estimates. It follows from (1.5) and the possibility of removing the driftterms in the case where α + β = H mentioned above thatlim ε ↓ ε H − α − β log P (cid:16) X ε , β , HT ≥ x ε α (cid:17) = lim ε ↓ ε H − α − β log P (cid:16) ε − α X ε , β , HT ≥ x (cid:17) = lim ε ↓ ε H − α − β log P (cid:16) X ε , α + β , HT ≥ x (cid:17) . (5.1)If α + β =
0, then Theorem 2.7 and the equality in (5.1) imply thatlim ε ↓ ε H log P (cid:16) X ε , β , HT ≥ x ε α (cid:17) = − I T ( x ) (to prove a similar equality in the case where 0 < α + β < H we use Corollary 3.5 and(5.1)).Let p > q > p + q =
1. Then C β , H , T ( ε , x ε α ) ≤ n E h(cid:12)(cid:12)(cid:12) S ε , β , HT (cid:12)(cid:12)(cid:12) p io p n P (cid:16) X ε , β , HT > x ε α (cid:17)o q .It can be seen from the previous estimate thatlim sup ε ↓ ε H − α − β log C β , H , T ( ε , x ε α ) ≤ p lim sup ε ↓ ε H − α − β log E h(cid:12)(cid:12)(cid:12) S ε , β , HT (cid:12)(cid:12)(cid:12) p i − q I T ( x ) .The rest of the proof of the upper estimate in part (i) of Theorem 5.2 follows the proofof a similar estimate in Corollary 31 in [25] (formula (81) in [25]). By reasoning as in thelatter proof we can establish that for all ε ∈ (
0, 1 ] , p >
1, and β ∈ [ H ) , E h(cid:12)(cid:12)(cid:12) S ε , β , HT (cid:12)(cid:12)(cid:12) p i ≤ s p (cid:26) E (cid:20) exp (cid:26)(cid:16) p − p (cid:17) ε H − β Z T σ ( ε H b B s ) ds (cid:27)(cid:21)(cid:27) . (5.2)In the proof of (5.2), the assumption that the asset price process S is a martingale is used.It was shown in [25] that there exists δ > E (cid:20) exp (cid:26) δ Z T b B s ds (cid:27)(cid:21) < ∞ . (5.3)Next, using the linear growth condition for σ , we prove that the inequality in (5.3) impliesthe existence of ε > δ > ε ∈ ( ε ] E (cid:20) exp (cid:26) δ Z T σ ( ε H b B s ) ds (cid:27)(cid:21) < ∞ . (5.4)Now, we can finish the proof of part (i) of Theorem 5.2 exactly as in [25], Corollary 31.Note that the following fact is needed here. Since the linear growth condition holds, thestochastic exponential t S ε , β , Ht defined in (1.4) is a martingale for every fixed ε , andtherefore 1 = E h S ε , β , HT i ≤ E h | S ε , β , HT | p i for all ε ∈ (
0, 1 ] and p > emark 5.3. It is interesting that if the volatility function σ grows at infinity a little faster thanlinearly, then the inequality in (5.4) may fail (see Theorem 6.7). This means that special methodsof establishing the upper LDP and MDP call price estimates, which are based on the finitenessof the exponential moments of the integrated variance can not be employed under even a slightlyweaker than the linear growth restriction on the volatility function σ (for more information see thediscussion in Section 6). It remains to characterize the asymptotic behavior of the call pricing function in theregime where α + β = H . We will first restrict ourselves to the case where α = β = H . Theorem 5.4.
Suppose α = and β = H. Suppose also that the conditions in Corollary 3.5 arevalid, and the function σ satisfies the linear growth condition. Then the following formula holds: lim ε ↓ C H , H , T ( ε , x ) = Z ∞ x e y ¯ N y √ T σ ( ) + √ T σ ( ) ! dy . (5.5) Remark 5.5.
The formula in (5.5) can be rewritten as follows: lim ε ↓ C H , H , T ( ε , x ) = C − ( x , √ T σ ( )) , (5.6) where the symbol C − ( k , ν ) stands for the call price in the Black-Scholes model as a function of thelog strike k ≥ and the dimensionless implied volatility ν (see the definition in formula (3.1) in [21] ). We leave the proof of the fact that the formulas in (5.5) and (5.6) are the same as an exercisefor the interested reader. It follows from [21] (see the second equality in formula (3.1) and formula(3.3) in [21] ) that for every fixed k, C − is a strictly increasing function of ν .Proof of Theorem 5.4. Set P α , H , T ε ( x ) = P (cid:18) − ε α Z T σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s ≥ x (cid:19) . (5.7)It is not hard to see that C H , H , T ( ε , x ) = E (cid:20)(cid:16) exp n X ε , H , HT o − e x (cid:17) + (cid:21) = Z ∞ x ( e y − e x ) d h − P H , T ε ( y ) i . (5.8)Our next goal is to estimate the distribution function P H , T ε ( y ) . It follows from (5.7),Chebyshev’s exponential inequality, and the Cauchy-Schwartz inequality that for every > P H , T ε ( y ) ≤ e − y E (cid:20) exp (cid:26) − Z T σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s (cid:27)(cid:21) ≤ e − y E (cid:20) exp (cid:26) Z T σ ( ε H b B s ) dZ s (cid:27)(cid:21) = e − y E (cid:20) exp (cid:26) − Z T σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s + Z σ ( ε H b B s ) ds (cid:27)(cid:21) ≤ e − y (cid:18) E (cid:20) exp (cid:26) − Z σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s (cid:27)(cid:21)(cid:19) × (cid:18) E (cid:20) exp (cid:26) Z T σ ( ε H b B s ) ds (cid:27)(cid:21)(cid:19) . (5.9)Now, using the linear growth condition for σ and the fact that the stochastic exponentialin (5.9) is a martingale (see Lemma 13 in [25]), we obtain P H , T ε ( y ) ≤ e − y (cid:18) E (cid:20) exp (cid:26) Z T σ ( ε H b B s ) ds (cid:27)(cid:21)(cid:19) ≤ e − y e c (cid:18) E (cid:20) exp (cid:26) c ε H Z T b B s ds (cid:27)(cid:21)(cid:19) .It follows from (5.3) that there exists ε > y and such thatsup < ε < ε P H , T ε ( y ) ≤ le − y , (5.10)for some constant l > y . It is not hard to see that (5.8), (5.10), and theintegration by parts formula imply the following: C H , H , T ( ε , x ) = Z ∞ x e y P H , T ε ( y ) dy . (5.11)Next, using (5.11), (4.13), (5.10), and the Lebesgue dominated convergence theorem, wesee that for all x >
0, the equality in (5.5) holds.We will next turn our attention to the case where α + β = H and β = H . This caseis exceptional. It exhibits a special discontinuity when compared with the neighboringregimes. Theorem 5.6.
Suppose α + β = H and β = H. Suppose also that the conditions in Corollary 3.5hold, and the function σ satisfies the linear growth condition. Then the following formula holds:C β , H , T ( ε , x ε α ) = ε α Z ∞ x ¯ N y √ T σ ( ) ! dy + o ( ε α ) as ε ↓ .Proof. It was established in the proof of Theorem 4.3 that for α = − ε α Z T σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s → − T σ ( ) + σ ( ) Z T n probability. Making slight modifications, we can prove that for α ∈ ( H ] , − ε α Z T σ ( ε H b B s ) ds + Z T σ ( ε H b B s ) dZ s → σ ( ) Z T in probability. Next, using the fact that convergence in probability implies convergencein distribution, we see thatlim ε ↓ P (cid:16) X ε , β , HT ≥ x ε α (cid:17) = ¯ N x √ T σ ( ) ! . (5.12)We have C β , H , T ( ε , x ε α ) = E (cid:20)(cid:16) exp n X ε , β , HT o − exp { x ε α } (cid:17) + (cid:21) = Z ∞ x ε α ( e y − exp { x ε α } ) d h − P β , H , T ε ( y ) i . (5.13)It is not hard to see, by reasoning as in the proof of (5.10) that there exists ε > < ε < ε P β , H , T ε ( y ) ≤ se − y , (5.14)for some constant s > y >
0. The estimate in (5.14) allows us to integrate byparts in (5.13). This gives C β , H , T ( ε , x ε α ) = Z ∞ x ε α P β , H , T ε ( y ) e y dy = ε α Z ∞ x P (cid:16) X ε , β , HT ≥ u ε α (cid:17) exp { u ε α } du . (5.15)Next, using (5.14) again, we can show that the dominated convergence theorem appliesto the integral in (5.15). Finally, taking into account (5.12), (5.14) and (5.15), we establishthe asymptotic formula in Theorem 5.6. Remark 5.7.
To the best of our knowledge, most of the methods, allowing to derive upper callprice estimates in stochastic volatility models from small-time and small-noise large deviationprinciples for the log-price, rely either on the finiteness of nontrivial exponential moments of theintegrated variance, or on the finiteness of the moments of the asset price for small values of thelarge deviation parameter. Some of the ideas used in such proofs go back to [16] , Corollary 2.1,and [42] , Subsection 5.1, where the Heston model was studied. It is worth mentioning that theexponential moment of order p of the log-price in a Heston model is equal to the exponentialmoment of order p ( p − ) of the integrated variance in the Heston model with a different drift inthe volatility equation (see formula (5.4) in [42] and the formula preceding it). The finiteness ofnontrivial exponential moments of the integrated variance plays an important role in some of theproofs of upper call price estimates in Gaussian stochastic volatility models. This can be seen,for instance, in the proof of Corollary 4.13 in [17] , that of Corollary 31 in [25] , and the proofof Theorem 5.2 in the present paper. However, Theorem 6.7 shows that this approach does notwork if the volatility function grows at infinity faster than linearly. Moreover, in uncorrelatedVolterra type models, the asset price process is a martingale, while all of its moments of ordergreater than one explode (see Theorem 6.11). An interesting method was used in [19] . It does notwork for uncorrelated models, if the volatility function grows faster than linearly, but works wellfor correlated models under Assumption (A2) formulated in the next section (see the discussionafter Corollary 6.10). . T HE LINEAR GROWTH CONDITION REVISITED
We have already advertized that the inequality in (5.4) may fail if the volatility function σ grows at infinity a little faster than linearly (see Remark 5.3). In the present section, weprove an assertion (Theorem 6.7), which makes the previous statement more precise. Thisassertion states that under the faster than linear growth condition in Definition 6.1 below,all the positive order exponential moments of the quadratic variation of the driftless log-price are infinite. In addition, for uncorrelated Volterra type Gaussian stochastic volatilitymodels, we establish that under the same growth condition, all the moments of ordergreater than one of the asset price are infinite (see part (i) of Theorem 6.11). Similar resultsfor correlated models are obtained in part (ii) of Theorem 6.11. Definition 6.1.
Let x > . A positive function f defined on [ x , ∞ ) grows faster than linearly ifthere exist x > x and a positive function g defined on [ x , ∞ ) , for which the following conditionshold: g ( x ) → ∞ as x → ∞ and f ( x ) ≥ xg ( x ) for all x > x . In the rest of this section, we will use certain results from the theory of slowly varyingfunctions (see the monograph [5] by Bingham, Goldie, and Teugels).
Definition 6.2.
Let l be a positive continuous function defined on some neighborhood of infinityand such that lim x → ∞ l ( λ x ) l ( x ) = for all λ > . Any function l satisfying the previous condition is called slowly varying. Definition 6.2 in a more general case of measurable functions can be found in Sub-section 1.2.1 of [5]. Slowly varying functions were originally introduced and studied byKaramata in [33]. The class of slowly varying functions is denoted by R .Smoothly varying functions with index 0 play an important role in the theory of slowvariation. Definition 6.3.
A positive function f defined on some neighborhood of infinity is called smoothlyvarying with index 0 if the function h ( x ) = log f ( e x ) is infinitely differentiable in a neighborhoodof infinity and such that h ( n ) ( x ) → as x → ∞ , for all integers n ≥ . The class of all functions of smooth variation with index 0 is denoted by SR . It isknown that SR ⊂ R . Moreover, the function f belongs to the class SR if and only iflim x → ∞ x n f ( n ) ( x ) f ( x ) = n ≥
1. Definition 6.3 in a more general case and the properties of functions ofsmooth variation formulated above can be found in Subsection 1.8.1 of [5].For positive continuous functions f and g defined on a neighborhood of infinity, thestandard symbol f ∼ g will be used when f ( x ) g ( x ) → x → ∞ . An important result in thetheory of slow variation is the Smooth Variation Theorem (see Theorem 1.8.2 in [5]). Wewill need only a special case of this theorem. A complete formulation can be found in [5]. Theorem 6.4.
Let f ∈ R . Then there exist g ∈ SR and h ∈ SR with g ∼ h and g ≤ f ≤ hin a neighborhood of infinity. The following statement follows from Theorem 6.4. orollary 6.5. Let f ∈ R be such that f ( x ) → ∞ as x → ∞ . Then there exist a functiong ∈ SR and a constant x > such that g is defined on ( x , ∞ ) , g ( x ) → ∞ as x → ∞ , andf ( x ) ≥ g ( x ) for all x > x . Let f ∈ R , and let the functions g and h be such as in Theorem 6.4. To prove Corollary6.5, we only need to establish that g ( x ) → ∞ as x → ∞ . It is easy to see that f ∼ g , andthe previous statement follows. Remark 6.6.
It was established by Bojanic and Karamata (see [6] , see also Problem 11 on p. 124in [5] ) that if < g ( x ) → ∞ as x → ∞ , then there exists l ∈ R such that < l ( x ) → ∞ asx → ∞ and g ( x ) ≥ l ( x ) for all x > x . It follows from the previous assertion that the functiong in Definition 6.1 can be replaced by a slowly varying function l such that < l ( x ) → ∞ asx → ∞ . Moreover, the function g can be replaced by a smoothly varying function h such that < h ( x ) → ∞ as x → ∞ (see Corollary 6.5). We will use the previous remarks in the sequel. Our next goal is to prove an assertion confirming what was said in Remark 5.3.
Theorem 6.7.
Suppose the volatility function σ in the model described in (1.1) satisfies a fasterthan linear growth condition introduced in Definition 6.1. Let b B be a nondegenerate centeredGaussian process. Then, for all t ∈ ( T ] and γ > , the following equality holds: E (cid:20) exp (cid:26) γ Z t σ (cid:16) b B s (cid:17) ds (cid:27)(cid:21) = ∞ . (6.2) The equality in (6.2) also holds with the function x σ ( − x ) instead of the function σ .Proof. Recall that we may assume that σ ( x ) ≥ xl ( x ) in a neighborhood of infinity, where l is a slowly varying function such that in Remark 6.6. It follows from the discussionabove that there exist a number x > x and a function h ∈ SR defined on ( x , ∞ ) andsuch that h ( x ) → ∞ as x → ∞ , and l ( x ) ≥ h ( x ) for all x > x . Define a function ˆ σ ( x ) on ( x , ∞ ) by ˆ σ ( x ) = x h ( x ) . Then σ ( x ) ≥ ˆ σ ( x ) for all x > x .Since h is a strictly positive function, we haveˆ σ ′ ( x ) h ( x ) = x (cid:18) + xh ′ ( x ) h ( x ) (cid:19) and ˆ σ ′′ ( x ) h ( x ) = + xh ′ ( x ) h ( x ) + x h ′′ ( x ) h ( x ) .for all x > x . Next, using the condition h ∈ SR , the previous formulas, and (6.1), wesee that there exists x > x such that the function ˆ σ is strictly increasing and convexin [ x , ∞ ) . Define a function ˜ σ on R by the following: ˜ σ ( x ) = ˆ σ ( x ) if x ≥ x , while˜ σ ( x ) = ˆ σ ( x ) if − ∞ < x < x . It is not hard to see that the function ˜ σ is convex in R .For every x ∈ R , we have σ ( x ) ≥ ˜ σ ( x ) − ˆ σ ( x ) . Therefore, to establish (6.2), it sufficesto prove that K = : E (cid:20) exp (cid:26) γ Z t ˜ σ (cid:16) b B s (cid:17) ds (cid:27)(cid:21) = ∞ .Since the function ˜ σ is convex, Jensen’s inequality implies that Z t ˜ σ (cid:16) b B s (cid:17) ds ≥ t ˜ σ (cid:18) t Z t b B s ds (cid:19) . (6.3) t follows that for all y > K ≥ e γ ty P (cid:18) ˜ σ (cid:18) t Z t b B s ds (cid:19) > y (cid:19) .The function ˜ σ is strictly increasing in ( x , ∞ ) . Therefore, for y > y , K ≥ e γ ty P (cid:18) Z t b B s ds > t [ ˜ σ ] − ( y ) (cid:19) ,and hence for u > u , K ≥ exp { γ t ˜ σ ( u ) } P (cid:18) Z t b B s ds > tu (cid:19) = exp { γ tu h ( u ) } P (cid:18) Z t b B s ds > tu (cid:19) . (6.4)The Riemann integral R t b B s ds of a nondegenerate continuous centered Gaussian process b B is a Gaussian random variable with mean zero and variance v = R t R t C ( u , s ) duds , where C is the covariance function of the process b B . Hence P (cid:18) Z t b B s ds > tu (cid:19) = √ π Z ∞ tu √ v exp (cid:26) − w (cid:27) dw with v >
0. Next, using the inequality Z ∞ z e − u du ≥ e − z z + √ z + P (cid:18) Z t b B s ds > tu (cid:19) ≥ √ √ π exp (cid:26) − t u v (cid:27) √ vtu + √ t u + v . (6.5)Finally, by taking into account (6.4), (6.5) and the fact that h ( u ) ↑ ∞ as u → ∞ , we seethat K = ∞ . To prove the last statement in the formulation of Theorem 6.7 we apply theformula in (6.2) to the process − b B .This completes the proof of Theorem 6.7.We will next show that if the volatility function grows at infinity faster than the thirdpower, then an assertion analogous to that in Theorem 6.7 can be established for the ab-solute value of the driftless log-price. Theorem 6.8.
Suppose the volatility function σ appearing in (1.1) is such that there exist a num-ber x > and a function g for which the following conditions hold: < g ( x ) → ∞ as x → ∞ and σ ( x ) ≥ x g ( x ) , x > x . (6.6) Let b B be a nondegenerate continuous Gaussian process adapted to the filtration {F t } ≤ t ≤ T . Then,for all t ∈ ( T ] and γ > , E (cid:20) exp (cid:26) γ (cid:12)(cid:12)(cid:12)(cid:12) Z t σ (cid:16) b B s (cid:17) dZ s (cid:12)(cid:12)(cid:12)(cid:12)(cid:27)(cid:21) = ∞ . (6.7) The equality in (6.7) also holds with the function x σ ( − x ) instead of the function σ . Remark 6.9.
Note that there is a gap between the linear growth condition in Definition 6.1 andthe cubic growth condition in (6.6). We do not know whether Theorem 6.8 holds true if condition(6.6) in it is replaced by the condition in Definition 6.1 . roof of Theorem 6.8. It is clear from Remark 6.6 that we can replace the function g in(6.6) by a function l ∈ R such that l ( x ) → ∞ as x → ∞ . In addition, with no loss ofgenerality, we may assume that E (cid:20) Z T σ (cid:16) b B s (cid:17) ds (cid:21) < ∞ . (6.8)(expand the exponential in (6.7) and use It ˆo’s isometry). Set M t = Z t σ (cid:16) b B s (cid:17) dZ s , 0 ≤ t ≤ T . (6.9)Then the process M is a martingale. Its quadratic variation is given by [ M ] t = Z t σ (cid:16) b B s (cid:17) ds , 0 ≤ t ≤ T .Denote by M ∗ the maximal function associated with the process M , that is M ∗ ( t ) = sup ≤ s ≤ t | M ( s ) | , 0 ≤ t ≤ T .Then, using the Burkholder-Davis-Gundy inequality, more precisely, the lower estimatein it, with the constants such as in [44], Theorem 2, we obtain the following: c √ n || [ M ] ( t ) || n ≤ || M ∗ ( t ) || n ,for all integers n ≥ c >
0. In addition, using Doob’s martin-gale inequality, we get || M ∗ ( t ) || n ≤ nn − || M ( t ) || n , n ≥ n ≥ c √ n || [ M ] ( t ) || n ≤ || M ( t ) || n ,with an absolute constants c > L γ , t the expectation in (6.7). Then the previous estimates give L γ , t ≥ E " ∞ ∑ n = c n γ n n ! n n (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n + γ E [ | M ( t ) | ] − c γ E [[ M ] ( t )] ≥ E " ∞ ∑ n = c n γ n n ! n n (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n − c γ E "(cid:18) Z T σ ( b B ) ds (cid:19) ≥ E " ∞ ∑ n = c n γ n n ! n n (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n − C , (6.10)where C > t . It follows from (6.10) that in order to prove the equality L γ , t = ∞ , it suffices to prove that K t : = E " ∞ ∑ n = c n γ n n ! n n (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n = ∞ . (6.11) t follows from Stirling’s formula that there exists c > n ≥ n n ≤ c n n !. Therefore, (6.11) implies that for some c > K t ≥ E " ∞ ∑ n = c n γ n ( n ! ) (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n .Let us set e K t : = E ∞ ∑ n = c n γ n n ! (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n , (6.12)where c = c . Next, applying H ¨older’s inequality with p = and q = e K t ≤ τ E " ∞ ∑ n = c n γ n ( n ! ) (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) n ≤ τ K t ,where τ = (cid:8) ∑ ∞ n = − n (cid:9) . Therefore, K t ≥ τ E " exp ( c γ (cid:18) Z t σ (cid:16) b B s (cid:17) ds (cid:19) ) , (6.13)for all t ∈ ( T ] . Note that we have not yet used the faster than cubic growth conditionfor the function σ . It follows from (6.13) that in order to finish the proof of Theorem 6.8,it suffices to show that e L γ , t = ∞ for all γ > t ∈ ( T ] , where e L γ , t is the expectationon the right-hand side of (6.13)The rest of the proof of Theorem 6.8 follows that of Theorem 6.7. We first choose afunction h ∈ SR such that h ( x ) → ∞ as x → ∞ , and moreover l ( x ) ≥ h ( x ) for all x > x . The function x x h ( x ) is strictly increasing and convex on [ x , ∞ ) . Set σ ( x ) = x h ( x ) { x > x } + x h ( x ) { x ≤ x } , x ∈ R . Then it is clear that the function σ is convexon R and σ ( x ) ≥ σ ( x ) − x h ( x ) for all x ∈ R . Next, using Jensen’s inequality as inthe proof of (6.3), we can estimate e L γ , t from below by an expression similar to the lastexpression in (6.4) with the function h instead of the function h . Finally, taking intoaccount Theorem 6.7, we establish the equality e L γ , t = ∞ .This completes the proof of Theorem 6.8. Corollary 6.10.
Suppose the conditions in Theorem 6.8 hold. Then, for every t ∈ ( T ] , at leastone of the next two conditions hold: E (cid:20) exp (cid:26) γ Z t σ (cid:16) b B s (cid:17) dZ s (cid:27)(cid:21) = ∞ for all γ >
0, (6.14) or E (cid:20) exp (cid:26) − γ Z t σ (cid:16) b B s (cid:17) dZ s (cid:27)(cid:21) = ∞ for all γ >
0. (6.15)
Proof.
Using (6.7) and the inequality e | u | ≤ e u + e − u , u ∈ R , we see that for every t ∈ ( T ] and γ > E (cid:20) exp (cid:26) γ Z t σ (cid:16) b B s (cid:17) dZ s (cid:27)(cid:21) = ∞ , (6.16) r E (cid:20) exp (cid:26) − γ Z t σ (cid:16) b B s (cid:17) dZ s (cid:27)(cid:21) = ∞ . (6.17)Fix t >
0. If there is no γ for which (6.16) holds, then (6.17) should hold for all γ > t . Otherwise set a t = inf { γ > } .It is easy to see using H ¨older’s inequality that (6.16) is valid for all γ > a t . Suppose a t = a t >
0, then (6.17) is true for all γ ∈ ( a t ) , andhence it is also true for all γ > S satisfies the following conditions:(iiia) S is a martingale;(iiib) For every 1 < γ < ∞ there exists t > E (cid:2) S γ t (cid:3) < ∞ (see Assumption 2.4 in Section 2 of [4]). It was shown in [4] that if conditions (iiia) and(iiib) hold, then upper large deviation style estimates for the call price follow from thecorresponding large deviation principle.An interesting improvement of the previous statement was obtained in [19], where thesame implication was obtained under weaker restrictions (see Assumption (A2) in [19]).In terms of the time parameter t , Assumption (A2) is as follows: There exists γ > t → E [ S γ t ] < ∞ . It is not hard to see that if condition (iiia) holds, thenAssumption (A2) follows from the following assumption: (i) There exist γ > t > E (cid:2) S γ t (cid:3) < ∞ .In the rest of the present section, we consider Volterra type Gaussian stochastic volatil-ity models. It will be shown next that for uncorrelated Volterra type models, Assumption(A2) does not hold if the volatility function σ grows faster than linearly. More precisely,all the moments of order greater than one or less than zero of the asset price explode (seeTheorem 6.11). Moreover, we prove in Theorem 6.11 that for correlated models ( ρ = γ ∈ ( − ∞ , 0 ) ∪ ( − ρ , ∞ ) . In the case, where γ = − ρ in a corre-lated model, the moment explosion results obtained in the present section are only partial(see Theorem 6.13). Note that in the assertions established in the rest of the present sec-tion, we only assume that the Volterra type process b B satisfies the conditions in Definition2.1. Let us also recall that with no loss of generality we are assuming that s = S in an uncorrelated Gaussian stochastic volatility model with s = S t = exp (cid:26) − Z t σ ( b B s ) ds + Z t σ ( b B s ) dW s (cid:27) , 0 ≤ t ≤ T ,Since Brownian motions W and B are independent, it is rather standard to prove, byconditioning on the path of the process B, that the process S is a martingale. Therefore,(iiia) holds true for the uncorrelated model. Moreover, for all t ∈ [ T ] , we have E [ S t ] =
1, 0 ≤ t ≤ T . (6.18) or negatively correlated models ( ρ < S is a martingale under the following additional condition: Condition (G):
For every a ∈ R , the function σ is bounded on ( − ∞ , a ] .It follows that the equality in (6.18) holds for any negatively correlated Volterra typemodel, in which the volatility function σ satisfies condition (G). On the other hand, forevery η ∈ R , the stochastic exponential t exp (cid:26) − η Z t σ ( b B s ) ds + η Z t σ ( b B s ) dB s (cid:27) is a strictly positive local martingale. Hence it is a supermartingale, and therefore E (cid:20) exp (cid:26) − η Z t σ ( b B s ) ds + η Z t σ ( b B s ) dB s (cid:27)(cid:21) ≤
1, (6.19)for every t ∈ ( T ] . Note that condition (G) is not needed to establish the validity of(6.19).The next assertion is one of the main results of the paper. It concerns moment ex-plosions in Gaussian stochastic volatility models (see the discussion in the introduction,where we compare Theorem 6.11 with the results obtained in [24]). Theorem 6.11. (i) Suppose the volatility function σ in an uncorrelated Volterra type Gaussianstochastic volatility model satisfies the faster than linear growth condition formulated in Definition6.1. Then the following equality holds: E (cid:2) S γ t (cid:3) = ∞ for all γ ∈ ( − ∞ , 0 ) ∪ ( ∞ ) and t ∈ ( T ] . (6.20) (ii) Suppose ρ = , and suppose also that the faster than linear growth condition holds in a Volterratype Gaussian stochastic volatility model. Then, the following equality is valid: E (cid:2) S γ t (cid:3) = ∞ for all γ ∈ ( − ∞ , 0 ) ∪ ( − ρ , ∞ ) and t ∈ ( T ] . (6.21) Proof.
It is clear that if ρ =
0, then E (cid:2) S γ t (cid:3) = E (cid:20) exp (cid:26) γ − γ Z t σ ( b B s ) ds (cid:27) exp (cid:26) − γ Z t σ ( b B s ) ds + γ Z t σ ( b B s ) dW s (cid:27)(cid:21) .Using the independence of W and B and conditioning on the path of the process B, weobtain E (cid:2) S γ t (cid:3) = E (cid:20) exp (cid:26) γ − γ Z t σ ( b B s ) ds (cid:27) E t , γ (cid:21) ,where E t , γ = E (cid:20) exp {− γ Z t σ ( b B s ) ds + γ Z t σ ( b B s ) dW s }| B s , 0 ≤ s ≤ t (cid:21) . (6.22)It is not hard to prove that for all t ∈ ( T ] and γ such as in part (i) of the theorem, wehave E t = E (cid:2) S γ t (cid:3) = E (cid:20) exp (cid:26) γ − γ Z t σ ( b B s ) ds (cid:27)(cid:21) . (6.23) ow suppose ρ =
0. Then the following equalities are true: E [( S t ) γ ] = E (cid:20) exp (cid:26) − γ Z t σ (cid:16) b B s (cid:17) ds + γ Z t σ (cid:16) b B s (cid:17) dZ s (cid:27)(cid:21) = E [ exp { γ ¯ ρ − γ Z t σ (cid:16) b B s (cid:17) ds + γρ Z t σ (cid:16) b B s (cid:17) dB s } exp {− γ ¯ ρ Z t σ (cid:16) b B s (cid:17) ds + γ ¯ ρ Z t σ (cid:16) b B s (cid:17) dW s } ] .By conditioning on the path of the process B and reasoning as in the proof of (6.23), weobtain the following generalization of the formula in (6.23): E [( S t ) γ ] = E (cid:20) exp (cid:26) γ ¯ ρ − γ Z t σ ( b B s ) ds + γρ Z t σ (cid:16) b B s (cid:17) dB s (cid:27)(cid:21) . (6.24)Let us first prove the equality in (6.20). By taking into account the restriction γ − γ > η ∈ R , and fix p > q > p + q =
1. Then (6.24), (6.19), and H ¨older’sinequality imply the following estimate: (cid:8) E (cid:2) S γ t (cid:3)(cid:9) p ≥ (cid:8) E (cid:2) S γ t (cid:3)(cid:9) p (cid:26) E (cid:20) exp (cid:26) − η Z t σ ( b B s ) ds + η Z t σ ( b B s ) dB s (cid:27)(cid:21)(cid:27) q ≥ E (cid:20) exp (cid:26)(cid:18) γ ¯ ρ − γ p − η q (cid:19) Z t σ ( b B s ) ds + (cid:18) γρ p + η q (cid:19) Z t σ ( b B s ) dB s (cid:27)(cid:21) . (6.25)Let us choose η = − γρ p − . Then γρ p + η q =
0. Next, using (6.25), we obtain (cid:8) E (cid:2) S γ t (cid:3)(cid:9) p ≥ E (cid:20) exp (cid:26)(cid:18) γ ¯ ρ − γ − η ( p − ) p (cid:19) Z t σ ( b B s ) ds (cid:27)(cid:21) . (6.26)It remains to choose p > l : = γ ¯ ρ − γ − η ( p − ) >
0. Then (6.26) and Theorem6.7 will show that the equality in 6.21 holds.We have l = γ ¯ ρ − γ − γ ρ p − . If γ > − ρ , then γ ¯ ρ − γ >
0, and the condition l > p > + γρ γ ¯ ρ − . Such a number p > γ <
0, then the condition l > p > + γ ρ γ ( − ρ )+ | γ | . As before, the previous condition allows us to easilychoose p . It follows that (6.21) holds.The proof of Theorem 6.11 is thus completed. Remark 6.12.
We are indebted to Paul Gassiat for a suggestion to use the semimartingale propertyin (6.19) in the proof of part (ii) of Theorem 6.11. Our original proof used condition (G) and themartingale property in (6.18).
The next theorem complements Theorem 6.11. However, the conclusion in this theoremis weaker than that in Theorem 6.11. heorem 6.13. Suppose the volatility function σ in a Volterra type Gaussian stochastic volatilitymodel satisfies the faster than cubic growth condition formulated in (6.6). Let ρ = , and let f M bethe Volterra type model, in which the volatility function and the volatility process are the same asin the given model, while the correlation parameter is − ρ instead of ρ . Denote by e S the asset priceprocess in the model f M . Then, for every t ∈ ( T ] , at least one of the following two conditionsholds: E (cid:20) ( S t ) − ρ (cid:21) = ∞ , or E (cid:20) ( e S t ) − ρ (cid:21) = ∞ . Proof.
It follows from (6.24) that if ρ =
0, then E (cid:20) ( S t ) − ρ (cid:21) = E (cid:20) exp (cid:26) ρ − ρ Z t σ (cid:16) b B s (cid:17) dB s (cid:27)(cid:21) .Similarly, we have E (cid:20) ( e S t ) − ρ (cid:21) = E (cid:20) exp (cid:26) − ρ − ρ Z t σ (cid:16) b B s (cid:17) dB s (cid:27)(cid:21) .Now, it is clear that Theorem 6.13 follows from Corollary 6.10. Remark 6.14.
The condition γ ≥ − ρ in the context of moment explosions appears in the paper [31] of Jourdain, where the Scott model is studied. In the Scott model, the volatility function is σ ( x ) = e x , while the volatility process is the Ornstein-Uhlenbeck process. To adapt Jourdain’sresults to our setting, we consider driftless Ornstein-Uhlenbeck processes. Jourdain proved thatfor the Scott model with the driftless Ornstein-Uhlenbeck process as the volatility process, equality(6.21) holds if ρ = . Moreover, he established that if ρ < , then for given t > and γ > , E [ S γ t ] < ∞ if and only if γ < − ρ (see Proposition 6 in [31] ). For ρ > , Jourdain proved that E [ S γ t ] = ∞ if γ ≥ − ρ , and mentioned that it does not seem easy to analyze whether E [ S γ t ] < ∞ if γ < − ρ (see Remark 7 in [31] ).
7. A
SYMPTOTIC BEHAVIOR OF THE IMPLIED VOLATILITY IN MIXED REGIMES
In this section, we describe small-noise asymptotic behavior of the implied volatility inthe mixed regimes considered in the previous section.The implied volatility can be determined from the equality C β , H , T ( ε , x ε α ) = C BS ( ε , x ε α ; σ = b σ β , H , T ( ε , x ε α ))= C − ( x ε α , √ ε b σ β , H , T ( ε , x ε α )) . (7.1)In the cases, where 0 ≤ α + β < H , Theorem 5.2 implies that L ( ε ) = : log 1 C − ( x ε α , √ ε b σ β , H , T ( ε , x ε α )) = J T ( x ) ε α + β − H + o (cid:16) ε α + β − H (cid:17) (7.2)as ε ↓
0. In the previous formula, the symbol J T stands for the rate function I T definedin (2.2), in the case where α + β = J T ( x ) = x T σ ( ) , in the case where 0 < α + β < H , and the ssumptions in part (ii) of Theorem 5.2 hold. In (7.2), the parametrized dimensionlessimplied volatility is given by ν ( ε ) = √ ε b σ β , H , T ( ε , x ε α ) . Moreover, we have k ( ε ) L ( ε ) = O (cid:16) ε H − α − β (cid:17) as ε →
0. Therefore, k ( ε ) L ( ε ) → ε →
0. This means that the formula in Remark 7.3 in[21] can be applied to characterize the asymptotic behavior of the dimensionless impliedvolatility ε ν ( ε ) in the mixed regime. In our case, the formula in [21], Remark 7.3, givesthe following: (cid:12)(cid:12)(cid:12)(cid:12) k ( ε ) L ( ε ) − ε b σ β , H ( ε , x ε α ) (cid:12)(cid:12)(cid:12)(cid:12) = o (cid:18) k ( ε ) L ( ε ) (cid:19) as ε ↓
0. It follows that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k ( ε ) p L ( ε ) − √ ε b σ β , H ( ε , x ε α ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = o k ( ε ) p L ( ε ) ! (7.3)as ε ↓
0. Next, taking into account (7.2), we obtain the following assertion.
Theorem 7.1. (i) Suppose the conditions in Theorem 2.7 hold. Suppose also that α + β = , andthe linear growth condition holds for the function σ . Then b σ β , H , T ( ε , x ε α ) = x p I T ( x ) ε H − β − + o (cid:16) ε H − β − (cid:17) as ε ↓ .(ii) Suppose the conditions in Corollary 3.5 hold. Suppose also that < α + β < H, and thelinear growth condition holds for the function σ . Then b σ β , H , T ( ε , x ε α ) = √ T σ ( ) ε H − β − + o (cid:16) ε H − β − (cid:17) as ε ↓ . Let α = β = H . Then, for the Black-Scholes model with σ = b σ H , H , T ( ε , x ) , theequality in (5.11) takes the following form: C BS ( ε , x ; b σ H , H , T ( ε , x ))= Z ∞ x e y ¯ N (cid:18) y √ ε b σ H , H , T ( ε , x )) + √ ε b σ H , H , T ( ε , x ) (cid:19) dy . (7.4)Using (7.1), (5.5), and (7.4), we obtain Z ∞ x e y ¯ N y √ T σ ( ) + √ T σ ( ) ! dy = lim ε ↓ Z ∞ x e y ¯ N (cid:18) y √ ε b σ H , H , T ( ε , x )) + √ ε b σ H , H , T ( ε , x ) (cid:19) dy , (7.5)for all x > ε j , j ≥
1, be a positive sequence such that ε j → j → ∞ , and the limit τ = lim j → ∞ p ε j b σ H , H , T ( ε j , x ) xists (finite or infinite). Applying Fatou’s lemma to the expression on the right-hand sideof (7.5) and taking into account the fact that the call price function C − is strictly increasingin ν (see Remark 5.5), we see that τ ≤ σ ( ) . Therefore, for j ≥ j , p ε j b σ H , H , T ( ε j , k = x ) ≤ C ,where C > j ≥ j " e y ¯ N y √ ε j b σ H , H , T ( ε j , x )) + p ε j b σ H , H , T ( ε j , x ) ! ≤ e y ¯ N (cid:16) yC (cid:17) .The previous estimate allows us to apply the dominated convergence theorem in for-mula (7.5) (along the sequence ε j ). This gives C − ( x , √ T σ ( )) = C − ( x , τ ) , and hence τ = √ T σ ( ) . Now, it is clear thatlim ε ↓ √ ε b σ H , H , T ( ε , x ) = √ T σ ( ) .Therefore, the following statement holds. Theorem 7.2.
Suppose α = and β = H. Then, under the assumptions in Corollary 3.5 and thelinear growth condition, b σ H , H , T ( ε , x ) = √ T σ ( ) ε − + o (cid:16) ε − (cid:17) as ε ↓ . We will next turn our attention to the only remaining case of the implied volatilityestimates in mixed regimes.
Theorem 7.3.
Suppose α + β = H and α ∈ ( H ] . Then, under the assumptions in Corollary 3.5and the linear growth condition, the following asymptotic formula holds for the implied volatility: b σ β , H , T ( ε , x ε α ) = x ε α − q α log ε + o ε α − q log ε (7.6) as ε ↓ .Proof. It follows from Theorem 5.6 that L ( ε ) = log 1 C β , H , T ( ε , x ε α ) = α log 1 ε − log Z ∞ x ¯ N y √ T σ ( ) ! dy + o ( ) as ε ↓
0. We also have k ( ε ) = x ε α , and hence k ( ε ) L ( ε ) → ε ↓
0. Next, applying the formulain Remark 7.3 in [21] (see (7.3) above), we derive (7.6).The proof of Theorem 7.3 is thus completed.A
CKNOWLEDGEMENTS
I thank Jean-Dominique Deuschel, Peter Friz, Josselin Garnier, Paul Gassiat, Stefan Ger-hold, Benjamin Jourdain, Barbara Pacciarotti, Paolo Pigato, and Knut Sølna for their at-tention to the paper and valuable remarks. EFERENCES [1] M. Abramovitz and I. A. Stegun (Eds.).
Handbook of Mathematical Functions.
Applied Mathematics Se-ries 55, National Bureau of Standards, Washington, 1972.[2] J.-M. Az¨ais and L.-V. Lozada-Chang. A toolbox on the distribution of the maximum of Gaussian pro-cesses. 2013. hal-00784874.[3] P. Baldi and L. Caramellino. General Freidlin-Wentzell large deviations and positive diffusions.
Statis-tics and Probability Letters , 81 (2011), 1218-1229.[4] C. Bayer, P. K. Friz, A. Gulisashvili, B. Horvath, and B. Stemper. Short-time near-the-money skew inrough fractional volatility models, (2018) Quantitative Finance, DOI: 10.1080/14697688.2018.1529420;available on arXiv:1703.05132, 2017.[5] N. H. Bingham, C. M. Goldie, and J. L. Teugels.
Regular Variation . Cambridge University Press, 1987.[6] R. Bojanic and J. Karamata.
On Slowly Varying Functions and Asymptotic Relations . Math. Research Cen-ter Tech. Report 432, Madison, Wis, 1963.[7] M. Cellupica and B. Pacchiarotti. Pathwise asymptotics for Volterra type stochastic volatility models.Submitted for publication, available on arXiv:1902.05896.[8] P. Cheridito, H. Kawaguchi, M. Maejima. Fractional Ornstein-Uhlenbeck processes.
Electron. J. Probab. ,8 (2003), 1-14.[9] G. Conforti, S. De Marco, and J.-D. Deuschel. On small-noise equations with degenerate limiting sys-tem arising from volatility models. In:
Large Deviations and Asymptotic Methods in Finance . P. K. Friz, J.Gatheral, A. Gulisashvili, A. Jacquer, J. Teichmann (Eds.), Springer International Publishing Switzer-land, 2015, 473-505.[10] L. Decreusefond. Regularity properties of some stochastic Volterra integrals with singular kernels.
Potential Analysis , 16 (2002), 139-149.[11] A. Dembo and O. Zeitouni.
Large Deviations Techniques and Applications . Springer-Verlag Berlin Heidel-berg, 2010.[12] J.-D. Deuschel and D. W. Stroock.
Large deviations . Academic Press, 1989[13] P. Eichelsbacher and M. L ¨owe. Moderate deviations for i.i.d. random variables.
ESAIM: Probability andStatistics , 7 (2003), 209-218.[14] S. El Rahouli. Financial modeling with Volterra processes and applications to options, interest ratesand credit risk. These, Universit´e de Lorraine, Universit´e du Luxembourg, 2014.[15] W. H. Fleming. Exit probabilities and optimal stochastic control.
Appl. Math. Optim. , 4 (1978), 329-346.[16] M. Forde and A. Jacquier. Small-time asymptotics for implied volatility under the Heston model.
Int.J. Theor. Appl. Finan. , 12 (2009), 861-876.[17] M. Forde and H. Zhang. Asymptotics for rough stochastic volatility models.
SIAM Journal on FinancialMathematics , 8 (2017), 114-145.[18] M. I. Freidlin and A. D. Wentzell.
Random Perturbations of Dynamical Systems . Springer-Verlag NewYork,1998.[19] P. K. Friz, P. Gassiat, and P. Pigato. Precise asymptotics: Robust stochastic volatility models, availableon arXiv:1811.00267, 2018.[20] P. K. Friz, S. Gerhold, and A. Pinter. Option pricing in the moderate deviations regime.
MathematicalFinance , 28 (2018), 962-988.[21] K. Gao and R. Lee. Asymptotics of implied volatility to arbitrary order.
Finance Stoch. , 18 (2014), 349-392.[22] J. Garnier and K. Sølna. Correction to Black-Scholes formula due to fractional stochastic volatility.
SIAM J. Financial Math. , 8 (2017), 560-588.[23] J. Garnier and K. Sølna. Option pricing under fast varying long-memory stochastic volatility. To bepublished in
Mathematical Finance , available on arXiv:1604.00105, 2017.[24] P. Gassiat. On the martingale property in the rough Bergomi model. Pre-print, avaliable onarXiv:1811.10935, 2018.[25] A. Gulisashvili. Large deviation principle for Volterra type fractional stochastic volatility models,
SIAM J. Financial Math. , 9 (2018), 1102-1136.
26] A. Gulisashvili, F. Viens, and X. Zhang. Small-time asymptotics for Gaussian self-similar stochasticvolatility models.
Appl. Math. Optim. (2018). https://doi.org/10.1007/s00245-018-9497-6, 41 p.; avail-able on arXiv:1505.05256, 2016.[27] A. Gulisashvili, F. Viens, and X. Zhang. Extreme-strike asymptotics for general Gaussian stochasticvolatility models.
Ann Finance (2018). https://doi.org/10.1007/s10436-018-0338-z, 43 p.; available onarXiv:1502.05442v3, 2017.[28] H. Hult. Approximating some Volterra type stochastic integrals with application to parameter estima-tion.
Stochastic Processes and their Applications , 105 (2003), 1-32.[29] H. Hult.
Extremal behavior of regularly varying stochastic processes . Doctoral Dissertation, Royal Instituteof Technology, Stockholm 2003.[30] E. A. Jaber, M. Larsson, and S. Pulido. Affine Volterra processes. Pre-print, available onarXiv:1708.08796v2, 2017.[31] B. Jourdain. Loss of martingality in asset price models with lognormal stochastic volatility. PreprintCERMICS 2004-267.[32] T. Kaarakka and P. Salminen. On fractional Ornstein-Uhlenbeck processes.
Communications on Stochas-tic Analysis , 5 (2011), 121-133.[33] J. Karamata. Sur un mode de croissance r´eguli`ere des fonctions.
Mathematica (Cluj) , 4 (1930), 38-53,[34] I. Karatzas and S. E. Shreve.
Brownain Motion and Stochastic Calculus , Second Edition, Springer-Verlag,1991.[35] A. N. Kolmogorov. Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum.
Doklady Acad. USSR , 26 (1940), 115-118.[36] P. L´evy. Wiener’s random functions, and other Laplacian random functions. Proc. Sec. Berkeley Symp.Math. Statist. Probab., Vol II, University of California Press, Berkeley, CA, 1950, pp. 171-186.[37] M. Lifshits.
Lectures on Gaussian Processes . Springer Verlag, 2012.[38] S. C. Lim and V. M. Sithi. Asymptotic properties of the fractional Brownian motion of Riemann-Liouville type.
Physics Letters A , 206 (1995), 311-317.[39] B. Mandelbrot and J. W. van Ness. Fractional Brownian motions, fractional noises and applications.
SIAM Review , 10 (1968), 422-437.[40] L. Mytnik and E. Neuman. Sample path properties of Volterra processes.
Communications on StochasticAnalysis , 6 (2012), 359-377.[41] G. Peˇ s kir. On the exponential Orlicz norm of stopped Brownian motion. Studia Mathematica
S´eminaire de Probabilit´es,Springer-Verlag , XLIII (2011), 3-70.[44] Y.-F. Ren. On the Burkholder-Davis-Gundy inequalities for continuous martingales.
Statistics and Prob-ability Letters , 78 (2008), 3034-3039.[45] D. Revuz and M. Yor.
Continuous Matringales and Brownian Motion . Springer-Verlag Berlin Heidelberg,1999.[46] S. Robertson. Sample Path Large Deviations and Optimal Importance Sampling for Stochastic Volatil-ity Models.
Stochastic Processes and Their Applications , 120 (2010), 66-83.[47] L. Scott. Option pricing when the variance changes randomly: theory, estimation, and an application.
Journal of Financial and Quantitative Analysis , 22 (1987), 419-438.[48] T. Sottinen and L. Viitasaari. Stochastic analysis of Gaussian processes via Fredholm representation.
International Journal of Stochastic Analysis , Volume 2016, Article ID 8694365, 15 pages.[49] M. Struwe.
Variational Methods: Applications to Nonlinear Partial Differential Equations and HamiltonianSystems , Fourth Edition, Springer-Verlag Berlin Heidelberg, 2008.[50] S. R. S. Varadhan.
Large Deviations and Applications . Society for Industrial and Applied Mathematics,Philadelphia, Pennsylvania, 1984.[51] A. D. Ventsel’ and M. I. Freidlin. On small random perturbations of dynamical systems.
Russian Math.Surveys , 25 (1970), 1-56.
52] A. D. Ventsel’ and M. I. Freidlin. Some problems concerning stability under small random perturba-tions.
Theory Prob. Appl. , 17 (1972), 269-283.[53] X. Zhang. Euler schemes and large deviations for stochastic Volterra equations with singular kernels.
Journal of Diff. Equations , 244 (2008), 2226-2250., 244 (2008), 2226-2250.