Optimal Kernel Estimation of Spot Volatility of Stochastic Differential Equations

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1 Optimal Kernel Estimation of Spot Volatility of Stocastic Differential Equations José Figueroa-López and Ceng Li Department of Matematics Wasington University in St. Louis St. Louis, MO 6313, USA Department of Statistics Purdue University West Lafayette, IN, 4797, USA Abstract: Te selections of te bandwidt and kernel function of a kernel estimator are of great importance in practice. Tis is not different in te context of spot volatility kernel estimators. In tis work, a feasible metod of bandwidt and kernel selection is proposed, under some mild conditions on te volatility process, wic not only cover classical Brownian motion driven dynamics but also some processes driven by long-memory fractional Brownian motions. Under te proposed unifying framework, we caracterize te leading order terms of te mean squared error, wic in turn enables us to determine an explicit formula for te leading term of te optimal bandwidt. Central limit teorems for te estimation error are also given. A feasible plug-in type bandwidt selection procedure is ten proposed, for wic, as a sub-problem, a new estimator of te volatility of volatility is developed. In addition, te optimal selection of te kernel function is also investigated. For Brownian Motion type volatilities, te optimal kernel turns out to be an exponential function. For fractional Brownian motion type volatilities, numerical results to compute te optimal kernel are devised and, for te deterministic volatility case, explicit optimal kernel functions of different orders are derived. Simulation studies furter confirm te good performance of te proposed metods. MSC 21 subject classifications: Primary 62M9, 62G5; secondary 6F5. Keywords and prases: spot volatility estimation, kernel estimation, bandwidt selection, kernel function selection, vol vol estimation. 1. Introduction Te estimation of te diffusive coefficient σ t of a dynamical stocastic system of te form dx t = µ t dt+σ t dw t driven by a Brownian motion W as received some renewed attention in te last few years. Tis researc as partly been pused by te advent of ig-frequency data (HFD) in several fields but more importantly in finance. In te latter context, te estimation of te spot volatility σ t elps Researc supported in part by te NSF Grants: DMS and DMS

2 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 2 market participants to better assess and caracterize te beavior of market volatility troug time and is needed for many problems of finance suc as option pricing and portfolio selection. In tis work, we revisit te problem of spot volatility estimation by kernel metods. Kernel estimation as a long istory and extensive treatments of te metod can be found in many textbooks, suc as, Tsybakov [21]. Te selection of te bandwidt and te kernel function are of great importance for te performance of te kernel estimator in a finite sample setting. Te problem as been extensively studied for density estimation and kernel regression (cf. [9], [17], [18], [3], [13], [5], [22]). However, in te context of spot volatility estimation, te literature related to tis problem is muc scarcer. In tis work, we put forward a unified framework to te problem tat allows us to deal not only wit well studied Brownian driven volatilities but also tose driven by oter Gaussian processes suc as geometric Brownian motions. Before going into furter details, we first review some relevant literature of te problem Literature review Foster and Nelson [8] studied a rolling window estimator, wic can be seeing as a kernel estimator wit a compactly supported kernel function. Tey establised te point-wise asymptotic normality of te estimator, and drew some conclusions about te optimal window lengt (i.e., bandwidt) and te optimal weigt functions (kernel functions). However, in spite of te non-parametric model setting, te volatility was constrained to ave a Brownian-like degree of smootness (see Assumption A (vii) and (viii) terein) and te selection of bandwidt and kernel function was not systematically studied, since it was assumed te strict relationsip 1 n n 1/2 between te window s lengt n and te sample size n (see Assumption D terein). Under suc a relationsip, tey obtained te optimal kernel weigts and separately determine te optimal constant c appearing in te formula n = cn 1/2, but only for te flat-weigts or uniform kernel case (see Teorem 4 terein). Fan and Wang [6] also sowed a point-wise asymptotic normality for a general kernel estimator under a specific constraint on te rate of convergence of te bandwidt (Condition A4 terein), wic allowed tem to neglect te error coming from approximating te spot volatility by a kernel weigted volatility (we refer te reader to Section 6 for details), but te acieved convergence rates are suboptimal. For a continuous Itô semimartingale wit volatility driven by a Brownian motion and jumps, Alvarez et al. [1] considered te estimation of σ p t by taking forward finite differences of te realized power variation process of order p, wic is equivalent to a forwardlooking kernel estimator wit uniform kernel. CLTs were also developed terein, wic allowed tem to argue tat te best possible rate of convergence of te estimation error is n 1/4 and tat tis is attained wen n 1/2 n c (, ), as n. More general results along te same vein ave also been developed 1 As usual, a n b n if ma n b n Ma n, for all n and some < m < M <.

3 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 3 in te monograp of Jacod and Protter [11] (see Capter 13 terein). More recently, Mancini et al. [15] as developed asymptotic normality for a more general class of spot volatility estimators, wic includes kernel estimators. Te closest work to ours may be Kristensen [14], wo, besides asymptotic normality, also studied optimal bandwidt selection metod, but under a strong pat-wise smootness condition, wic as several practical and teoretical drawbacks. Indeed, even for simple volatility processes, it is not possible to verify te Hölder continuity needed for a central limit teorem wit optimal rate. Furtermore, even toug an optimal bandwidt is deduced in closed form terein, tis is not well-defined if we want to attain optimal convergence rates for te estimation error (see Remark 2.1 below for furter details). Based on a euristic argument, an alternative cross-validation metod of bandwidt selection was also proposed by [14], but tis algoritm as ig computational complexity and te asymptotic properties are not known Our contributions Having discussed some previous work on te kernel estimation of spot volatility, we now mention some motivating factors and objectives of te present work. To begin wit, we wis to adopt easily verifiable and general enoug conditions to cover a wide range of frameworks witout imposing strong constraints on te degree of smootness of te volatility process. From a teoretical point of view, we also aim to provide a formal justification of te optimal convergence rate of te kernel estimator and to establis central limit teorems (CLT) and asymptotic estimates of te mean square errors wit optimal rates. From te practical side, te two factors tat affects te performance of te estimator, bandwidt and kernel function, ougt to be optimized jointly, not separately, and meanwile, te proposed metod sould remain feasible and sufficiently efficient to be implementable for HFD. Te key to our unifying treatment of te problem is a relatively mild local scaling property of te covariance structure of te volatility process. Tis assumption covers a wide range of frameworks including deterministic differentiable volatility processes and volatilities driven by Brownian Motion, longmemory fractional Brownian Motion, and more generally, functions of suitable Gaussian processes. Under te referred assumption, we caracterize te leading order terms of te Mean Squared Error (MSE) and, as a byproduct, we derive an approximated optimal bandwidt in closed form, wic is sown to be asymptotically equivalent to te true optimal bandwidt. From tis, te teoretical optimal convergence rate for te estimation error is identified. We ten proceed to sow tat our optimal bandwidt formulas are feasible by proposing an iterated plug-in type algoritm for teir implementation. An important intermediate step is to find an estimate of te Integrated Volatility of Volatility (IVV), for wic we propose a new estimator based on te two-time scale realized variance of Zang et al. (25). Consistency and convergence rate of our vol vol estimator are also establised. Te estimation of te IVV as also been addressed in Vetter [23] and Barndorff-Nielsen and Veraart [2].

4 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 4 Equipped wit an explicit formula for te asymptotically optimal MSE, we proceed to setup a well posed problem for optimal kernel selection, wic was not considered before in te literature in te generality of te present work. Concretely, for Brownian motion driven volatilities, we prove tat te optimal kernel function is te exponential kernel: K(x) = 2 1 exp( x ). Suc a result formalizes and extends a previous result of Foster and Nelson (1994), were only kernels of bounded support were considered. We also sow tat, due to te nature of te data we are analyzing (namely, HFD), exponential kernel function enjoys outstanding computational advantages, as it reduces te time complexity for estimating te wole pat of te volatility on all grid points from O(n 2 ) to O(n). We also consider te volatility processes driven by te long-memory fractional Brownian motion and, in suc a case, we provide numerical scemes to compute te optimal kernel function. For sufficiently smoot volatilities, we also consider iger order kernel functions and, by using calculus of variation wit constraints, we obtain optimal kernel functions of different orders. Te second order optimal kernel is exactly Epanecnikov (1969) kernel and, for iger order cases, we provide ways to calculate tose optimal kernel functions. To complement our asymptotic results based on MSE, asymptotic normality of te kernel estimators is also establised for two broad types of volatility processes: Itô processes and continuous function of some Gaussian processes. In tis way, our results cover volatility processes wit flexible degrees of smootness. Te results are consistent wit te leading order approximation of te MSE, so tat CLT s wit te optimal convergence rate are obtained. By contrast, as mentioned above, te CLT s of Fan and Wang (28) and Kristensen (21) ave suboptimal convergence rate, wile te analogous result of Foster and Nelson (1994) is limited to a specific smootness order and strong constraints on te kernel function and bandwidt. In te case of Itô volatility processes, we generalize te CLT of Alvarez et al. [1] and Jacod and Protter [11], from uniform to general forward looking kernels. Te rest of te paper is organized as follows. In Section 2, we introduce te kernel estimator and our assumptions, and verify tat common volatility processes satisfy our assumptions. In Section 3, we deduce te leading order approximation of te MSE of te kernel estimator and solve te optimal bandwidt selection problem. Ten, in Section 4, we deal wit te optimal kernel function selection problem for different types of volatility processes. A feasible implementation approac of te optimal bandwidt is discussed in Section 5, were we also introduce te two-scale estimator of te IVV. Central Limit Teorems of te kernel estimator are discussed in section 6. Finally in Section 7, we perform Monte Carlo studies. Te proofs of te main results are provided in Appendix A wile te proofs of some tecnical lemmas and supporting propositions are deferred to te supplemental article [7]. 2. Kernel Estimators and Assumptions In tis section, we first introduce te classical kernel estimator for te spot volatility. We ten discuss some needed smootness conditions on te volatility

5 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 5 processes and verify tat most common volatility processes used in te literature indeed satisfy our assumptions. Finally, we discuss some regularity conditions on te kernel function and state some needed tecnical lemmas Framework and Estimators Trougout te paper, we consider te following stocastic differential equation (SDE): dx t = µ t dt + σ t db t, (2.1) were all stocastic processes (µ := {µ t } t, σ := {σ t } t, B := {B t } t, etc.) are defined on a complete filtered probability space (Ω, F, F, P), were F = {F t } t. We also assume tat µ and σ are adapted to te filtration F and B := {B t } t is a standard Brownian Motion (BM) adapted to F. We suppose trougout te paper tat we observe te process X at te times t i := t i,n := i n, i n, were n := T/n. We will use n i Z := Z t i 1 := Z ti Z ti 1 to denote te increments of a process Z and n = T/n to denote te time increments. In tis paper, we study te problem of estimating te spot volatility σ τ, at a given time τ (, T ), by te kernel estimator (cf. [6] and [14]), ˆσ 2 τ,n, := n K (t i 1 τ)( n i X) 2, (2.2) were K (x) = K(x/)/. As is often te case wit kernel estimation, te selections of te bandwidt and kernel function K of (2.2) are of great importance in practice, especially for te finite sample settings commonly encountered in econometric applications. As explained in te introduction, te literature on bandwidt and kernel selection for te spot volatility estimator (2.2) is rater scarce. In tis work, we go furter wit better crafted conditions tat allow us to give a unified treatment to te problem for most common volatility processes, including not only Brownian driven volatilities but also tose driven by more general Gaussian processes Assumptions on te Volatility Process Our first assumptions, wic is also imposed in [14], is a non-leverage condition. Toug not ideal for financial applications, tis is imposed for tractability reasons and, in order to allow a unified treatment of volatilities driven by a large spectrum of noises beyond Brownian motion. In fact, if we were dealing wit, say, volatilities driven by fractional Brownian motion, it is not clear ow to incorporate leverage in suc a context. Our simulations in Section 7 sow tat tis assumption may not be crucial. Indeed, for Brownian-driven volatilities, tis assumption is not needed to obtain a central limit teorem for forward looking kernel functions (see Section 6 for details).

6 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 6 Assumption 1. (µ, σ) is independent of B. Anoter assumption tat we need later is te boundedness of some moments of µ and σ. Assumption 2. Tere exists M T t T. > 1 suc tat E[µ 4 t + σ 4 t ] < M T, for all Te following is our key assumption. Tis imposes some local scaling condition on te covariance function of te variance process t σ 2 t. Assumption 3. Suppose tat for γ > and certain functions L : R + R + and C γ : R R R, suc tat C γ is not identically zero and C γ (r, s) = γ C γ (r, s), for r, s R, R +, (2.3) te variance process V := {V t = σ 2 t : t } satisfies E[(V t+r V t )(V t+s V t )] = L(t)C γ (r, s) + o((r 2 + s 2 ) γ/2 ), r, s. (2.4) As sown in te next section, Assumption 3 is satisfied by a large spectrum of volatility models. Remark 2.1. We now draw some connections wit te assumptions and work in [14]. Terein, te variance process {V t } t is assumed to satisfy te following patwise condition V t+δ V t L(t, δ ) δ γ + o( δ γ ), δ, (2.5) were L(t, ) is a slowly varying random function. To gain some intuition about te plausibility of tis assumption, let us suppose tat {V t } is a Brownian motion. In tat case, te above olds for all γ < 1/2, but suc coices of γ can only produce suboptimal convergence rate of te kernel estimator. On te oter, in ligt of Lévy s modulus of continuity, te condition (2.5) olds for γ = 1/2, but only if L(t, δ), as δ. But, in tat case, te optimal bandwidt selection formulas obtained in [14] are not well defined as tey presume tat lim δ L(t, δ) =: L(t, ) is finite. A function C γ satisfying te condition (2.3) is said to be omogeneous of order γ. Te index γ determines te degree of smootness of te volatility pats t σ t. It is easy to ceck (see details in te supplemental article [7]) tat C γ (r, s; t) := L(t)C γ (r, s) is unique and satisfies te following non-negative definiteness property: K(x)K(y)C(x, y)dxdy. (2.6) We sall see in te next section tat most volatility processes tat are studied in te literature satisfy Assumption 3 wit a function C γ of te form: C γ (r, s) = 1 2 ( r γ + s γ r s γ ), (2.7)

7 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 7 for some γ [1, 2]. Te case of γ = 1 covers volatility processes driven by BM, wile γ (1, 2) corresponds to volatility processes driven by fractional Brownian Motions (fbm) wit Hurst parameter H > 1/2. Deterministic and sufficiently smoot volatility processes can also be incorporated by taking γ = 2. Matematically one can also consider more general models as long as Assumption 3 is satisfied. For instance, we will see in te next section tat for a suitable Gaussian processes {Z t } t and a smoot function f : R R, V t := f(z t ) satisfies Assumption 3. Furtermore, for any valid non-negative definite symmetric function C γ tat is omogeneous to order γ, one can define a zero-mean continuous Gaussian process {Z t } t suc tat E[Z t Z s ] = C γ (t, s). In suc a case, if we define V t = σ 2 t as a stocastic integral wit respect to {Z t } t, ten generally {V t } t would satisfy Assumption Common Volatility Processes In tis section, we demonstrate tat many popular volatility processes satisfy Assumption 3. We consider four fundamental cases. Te simplest case is wen te volatility process is deterministic and is differentiable. Te second and tird cases are te solutions to SDEs driven by BM and long-memory fbm, respectively. Finally, we consider a volatility tat is a smoot function of a Gaussian process satisfying te Assumption 3. Te proofs of te following propositions are deferred to te supplemental article [7]. Te following proposition considers a deterministic volatility process. Proposition 2.1. Suppose te squared volatility process is given by a deterministic function f(t) = σ 2 t, t T, suc tat, for some m 1, f is m t -times differentiable at τ (, T ), f (i) (τ) =, for 1 i < m, and f (m) (τ). Ten, f satisfies Assumption 3 wit γ = 2m and C 2m (r, s) := r m s m. We next consider te solutions of a standard SDE driven by BM. Tis is one of te most popular approaces to generalize te Black-Scoles-Merton model to non-constant volatility and is widely used in practice. Proposition 2.2. Consider a complete filtered probability space (Ω, F, F = {F t } t, P) and an Itô process V t = σ 2 (t, ω) tat satisfies te SDE dv t = f(t, ω)dt + g(t, ω)dw t, t [, T ], (2.8) were {W t } t is a standard Wiener process adapted to F. Assume tat f(t, ω) and g(t, ω) are adapted and progressively measurable wit respect to F, E [ f 2 (t, ω) ] < M, for t [, T ], and E [ g 2 (t, ω) ] is continuous for t [, T ]. Ten, Assumption 3 is satisfied wit γ = 1, C 1 (r, s) = min{ r, s }1 {rs }, and L(t) = E[g 2 (t, ω)]. Furtermore, C 1 (r, s) is an integrable positive definite function; i.e., we ave strict inequality in (2.6) for all K : R R suc tat K(x) dx >. In wat follows, we sow tat some processes defined as integrals wit respect to a two-sided fbm B (H) = {B (H) t : t R} (see [2] for a detailed survey of

8 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 8 fbm) satisfy Assumption 3. A prototypical example is te fractional Ornstein- Ulenbeck process (fou) (cf. [4]), Y (H) t = σ t e λ(t u) db (H) u, t. (2.9) wic is frequently used to model volatility processes. It is wort mentioning tat, wen H 1/2, te fbm is not a semimartingale and te problem of defining te stocastic integral wit respect to fbm is more subtle. Tere are several approaces to tis problem. In our paper, we only focus on integrals of deterministic functions f for wic te integral can be defined on a pat-wise sense under te following condition (cf. [2]): f(u)f(v) u v 2H 2 dudv <. (2.1) Since tere is no guarantees tat te stocastic integral of f wit respect to fbm is nonnegative, wic is a requirement of a volatility process, we also consider te exponential of suc a process. Proposition 2.3. Let Y (H) t = t f(u)db(h) u were f( ) is a deterministic continuous function tat satisfies (2.1) and {B (H) t } t R is a (two-sided) fbm wit Hurst parameter H ( 1 2, 1) defined on a filtered probability space (Ω, F, F = {F t } t, P ). Ten, te processes Y (H) and {exp(y (H) t )} t satisfy Assumption 3 wit γ = 2H (1, 2) and C γ given by (2.7). For our final case, we sow tat if a Gaussian process satisfies Assumption 3, so does a suitable smoot function of te process. Proposition 2.4. Assume tat (Z t ) t is a Gaussian process tat satisfies Assumption 3 uniformly over (, T ), 2 wit γ (Z) [1, 2), L( ), and C γ (Z) (, ) defined as in (2.4). For eac fixed τ (, T ) and a function f C 2 (R), furter assume te following: (a) E[(Z τ+r Z τ )Z τ ] = O( r ), E[Z τ+r ] E[Z τ ] = O( r ), as r. (b) E[(f (Z τ )) 4 ] <, E[sup t (τ ɛ,τ+ɛ) (f (Z t )) 4 ] < for some ɛ >. Ten, te process V t := f(z t ), t, satisfies Assumption 3 wit γ (V ) = γ and C γ (V ) = E[(f (Z t )) 2 ]C γ (Z). Remark 2.2. Note tat te condition (a) in Proposition 2.4 is not a consequence of Assumption 3. Tis is satisfied by a large class of Gaussian processes, suc as a fbm wit zero mean and covariance structure given by (2.7). Intuitively, tis condition states tat, altoug Z τ and Z τ+r Z τ may not be independent, its correlation coefficient vanises, as r, fast enoug as compared wit standard deviation of Z τ+r Z τ. 2 Te Assumption 3 is satisfied uniformly over (, T ) if sup τ (,T ) (r 2 + s 2 ) γ/2 E[(V τ+r V τ )(V τ+s V τ )] L(τ)C γ(r, s), as r, s, and, also, sup τ (,T ) L(τ) <. Tis implies te existence of a positive constant C suc tat E[(Z t Z s) 2 ] C t s γ, for all t, s (, T ).

9 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility Conditions on te Kernel In tis part, we introduce te assumptions needed on te kernel function, togeter wit some required lemmas. Assumption 4. Given γ > and C γ as defined in Assumption 3, we assume tat te kernel function K : R R satisfies te following conditions: (1) K(x)dx = 1; (2) K is Lipscitz and piecewise C 1 on its support (A, B), were A < < B ; (3) (i) K(x) x γ dx < ; (ii) K(x)x γ+1, as x ; (iii) K (x) dx <, (iv) V ( K ) <, were V ( ) is te total variation; (4) K(x)K(y)C γ (x, y)dxdy >. Remark 2.3. Note tat (4) above does not put substantial restriction on K since, in any case, C γ is non-negative definite as sown by (2.6) and, furtermore, C γ is strictly positive definite in some important cases suc as BM type volatilities. Te condition (4) above is made in consideration of te superefficiency penomenon mentioned in Section of [21]. Indeed, tere are two distinct situations. For BM type volatility, it is not possible to find a iger order kernel K to furter reduce te MSE of te kernel estimator. Te case of deterministic volatility is different since, in tis case, it is possible to find K suc tat K(x)K(y)C γ (x, y)dxdy =, wic actually will lead to even a faster rate of convergence of te estimation mean-squared error. Tis will be discussed furter in Section Approximation of MSE and Optimal Bandwidt Selection In tis section, we assume tat te processes µ, σ, and, B satisfy Assumptions 1-3, and we consider a kernel function K tat satisfies Assumption 4. In wat follows, we first deduce an explicit leading order approximation (up to O( ) and O( γ ) terms) of te MSE = MSE n, = E[(ˆσ 2 τ,n, σ2 τ ) 2 ]. After tis, we proceed to study te approximated optimal bandwidt, wic is defined as te bandwidt tat minimizes te leading order approximation of te MSE. Finally, we prove tat our approximated optimal bandwidt is asymptotically equivalent to te true optimal bandwidt tat minimizes te true MSE Approximation of te Mean Squared Error Te following result gives an explicit asymptotic expansion for te MSE of te kernel estimator (2.2). Te proof is deferred to te Appendix A. Teorem 3.1. For te model (2.1) wit µ and σ satisfying Assumptions 1-3, and a kernel function K satisfying Assumption 4, let MSE a τ,n, := 2 E[σ4 τ ] K 2 + γ L(τ) K(x)K(y)C γ (x, y)dxdy. (3.1)

10 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 1 Ten, for any τ (, T ), MSE τ,n, = E[(ˆσ 2 τ σ 2 τ ) 2 ] = MSE a τ,n, + o ( ) + o ( γ ). (3.2) It is not ard to see from te proof of te previous result tat all o( ) terms are uniform for τ (, T ) if te condition given by (2.4) is satisfied uniformly in t, and, terefore, te following explicit asymptotic expansion for te integrated mean-squared error (IMSE) olds. Corollary 3.1. Let MSE a n,(a, b) := 2 b a E[σ 4 t ]dt K 2 + γ b a L(t)dt K(x)K(y)C γ (x, y)dxdy. (3.3) Ten, for te model (2.1) wit µ and σ satisfying Assumptions 1-3, so tat te term o((r 2 + s 2 ) γ/2 ) in Eq. (2.4) is uniform in t, and a kernel function K satisfying Assumption 4, we ave, for any < a < b < T, IMSE n, := b a E[(ˆσ 2 t σ 2 t ) 2 ]dt = MSE a n,(a, b) + o 3.2. Approximated Optimal Bandwidt ( ) + o( γ ). (3.4) Based on te approximations above, it is natural to analyze te beavior of te approximated MSE of te kernel estimator. We focus on te integrated MSE (3.4) but an analogous analysis can be made for te local MSE (3.2). Note tat, by Assumption 4, we ave tat K(x)K(y)C γ (x, y)dxdy >. We ten obtain te following approximated optimal bandwidt: Proposition 3.1. Wit te same assumptions as Corollary 3.1, te approximated optimal omogeneous bandwidt, denoted by n, wic is defined to a,opt minimize te approximated IMSE given by (3.3), is given by a,opt n =n 1/(γ+1) [ 2T b a E[σ4 t ]dt ] 1/(γ+1) K 2 (x)dx γ b a L(t)dt, (3.5) K(x)K(y)C γ (x, y)dxdy wile te attained minimum of te approximated IMSE is given by IMSE a,opt n (a, b) = n γ/(1+γ) ( γ ) ( 2T ( γ b a L(t)dt b a E[σt 4 ]dt K 2 (x)dx ) γ/(1+γ) K(x)K(y)C γ (x, y)dxdy) 1/(1+γ).

11 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 11 A direct consequence of te previous result is te following proposition about te optimal convergence rate. Tis provides a rigorous justification of te optimal convergence rate of te kernel estimator. Proposition 3.2. Wit te same assumptions as tose in Corollary 3.1, te optimal convergence rate of te kernel estimator is given by n γ/(1+γ). Tis is attainable if te bandwidt is cosen as n = cn 1/(γ+1) for some constant c (, ). It is wortwile to draw some connections wit [14] by considering te case of γ = 2, wic corresponds to a deterministic variance function σ 2 t = f(t) tat is continuously differentiable at τ and suc tat f (τ). In tat case, te approximated MSE (3.1) is given by MSE a τ,n, = 2 f 2 (τ) ( K 2 (x)dx + f (τ) K(x)xdx) 2, (3.6) a,opt n wic coincides wit te approximation obtained in [14]. However, it is evident tat, in te case tat te volatility is stocastic and non-smoot, our results are different from tose in [14]. In Section 4.4, we will see tat in te case of deterministic and smoot volatility, we are able to use iger order kernels to improve te rate of convergence of te kernel estimator, but in oter situations, for example BM type volatility, tis is not possible. An important problem is to formalize te connection between te approximate optimal bandwidt (respectively, a,opt n ), wic is defined as te minimizer of te MSE (3.3) (respectively, (3.1)), and te true optimal bandwidt, wenever it exists, wic is denoted by n (respectively, n) and is defined as a value of te bandwidt tat minimizes te actual IMSE (respectively, MSE) of te kernel estimator. In te supplemental article [7], we sow tat, under a mild additional condition, tey are equivalent in te sense tat n = a,opt n + o( a,opt n 4. Kernel Function Selection ) and n = a,opt + o( a,opt n ). n As an important application of te optimal bandwidt selection problem defined in Section 3, we now study te problem of selecting an optimal kernel function by minimizing te optimal IMSE attained by (3.5). As sown terein, te optimal kernel function only depends on te covariance structure, C γ (, ). Tere are two possible situations. Te first one is wen C γ is positive definite. In suc a case, we cannot improve te rate of convergence of te IMSE, but we can minimize te constant appearing before te asymptotic IMSE. Anoter situation is wen C γ is simply non-negative definite. In suc a case, if we relax (4) of Assumption 4, it is possible to improve te rate of convergence of te IMSE by coosing a so-called iger order kernel function. In tis section, we focus on te covariance function C γ defined in Eq. (2.7). For future reference, let us first point out some useful properties of C γ. It is

12 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 12 possible to write C γ in te integral form C γ (x, y) = F γ (x, u)f γ (y, u)du, wit ) F γ (x, y) := m ( x y γ 1 γ 1 2 sgn(x y) + y 2 sgn(y), for a certain constant m (see [16] for details). We can ten easily ceck tat, for an arbitrary kernel function K : R R, its symmetrization, K s (x) := (K(x) + K( x))/2, is suc tat K(x)K(y)C γ (x, y)dxdy K s (x)k s (y)c γ (x, y)dxdy = 1 [ 2 [K(x) K( x)]f γ (x, u)dx] du. 4 (4.1) Te previous relation implies tat in order to minimize te constant appearing before te asymptotic IMSE in (3.5), it suffices to consider symmetric kernel functions K Optimal Kernel Selection for BM driven Volatilities Consider a BM type volatility wit γ = 1 and C 1 (r, s) = 1 {rs>} min( r, s ). We will sow tat te exponential kernel function is te optimal kernel function. Exponential kernel function as also been sown to be optimal in oter contexts (e.g., [22] for te density estimation problem). Foster and Nelson (1994) also argued tat te exponential kernel is te optimal kernel function to estimate spot volatility, under similar but different assumptions as we ave. However, teir proof lacks rigor, due to teir bounded support assumption on te kernel function. From (3.1), te objective function tat we need to minimize is I(K) = K 2 (x)dx [K(x)K(y) + K( x)k( y)] min(x, y)dxdy, wit te restriction K(x)dx = 1. As implied by te relation (4.1), we only need to consider symmetric kernel functions K. Terefore, it suffices to minimize 1 4 I(K) = K 2 (x)dx K(x)K(y) min(x, y)dxdy. Next, we define L(u) = K(x)dx and note tat, by definition of K, tere are u only finite many points were L does not exist and, furtermore, 1 4 I(K) = [L (x)] 2 dx [L(x)] 2 dx =: I (L). Te problem is ten canged to minimize I (L) for functions L wit te following restrictions: (1) L is continuous and piece-wise twice differentiable on R +.

13 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 13 (2) L() = 1 2 and lim x + L(x) =. Finally, using Caucy-Scwartz inequality, note tat ( 2 ( ( 2 ) 2 I (L) L (x)l(x)dx) = L(x)dL(x)) = udu = 1 1/2 64, were te first inequality becomes equality if and only if tere exist non-zero constants C 1 and C 2 suc tat C 1 L (x) + C 2 L(x), for all x R +. We ave two possible cases: (1) tere exists x >, suc tat L(x) >, for all x [, x ) and L(x ) = ; (2) L(x) >, for all x R +. For te first case, we ave tat L (x)/l(x) = C 2 /C 1, for x (, x ), wose solution is L(x) = 1 2 ebx and it is ten impossible tat L(x ) =. Terefore, only te second case is possible and, by solving te same differential equation, we ave te following. Teorem 4.1. For te model (2.1) wit µ and σ satisfying Assumptions 1-3, were C γ is given by (2.7) wit γ = 1, and for a kernel function K satisfying Assumption 4, we ave tat te optimal kernel function tat minimizes te first order approximation of te MSE of te kernel estimator is te exponential kernel function: K opt (x) = 1 2 e x, x R. Remark 4.1. We can easily demonstrate to wat extent te exponential kernel decreases te MSE. As seen from (3.1), IMSEn a,opt = C I(K), were te constant C does not depend on te kernel function K. Below, we sow te value of I(K) for te exponential, uniform, triangular, and te Epanecnikov kernels: I(.5 e x ) = , I(.5 1 { x <1}) = , I( 1 x 1 { x <1} ) = , I(.75 (1 x2 )1 { x <1} ) = Interestingly enoug, te triangle kernel performs better tan Epanecnikov kernel and Epanecnikov kernel performs better tan te uniform kernel. Te intuition beind tis is tat a kernel function wit a sape more similar to te exponential kernel generally performs better Efficient Implementation of te Uniform and Exponential Kernel In tis subsection, we demonstrate tat te exponential kernel function not only minimizes te MSE of te kernel estimator, but also enables us to substantially reduce te computational complexity of te volatility estimation. In general, te evaluation of ˆσ τ for a fixed time τ (, T ) requires O(n) (respectively, O(n)) computations for a kernel function wit unbounded (respectively,

14 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 14 bounded) support. Tus, if we ope to get an estimation of {σ ti } i=,...,n and use te approximated optimal bandwidt given by (3.5), te best possible computational complexity is in general O(n 2 1/(γ+1) ). We now sow tat, for te exponential kernels, we can do substantially better. Let us start by introducing te following notations: K exp (x) = 1 2 e x /, ˆσ 2 τ, = i<i K exp (t i 1 τ)( i X) 2, ˆσ 2 τ, = K exp (t i 1 τ)( i X) 2, ˆσ 2 τ,+ = i>i K exp (t i 1 τ)( i X) 2, were t i 1 τ < t i. Ten, we can use te following recurrent algoritm to compute ˆσ 2 τ,exp: ˆσ 2 τ+, =e / [ˆσ 2 τ, + ˆσ 2 τ, ], ˆσ 2 τ+, =K exp (t i (τ + ))( i+1x) 2, ˆσ 2 τ+,+ =e / [ˆσ 2 τ,+ K exp (t i τ)( i+1x) 2]. (4.2) It is now clear tat, in order to estimate {σ ti } i=,...,n, using an exponential kernel, we need a time of O(n), instead of O(n 2 ) or O(n 2 ). Te difference between te two time complexities mentioned above is quite significant, since we are considering ig frequency data. In practice, kernel estimators suffer of biases at times closer to te boundary. As proposed in [14], we can correct suc boundary effects by using te following estimator: n K (t i 1 τ)( n i X)2 ˆσ b τ,n, = n K (t i 1 τ). (4.3) were te superscript denotes boundary effect. We can still implement our fast estimation algoritm to calculate tis estimator since we only need to calculate n K (t i 1 τ), wic can be calculated similarly as (4.2) except tat all ( i X) 2 are replaced by Optimal Kernel Function for a fbm driven Volatility In tis section, we now consider a general fbm covariance structure, i.e. γ (1, 2) and C γ given by (2.7). From (3.1), our goal is to minimize ( ) γ I(K) = K 2 (x)dx K(x)K(y)C γ (x, y)dxdy. (4.4) Again, as implied by te relation (4.1), we only need to consider symmetric kernel functions. Unfortunately, te problem of finding an explicit form for te optimal kernel function is muc more callenging in tis case. Terefore, we instead seek for a numerical metod to find te optimal kernel function, for wic, we consider two approximation. First, since all unbounded support kernels can

15 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 15 be approximated by a kernel wit a bounded support and te optimization problem is uncanged wit K(x) scaled by a small bandwidt, we will limit te support of K(x) to be [, 1]. Second, we approximate te kernel function K by step functions of te form K m (x) = 1 n a i m a i 1 [ i 1 m, i m )( x ), x [ 1, 1], a i R, i = 1,..., n, and ten use gradient descent to directly optimize (4.4) over all valid values of (a 1,..., a m ). In spite of te ig dimensionality of te optimization problem, tis is still tractable, since te gradient can be calculated explicitly. Figure 1 sows te resulting optimal kernels for γ = 1., 1.3, 1.6, 1.9. Note tat te resulting approximated optimal kernel for γ = 1 is consistent wit true optimal kernel tat was proved to be exponential in Section 4.1. We also observe from Figure 1 tat, as γ increases, te optimal kernel function becomes flatter and less convex. Tis indeed makes sense, since a iger γ indicates less caos of te volatility, and tus more weigts sould be given to data farter from te estimated point. Fig 1. Optimal Kernel Functions for Different γ 4.4. Optimal Kernel for a Deterministic Volatility Function Lastly, we consider te case σt 2 = V t = f(t), for a deterministic differentiable function f, in wic case, γ = 2 and C 2 (r, s) = rs. Obviously, suc a C 2 is not strictly positive definite, so we can consider iger order kernels to improve te convergence rate of te estimation MSE. Specifically, we generalize te condition (4) of Assumption 4 as follows: K(x)x i dx =, i = 1, 2,..., p 1, and K(x)x p dx.

16 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 16 Suc a kernel is said to be of order p. We refer te reader to [21, Section 1.2.2] for te construction of suc iger order kernels. We also need to extend Assumption 3 as follows: E[(V t+r V t )(V t+s V t )] = 2p 1 L i (t)c i+1 (r, s) + o((r 2 + s 2 ) p ), r, s, were te function C i is suc tat C i (r, s) = i C γ (r, s), for any r, s R, R +, and i 1. In tat case, wit a similar procedure as tat of Section 3, we can prove tat te approximated MSE (3.1) is given by MSE a τ,n, = 2 2 E[σ4 τ ] K 2 (x)dx + (f 2p (p) (τ) K(x)x dx) p. We can ten select an optimal bandwidt. Te optimal bandwidt and resulting optimal convergence rate are te same as tose obtained in [14]. An interesting application, wic was not considered in [14], is to find te kernel tat minimizes te resulting optimal approximated MSE. Concretely, if we limit ourselves to symmetric kernels of order p, te goal is to minimize ( ) p I p (K) = K 2 (x)dx K(x)x p dx, (4.5) subject to te constrain K(x)dx = 1/2. By limiting ourselves to kernel functions wit support [, 1] and using calculus of variation, we can sow tat optimal kernel function is of te form K p (x) = (2p 1 )(p + 1)(1 x p )1 [ 1,1] (see supplemental article [7]). It is wort mentioning tat wen p = 2, we ave K 2 (x) = 3 4 (1 x p )1 [ 1,1], wic is exactly te Epanecnikov kernel. Tere is still a problem for suc a kernel. Take p = 4 as an example. Altoug K 4 minimizes (4.5), it does not satisfy 1 1 K(x)x2 dx =. Terefore, we propose to consider instead te following optimization problem: ( 1 ) 2q 1 minimize I 2q (K) = K 2 (x)dx K(x)x 2q dx, subject to 1 K(x)x 2r dx =, for < r < q, and 1 K(x)dx = 1 2. Again, using calculus of variation, we can sow (see supplemental article [7]) tat optimal kernel function is of te form K(x) = ax 2q + a q 1 i= λ ix 2i, were a and λ,..., λ q 1 satisfies te equations: ( ) q 1 1 = (4q + 1)a 4q λ i + λ 2(q + i) + 1 2, i= q 1 1 = 2(q + r) λ i 2(i + r) + 1, < r < q, i= ( ) q = a 1 2q λ i. 2i + 1 i=

17 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 17 Te system above can easily be solved in particular cases. 5. Plug-In Bandwidt Selection Metods In tis section we propose a feasible plug-in type bandwidt selection algoritm, for wic, as a sub-problem, we also develop a new estimator of te volatility of volatility based on te kernel estimator of te spot volatility and a type of twotime scale realized variance estimator. We sall focus on te case of a BM type volatility, wile similar metods can be developed for oter types of volatility structures. To implement te approximated optimal bandwidt formula (3.5), it is natural to first use te integrated quarticity of X, IQ(X) = T σ4 τ dτ, and te quadratic variation of σ 2, IV (σ 2 ) = T g2 (τ)dτ, instead of teir expected values. A popular estimate for T σ4 τ dτ is te so-called realized quarticity, wic is defined by ÎQ = (3 ) 1 n ( ix) 4. Te estimation of T g2 (τ)dτ is a more subtle problem and, below, we propose an estimator, wic we call Two-time Scale Realized Volatility of Volatility (TSRVV) and is ereafter denoted by IV (σ 2 ) tsrvv. Wit tese estimators, te final bandwidt can ten be written as a,opt n = 2T IQ(X) K 2 (x)dx niv (σ 2 ) tsrvv K(x)K(y)C 1 (x, y)dxdy 1/2. (5.1) Iterative Algoritm Te previous bandwidt estimator involves te spot volatility itself, troug IV (σ 2 ) tsrvv, wic, of course, we do not know in advance. To deal wit tis problem, we propose to use an iterative algoritm in te same spirit of a fixedpoint type of procedure. Concretely, we start wit an initial guess for te bandwidt. For example, we can take (5.1) and simply set IQ(X)/ IV (σ 2 ) tsrvv = 1, wic gives: [ init 2T K 2 ] 1/2 (x)dx n = n. (5.2) K(x)K(y)C 1 (x, y)dxdy Wit suc a bandwidt, we can obtain initial estimates of te spot volatility at all te grid points. Suc an initial spot volatility estimation can ten be applied to compute IV (σ 2 ) tsrvv, wic, in turn, can be used to obtain anoter estimation of te optimal bandwidt. Tis procedure is continued iteratively until a predetermined stopping criteria is met. In reality, our simulations sow tat one or two iterations are enoug to obtain a satisfactory result and more iterations do not generally produce any improvement.

18 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility A Two-time Scale Estimator of Integrated Volatility of Volatility In tis subsection, we propose an estimator of te quadratic variation of σ 2, IV (σ 2 ) = T g2 (τ)dτ, wic is often referred to as te Integrated Volatility of Volatility (IVV) of X. Te idea is to note tat, at eac observation time t i, te estimated spot volatility can be written as ˆσ t 2 i = σt 2 i + e ti, were e ti is te estimation error. Tis suggests to make an analogy wit te problem of estimating te realized quadratic variation of a semimartingale Y based on discrete observations of Y exposed to market microstructure. So, we can apply any of te different tecniques to tackle tis problem suc as te Two-time Scale Realized Volatility (TSRV) estimator of [25]. One important difference between our problem and tat in [25] is tat our estimation errors are not independent. In fact, tey are correlated. Suc a correlation becomes more significant wen we take te difference iˆσ 2 = ˆσ t 2 i+1 ˆσ t 2 i. To alleviate suc a problem, we propose to use one-sided kernel estimators and take te difference between te rigt and left side estimators to find iˆσ 2. Concretely, let ˆσ l,t 2 i and ˆσ r,t 2 i be te left and rigt side estimator of σt 2 i, respectively, defined as j>i ˆσ l,t 2 i = K (t j 1 t i )( n j j i X)2 j>i K, ˆσ r,t 2 (t j 1 τ) i = K (t j 1 t i )( n j X)2 j i K. (t j 1 τ) (5.3) Next, we define te following two difference terms: iˆσ 2 = ˆσ r,t 2 i+1 ˆσ l,t 2 i, (k) i ˆσ 2 = ˆσ r,t 2 i+k ˆσ l,t 2 i. Finally, we can construct te following Two-time Scale Realized Volatility of Volatility (TSRVV): ÎV V (tsrvv) T = 1 k n k b i=b ( (k) i ˆσ 2 ) 2 n k + 1 nk n k b i=b+k 1 ( iˆσ 2 ) 2. (5.4) Here, b is a small enoug integer, wen compared to n. Te purpose of introducing suc a number b is to alleviate te boundary effect of te one sided estimators. It is wort to noting tat ÎV V (tsrvv) T case, a possible solution is to take simple ÎV V (tsrvv) T migt become negative. In tis = 1 k n k b i=b ( (k) i ˆσ 2 ) 2. Similar to [25], we can take k = n 2/3 in our case. Tere is some work to do if one wants to optimize suc a TSRVV estimator, but tis is outside te scope of te present work. Neverteless, as our simulations in Section 7 sow, te accuracy of spot volatility is good enoug even witout refining suc a TSRVV estimator. Te following result sows te consistency of our estimator and sed some ligt on te rate of convergence. Its proof is deferred to Appendix A. Teorem 5.1. Fix a t b (, T/2). Ten, for te model (2.1) wit µ and σ satisfying Assumptions 1 and 2 and σ being a squared integrable Itô process as in Eq. (2.8) (tus satisfying Assumption 3), and a kernel function K satisfying

19 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 19 Assumption 4, (5.4) is a consistent estimator of T t b t b gt 2 dt wit b = t b /. Furtermore, te convergence rate is given by O p ( n1/4 k ) + O k 1/2 p ( n ) and, tus, k can be cosen to be of te form Cn 3/4 so tat to attain te optimal convergence rate n 1/8. Remark 5.1. Based on a different idea, Vetter [24] proposed a similar estimator for te IVV, but taking a rigt-sided uniform kernel wen computing te difference iˆσ 2 of te estimated volatility and also applying a different bias correction tecnique from ours. It is sown terein tat is estimator attains te optimal rate of convergence of n 1/4. Simulations, tat are not sown ere for te sake of space, indicate tat our TSRVV using te optimal exponential kernel as better performance tan [24] at least for te cosen parameter coices. Tis suggests tat tere may be some room for improvement of te convergence rate of (5.4) stated in Teorem 5.1. On te oter and, te observed improved performance of our TSRVV may be a consequence of te fact tat we are using an exponential kernel, wile te estimator in [24] uses te suboptimal uniform kernel. As sown in Table 2 below, tere is an expected 1% performance improvement wen using an exponential kernel over te uniform. 6. Central Limit Teorems In tis section, we aim to caracterize te limiting distribution of te estimation error of te kernel estimator by proving a Central Limit Teorem (CLT). Te starting point is te following decomposition: ˆσ 2 τ σ 2 τ = n T + T K (t i 1 τ)( i X) 2 K (t τ)σt 2 dt K (t τ)(σ 2 t σ 2 τ )dt + o p ( γ ), (6.1) were te last term on te rigt-and side above follows from Assumption 4. Two general type of results can be found in te literature to deal wit te estimation error of te kernel estimator: (1) One approac consists of using a suboptimal bandwidt so tat only te first error term in (6.1) is significant. Tis would be te case if, for instance, we coose = o( 1/(γ+1) ), in wic case te order of te second term in (6.1), wic, as seeing below, is O p ( γ/2 ), is negligible compared to te order of te first term, wic we will see is O p (( /) 1/2 ). Instances of tis type of results can be found in [6] (see Assumption A4 and Teorem 1 terein), [14] (Teorem 3 terein), and [15] (Teorem 2.7 terein). (2) In te case tat σt 2 follows an Itô process, [8] obtained a CLT for te kernel estimator ˆσ τ 2 wit optimal convergence rate but under a number of stringent conditions. In particular, only kernels wit bounded support were considered. More recently, under relatively mild assumptions in te

20 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 2 Itô dynamics of X and σ, [1] and [11] also obtained a CLT wit optimal convergence rate but only for a forward-looking uniform kernel function. Te two previous approaces ave some obvious limitations. Te first approac can only yield results wit suboptimal convergence rates, wile te second type of results only deal wit one level of smootness in te volatility process. In tis section, we obtain a CLT for two broad frameworks: (i) Itô type volatilities and (ii) deterministic functions of certain Gaussian processes. Tese cover all te examples mentioned in Section 2.3. For te framework (i), we consider two cases: 1) A general kernel but no leverage; 2) Leverage but only forward looking kernel as in [1] and [11], even toug tese two works only consider uniform kernels, wile we consider ere a general forward-looking kernel function. Te second framework (ii) covers a wide range of models of different smootness order, toug witout leverage, wic, in any case, is not clear ow to incorporate in suc a setting. In wat follows, we replace Assumption 1 wit te following: Assumption 5. Te processes µ and σ are adapted cádlág. We begin wit an analysis of te first error term in (6.1), wic, in te nonleverage case was already studied in [14]. Its proof uses te CLT for martingale differences and is similar to tat of Teorem 2.7 in [15], but, since te proof in [15] is for a more general class of estimators and requires more tecnical analysis, we give a simpler proof in te supplemental article [7]. Te proof is also embedded in te proof of Teorem 6.3 below. Teorem 6.1. For te model (2.1) wit µ and σ satisfying Assumption 5, and a kernel function K satisfying Assumption 4, we ave, for any τ (, T ), ( ) [ 1/2 n ] T K (t i 1 τ)( i X) 2 K (t τ)σt 2 dt D δ 1 N(, 1), (6.2) were δ1 2 = 2στ 4 K 2 (x)dx. Next, we consider te second error term in (6.1), wic only involves properties of te volatility process σ and not te interaction between X and σ. Te proof is provided in Appendix A. Teorem 6.2. Let K be a kernel function satisfying Assumption 4 and fix a τ (, T ). Additionally, suppose tat eiter one of te following conditions olds: (1) {σt 2 } t is an Itô process given by σt 2 = σ 2 + t f sds + t g sdw s wit adapted cádlág processes {f t } t and {g t } t. (2) σt 2 := f(z t ), t [, T ], for a deterministic function f : R R and a Gaussian process {Z t } t satisfying all requirements of Proposition 2.4. Ten, on an extension ( Ω, F, P) of te probability space (Ω, F, P), equipped wit

21 J.E. Figueroa-López and Ceng Li/Optimal Kernel Estimation of Spot Volatility 21 a standard normal variable ξ independent of g τ in (1) or Z τ in (2), we ave: ( ) T γ/2 K (t τ)(σt 2 στ 2 )dt D δ 2 ξ, (6.3) were, under te condition (1) above, δ2 2 = gτ 2 K(x)K(y)C(x, y)dxdy, wile, under te condition (2), δ2 2 = [f (Z τ )] 2 L (Z) (τ) K(x)K(y)C γ (Z) (x, y)dxdy. As a byproduct of Teorems 6.1 and 6.2 and in accordance wit our former Proposition 3.2, we deduce tat te optimal convergence rate is n γ/(1+γ) and tat tis would be attained if n = c n 1/(γ+1) for any constant c (, ). In tat case, te following result sows a CLT for ˆσ τ 2 under te non-leverage Assumption 1 (its proof is omitted for te sake of brevity and presented in te supplemental article [7]): Corollary 6.1. Suppose te assumptions of Teorems 6.1 and 6.2 are satisfied as well as te nonleverage Assumption 1. Ten, for te optimal bandwidt n = 1/(γ+1), we ave γ 2(1+γ) 2 (ˆσ τ στ 2 ) D δ1 2 + δ2 ξ, 2 were δ 1 and δ 2 are defined in Teorems 6.1 and 6.2, respectively, and ξ is a standard normal random variable independent from g t, under te setting (1) of Teorem 6.2, or from Z t under te setting (2) of Teorem 6.2. Our final result sows a CLT for ˆσ τ 2 wen n = cn 1/(γ+1) for general Itô volatilities (as in te setting (1) of Teorem 6.2), but only forward looking kernels. Tis generalizes results of [1] and [11], were only uniform forward kernels were considered. Teorem 6.3. Consider te model (2.1) wit a cádlag process µ and an Itô process σ given by σt 2 = σ 2 + t f sds + t g sdw s were W is a Brownian motion suc tat E(dB t dw t ) = ρdt and {f t } t and {g t } t are adapted cádlág processes. Let K be a kernel function satisfying Assumption 4 and, in addition, K(x) = for all x <. Ten, te conclusion of Corollary 6.1 olds true wit γ = Simulation Results In tis section, we sow some simulations to furter investigate te performance of te plug-in metod tat we developed in Sections 3 and 5 and compare it wit te cross-validation metod proposed in [14]. Trougout, we will consider te Heston model [1]: dx t = µ t dt + V t db t, dv t = κ(θ V t )dt + ξ V t dw t. (7.1) As to te parameters values, we adopt te following setting, used, e.g., in [25]: κ = 5, θ =.4, ξ =.5, µ t =.5 V t /2.