Optimal Kernel Estimation of Spot Volatility

Transcription

1 Optimal Kernel Estimation of Spot Volatility José E. Figueroa-López Department of Mathematics and Statistics Washington University in St. Louis Joint work with Cheng Li from Purdue U. and Bei Wu from WUSTL INFORMS Annual Meeting Phoenix, AZ Nov. 4, 2018 José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 1 / 48

2 Overview 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 2 / 48

3 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 3 / 48

4 Motivation 1 Spot volatility estimation has at times being presumed a simple byproduct of integrated variance estimation [X, X ] t = t 0 σ2 s ds: σt 2 1 ( [X, X ]t+h h [X ), X ] t 2 However, in a finite sample setting, bandwidth h affects the performance of the estimators and it is important to determine suitable values for h. 3 A natural approach is to assume that h = h n α β n, where n is the time mesh between observations, and determine the optimal α and β for large n. 4 Another important questions is whether a different type of localization would be more accurate: σt 2 K h (s t)d [X, X ] s, K h (u) = 1 h K(u h ). José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 4 / 48

5 Brownian Driven Volatilities Aït-Sahalia & Jacod (2014) takes h n α 1/2 n and K with support on [0, 1] and identifies two asymptotic regimes for ˆσ 2 t := K h (s t)d [X, X ] s as n 0: 1 h n 1/2 n 0: 2 h n 1/2 n ( hn n ( hn n α (0, ): ) 1/2 (ˆσ 2 t σ 2 t ) N ( 0, 2σ 4 t K 2 ). ) 1/2 ( ) 2 (ˆσ t σt 2 N 0, 2σt 4 K 2 + αgt 2 where g t = d dt σ2 t is the spot vol vol. 1 K(u)du 2 ), José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 5 / 48

6 Comments and Key Motivating Question Note that, comparing the two cases, the optimal rate of 1/4 n is obtained in the second case when bandwidth h n α 1/2 n for α (0, ). In spite of this, Aït-Sahalia & Jacod (2014) (see also Jacod & Protter (2012)) state the following: For practical purposes, we also need an estimator for the asymptotic variance. There is no reasonable way to estimate g t, hence, in practice, choosing α > 0, should be avoided. The case α > 0 leads to unfeasible CLT and, so, it seems that spot volatility estimation with constant kernel actually perform better than general kernel estimators. Is these two claims really the case? José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 6 / 48

7 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 7 / 48

8 The Problem I We consider the SDE dx t = µ t dt + σ t db t (1) where B := {B t } t 0 is a standard Brownian Motion (BM). Spot volatility σ t is crucial in asset portfolio management, option pricing, risk management, etc. We revisit the problem of estimating the spot volatility σ t at a fixed time t (0, T ) based on a discrete record of observations X ti, where t i = i = Ti/n, i = 0, 1,..., n. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 8 / 48

9 The Problem II We consider the class of kernel estimators: n ˆσ t,n,h 2 := K h (t i 1 t)( i X ) 2, i=1 where i X := X ti X ti 1, K h (x) := 1 h K ( x h ), for a suitably chosen bandwidth h := h n and a kernel function K : R R such that K(x)dx = 1. In this talk, we focus on the problem of optimal bandwidth and kernel selection; i.e., how to choose the bandwidth h and the kernel K to minimize certain loss function of estimation. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 9 / 48

10 Literature Review I Consistency and Asymptotic Normality: For Brownian-driven volatilities {σ t } t, Foster & Nelson (1994) studied a rolling window estimator, which can be seeing as a kernel estimator with a compactly supported kernel. Fan & Wang (2008), Kristense (2010), and Mancini, Mattiussi, & Reno (2015) obtained CLT s with suboptimal rates under a Hölder type condition on σ; Jacod and Protter (2012) and Alvarez et al. (2012) obtained CLT s with optimal rates for Brownian-driven volatilities and one-sided uniform kernel functions K(x) = 1 [0,1] (x) or K(x) = 1 [ 1,0] (x). José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 10 / 48

11 Literature Review II Optimal bandwidth and Kernel Selection: Foster and Nelson (1996) assumes that h n = c n 1/2 and finds the optimal constant c; argues that K(x) = 1 2 e x is optimal; Kristensen (2010) considers the problem of optimal bandwidth selection under a path-wise Hölder condition of the form: P a.e. ω : σ 2 t+δ(ω) σ 2 t (ω) 2 = L t (δ; ω)δ γ + o(δ γ ), t, ( ) where γ (0, 2] and δ L t (δ) slowly varying at 0; Under ( ), he proposes the following optimal bandwidth: ( ) h opt n,t = n 1 γ+1 2T σ 4 t K 2 1 γ+1 1 ; γl t (0 + ) However, ( ) is hard to verify with L t (0 + ) (0, ). José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 11 / 48

12 Assumptions on the Volatility Process I Our first assumption is a simplifying non-leverage condition (this can be relaxed for Brownian-driven volatilities): Assumption 1 σ is independent of B. Some mild moments conditions on µ and σ: Assumption 2 There exists M T > 1 such that E[µ 4 t + σ 4 t ] < M T, for all 0 t T. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 12 / 48

13 Assumptions on the Volatility Process II The following is the key assumption that allows to cover a large ranger of models at once: Assumption 3 ( ) The variance process V := {V t = σ 2 t : t 0} 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 0, ( ) for some γ > 0 and certain locally bounded function L : R + R + and a function C γ : R R R that is not identically zero and satisfies the scaling property: C γ (hr, hs) = h γ C γ (r, s), for r, s R, h R +. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 13 / 48

14 Examples I: Differentiable Volatility Processes Proposition 1 If V t = σt 2 = f (t) for a differentiable function f s.t. f (t) 0, then E[(V t+r V t )(V t+s V t )] = E[(f (t + r) f (t))(f (t + s) f (t))] = E[(f (t)r + o(r)) (f (t)s + o(s))] = E[(f (t)) 2 ]rs + o((r 2 + s 2 ) γ/2 ), and ( ) holds with γ = 2, C γ (r, s) = rs, L(t) = E[(f (t)) 2 ]. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 14 / 48

15 Example II: BM Type Volatility Processes Proposition 2 Suppose that V t = σ 2 t satisfies the SDE: dv t = f (t)dt + g(t)dw t, t [0, T ], where W is a standard Brownian Motion. Then, E[(V t+r V t )(V t+s V t )] = L(t)C γ (r, s) + o((r 2 + s 2 ) γ/2 ), ( ) with γ = 1, L(t) = E[g 2 (t)], C γ (r, s) := min{ r, s }1 {rs 0}. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 15 / 48

16 Example III: fbm Type Volatility Proposition 3 Consider a process {Yt H } t 0 that satisfies t Yt H = f (u)dbu H, where {Bu H } u R is a (two-sided) fractional Brownian motion with Hurst parameter H ( 1, 1). 2 Then, for any C 2 -function g, V t = g(yt H ) satisfies the key Assumption ( ) with γ = 2H (1, 2) and C γ (r, s) := E[B H r B H s ] = 1 2 ( r 2H + s 2H r s 2H ), r, s R. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 16 / 48

17 Conditions on the Kernel Function Assumption 4 Given γ and C γ as defined in our key Assumption ( ), we suppose that the kernel function K : R R satisfies the following: (1) K(x)dx = 1, (2) K(x) is piece-wise continuously differentiable, (3) K(x) x γ dx <, K(x)x γ+1 0, x, (4) K(x)K(y)C γ (x, y)dxdy > 0. The condition (4) above does not put substantial restriction on K since C γ is already integrally non-negative definite. In the case of Brownian driven volatilities (when C γ (r, s) := ( r s )1 {rs 0} ), condition (4) holds for any nonzero kernel K; José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 17 / 48

18 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 18 / 48

19 (Integrated) Mean Square Error Define the MSE and IMSE of the kernel estimator as Problem: MSE n (h; τ) = E[(ˆσ 2 τ σ 2 τ) 2 ] IMSE n (h) = T 0 E[(ˆσ 2 τ σ 2 τ) 2 ]dτ. Choose bandwidth h = h n and kernel K to minimize MSE n or IMSE n. Approach: We solve this problem in two-steps: 1 First, minimizing (I)MSE in the bandwidth h for a fixed kernel K; 2 Then, minimizing the resulting minimal (I)MSE in the kernel K; José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 19 / 48

20 Main Result Theorem 1 (F-L & Li 2017) For µ and σ satisfying Assumptions 1, 2, and 3 and a kernel function K satisfying Assumption 4, [ MSE τ,n (h) = E (ˆσ 2 τ,n,h στ 2 ) ] 2 = 2 h E[σ4 τ ] K 2 (x)dx + h γ L(τ) K(x)K(y)C γ (x, y)dxdy + o( h ) + o(hγ ); with an analogous asymptotic expansion for IMSE, but replacing E[στ] 4 and L(τ) with its integrated versions T 0 E[σ4 τ]dτ and T L(τ)dτ. 0 José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 20 / 48

21 Approximation of the Optimal Local Bandwidth Proposition 4 (F-L & Li 2017) The approximated local optimal bandwidth, which, by definition, minimizes the leading order terms of the MSE, is given by [ hn,τ a,opt = n 1 γ+1 2T E[σ 4 τ] K 2 1 γl(τ) K(x)K(y)C γ (x, y)dxdy ] 1 γ+1, resulting in an approximation of the minima MSE of the form: ( MSE a,opt n =n 1 γ ) (2T ) E[σ 4 γ τ] K 2 γ γ+1 1 ( γl(τ) ) 1 γ+1 K(x)K(y)C γ (x, y)dxdy José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 21 / 48

22 Examples I When σ 2 t = f (t) is deterministic and smooth (γ = 2, L(t) = (f (t)) 2, and C γ (r, s) = rs), h opt n,τ = n 1 3 MSE opt n = 3 2 n 1 3 ( ) Tf (t) 2 K , κ(k) := K(x)xdx 0; f (τ) 2 κ 2 (K) ( ) 2Tf (t) 2 K 2 2 ( 3 1 2f (t) 2 κ 2 (K) ) o(n 1 3 ); In particular, one can improve the rate of convergence n 1 3 of the MSE by choosing K such that κ(k) = K(x)xdx = 0; José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 22 / 48

23 Examples II For B.M.-driven volatilities dσ 2 t = f (t)dt + g(t)dw t, [ ] hn,τ a,opt = n 1 2T E[σ 4 2 τ ] K , E [g 2 (τ)] κ BM (K) κ BM (K) = 0 0 [K(x)K(y) + K( x)k( y)] min(x, y)dxdy, where the latter is always positive (regardless K 0); The leading term of the optimal MSE takes the form ( MSE opt n = 2 3/2 n 1 2 T E[σ 4 τ ]E[g 2 (τ)] K 2 1 κ BM (K) ) o(n 1/2 ); Its optimal rate of convergence, n 1/2, cannot be improved regardless of the choice of K; José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 23 / 48

24 Integrated MSE In the case that we consider the Integrated MSE (IMSE) IMSE n (h) = T 0 E[(ˆσ 2 τ,n,h σ 2 τ) 2 ]dτ the optimal (uniform) bandwidth takes the form: h a,opt n = n 1 γ+1 [ 2T T 0 E[σ4 τ]dτ K 2 1 γ T 0 L(τ)dτ K(x)K(y)C γ (x, y)dxdy ] 1 γ+1 José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 24 / 48

25 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 25 / 48

26 Approach Recall that the approximated optimal MSE takes the form: ( MSE a,opt n (K) =n γ 1+γ ) ( 2T E[σ 4 γ τ] ( γl(τ) ) γ K 2 1+γ (x)dx ) 1 1+γ K(x)K(y)C γ (x, y)dxdy. This leads to consider the following calculus of variation problem: ( ) γ min K 2 (x)dx K(x)K(y)C γ (x, y)dxdy, K subject to the restriction K(x)dx = 1. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 26 / 48

27 BM Volatility Case Theorem 2 (F-L & Li 2017) For Brownian-driven volatilities, when C γ (r, s) := ( r s )1 {rs 0}, the optimal kernel function is the exponential kernel: K exp (x) = 1 2 e x, x R. Remark: Other two common kernels are the uniform K 0 (x) = { x <1} and the Epanechnikov K 2 (x) = 3 4 (1 x 2 )1 { x <1} kernels; As it turns out MSE a,opt MSE a,opt a,opt n (K exp ) n (K 0 ) = 0.86; MSEn (K exp ) MSE a,opt n (K 2 ) = 0.93; José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 27 / 48

28 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 28 / 48

29 Main Result: Brownian-Drive Volatilities For Brownian driven volatilities dσ 2 t = f (t)dt + g(t)dw t, the optimal rate is attained when ( /h) 1/2 h 1/2 (i.e., h 1/2 ). If h = 1/2, then ) (ˆσ τ στ 2 D ξ1 2 + ξ2 2 N (0, 1), with ξ 2 1 := 2σ 4 τ K 2 1, ξ 2 2 := g 2 τ xy 0 K(x)K(y)( x y )dxdy. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 29 / 48

30 Main Results: General Volatilities Consider the decomposition ˆσ 2 τ σ 2 τ = n K h (t i 1 τ)( i X ) 2 i=1 T + 0 T K h (t τ)(σ 2 t σ 2 τ)dt + o P (h γ ) =: P 1 + P 2 + o P (h γ ). 0 K h (t τ)σ 2 t dt For Brownian driven volatilities dσ 2 t = f (t)dt + g(t)dw t, we have: ( ) 1/2 P 1 D ξ 1 N (0, 1), h γ/2 P 2 D ξ 2 N (0, 1), h with ξ1 2 := 2στ K 4 2 1, ξ2 2 := gτ 2 K(x)K(y)C(x, y)dxdy. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 30 / 48

31 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 31 / 48

32 Practical Bandwidth Selection Methods Generally, there are two types of bandwidth selection methods. Cross-validation methods: Advantages: yield relatively good results for a wide range of volatility processes (regardless of γ); Disadvantages: time consuming, hard to implement. Plug-in type methods: Advantages: usually faster and have better accuracy. Disadvantages: non adaptive (in our case, this means that it is tailored to some specific γ). In Kristensen (2010), a leave-one-out cross validation method is proposed, though their asymptotic properties were not studied. Here, we consider a plug-in type estimation and show that it is indeed more accurate than cross-validation. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 32 / 48

33 Plug-In Type Bandwidth Selection Methods I The idea of the plug-in method is to estimate all the parameters encountered in the explicit approximated optimal bandwidth; Consider the BM type volatility processes: dv t = f (t)dt + g(t)dw t. The optimal bandwidth that minimizes the approximate IMSE is given by h a,opt n = [ 2T T 0 E[σ4 t ]dt ] 1/2 K 2 (x)dx n T L(t)dt. K(x)K(y)C 0 1 (x, y)dxdy We need to estimate T 0 E[σ4 t ]dt and T L(t)dt = T E[g 2 (t)]dt, the expected quarticity and 0 0 integrated vol vol. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 33 / 48

34 Plug-In Type Bandwidth Selection Methods II Given that we have at hand only one realization of X, it is natural to estimate these two quantities with T 0 σ4 t dt and T g 2 (t)dt; 0 T 0 σ4 t dt can be estimated by the Realized Quarticity: ÎQ = (3 ) 1 n ( i X ) 4. Estimation of T 0 g 2 (t)dt, which is just the quadratic variation of σ 2, σ 2, σ 2 T, is more involved. σ 2, σ 2 T = T 0 g 2 (t)dt is sometimes called volatility of volatility or simply vol vol. i=1 José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 34 / 48

35 Two-time Scale Realized vol vol (TSRVV) I Zhang et al. (2005) proposed a Two-time Scale Realized Volatility (TSRV) estimator of the quadratic variation Y, Y T of a process Y in the presence of market micro-structure noise: TSRV = 1 k n k (Y ti+k Y ti ) 2 n k + 1 nk i=0 n 1 (Y ti+1 Y ti ) 2. i=0 Inspired by this, we propose the following estimator: IVV (tsrvv) T = 1 n k b (ˆσ t 2 k i+k ˆσ t 2 i ) 2 n k + 1 nk i=b n k b i=b+k 1 (ˆσ 2 t i+1 ˆσ 2 t i ) 2. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 35 / 48

36 Two-time Scale Realized vol vol (TSRVV) II Theorem 3 (Consistency of TSRVV, F-L & Li 2017) For any fixed t b (0, T /2), the TSRVV is a consistent estimator of T tb t b gt 2 dt. ( ( ) n The convergence rate is given by O 1/4 k P + O P. n k 1/2 ) José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 36 / 48

37 Iterative Plug-in Bandwidth Selection The TSRVV involves the estimation of spot volatility, which we do not know in advance, so it is natural to consider the following iterative algorithm: The Iterative Plug-in Bandwidth Selection Algorithm: Data: n 1X = X t1 X t0,..., n nx = X tn X tn 1 ; Set an initial value of h: ; while Stopping criteria not met do Estimate ˆσ 2 t i for all 0 i n based on the bandwidth h ; Estimate the integrated vol vol σ 2, σ 2 using the TSRVV; Update the approximated optimal bandwidth h; end In our simulations, two iterations are typically enough for satisfactory result, even with bad initial guess. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 37 / 48

38 1 Introduction 2 Setting, Estimator, and Assumptions 3 Optimal Bandwidth Selection 4 Optimal Kernel Selection 5 Central Limit Theorem 6 Implementation of the Bandwidth Selection Method 7 Simulation Study José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 38 / 48

39 Simulation Study We consider Heston model dx t =µ t dt + V t db t, dv t =κ(θ V t )dt + ξ V t dw t, with the following parameter settings: 1 T = 5 or 21 days, = 1 or 5 minute, 6.5 trading hours. 2 µ t = V t, σ 0 = 0.2, κ = 5, θ = 0.04, ξ = The leverage is taken to be 0 and 0.5. Except when we compare different kernel functions, we use the exponential kernel function. We will estimate the sample Mean of the Average Squared Error of the estimators based on 2000 simulations: MASE := 1 n 2l + 1 n l (ˆσ t 2 i σt 2 i ) 2, i=l l = 0.1n. (2) José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 39 / 48

40 Plug-In v.s. Cross-Validation 5 Days Data ndata/h ρ MASE PI MASE CV MASE oracle E E E E E E E E E E E E Days Data ndata/h ρ MASE PI MASE CV MASE oracle E E E E E E E E E E E E-09 José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 40 / 48

41 Estimation of Volatility of Volatility ndata/h ρ ξ Bias Std MSE Table: Estimation of Volatility of Volatility by TSRVV (1 month data, sample paths) José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 41 / 48

42 Comparison of Different Kernels We consider four different kernels: K exp (x) = 1 2 e x, K 0 (x) = { x <1} K 1 (x) = 1 x 1 { x <1}, K 2 (x) = 3 4 (1 x 2 )1 { x <1} length ρ K exp K 0 K 1 K 2 5 days E E E E-05 5 days E E E E days E E E E days E E E E-05 Table: Comparison of Different Kernel Functions (5 min data, 2000 sample paths) José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 42 / 48

43 Conclusions 1 An optimal bandwidth selection method is proposed under a mild scaling condition on the local behavior of the covariance function of the variance process. 2 The considered framework covers a wide range of models including volatility models driven by BM and fbm 3 The problem of optimal kernel selection is also considered: it is shown that an exponential kernel is the optimal kernel function for B.M.-driven volatility models 4 Fast iterated plug-in type algorithms are also devised as a way to implement the proposed optimal selection methods José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 43 / 48

44 Future Work - Market Micro-Structure Noise I Assume the log price with additive noise is observed: dx t = µ t dt + σ t db t, Y t = X t + ε t, where {ε t } t are iid with mean zero and variance ω 2. A natural way to handle such a case is to replace the quadratic variation [X, X ] t = t 0 σ2 s ds below σ 2 t = t 0 K h (s t)d[x, X ] s, with an integrated variance estimator [X, X ] t that is robust against microstructure noise. José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 44 / 48

45 Future Work - Market Micro-Structure Noise II For instance, if we use Zhang et al. (2005) two-time scale estimator of [X, X ] t, we obtain the estimator: ˆσ τ 2 = 1 n k K h (t i τ)(x ti+k X ti ) 2 k i=0 n k + 1 nk n 1 K h (t i τ)(x ti+1 X ti ) 2. i=0 Mancini at al. (2015) consider a similar estimator and obtained a CLT with suboptimal rate; Zu and Boswijk (2014) obtained a CLT with optimal rate but only considered uniform one-sided kernel K(x) = 1 [0,1] (x). José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 45 / 48

46 References I Y. Aït-Sahalia and J. Jacod. High-Frequency Financial Econometrics. Princeton University Press, A. Alvarez, F. Panloup, M. Pontier, and N. Savy. Estimation of the instantaneous volatility. Statistical Inference For Stochastic Processes, 15(1):27 59, J. Fan and Y. Wang. Spot volatility estimation for high-frequency data. Statistics and its Interface, 1(2): , D. Foster and D. Nelson. Continuous record asymptotics for rolling sample variance estimators. Econometrica, 64(1): , José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 46 / 48

47 References II J. Jacod and P. Protter. Discretization of Processes. Springer-Verlag Berlin Heidelberg, D. Kristensen. Nonparametric filtering of the realized spot volatility: A kernel-based approach. Econometric Theory, 26(01):60 93, C. Mancini, V. Mattiussi, and R. Renò. Spot volatility estimation using delta sequences. Finance & Stochastics, 19(2): , L. Zhang, P. Mykland, and Y. Aït-Sahalia. A tale of two time scales. Journal of the American Statistical Association, 100(472), José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 47 / 48

48 References III José E. Figueroa-López (WashU) Kernel Estimation of Spot Volatility 48 / 48