Short dated smile under Rough Volatility: asymptotics and numerics
SSHORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS
P. K. FRIZ, P. GASSIAT, AND P. PIGATO
Abstract.
In [Precise Asymptotics for Robust Stochastic Volatility Models; Ann. Appl. Probab. 2020] we introducea new methodology to analyze large classes of (classical and rough) stochastic volatility models, with special regardto short-time and small noise formulae for option prices, using the framework [Bayer et al; A regularity structure forrough volatility; Math. Fin. 2020]. We investigate here the fine structure of this expansion in large deviations andmoderate deviations regimes, together with consequences for implied volatility. We discuss computational aspectsrelevant for the practical application of these formulas. We specialize such expansions to prototypical rough volatilityexamples and discuss numerical evidence. Introduction
In [21], precise short-time asymptotics were established for call and put option prices under stochastic volatility,under a set of abstract conditions satisfied by most classical and rough volatility (RoughVol) models. These resultsare refinements of large deviation statements, providing the higher order, algebraic term in an asymptotic expression,known as Laplace expansion. For RoughVol models, short dated large deviation pricing is due to Forde and Zhang[20], as is the induced implied volatility expansion (FZ expansion), which can be seen as a “rough” BBF (Berestycki-Busca-Florent [12]) formula. Our precise asymptotics provide a mechanism to compute refined implied volatilityexpansions, for log-strike k t = xt / − H , of the form(1.1) σ BS ( t, k t ) = Σ ( x ) + t H a ( x ) + o ( t H ) as t ↓ , where the zero-order Σ( x ) term corresponds to the rough BBF formula in [20]. The next-order term is seen oforder t H and hence increasingly important for small Hurst parameter H , the basic premise of RoughVol modelling.Inclusion of this term hinges on an accurate evaluation of a . In this paper, we assume that the volatility processis of the form σ ( (cid:99) W t , t H ), where (cid:99) W is a fractional Brownian motion (fBM) with Hurst exponent H ∈ (0 , / ρ -correlated with the Brownian driving the asset.The functions Σ( x ) and a ( x ) do not have explicit expressions and we discuss how to compute them numerically.Following [20], Σ( x ) can be computed using the Ritz method. Moreover, we propose a method for computing a ( x )based on a Karhunen-Love (KL) decomposition of the Brownian motions. (This entails a numerical approximationto an infinite-dimensional Carleman-Fredholm determinant.)We also derive near-the-money (meaning, as x →
0) expansions of Σ( x ) and of the term structure a ( x ) whichcan alternatively be used for numerics (and have the advantage of being explicit functions of model parameters).From these asymptotics, we derive consequences for at-the-money (ATM) implied skew and curvature. We alsorefine some moderate deviation asymptotics for call prices and implied volatilities, cf. [22, 6, 31, 36].Being able to evaluate Σ( x ) and a ( x ) allows us to test the accuracy of the short-time asymptotics in practice.We do so with a numerical case study of the rough Bergomi (rBergomi) model. To exploit our general framework, TU and WIAS BerlinU Paris Dauphine, PSL UniversityU Rome Tor Vergata, Department of Economics and Finance
Date : September 21, 2020.2010
Mathematics Subject Classification.
Key words and phrases. rough volatility, European option pricing, implied volatility, small-time asymptotics, rough paths, regularitystructures, Karhunen-Love.We are grateful to C. Bayer and M. Fukasawa for discussion and to M. Pakkanen for the R code for simulating the rough Bergomimodel. PKF and PP gratefully acknowledge financial support from European Research Council Grant CoG-683164 and German sciencefoundation (DFG) via the cluster of excellence MATH+, project AA4-2. PG acknowledges financial support from the French ANR viathe project ANR-16-CE40- 0020-01. a r X i v : . [ q -f i n . C P ] S e p PETER K. FRIZ, PAUL GASSIAT, AND P. PIGATO we add one parameter to the classical rBergomi formulation and look at a volatility given by(1.2) σ ( (cid:99) W t , t H ) = σ exp (cid:18) η (cid:99) W t − θη t H (cid:19) , so that for θ = 1 we get the rBergomi model considered in [4, 11], for θ = 0 the modified version in [6, 20]. Wecompare our approximation to the FZ expansion from [20] and to the Edgeworth asymptotics in [14]. We considerhow smiles vary as θ varies in (1.2) and as expiry t increases. We discuss and test the volatility term structureand its slope ATM, and observe how the term a ( x ) t H improves the asymptotics as H decreases. We observe thesame feature when we implement the moderate deviation asymptotics for implied volatility, where for H small theinclusion of the term structure correction a (0) t H significantly improves on the numerical results presented in [6].Proofs rely on stochastic Taylor expansions, rate function representations in [20, 6] and on the local analysison the Wiener space introduced in [21, 5]. The classical Gao-Lee results [27] are used to go from option prices toimplied volatility asymptotics both in large and moderate deviation regimes. Rough Volatility.
It has been shown in recent years that RoughVol models provide great fits to observed volatilitysurfaces [4] capturing fundamental stylized facts of implied volatility in a parsimonious way. Specifically, this classof models can reproduce the steep short end of the smile, displaying exploding implied skew [3, 23, 24], and theyare the only models consistent with the power law of the skew [4, 40] not admitting arbitrage [25]. RoughVol is alsosupported by statistical and time series analysis [28, 26, 10] and by market microstructure considerations [15]. Manyauthors have even argued that H ≈
0, such as to be consistent with a skew explosion close to t − / [4, 8]. One mainaspect of RoughVol is non-Markovianity. This is a serious complication when it comes to pricing, as Monte Carlomethods become more expensive and PDE methods are not available. For this reason, efficient simulation schemeshave been proposed [7, 11, 42]. Fourier based methods are available for the rough Heston model [16]. Deep andmachine learning approaches have also recently been discussed in [9, 30]. Small maturity approximations are usedin this context to obtain starting points for calibration procedures, which are then based on numerical evaluations. Asymptotic option pricing.
Classical motivation for (semi-closed form) asymptotic pricing includes fast cali-bration, and a quantitative understanding of the impact of model parameters on relevant quantities such as impliedskew and curvature/convexity along the moneyness dimension or slope along the term-structure dimension. Explicitexpressions for such quantities (that follow in this setting from our expansion) and their shape characteristics arealso used to choose the most appropriate model to be fitted to data [1], leave alone being the origin of some widelyused parametrisations of the volatility surface. An interesting, if recent, addition to this list comes from a machinelearning perspective: the form of an expansion such as (1.1) may be viewed as expert knowledge , which significantlynarrows the learning task to finer information such as the error in that expansions; it is equally conceivable to learn a = a ( x ) and other components in the expansion.Under Markovian stochastic volatility, expansion (1.2) is analogous, e.g., to the result derived in [19] for theHeston model. There, the term structure is a ( x ) t (due to the diffusive scaling of the volatility), whereas here thecorrection term is a ( x ) t H (due to the rough scaling of the volatility). Similar expansions are derived also in [46],for more general Markovian models, and (formally) in [43, 44] for Markov stochastic volatility models with jumps.In recent years several authors have studied the short-time behavior of RoughVol models. Theoretical results onshort-time skew and curvature are given in [24, 2]. A second order short-time expansion is given in [14] for general(rough) stochastic volatility models. In [35], the pathwise large deviation behavior under rBergomi dynamics isstudied. Pathwise large and moderate deviation principles for (possibly rough) Gaussian stochastic volatility modelsare established in [31, 32], together with asymptotic results at the central limit (Edgeworth) regime. For the roughHeston model, the recent work [17] provides call expansions of the same type as ours, involving the energy functionand the first order algebraic term, at the same large deviations regime k t = xt / − H . (The rigid infinite-dimensionalaffine structure which underlies [17] is not available for rBergomi type models as considered in this work.) As alreadymentioned, our work builds on the large deviations principle proved in [20] for models with volatility σ ( (cid:99) W t ), and on[6], where the at-the-money behavior of the Forde-Zhang rate function is used to prove moderate deviation priciplesand implied volatility asymptotics for the same type of models. The theoretical foundations of the present paperare given in [21].In Section 2 we explain our RoughVol setting. In Section 3 we state and comment our results. In Section 4 wediscuss and implement our results in the case of the rBergomi model. In Section 5 we show how Σ and a can becomputed using Ritz method and KL decomposition. We collect all the proofs in Section 6. HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 3 Preliminaries on rough volatility
We consider the following RoughVol model, with H ∈ (0 , / r = 0 and S = 1(2.1) dS t S t = σ ( (cid:99) W t , t H ) d ( ρW t + ρW t )where W, W are independent Brownian motions (BM) and ρ ∈ ( − , ρ + ρ = 1. We also write (cid:102) W = ρW + ρW .Moreover, (cid:99) W = ( (cid:99) W t ) t ≥ is a Gaussian Volterra process of the form(2.2) (cid:99) W t = ( K ∗ ˙ W ) t = (cid:90) t K ( t, s ) dW s , for a kernel K ( t, s ) such that (cid:99) W is self-similar with exponent H ∈ (0 , / (cid:99) W ε t : t ≤ t ) = ε H Law( (cid:99) W t : t ≤ t ) , for some t > . The BM W drives the stochastic “rough” volatility, meaning (with abusive notation) that σ ( t, ω ) = σ ( (cid:99) W t , t H ),where σ ( x, y ) is a smooth deterministic real-valued function. We denote σ (cid:48) ( x, y ) = ∂ x σ ( x, y ), σ (cid:48)(cid:48) ( x, y ) = ∂ xx σ ( x, y ),˙ σ ( x, y ) = ∂ y σ ( x, y ). We also denote σ = σ (0 ,
0) the spot volatility and(2.4) σ (cid:48) = σ (cid:48) (0 , , σ (cid:48)(cid:48) = σ (cid:48)(cid:48) (0 , , ˙ σ = ˙ σ (0 , t H in σ ( · ), becausethis is the scaling of the variance of the fBm at time t . For this reason, this is the scaling of the time-dependent termin the rBergomi model, and also the scaling such that we observe a dependence in ˙ σ in our precise asymptotics. Weapply the abstract results proved in [21] for K ( t, s ) = const × ( t − s ) H − / . However, we expect these approximationsto hold in greater generality: the same type of expansions should hold for other kernels such that (cid:99) W in (2.2) satisfies(2.3). Self-similarity is equivalent to the fact that K can be written in the following form(2.5) K ( t, s ) = ( t − s ) H − / f K ( s/t ) , for a suitable function f K (see [38, Lemma 2.4]), so that all such kernels can be seen as a perturbation of ( t − s ) H − / .Two classical processes of this form are the Mandelbroth-Van Ness and the Riemann-Liouville fBMs (see AppendixA). Without loss of generality, we also assume K ( t, s ) = 0 for t < s .A similar setting has been considered in [20, 6]. The main difference in the structure of the model is that here weallow for a direct dependence on time in σ ( t, ω ) = σ ( (cid:99) W t , t H ), whereas in [20, 6] the volatility function depends onlyon the fBM, so σ ( t, ω ) = σ ( (cid:99) W t ). As mentioned in the introduction, assuming that the volatility is a deterministicfunction only of the fBM rules out the rBergomi model σ ( (cid:99) W t , t H ) = σ exp( η (cid:99) W t / − η t H / σ ( (cid:99) W t , t H ), one can write the dynamics of the log-price X = log S as(2.6) X t = (cid:90) t σ (cid:16)(cid:99) W s , s H (cid:17) d ( ρW + ρW ) s − (cid:90) t σ ( (cid:99) W s , s H ) ds. In this case, a LDP holds, writing (cid:98) ε = ε H , for(2.7) X ε = (cid:90) σ (cid:16)(cid:98) ε (cid:99) W t , (cid:98) ε t H (cid:17) (cid:98) εd ( ρW + ρW ) t − ε (cid:98) ε (cid:90) σ (cid:16)(cid:98) ε (cid:99) W t , (cid:98) ε t H (cid:17) dt , with speed (cid:98) ε and rate function(2.8) Λ( x ) := inf (cid:26) (cid:107) h, h (cid:107) H : (cid:90) σ (cid:0) (cid:98) h, (cid:1) d (cid:0) ρh + ρh (cid:1) = x (cid:27) ≡ (cid:107) h x , h x (cid:107) H , where (cid:98) h t = ( K ∗ ˙ h ) t and (cid:107) · (cid:107) H is the Cameron-Martin norm. This result was proved for σ ( (cid:99) W t , t H ) = σ ( (cid:99) W t ) in[20] and then extended to possible dependence in t H in [21, Section 7.3]. In general, when looking only at large(or moderate) deviations, the t H -dependence in σ ( · ) does not affect the analysis, and the large (or moderate)deviations behavior is the same one would get with volatility σ ( (cid:99) W t , t H -dependence actually affects the asymptotics. In PETER K. FRIZ, PAUL GASSIAT, AND P. PIGATO the present paper we provide computationally relevant results that allow for the practical usage of such refinedpricing asymptotics and discuss their consequences on the Black-Scholes implied volatility.3.
Results
We consider small-noise call and put prices under model (2.6), i.e. c ( t, k ) = E [(exp X t − exp k ) + ] , p ( t, k ) = E [(exp k − exp X t ) + ] , where k is the log-strike (or log-moneyness). In [21, Theorem 1.1] we obtain precise small-noise price expansionsfor generic (classical and rough) volatility dynamics. As in the classical Brownian case, such small noise results canbe translated into short-time results writing t = ε . In this paper, we focus on the short-time setting. We write ∼ for asymptotic equivalence, f t ∼ g t if f t /g t → t →
0, and “ ≈ ” for “is close to” in informal terms. We alsowrite σ x = 2Λ( x ) / Λ (cid:48) ( x ) . In short-time, [21, Theorem 1.1] reads as follows: Theorem 3.1.
Let H ∈ (0 , / and k t = xt / − H . Assume that a LDP holds for c, p above, with a non degenerateminimizer and the existence of + moments for exp X t . Then, for x > small enough, the rate function Λ = Λ( x ) is continuously differentiable at x and c ( t, k t ) ∼ exp (cid:18) − Λ( x ) t H (cid:19) t / H A ( x )(Λ (cid:48) ( x )) σ x √ π as t ↓ ,for some function A ( x ) with A ( x ) → as x ↓ . Similarly, for x < , close enough to , we have p ( t, k t ) ∼ exp (cid:18) − Λ( x ) t H (cid:19) t / H A ( x )(Λ (cid:48) ( x )) σ x √ π as t ↓ ,for some function A ( x ) with A ( x ) → as x ↑ . Moreover, such A can be expressed as (3.1) A ( x ) = (cid:40) E (cid:2) exp(Λ (cid:48) ( x )∆ x ) (cid:3) , if H < / e x E (cid:2) exp(Λ (cid:48) ( x )∆ x ) (cid:3) , if H = 1 / where ∆ x is a certain quadratic Wiener functional (specified in [21, Equation (7.5)] ). We write Kf ( t ) = (cid:82) t K ( t, s ) f ( s ) ds , K f ( t ) = (cid:82) t K ( t, s ) f ( s ) ds and (cid:104)· , ·(cid:105) for the inner product in L [0 , K the adjoint of K in L [0 ,
1] so that K u ) = (cid:82) u K ( t, u ) dt . Fully explicit expressions are computablein the case of the Riemann-Liouville fBM (Appendix A) and in particular in the case of standard BM (this is theclassical case of Markovian stochastic volatility). We denote C K,ρ = (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) + ρ (cid:18) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) − (cid:104) K , K (cid:105) (cid:19) ,C K,ρ = (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) . Lemma 3.2 (Fine structure of A ) . For H ∈ (0 , / , the following expansion holds for A ( x ) as x → : (3.2) A ( x ) = 1 − x ρσ (cid:48) (cid:104) K , (cid:105) σ + x (cid:18) ( σ (cid:48) ) σ C K,ρ + σ (cid:48)(cid:48) σ C K,ρ + ˙ σ (2 H + 1) σ (cid:19) + (cid:18) x x (cid:19) { H =1 / } + O ( x )As a consequence of Theorem 3.1 the following expansion holds for the Black-Scholes implied volatility (by astandard application of Gao-Lee [27], detailed in [21, Appendix D]). Corollary 3.3 (Asymptotic smile and term structure at the large deviations regime) . Writing k t = xt / − H , wehave the following expansion, for x ∈ R s.t. Theorem 3.1 holds. (3.3) σ BS ( t, k t ) = Σ ( x ) + t H a ( x ) + o ( t H ) as t ↓ , where (3.4) Σ( x ) = | x | (cid:112) x ) HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 5 and (3.5) a ( x ) = x x ) log (cid:18) A ( x )Λ( x )Λ (cid:48) ( x ) x (cid:19) if H < / x x ) log (cid:18) A ( x )Λ( x )Λ (cid:48) ( x ) x exp( x/ (cid:19) if H = 1 / Remark . In general, from a LDP for call prices follows the celebrated BBF formula for implied volatility(Berestycki-Busca-Florent [12], see also also Pham [47] for a derivation). Under RoughVol pricing with σ ( ω, t ) = σ ( (cid:99) W t ), this has been extended in [20] to(3.6) σ BS ( t, k t ) ∼ x x ) , holding for fixed x , in short-time, with k t = xt / − H . Thanks to the A -term in (3.1), we can extend this approxi-mation, adding the term structure t H a ( x ). Note that the expansions hold for H ∈ (0 , / H = 1 / A ( x ) and in the term structure of the Black-Scholesimplied volatility a ( x ).We denote now D K,ρ = (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) + ρ (cid:0) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) − (cid:104) K , K (cid:105) (cid:1) ,D K,ρ = (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) . (3.7)The short-time implied volatility coefficients in the previous statement can be expanded as follows near-the-money. Theorem 3.5 (At-the-money expansion of the coefficients) . For x → , the Σ coefficient has the following expan-sion: Σ( x ) = σ + x Σ (cid:48) (0) + x Σ (cid:48)(cid:48) (0)2 + O ( x )(3.8) where Σ (cid:48) (0) = ρσ (cid:48) (cid:104) K , (cid:105) σ Σ (cid:48)(cid:48) (0)2 = ( σ (cid:48) ) σ (cid:26) − ρ (cid:104) K , (cid:105) + ρ (cid:104) ( K , (cid:105) + 12 (cid:104) ( K , (cid:105) + ρ (cid:104) K , K (cid:105) (cid:27) + σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105) The term structure coefficient, at the first order in x at , is a ( x ) = a + O ( x ) , (3.9) with a = ( σ (cid:48) ) D K,ρ + σ σ (cid:48)(cid:48) D K,ρ + σ ˙ σ H + 1 / ρσ (cid:48) σ (cid:104) K , (cid:105) { H =1 / } Remark . From definition (3.4)-(3.5) and from the fact that Λ is quadratic in x we see that (3.9) implies arelation between A and Λ for x → Remark . Implied variance expansion (3.3) reads as follows on implied volatility(3.10) σ BS ( t, k t ) ≈ | x | (cid:112) x ) + t H a ( x ) | x | (cid:114) Λ( x )2 . In order to implement these expansions, one can use the methods discussed in Section 5, computing numericallythe rate function Λ( x ) and Σ( x ) using FZ expansion, and then computing a ( x ) using KL. However, this last stepcan be computationally expensive, since a large number of basis functions are needed for the KL decomposition tobe accurate, for H close to 0. As an alternative, one can use approximation(3.11) σ BS ( t, k t ) ≈ Σ( x ) + t H a σ = | x | (cid:112) x ) + t H a σ , PETER K. FRIZ, PAUL GASSIAT, AND P. PIGATO for implied volatility, which follows from implied variance expansion (3.3) and (3.9). If the rate function cannot becomputed, we can use (3.8) to expand the implied volatility as(3.12) σ BS ( t, k t ) ≈ Σ(0) + Σ (cid:48) (0) x + Σ (cid:48)(cid:48) (0)2 x + t H a σ . In particular, we get the following explicit expansion for the ATM term structure:(3.13) σ BS ( t,
0) = σ + t H a σ + o ( t H ) . Remark . From the expansion of the ATM term structure (3.13) wealso see, in the short end, that σ BS ( t,
0) is increasing in t if a > a <
0. This may be comparedwith a large body of literature concerning monotonicity properties of the term structure of implied volatility, seee.g. [13, 33, 39, 48].
Corollary 3.9 (Skew and curvature at the large deviation regime) . Let k t = xt / − H , for x ∈ R . Then, if H < / ,for t ↓ σ BS ( t, k t ) − σ BS ( t, − k t )2 k t ∼ Σ( x ) − Σ( − x )2 x t H − / (3.14) σ BS ( t, k t ) + σ BS ( t, − k t ) − σ BS ( t, k t ∼ Σ( x ) + Σ( − x ) − x t H − (3.15) Remark . The quantities in the rhs of the equivalences converge as x ↓ (cid:48) (0) , Σ (cid:48)(cid:48) (0) given in Theorem3.5. The quantities in the lhs of the equivalences are finite difference approximations of ATM implied volatilityskew ∂ k σ BS ( t,
0) and curvature ∂ kk σ BS ( t,
0) . Such finite differences are relevant because only a finite number ofprices are observable on real markets. They give skew and curvature at the large deviation regime, a result thatcomplements [24, 14] (skew and curvature at central limit regime), [6] (skew at moderate deviation regime), [17](skew and curvature at large deviations regime for rough Heston), [2] (true skew and curvature).From these formulas, we also infer the sign of implied skew and of implied curvature (convexity). Indeed, if σ , σ (cid:48) (cid:54) = 0, it is clear that s gn (Σ (cid:48) (0)) = s gn ( ρ ) and that(3.16) Σ (cid:48)(cid:48) (0) = 0 iff ρ = (cid:104) ( K , (cid:105) (cid:104) K , (cid:105) −(cid:104) ( K , (cid:105)− (cid:104) K ,K (cid:105)− σ (cid:48)(cid:48) σ σ (cid:48) (cid:104) ( K , (cid:105) , Σ (cid:48)(cid:48) (0) < ρ > (cid:104) ( K , (cid:105) (cid:104) K , (cid:105) −(cid:104) ( K , (cid:105)− (cid:104) K ,K (cid:105)− σ (cid:48)(cid:48) σ σ (cid:48) (cid:104) ( K , (cid:105) > , Σ (cid:48)(cid:48) (0) > Theorem 3.11 (Moderate deviations) . Assume that Λ is i ∈ N times continuously differentiable. Let H ∈ (0 , / , β > and n ∈ N such that β ∈ ( Hn +1 , Hn ] . Set k t = xt / − H + β . Then c ( t, k t ) ∼ exp (cid:32) − n (cid:88) i =2 Λ ( i ) (0) i ! x i t iβ − H (cid:33) t / H − β σ x √ π . Moreover σ BS ( t, k t ) = n − (cid:88) j =0 ( − j j σ j +1)0 (cid:32) n (cid:88) i =3 Λ ( i ) (0) i ! x i − t ( i − β (cid:33) j + o ( t H − β ) . (3.17) Remark . An implied volatility expansion similar to (3.17) was proved in [6], in the case σ ( t, ω ) = σ ( (cid:99) W t ), for β ∈ [ Hn +1 , Hn ). with reminder of order max( t H − β − ε , t ( n − β ). The derivatives of the rate function were computeduntil Λ (cid:48)(cid:48)(cid:48) (0), here we also computed Λ (4) (0) (cf. Lemma 6.1). This allows us to use the second order moderatedeviation (instead of first order as in [6]) σ ( t, k t ) = Σ(0) + Σ (cid:48) (0) xt β + Σ (cid:48)(cid:48) (0)2 x t β + o ( t H − β )Moreover, even if it does not show up in the asymptotics, the term structure can be incorporated as follows σ ( t, k t ) ≈ Σ(0) + Σ (cid:48) (0) xt β + Σ (cid:48)(cid:48) (0)2 x t β + a σ t H , HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 7 and this provides a sensible improvement in the implementation of such short-time result (cf. Figure 5).4.
A case study: the rough Bergomi model
The rough Bergomi model.
Introduced in [4], as a modification of the classical Bergomi model where theexponential (Ornstein-Uhlenbeck) kernel is replaced by a power-law kernel, the rBergomi model provides great fitsof empirical implied volatility surfaces with a very small number of parametres. In such model, the volatility isgiven by the “Wick” exponential of a fBM(4.1) σ ( t, ω ) = σ exp (cid:18) η (cid:99) W t − η t H (cid:19) . In the most general framework [4], the constant σ is replaced by the forward variance curve, which is a function oftime observable on the market (so it plays the role of an initial condition). The specific volatility in (4.1) did notfit in the framework of [20, 6], as in these papers the volatility is assumed to be σ ( (cid:99) W t ). For this reason, in [6], thefollowing version of the rBergomi model is considered(4.2) σ ( t, ω ) = σ exp (cid:16) η (cid:99) W t (cid:17) . In this work we consider (1.2), a version of the rBergomi model with one additional parameter θ ∈ R , that includesboth the previous ones (for θ = 0 , σ ( x, y ) = σ exp (cid:18) η x − θη y (cid:19) The interpretation of the parameters is the following: σ is the spot volatility and η represents the volatility ofvolatility. The parameters of the driving noise are the Hurst exponent H of (cid:99) W and the correlation parameter ρ between the BM (cid:102) W driving the asset and W in (2.2). We can interpret the newly introduced θ parameter as adamping coefficient of the volatility.Coming now to short-time pricing, Lemma 6.1 holds for the general model in (1.2), so that we are able to compareour asymptotics with large or moderate deviations results for the different versions of rBergomi in [6, 20, 35].However, in Corollary 3.3, Σ ( x ) is not affected by the value of θ , but the term structure a ( x ) is.From the volatility function (4.3) we get σ = σ , σ (cid:48) = σ η , σ (cid:48)(cid:48) = σ η , ˙ σ = − θη σ so all constants can be simplified. In particular condition (3.16) for the convexity of the short-time smile (with σ , η (cid:54) = 0) simplifies to Σ (cid:48)(cid:48) (0) = 0 iff ρ = (cid:104) ( K , (cid:105) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105)− (cid:104) K ,K (cid:105) , Σ (cid:48)(cid:48) (0) < ρ > (cid:104) ( K , (cid:105) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105)− (cid:104) K ,K (cid:105) > (cid:48)(cid:48) (0) > H , trough K , and ρ ). On calibrated parameters (for example in [4]) we have thatthe condition for vanishing second derivative is almost satisfied. This means that the short-time ATM curvature isvery close to 0, and indeed observed smiles are almost linear ATM.All the constants in previous expansions depend on the kernel K . For the Riemann-Liouville kernel (A.4) the K -functionals involved are explicit, given in (A.5).4.2. Implementation of Rough Bergomi.
Our goal in this section is to compare expansion (3.10) with otherknown implied volatility expansions under RoughVol. We consider: • Implied volatility from Monte Carlo pricing, using the hybrid scheme for rBergomi in [11] with κ = 2 (notethat a slight modification of the implementation is necessary for θ (cid:54) = 1). • Our implied volatility expansion, where the term structure coefficient a ( x ) is computed using KL, so thatwe have (3.10), or where a ( x ) is expanded at 0, so that we have (3.11). PETER K. FRIZ, PAUL GASSIAT, AND P. PIGATO • The FZ expansion (3.6). In [20], Forde and Zhang show that this asymptotics holds for volatilities of type σ ( t, ω ) = σ ( (cid:99) W t ), with no direct dependence on t , so this applies to (4.3) for θ = 0. However, as we haveshown in [21, Section 7.3], the same large deviation behavior holds when θ (cid:54) = 0. Therefore, the FZ expansiongives the same asymptotic smile, independently of the choice of θ . • Expansion (3.11), with ATM expansion of Σ as in (3.12) (so, rate function is expanded as well). In case θ = 1, one can check that this approximation is consistent with the expansion in [14, Section 5], that werefer to as “EFGR expansion”. These two mathematical results ar different, since log-strikes are in our case(large deviation regime) k t = xt / − H and in [14] (central limit regime) k t = xt / . However, when plottingfor finite k and t the approximate implied volatility, the two curves are the same.We first use the numerical methods detailed in next Section 5 to compute Σ( x ) and a ( x ). In Figure 1 we displayimplied volatility smiles in the rBergomi model with θ = 1, for varying t , where the rate function is computed usingthe Ritz method in Section 5.1 and the coefficient a ( x ) is computed using the KL decomposition from Section 5.3. Forcomparison, we also use approximation a ( x ) ≈ a , and show (3.11). We notice that both implementations performwell, and the use of KL decomposition gives a better approximation of the right wing. On several simulations, thisimprovement of KL over expansion a ( x ) ≈ a is more evident when taking θ = 1, less when θ = 0. Practically, −0.2 −0.1 0.0 0.1 . . . . H = = q = k t s BS Monte Carloour expansion − KLour expansion − a(0)FZ expansionwith ATM energy expansion,same as EFGR expansion −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 . . . . H = = q = k t s BS Monte Carloour expansion − KLour expansion − a(0)FZ expansionwith ATM energy expansion,same as EFGR expansion
Figure 1.
Implied volatility smile approximations for the rBergomi model with parameters θ =1 , σ = 0 . , η = 1 . , ρ = − . , H = 0 .
3, for expiry t = 0 . , . ,
2. The Monte Carlo price iscomputed via the hybrid scheme for rBergomi in [11] with κ = 2, with 10 simulations and 500time steps. The rate function is computed using the Ritz method in Section 5.1 with N = 8Haar basis functions, the coefficient a ( x ) is computed using the Karhunen-Love decompositionwith N = 300 Haar basis functions (KL). We also compare with a ( x ) expanded at 0 ( a ( x ) ≈ a (0)).implementation of the KL formula requires to approximate the infinite product (5.8), and we observed that forsmaller values of H the convergence of this product was much slower, requiring a prohibitively large number ofbasis functions, which is why we present these results for H = 0 .
3. We leave the numerically efficient implementationof the KL decomposition method for small values of H as a topic for future research. In what follows we will considerthe approximation a ( x ) ≈ a (0), which is faster while still producing accurate smiles. First, in Figure 2 we showimplied volatilities under model (1.2), with realistic parameters (close to the calibrated parameter to the SPXvolatility on February 4, 2010, see [4]), varying θ from 0 to 1. We note how our approximation is general enoughto be applicable for any θ , improving previous asymptotic in all cases. We also note a slight deterioration of the HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 9 quality of the approximation in the right wing as θ →
1, that could be improved using KL to compute a ( x ). Then, −0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion−0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.3 −0.2 −0.1 0.0 0.1 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion,same as EFGR expansion
Figure 2.
Implied volatility smile approximation for the rBergomi model with parameters σ =0 . , η = 1 . , ρ = − . , H = 0 .
07, for expiry t = 0 .
05. The Monte Carlo price is computed via thehybrid scheme for rBergomi in [11] with κ = 2, with 10 simulations and 500 time steps. The ratefunction is computed using the Ritz method with N = 9 Fourier basis functions.instead of varying θ , we fix θ = 0 and show in Figure 3 the comparison with the same approximations as before,when the expiry t increases. We see how our expansion lifts the FZ expansion, improving the approximation of theMonte Carlo price. The difference between the two approximations is due to the term structure correction a t H .Clearly, the effect of this correction becomes more evident as t increases. On a number of numerical experiments, itis also clear that this correction becomes more and more important as H →
0, not surprisingly since t H is larger,for small t , when H vanishes.Now we check how our approximations behave as time increases. To do so, we take parameters as in [6, Section4]. In Figure 4 we show the ATM term structure of implied volatility, comparing ATM implied volatilities computedusing Monte Carlo simulations and expansion (3.13), for rBergomi with θ = 0 and θ = 1. We see how the termstructure is increasing in one case and decreasing in the other. This is always the case: a in (3.9) is alwayspositive for θ = 0, always negative for θ = 1 (cf. Remark 3.8). Note that if the forward variance curve were takennon-constant, the slope of the term structure would also be affected.Finally, as in Remark 3.12, we consider moderate deviations. Figure 5 is as in [6, Figure 1], the “very rough”case H = 0 . k t = xt / − H + β , where β = 0 . σ ( t, k t ) ≈ Σ(0) + Σ (cid:48) (0) xt β + Σ (cid:48)(cid:48) (0)2 x t β + a σ t H , stopping at first order moderate deviation t β , second order moderate deviation t β , and considering also the termstructure t H . We see how the term structure term improves the moderate deviation pricing. This also explainswhy, in [6], the moderate deviation pricing gets worse as H ↓
0, since the distance of such price from the real (MonteCarlo) one is of order t H . We also see that using the second order moderate deviation actually does not improve −0.20 −0.15 −0.10 −0.05 0.00 0.05 . . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.4 −0.2 0.0 0.2 . . . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion −0.6 −0.4 −0.2 0.0 0.2 0.4 0.6 . . . . . H = = q = k t s BS Monte Carloour expansionFZ expansionwith ATM energy expansion
Figure 3.
Implied volatility smile approximation for the rBergomi model with parameters θ =0 , σ = 0 . , η = 1 . , ρ = − . , H = 0 .
07, for expiry t = 0 . , . , .
2. The Monte Carlo price iscomputed via the hybrid scheme for rBergomi in [11] with κ = 2, with 10 simulations and 500 timesteps. The rate function is computed using the Ritz method with N = 9 Fourier basis functions. . . . . H=0.1; q =0 t s BS Monte Carloasymptotic term structure 0.000 0.002 0.004 0.006 0.008 0.010 . . . H=0.1; q =1 t s BS Monte Carloasymptotic term structure
Figure 4.
Term structure of volatility for the rBergomi model with parameters σ = 0 . , η =0 . , ρ = − . , H = 0 .
1. The Monte Carlo prices are computed via the hybrid scheme in [11]with κ = 2, with 10 simulations and 500 time steps.much, and this follows from the fact that the curvature is almost 0 with such choice of parameters (cf. Remark3.12). Remark . In [20, Section 4.5] asymptotics for model (2.1) with volatility driven by a Mandelbroth-Van Ness fBm(A.2) are implemented. Without being completely rigorous, we have applied our expansion also in this case. Wecomputed the K -functional numerically, as in this case no explicit formulas are available. Also in this case the term a t H lifts the smile, which gets closer to the real (Monte Carlo) implied volatility, for small | x | , with respect to thesole FZ expansion. HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 11 . . . . . H=0.1; b =0.06 t s BS Monte Carlofirst order moderate deviationssecond order moderate deviationsmoderate deviations and term structure
Figure 5.
Moderate deviation with β = 0 .
06 of implied volatility in rBergomi model with σ =0 . , η = 0 . , ρ = − . , H = 0 . , θ = 0. Simulation parameters: 10 simulation paths, 500time steps. Time interval [0 , . Computing the coefficients via projections
Computing Σ( x ) using the Ritz method. In order to use (3.10), the first challenge is the computation ofthe rate function. A numerical approximation to Λ can be obtained as described in [29, Section 40], using the Ritzmethod, as is done in [20]. Natural choices for the orthornormal basis (ONB) { e i } i ≥ of H are the Fourier Basis,(5.1) ˙ e ( s ) = 1 , ˙ e n ( s ) = √ πns ) , ˙ e n +1 ( s ) = √ πns ) , for n ∈ N \ { } or the Haar basis,(5.2) ˙ e ( s ) = 1 , ˙ e k + l ( s ) = 2 k/ (cid:16) l − k +1 , l − k +1 ]( s ) − l − k +1 , l k +1 ]( s ) (cid:17) for k ≥ , ≤ l ≤ k . We consider functions h ∈ H with h (0) = 0 so that˙ h ( s ) = N (cid:88) n =1 a n ˙ e n ( s )for N ∈ N fixed. Then we minimize Λ( x ) = ( x − ρG ( h )) ρ F ( h ) + (cid:104) ˙ h, ˙ h (cid:105) , with F ( h ) = (cid:104) σ ( (cid:98) h, , (cid:105) , G ( h ) = (cid:104) σ ( (cid:98) h, , ˙ h (cid:105) (5.3)over the Fourier coefficients ( a n ) n . This representation of the energy function is also taken from [20] (see notationin [6, Proposition 5.1]). The minimizing value for Λ( x ) is therefore our approximation for the energy and thecorresponding function (cid:98) h x is the approximate most likely path for the fBm W H associated with final condition x .5.2. A stochastic Taylor development.
The following stochastic Taylor expansion is sketched in [21, Section7.2] for σ ( ω, t ) = σ ( W Ht ). As discussed in Section 2 and [21, Section 7.3], our expansions can actually be carried outin the more general setting σ ( ω, t ) = σ ( W Ht , t H ). Under such volatility dynamics, the (rescaled) log-price processis as in (2.7). As in [21, Section 7.2], we can shift the dynamics via (cid:98) ε ( W, W ) (cid:55)→ ( (cid:98) εW + h, (cid:98) εW + h ), and applyGirsanov theorem in order to center Brownian fluctuations in the minimizer. Then, a stochastic Taylor expansiongives, (cid:90) σ (cid:16)(cid:98) ε (cid:99) W t + (cid:98) h t , (cid:98) ε t H (cid:17) d [ (cid:98) ε (cid:102) W + (cid:101) h ] t − ε (cid:98) ε (cid:90) σ (cid:16)(cid:98) ε (cid:99) W t + (cid:98) h t , (cid:98) ε t H (cid:17) dt ≡ g + (cid:98) εg ( ω ) + (cid:98) ε g ( ω ) + r ( ω ) , where r ( ω ) is small , with(5.4) g = (cid:90) σ (cid:48) ( (cid:98) h xs , (cid:99) W s d (cid:101) h xs + (cid:90) σ ( (cid:98) h xs , d (cid:102) W s , (cf. [21, Section 7.2]) and(5.5) g = (cid:40) (cid:82) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:99) W s d (cid:101) h xs + (cid:82) σ (cid:48) ( (cid:98) h xs , (cid:99) W s d (cid:102) W s + (cid:82) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs if H < / (cid:82) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:99) W s d (cid:101) h xs + (cid:82) σ (cid:48) ( (cid:98) h xs , (cid:99) W s d (cid:102) W s + (cid:82) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs − (cid:82) σ ( (cid:98) h xs , ds if H = 1 / . The following formula for ∆ follows as [21, Equation 7.5](5.6) ∆ = (cid:40) (cid:82) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) h xs + (cid:82) σ (cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) V s , + (cid:82) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs if H < / (cid:82) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) h xs + (cid:82) σ (cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) V s + (cid:82) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs − (cid:82) σ ( (cid:98) h xs , ds if H = 1 / v t = E [ W t g ] /E [ g ] , v t = E [ W t g ] /E [ g ] , (cid:101) v t = ρv t + ρv t , (cid:98) v = K ˙ v and (cid:101) V t = (cid:102) W t − (cid:101) v t g , (cid:98) V t = (cid:99) W t − g (cid:98) v t . Computing a ( x ) using Karhunen-Love decomposition. Assume we are given h x computed by the Ritzmethod. Note then that h x is obtained from h x via the following formula(5.7) ˙ h x ( s ) = x − ρG ( h x ) ρF ( h x ) σ ( (cid:98) h x ( s ))with G ( h ) , F ( h ) as in (5.3), as can be seen by optimizing over h for fixed h in the definition (2.8) of the rate function.Then we assume a Karhunen-Love (KL) decomposition of W (see also [18] for other applications of a similar KLdecomposition): W = (cid:88) i γ i e i , W = (cid:88) i γ i e i , where { e i } i is the ONB in (5.1) or (5.2) and γ i , γ i are i.i.d. standard Gaussians. This implies (cid:99) W = (cid:88) i γ i (cid:98) e i , with (cid:98) e i ( t ) = ( K ∗ ˙ e i ) t . This yields g = (cid:88) i g i γ i + g i γ i , where g i = (cid:90) σ (cid:48) ( (cid:98) h xs , (cid:98) e i ( s ) d (cid:101) h x ( s ) + ρ (cid:90) σ ( (cid:98) h xs , de i ( s ) , g i = ρ (cid:90) σ ( (cid:98) h xs , de i ( s ) . In particular σ x = (cid:88) i g i + g i . Note then that v ( t ) = (cid:88) i g i e i ( t ) /σ x , v ( t ) = (cid:88) i g i e i ( t ) /σ x , (cid:98) v ( t ) = (cid:88) i g i (cid:98) e i ( t ) /σ x , We then can write all the terms in ∆ as follows. We denote α ij = (cid:90) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:98) e i ( s ) (cid:98) e j ( s ) d (cid:101) h xs , β ij = (cid:90) σ (cid:48) ( (cid:98) h xs , (cid:98) e i de j , The precise control of this remainder is detailed in [21] and requires the sophisticated mathematical framework of regularitystructures, that we do not intend to introduce in this paper. The interested reader is referred to [5, 21].
HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 13 δ ij = i = j and (cid:101) g i = ρ g i + ρ g i . Now, expanding (5.6) with some long but standard computations we get to∆ = (cid:88) ij ( γ i γ j − δ ij ) η ij + γ i γ j η ij + ( γ i γ j − δ ij ) η ij + C =: ∆ (2)2 + C, where η ij = 12 α ij − σ x g i (cid:88) k g k α jk + 12 σ x g i g j (cid:32)(cid:88) kl g k g l α kl (cid:33) + ρβ ij − σ x g i (cid:88) k (cid:101) g k β jk − ρσ x g i (cid:88) k g k β kj + 1 σ x g i g j (cid:88) k,l g k (cid:101) g l β kl η ij = ρβ ij − σ x g j (cid:88) k g k α ik + 12 σ x g i g j (cid:88) k,l g k g l α k,l − σ x g j (cid:88) k (cid:101) g k β ik − ρσ x g i (cid:88) k g k β kj − ρσ x g j (cid:88) k g k β ki + 1 σ x g i g j (cid:32)(cid:88) k (cid:88) l g k (cid:101) g l β kl (cid:33) η ij = 12 σ x g i g j (cid:88) k,l g k g l α k,l − ρσ x g i (cid:88) k g k β kj + 1 σ x g i g j (cid:88) k,l g k (cid:101) g l β k,l and C = (cid:90) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs + 12 (cid:88) i α ii − σ x (cid:88) i,k g i g k α ik − σ x (cid:88) i,k g i (cid:101) g k β ik . Recall that one has A ( x ) = e Λ (cid:48) ( x ) C E exp(Λ (cid:48) ( x )∆ (2)2 ) , where Λ (cid:48) ( x ) = s gn ( x ) (cid:115) x ) σ x , and since ∆ (2)2 is an element of the homogeneous Wiener chaos of order 2, the expectation above can be computedas the Carleman-Fredholm determinant det ( I − M ) − / , where M is the symmetric matrix M = Λ (cid:48) ( x ) (cid:18) ( η + η t ) η ( η ) t ( η + η t ) (cid:19) . Namely one has(5.8) E exp(Λ (cid:48) ( x )∆ (2)2 ) = Π k ≥ (1 − λ k ) − / e − λ k where ( λ k ) k are the eigenvalues of M (note that the fact that all λ k < / M is diagonal, and the general case follows bydiagonalisation, cf e.g. [34, Remark 5.5] or [37, p.78].Of course, in practice we consider approximations W N , W N obtained by truncating the sums to only keep indices i ≤ N , where N is fixed, so that all the sums above are then replaced by finite sums. One also needs to computenumerically the integrals appearing in the definition of the coefficients g, α , β . We have found the Haar basis to bemore convenient than the Fourier basis for this purpose since the (cid:98) e i ’s have explicit expressions in that case.6. Proofs
Energy expansion.Lemma 6.1 (Fourth order energy expansion) . Consider a stochastic volatility model following dynamics (2.1) andthe associated energy function in (2.8) . Let Λ( x ) be the energy function in (2.8) . Then Λ( x ) = Λ (cid:48)(cid:48) (0)2 x + Λ (cid:48)(cid:48)(cid:48) (0)3! x + Λ (4) (0)4! x + O ( x ) where (6.1) Λ (cid:48)(cid:48) (0) = 1 σ , Λ (cid:48)(cid:48)(cid:48) (0) = − ρσ (cid:48) σ (cid:104) K , (cid:105) , and Λ (4) (0) = 12 ( σ (cid:48) ) σ (cid:8) ρ (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) − (cid:104) ( K , (cid:105) − ρ (cid:104) K , K (cid:105) (cid:9) − σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105) . Remark . In this lemma we expand the rate function Λ( x ), which has been studied first in [20]. The second andthird order terms in (6.1) have been computed in [6, Theorem 3.4]. In both these papers, the volatility functionis supposed to be σ ( W Ht ), but adding the dependence σ ( W Ht , t H ) does not change the large deviations behavior,meaning that the rate function is the same as the one of the model given by σ ( W Ht , Proof.
We have the following development for the minimizer h x · in (2.8), for x → h xt = α t x + β t x γ t x O ( x ) , with α t = ρσ t,β t = 2 σ (cid:48) σ [ ρ (cid:104) K , [0 ,t ] (cid:105) + (cid:104) K [0 ,t ] , (cid:105) − ρ t (cid:104) K , (cid:105) ] , where α, β have been also computed in [6]. We make here the ansatz that the expansion goes on one more orderwith γ , that we do not actually need to compute. The existence of such γ follows from the smoothness of σ ( · , · ) (cf.[21] and [6, Section 5.2]). We can compute, using (cid:104) K ( K , (cid:105) = (cid:104) K , K (cid:105) and (cid:104) K ( K , (cid:105) = (cid:104) ( K , (cid:105) , K ˙ β = 2 σ (cid:48) σ [ ρ K ( K
1) + K ( K − ρ (cid:104) K , (cid:105) K (cid:104) K ˙ β, (cid:105) = 2 σ (cid:48) σ [ ρ (cid:104) K , K (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) ] (cid:104) K , ˙ β (cid:105) = 2 σ (cid:48) σ [ ρ (cid:104) ( K , (cid:105) + (cid:104) K , K (cid:105) − ρ (cid:104) K , (cid:105) ]We also have(6.3) σ ( (cid:98) h xs ,
0) = σ + x σ (cid:48) σ ρK s ) + (cid:18) σ (cid:48)(cid:48) σ ρ ( K ( s ) + σ (cid:48) K ˙ β ( s ) (cid:19) x O ( x )We use now (5.3) and compute F ( h x ) = σ + x ρσ (cid:48) (cid:104) K , (cid:105) + x (cid:26)(cid:18)(cid:18) σ (cid:48) σ (cid:19) + σ (cid:48)(cid:48) σ (cid:19) ρ (cid:104) ( K , (cid:105) + σ σ (cid:48) (cid:104) K ˙ β, (cid:105) (cid:27) + O ( x ) G ( h x ) = ρx + x (cid:18) σ (cid:48) σ ρ (cid:104) K , (cid:105) + σ β (cid:19) + x (cid:18) σ γ + σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105) + ρ ( σ (cid:48) ) σ (cid:2) ( ρ + 1) (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) (cid:3)(cid:19) + O ( x )(6.4) HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 15 from which we get x − ρG ( h x ) = (1 − ρ ) x − x ρ σ (cid:48) σ (1 − ρ ) (cid:104) K , (cid:105) − x ρ (cid:18) σ γ + σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105) + ρ ( σ (cid:48) ) σ (cid:2) ( ρ + 1) (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) (cid:3)(cid:19) + O ( x )1 F ( h x ) = 1 σ − x ρ σ (cid:48) σ (cid:104) K , (cid:105) − x (cid:26) σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105) + (cid:18) σ (cid:48) σ (cid:19) (cid:18) ρ (cid:104) ( K , (cid:105) + 2 (cid:104) ( K , (cid:105) + 2 ρ (cid:104) K , K (cid:105) − ρ (cid:104) K , (cid:105) (cid:19)(cid:27) + O ( x )( x − ρG ( h x )) − ρ = (1 − ρ ) x − x ρ σ (cid:48) σ (1 − ρ ) (cid:104) K , (cid:105) + x (cid:20) ρ ( σ (cid:48) ) σ (1 + 11 ρ ) (cid:104) K , (cid:105) − σ (cid:48)(cid:48) σ ρ (cid:104) ( K , (cid:105)− ρσ γ − ρ ( σ (cid:48) ) σ (cid:2) ( ρ + 1) (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) (cid:3)(cid:19)(cid:21) + O ( x )( x − ρG ( h x )) − ρ ) F ( h x ) = ( . . . ) x + ( . . . ) x − x ρ σ γ + x (cid:26) − ρ σ (cid:48)(cid:48) σ (cid:104) ( K , (cid:105) − ρ ( σ (cid:48) ) σ (cid:104) K , K (cid:105)− ( σ (cid:48) ) σ ρ + ρ (cid:104) ( K , (cid:105) − ( σ (cid:48) ) σ (cid:104) ( K , (cid:105) + ( σ (cid:48) ) σ ρ (5 − ρ ) (cid:104) K , (cid:105) (cid:27) + O ( x )We also have, from (6.2) (cid:104) ˙ h x , ˙ h x (cid:105) = · · · + x (cid:18) ρ σ γ + (cid:0) σ (cid:48) σ (cid:1) (cid:2) ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) +2 ρ (cid:104) K , K (cid:105) +3 ρ (cid:104) K , (cid:105) − ρ (cid:104) K , (cid:105)(cid:104) K , (cid:105) (cid:3)(cid:19) + O ( x )Now we write, from [6, Proposition 5.1], Λ( x ) = ( x − ρG ( h x )) ρ F ( h x ) + (cid:104) ˙ h x , ˙ h x (cid:105) x ) follows. (cid:3) Proof of Lemma 3.2.
Let us take x (cid:54) = 0.STEP 1: We first need to expand h xt in (2.8), for small x (an expansion of h x was computed in [6]). We write(6.5) Φ ( W, W ) = X for the Itˆo Map associated with the RoughVol model (2.6). Computing the Frechet derivative of Φ with respectto the second component at h = ( h, h ) in the direction f we get (cf. (5.4))(6.6) (cid:104) D Φ (h) , (0 , f ) (cid:105) = (cid:104) D Φ (h) , f (cid:105) = ddδ Φ ( h, h + δf ) = ρ (cid:90) σ ( (cid:98) h, df From the first order optimality condition [21, Appendix B], we get that for h x minimizer and any f in the Cameron-Martin space H , (cid:104) h xt , f t (cid:105) H = Λ (cid:48) ( x ) (cid:104) D Φ , f (cid:105) . Let f be the second component of f. Using (6.6) we get (cid:90) ˙ h xt ˙ f t dt = (cid:104) h xt , f t (cid:105) H = Λ (cid:48) ( x ) (cid:104) D Φ , f (cid:105) = ρ Λ (cid:48) ( x ) (cid:90) σ ( (cid:98) h xt ,
0) ˙ f t dt. Now, from (6.1) we derive that, for x → (cid:48) ( x ) = xσ − x ρσ (cid:48) σ (cid:104) K , (cid:105) + O ( x )We get h xt = x ρσ t + O ( x ) We also have(6.8) h xt = x ρσ t + O ( x ) , (cid:101) h xt = ρh xt + ρh xt = x tσ + O ( x ) , (cid:98) h xt = ( K ˙ h x ) t = x ρσ K t )and σ ( (cid:98) h x ,
0) = σ + x ρσ (cid:48) σ K O ( x ) , σ (cid:48) ( (cid:98) h x ,
0) = σ (cid:48) + x ρσ (cid:48)(cid:48) σ K O ( x ) , ˙ σ ( (cid:98) h x ,
0) = ˙ σ + O ( x )STEP 2: We recall here, from [21], the definition of some quantities needed to compute A ( x ). Let g be as in (5.4)and let us write σ x = V ar ( g ) for its variance. We recall, again from [21, Equation (6.3)], σ x = 2Λ( x ) / Λ (cid:48) ( x ) , fromwhich we get(6.9) σ x = σ + 4 ρσ (cid:48) (cid:104) K , (cid:105) x + O ( x )From (5.4) we define and compute v t = E [ W t g ] E [ g ] = 1 σ x (cid:18) ρ (cid:90) t σ ( (cid:98) h xs , ds + (cid:90) σ (cid:48) ( (cid:98) h xs , K [0 ,t ] ( s ) d (cid:101) h xs (cid:19) ,v t = E [ W t g ] E [ g ] = 1 σ x ρ (cid:90) t σ ( (cid:98) h xs , ds. (6.10)(Note that v, v are in the Cameron-Martin space). From (3.1) we have that A ( x ) in Theorem 3.1 is A ( x ) = E [exp(Λ (cid:48) ( x )∆ )] . where ∆ is given in (5.6).STEP 3: We can expand now such quantity, for x → A ( x ) = 1 + x Λ (cid:48)(cid:48) (0) E (cid:2) ∆ (cid:3) + x (cid:18) Λ (cid:48)(cid:48)(cid:48) (0) E (cid:2) ∆ (cid:3) + Λ (cid:48)(cid:48) (0) E (cid:2) (∆ ) (cid:3) (cid:48)(cid:48) (0) E (cid:2) ∂ x (cid:12)(cid:12) x =0 ∆ (cid:3) (cid:19) + O ( x )(6.11)where ∆ denotes ∆ (cid:12)(cid:12) x =0 . The statement of the theorem follows from the computation of the quantities in (6.11).STEP 4: We compute ˙ v t = 1 σ x (cid:18) ρσ ( (cid:98) h xt ,
0) + (cid:90) t σ (cid:48) ( (cid:98) h xs , K ( s, t ) d (cid:101) h xs , (cid:19) and we obtain, also using (6.10), σ x v t = ρσ t + x σ (cid:48) σ (cid:0) ρ (cid:104) K , [0 ,t ] (cid:105) + (cid:104) K [0 ,t ] , (cid:105) (cid:1) + O ( x ) σ x (cid:101) v t = σ t + x σ (cid:48) σ ρ (cid:0) (cid:104) K , [0 ,t ] (cid:105) + (cid:104) K [0 ,t ] , (cid:105) (cid:1) + O ( x ) σ x (cid:98) v t = ρσ K t ) + x σ (cid:48) σ (cid:0) ρ K ( K t ) + K ( K t ) (cid:1) + O ( x )(6.12)where we have used (cid:104) K [0 ,t ] , (cid:105) = (cid:82) t K u ) du. We have σ x (cid:98) v t d (cid:101) v t = ρσ K t ) dt + xσ (cid:48) (cid:16) ρ K ( K t ) + K ( K t ) + ρ K t ) (cid:0) K t ) + K t ) (cid:1)(cid:17) dt Putting together the previous expressions and using (cid:104) K ( K , (cid:105) = (cid:104) K , K (cid:105) and (cid:104) K ( K , (cid:105) = (cid:104) ( K , (cid:105) we get σ x (cid:90) (cid:98) v t d (cid:101) v t = ρσ (cid:104) K , (cid:105) + xσ (cid:48) (cid:18) ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) + 2 ρ (cid:104) K , K (cid:105) (cid:19) + O ( x ) σ x (cid:90) K t ) (cid:98) v t d (cid:101) v t = ρσ (cid:104) ( K , (cid:105) + O ( x ) σ x (cid:90) (cid:98) v t dt = ρ σ (cid:104) ( K , (cid:105) + O ( x )(6.13)This implies, toghether with (6.9),(6.14) ∂ x (cid:18) σ x (cid:90) (cid:98) v t d (cid:101) v t (cid:19) = σ (cid:48) σ (cid:18) ρ (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) (cid:19) HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 17
We can now compute E (cid:90) (cid:18) (cid:98) V s d (cid:101) V s (cid:19) = E (cid:90) (cid:99) W s d (cid:102) W s + E [ g ] (cid:90) (cid:98) v s d (cid:101) v s − E (cid:20) g (cid:18) (cid:90) (cid:99) W s d (cid:101) v s + (cid:98) v s d (cid:102) W s (cid:19)(cid:21) where E (cid:2) g (cid:90) (cid:99) W s d (cid:101) v s (cid:3) = (cid:90) (cid:90) s K ( s, u ) dE (cid:2) g W u (cid:3) d (cid:101) v s = σ x (cid:90) (cid:98) v s d (cid:101) v s E (cid:2) g (cid:90) (cid:98) v s d (cid:102) W s (cid:3) = (cid:90) (cid:98) v s dE (cid:2) g (cid:102) W s (cid:3) = σ x (cid:90) (cid:98) v s d (cid:101) v s so that E (cid:90) (cid:18) (cid:98) V s d (cid:101) V s (cid:19) = − σ x (cid:90) (cid:98) v s d (cid:101) v s We also compute (cid:90) E [ (cid:98) V s ] ds = (cid:104) K , (cid:105) − σ x (cid:90) (cid:98) v s ds (cid:90) K s ) E [ (cid:98) V s d (cid:101) V s ] = − σ x (cid:90) K s ) (cid:98) v s d (cid:101) v s and all these quantities can be expansionded in x using (6.13). Now we use (5.6) to write, in the case H < / E ∆ = σ (cid:48) E (cid:90) (cid:98) V s d (cid:101) V s = − ρσ (cid:48) (cid:104) K , (cid:105) . Moreover, using (6.8), ∂ x (cid:12)(cid:12) x =0 (cid:90) ˙ σ ( (cid:98) h xs , s H d (cid:101) h xs = ˙ σ (2 H + 1) σ Now, also using (5.6) and (6.13) we get ∂ x E ∆ (cid:12)(cid:12) x =0 = σ (cid:48)(cid:48) σ (cid:90) E [( (cid:98) V s ) ] ds + ρ σ (cid:48)(cid:48) σ (cid:90) K s ) E [ (cid:98) V s d (cid:101) V s ] + σ (cid:48) (cid:90) ∂ x E [ (cid:98) V s d (cid:101) V s ] (cid:12)(cid:12) x =0 + ˙ σ (2 H + 1) σ = σ (cid:48)(cid:48) σ (cid:18) (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) (cid:19) − (cid:18) σ (cid:48) σ (cid:19) (cid:18) ρ (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) (cid:19) + ˙ σ (2 H + 1) σ STEP 5: We need now to compute E [(∆ ) ] = ( σ (cid:48) ) E (cid:18) (cid:90) (cid:98) V s d (cid:101) V s (cid:19) , where (using definitions and (6.12)) (cid:98) V s = (cid:99) W s − ρK s ) (cid:102) W and d (cid:101) V s = d (cid:102) W s − (cid:102) W ds We can rewrite (cid:90) (cid:98) V s ds (cid:102) W = ρ (cid:90) K u ) (cid:102) W u dB u + (cid:90) (cid:18) ρ (cid:0) K u ) − (cid:104) K , (cid:105) (cid:1)(cid:102) W u + (cid:90) u K s ) dW s (cid:19) d (cid:102) W u and, differentiating the product (cid:82) K u ) d (cid:102) W u , (cid:102) W (cid:90) (cid:98) V s d (cid:101) V s = − ρ (cid:90) K u ) (cid:102) W u dB u + (cid:90) (cid:18) (cid:98) V u − (cid:90) u K s ) dW s + ρ (cid:0) (cid:104) K , (cid:105) − K u ) (cid:1)(cid:102) W u (cid:19) d (cid:102) W u = − ρ (cid:90) K u ) (cid:102) W u dB u − ρ (cid:90) K u ) du + (cid:90) (cid:18)(cid:99) W u − (cid:90) u K s ) dW s + ρ (cid:0) (cid:104) K , (cid:105) − K u ) − K u ) (cid:1)(cid:102) W u − ρ (cid:90) u K s ) d (cid:102) W s (cid:19) d (cid:102) W u with (cid:102) W independent of B . Therefore, by Itˆo isometry, E (cid:20)(cid:18) (cid:90) (cid:98) V s d (cid:101) V s (cid:19) (cid:21) = ρ (cid:90) K u ) udu + ρ (cid:18) (cid:90) K u ) du (cid:19) + E (cid:90) (cid:18)(cid:99) W u − (cid:90) u K s ) dW s + ρ (cid:0) (cid:104) K , (cid:105) − K u ) − K u ) (cid:1)(cid:102) W u − ρ (cid:90) u K s ) d (cid:102) W s (cid:19) du. We can apply again Itˆo isometry to compute the last expectations, and E (cid:20)(cid:18) (cid:90) (cid:98) V s d (cid:101) V s (cid:19) (cid:21) = ρ (cid:90) (cid:90) u (cid:0) K ( u, s ) − K s ) + 2 (cid:104) K , (cid:105) − K u ) − K u ) − K s ) (cid:1) dsdu + ρ (cid:18) (cid:90) K u ) du (cid:19) + ρ (cid:90) (cid:90) u ( K ( u, s ) − K s )) dsdu + ρ (cid:90) K u ) udu. At this point it is a (long) calculus excercise (noting (cid:104) K , (cid:105) = (cid:104) K , (cid:105) ) to show that E (cid:20)(cid:18) (cid:90) (cid:98) V s d (cid:101) V s (cid:19) (cid:21) = ρ (3 (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) − (cid:104) K , K (cid:105) ) + (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) (6.16)STEP 6: Substituting in (6.11) we get A ( x ) = 1 − x ρσ (cid:48) σ (cid:104) K , (cid:105) + x (cid:26) ( σ (cid:48) ) σ (cid:18) ρ (cid:104) K , (cid:105) + 12 E (cid:20)(cid:16) (cid:90) (cid:98) V s d (cid:101) V s (cid:17) (cid:21)(cid:19) + σ (cid:48)(cid:48) σ (cid:18) (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) (cid:19) − ( σ (cid:48) ) σ (cid:18) ρ (cid:104) K , K (cid:105) + ρ (cid:104) ( K , (cid:105) + (cid:104) ( K , (cid:105) − ρ (cid:104) K , (cid:105) (cid:19) + ˙ σ (2 H + 1) σ (cid:27) + O ( x )=1 − x ρσ (cid:48) σ (cid:104) K , (cid:105) + x (cid:26) ( σ (cid:48) ) σ (cid:18) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) + ρ (cid:16) (cid:104) K , (cid:105) − (cid:104) ( K , (cid:105) − (cid:104) K , K (cid:105) (cid:17)(cid:19) + σ (cid:48)(cid:48) σ (cid:18) (cid:104) K , (cid:105) − ρ (cid:104) ( K , (cid:105) (cid:19) + ˙ σ (2 H + 1) σ (cid:27) + O ( x )and we get Theorem 3.2.STEP 7: When H = 1 /
2, ∆ in (5.6) has an additional summand. Let us write (cid:101) ∆ = 12 (cid:90) σ (cid:48)(cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) h xs + (cid:90) σ (cid:48) ( (cid:98) h xs , (cid:98) V s d (cid:101) V s + (cid:90) ˙ σ ( (cid:98) h xs , d (cid:101) h xs , so that (cid:101) ∆ has the same expression as ∆ in the rough case H < /
2. For H = 1 / = (cid:101) ∆ − (cid:90) σ ( (cid:98) h xs , ds, so that(6.18) ∆ = (cid:101) ∆ − σ ∂ x (cid:12)(cid:12) x =0 ∆ = ∂ x (cid:12)(cid:12) x =0 (cid:101) ∆ − σ (cid:48) ρ (cid:104) K , (cid:105) Now, A ( x ) in Theorem 3.1 is A ( x ) = e x E [exp(Λ (cid:48) ( x )∆ )]with ∆ as above. Expanding in x we find A ( x ) = 1 + x (Λ (cid:48)(cid:48) (0) E [∆ ] + 1) + x (cid:18) Λ (cid:48)(cid:48)(cid:48) (0) E [∆ ] + E (cid:2) (Λ (cid:48)(cid:48) (0)∆ + 1) (cid:3) (cid:48)(cid:48) (0) E (cid:2) ∂ x (cid:12)(cid:12) x =0 ∆ (cid:3) (cid:19) + O ( x )= 1 + x Λ (cid:48)(cid:48) (0) E [ (cid:101) ∆ ] + x (cid:18) Λ (cid:48)(cid:48)(cid:48) (0) E [ (cid:101) ∆ ] + Λ (cid:48)(cid:48) (0) E [( (cid:101) ∆ ) ]2 + Λ (cid:48)(cid:48) (0) E [ ∂ x (cid:12)(cid:12) x =0 (cid:101) ∆ ] (cid:19) + x x O ( x )(6.20)(we have used (6.1) and (6.15)). HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 19
Proof of Theorem 3.5.
A Talyor expansion givesΣ( x ) = x (cid:112) x ) = 1 (cid:112) Λ (cid:48)(cid:48) (0) (cid:18) − Λ (cid:48)(cid:48)(cid:48) (0)6Λ (cid:48)(cid:48) (0) x + Λ (cid:48)(cid:48)(cid:48) (0) − Λ (cid:48)(cid:48) (0)Λ (4) (0)24Λ (cid:48)(cid:48) (0) x (cid:19) + O ( x ) . The explicit expressions for the three terms now follow from Lemma 6.1. Let us compute a ( x ). The rate functionis quadratic and Λ(0) = Λ (cid:48) (0) = 0. Then, using Taylor developments of Λ and x → x we get2Λ( x ) x Λ (cid:48) ( x ) = 1 − x (cid:48)(cid:48)(cid:48) (0)Λ (cid:48)(cid:48) (0) + x (cid:40)(cid:18) Λ (cid:48)(cid:48)(cid:48) (0)Λ (cid:48)(cid:48) (0) (cid:19) − Λ (4) (0)Λ (cid:48)(cid:48) (0) (cid:41) + O ( x ) = 1 + (cid:112) Λ (cid:48)(cid:48) (0) (cid:18) x Σ (cid:48) (0) + x Σ (cid:48)(cid:48) (0) (cid:19) + O ( x )From Lemma 6.1, 2Λ( x ) x Λ (cid:48) ( x ) = 1 + x ρσ (cid:48) (cid:104) K , (cid:105) σ + x (cid:112) Λ (cid:48)(cid:48) (0)Σ (cid:48)(cid:48) (0) + O ( x )and, with A ( x ) given in Lemma 3.2, we have when H < / A ( x )Λ( x ) x Λ (cid:48) ( x ) = 1 + x a + O ( x )with a = ( σ (cid:48) ) σ C K,ρ + σ (cid:48)(cid:48) σ C K,ρ + ˙ σ (2 H + 1) σ + (cid:112) Λ (cid:48)(cid:48) (0) v (cid:48)(cid:48) (0) − ρ ( σ (cid:48) ) (cid:104) K , (cid:105) σ = ( σ (cid:48) ) σ D K,ρ σ (cid:48)(cid:48) σ D K,ρ σ (2 H + 1) σ , with D K,ρ , D
K,ρ , defined in (3.7). Now, as a consequence of Lemma 6.1, we have x x ) ∼ (cid:48)(cid:48) (0) x = 2 σ x and the expansion of a ( x ) follows. When H = 1 / A ( x )Λ( x ) x Λ (cid:48) ( x ) exp( x/
2) = 1 + x a + O ( x )with a = ( σ (cid:48) ) σ D K,ρ σ (cid:48)(cid:48) σ D K,ρ σ (2 H + 1) σ + ρσ (cid:48) σ (cid:104) K , (cid:105) . We conclude as in the case
H < / Proof of Theorem 3.11.
The call asymptotics is a corollary of Theorem 3.1, taking into consideration thatΛ( x t ) = n (cid:88) i =2 Λ ( i ) (0) i ! x i t iβ + O ( t ( n +1) β )and that O ( t ( n +1) β − H ) → β ∈ ( Hn +1 , Hn ]. Recall Λ (cid:48)(cid:48) (0) = σ − and the first statement follows.Let us write α = 2 H − β , δ = 1 / H − β , γ = 1 / − H + β and M ( t, x ) = (cid:80) ni =3 Λ ( i ) (0) i ! x i t iβ − H . We intendto apply [27, Corollary 7.1, Equation (7.2)], where G − ( k, u ) denotes √ √ u + k − √ u ) and V denotes √ tσ BS . Todo so, we notice that(6.21) G − ( k, u ) = k u + o (cid:0) k u (cid:1) when k ↓ u → ∞ . In the notation of [27], we have L t = − log c ( t, k t ) , and G will be computed for u = L t − log L t + log( k t √ π ), so let us compute L t −
32 log L t + log( k t √ π )= x σ t α + M ( t, x ) − log σ x √ π − δ log t −
32 log L t + log (cid:0) k t √ π (cid:1) + o (1)and take care of the logarithmic terms in t . For t ↓ − δ log t −
32 log L t + log (cid:0) k t √ π (cid:1) = ( − δ + 32 α + γ ) log t −
32 log( x σ ) + log (cid:0) x √ π (cid:1) + o (1)= −
32 log( x σ ) + log (cid:0) x √ π (cid:1) + o (1)So(6.22) L t −
32 log L t + log (cid:0) x √ π (cid:1) = x σ t α + M ( t, x ) + o (1)Equations (6.21) and (6.22) tell us that1 t G − (cid:0) k t , L t −
32 log L t + log (cid:0) k t √ π (cid:1)(cid:1) = σ
11 + σ M ( t,x ) x t α + o ( t α ) + o ( t α )(6.23)The proof now boils down to writing the development of this factor using the Taylor developement of u , with u = 2 σ M ( t, x ) t α /x + o ( t α ). We have, for j ∈ N , u j = (cid:18) σ M ( t, x ) x t α (cid:19) j + o ( t α )using M ( t, x ) p − t jα = o ( t α ) for j ≥ p ≥
1. Also notice u n − = O (( M ( t, x ) t α ) n − ) = o ( t α ) because β ∈ ( Hn +1 , Hn ].We have 11 + u = n − (cid:88) j =0 ( − j u j + O ( u n − ) = n − (cid:88) j =0 ( − j (cid:18) σ M ( t, x ) x t α (cid:19) j + o ( t α )(6.24)So from (6.23) and (6.24)1 t G − ( k t , L t −
32 log L t + log( k t √ π )) = n − (cid:88) j =0 ( − j j σ j +1)0 (cid:18) M ( t, x ) x t α (cid:19) j + o ( t α )(6.25)We apply now [27, Corollary 7.1, Equation (7.2)]: (cid:12)(cid:12)(cid:12)(cid:12) t G − ( k t , L t −
32 log L t + log( k t √ π )) − σ BS ( k t ) (cid:12)(cid:12)(cid:12)(cid:12) = o (cid:0) k t tL t (cid:1) = o (cid:0) t α (cid:1) and obtain expansion (3.17). Appendix A. Fractional Brownian motion
The fBM is a “rough” continuous-time Gaussian process in that, depending on a parameter H ∈ (0 , H . Unlike classical BM, the increments offBm are not independent if H (cid:54) = 1 /
2. The fBM was introduced for the first time by Mandelbrot and Van Ness in[41] as the following stochastic integral, for t ≥ Z Ht = c H (cid:20) (cid:90) t −∞ ( t − s ) H − / dZ s − (cid:90) −∞ ( − s ) H − / dZ s (cid:21) , where Z is a BM and c H = (cid:0) (cid:82) ∞ [(1 + s ) / − H − s / − H ] ds + H (cid:1) / . Such process is Gaussian with covariance(A.1) E [ Z Ht Z Hs ] = 12 ( | t | H + | s | H − | t − s | H ) . HORT DATED SMILE UNDER ROUGH VOLATILITY: ASYMPTOTICS AND NUMERICS 21
It can also be represented as a Volterra integral on the interval [0 , t ]:(A.2) Z Ht = (cid:90) t K H ( s, t ) dB s , with K H as in [45] or [20, Section 3.1]). One can consider the following variant of fBM, known as Riemann-Liouvilleprocess [41], introduced in 1953 by L´evy. This process is also represented as Volterra integral as(A.3) (cid:98) B Ht = (cid:90) t K ( t, s ) dB s , with a simpler kernel(A.4) K ( t, s ) = √ H ( t − s ) H − / , for H ∈ (0 , . It is still self-similar, but stationarity of increments does not hold. Moreover, the covariance structure is morecomplicated than (A.1). It can be expressed using hypergeometric functions (see [6, Lemma 4.1]). The K -functionalsthat we find in our expansion can be computed in this case as (cid:104) K , (cid:105) = √ H ( H + 1 / H + 3 / (cid:104) K , (cid:105) = 12 H + 1 (cid:104) ( K , (cid:105) = (cid:104) ( K , (cid:105) = H ( H + 1)( H + 1 / (cid:104) K , K (cid:105) = 2 H ( H + 1 / β ( H + 3 / , H + 3 / β is the beta function. In the case K ≡ References [1] Y. Ait-Sahalia, C. Li, and C. X. Li. Implied Stochastic Volatility Models.
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