Optimally Thresholded Realized Power Variations for Stochastic Volatility Models with Jumps
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1 Optimally Thresholded Realized Power Variations for Stochastic Volatility Models with Jumps José E. Figueroa-López 1 1 Department of Mathematics Washington University ISI 2015: 60th World Statistics Conference Rio de Janeiro July 29, 2015 (Joint work with Jeff Nisen, Cheng Li) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
2 Introduction The Framework and the Statistical Problems Framework 1 Finite-Jump Activity (FJA) Itô Semimartingales: dx t = γ t dt + σ t dw t + dj t t W t is a standard Brownian motion; t J t := N t j=1 ζ j: t N t is the counting process of jumps s.t. N t <, for all t > 0 {ζ j } j are the jump sizes; t γ t and t σ t are the drift and volatility functions; Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
3 Introduction The Framework and the Statistical Problems Framework 1 Finite-Jump Activity (FJA) Itô Semimartingales: dx t = γ t dt + σ t dw t + dj t t W t is a standard Brownian motion; t J t := N t j=1 ζ j: t N t is the counting process of jumps s.t. N t <, for all t > 0 {ζ j } j are the jump sizes; t γ t and t σ t are the drift and volatility functions; 2 FJA Lévy Model: X t = γt + σw t + N t j=1 ζ j {N t} t 0 is a homogeneous Poisson process with jump intensity λ; {ζ j } j 0 are i.i.d. with density f ζ : R R +; the triplet ({W t}, {N t}, {ζ j }) are mutually independent. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
4 Introduction The Framework and the Statistical Problems Statistical Problems Given a finite discrete record of observations, X t0, X t1,..., X tn, π : 0 = t 0 < t 1 < < t n = T, the following problems are of interest under a high-frequency sampling setting (i.e., mesh(π) := max i {t i t i 1 } 0): 1 Estimating the integrated variance: σ 2 T := T 0 σ 2 t dt. 2 Estimating the jump features of the process: Jump times: say, {τ 1 < τ 2 < < τ NT } if N T 1, or, otherwise. Corresponding jump sizes: {ζ 1, ζ 2,..., ζ NT } if N T 1, or, otherwise. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
5 Introduction The Main Estimators Two main classes of estimators 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard (2004)): n 1 BPV (X) T := Xti+1 X ti Xti+2 X ti+1, i=0 n k MPV (X) T := Xti+1 X ti r 1... Xti+k X ti+k 1 r k, (r r k = 2). i=0 2 Threshold Quadratic Realized Variations (Mancini (2001, 2003)): n 1 ( TRV (X)[B] π T := Xti+1 X ) 2 t i 1{ Xti+1 Xti B}, (B (0, )). i=0 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
6 Introduction The Main Estimators Advantages and Drawbacks of TRV 1 Pros: Can exhibit reduced bias for estimating σ 2 T, in the presence of jumps Can be adapted for estimating the process jump features: e.g., π n 1 N[B] T := 1 { } mesh 0 >B Xti+1 N X T. ti i=0 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
7 Introduction The Main Estimators Advantages and Drawbacks of TRV 1 Pros: Can exhibit reduced bias for estimating σ 2 T, in the presence of jumps Can be adapted for estimating the process jump features: e.g., π n 1 N[B] T := 1 { } mesh 0 >B Xti+1 N X T. ti i=0 2 Cons: Performance strongly depends on a good" selection of the threshold B; Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
8 Introduction The Main Estimators Advantages and Drawbacks of TRV 1 Pros: Can exhibit reduced bias for estimating σ 2 T, in the presence of jumps Can be adapted for estimating the process jump features: e.g., π n 1 N[B] T := 1 { } mesh 0 >B Xti+1 N X T. ti i=0 2 Cons: Performance strongly depends on a good" selection of the threshold B; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), F-L & Nisen (SPA, 2013): (i) TRV (X)[B n] π T is consistent for σ 2 T iff hn B n n and B n 0; Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
9 Introduction The Main Estimators Advantages and Drawbacks of TRV 1 Pros: Can exhibit reduced bias for estimating σ T 2, in the presence of jumps Can be adapted for estimating the process jump features: e.g., π n 1 N[B] T := 1 { } mesh 0 >B Xti+1 N X T. ti i=0 2 Cons: Performance strongly depends on a good" selection of the threshold B; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), F-L & Nisen (SPA, 2013): (i) TRV (X)[B n] π T is consistent for σ 2 T iff (ii) hn B n n and B n 0; [ ] E TRV (X)[B n] π ( T σ T 2 Th n γ 2 λσ 2) ( 2T σφ ) B n σ hn B n + 2T λb3 n C(f ζ ) h n 3 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
10 Introduction The Main Estimators Popular thresholds B Power Threshold (Mancini (2003)) B Pow α,ω := α mesh(π) ω, for α > 0 and ω (0, 1/2). Bonferroni Threshold (Bollerslev et al. (2007) and Gegler & Stadtmüller (2010)) ( B BF σ,c := σmesh(π)1/2 Φ 1 1 C mesh(π) ), for C > 0 and σ > 0. 2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
11 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Optimally Thresholded Realized Variations 1 Aims Introduce an optimal" selection criterion for the threshold B, that minimizes a suitable loss function of estimation. Develop a feasible implementation method for the optimal threshold B. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
12 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Optimally Thresholded Realized Variations 1 Aims Introduce an optimal" selection criterion for the threshold B, that minimizes a suitable loss function of estimation. Develop a feasible implementation method for the optimal threshold B. 2 Loss Function Loss n (B) := E n i=1 ( ) 1 [ n i X B, n N 0] + 1 i [ n i X >B, n N=0], i where, as usual, n i X := X t i X ti 1 and n i N := N t i N ti 1. 3 Interpretation Loss n (B) represents the Total Number of Jump Miss-Classifications: fail to identify the occurrence of a jump during [t i 1, t i ]. to flag that a jump occurred during [t i 1, t i ), when no jump occurred, Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
13 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (SPA, 2013)) Suppose that t i+1 t i =: h n = T /n for any i and X is a Finite Jump Activity Lévy Model with jump density f ζ, jump intensity λ, and volatility σ: Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
14 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (SPA, 2013)) Suppose that t i+1 t i =: h n = T /n for any i and X is a Finite Jump Activity Lévy Model with jump density f ζ, jump intensity λ, and volatility σ: 1 For n large enough, the loss function Loss n (B) is quasi-convex and, moreover, possesses a unique global minimum B n. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
15 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (SPA, 2013)) Suppose that t i+1 t i =: h n = T /n for any i and X is a Finite Jump Activity Lévy Model with jump density f ζ, jump intensity λ, and volatility σ: 1 For n large enough, the loss function Loss n (B) is quasi-convex and, moreover, possesses a unique global minimum B n. 2 As n, the optimal threshold sequence (Bn) n is such that ( ) ( ) 1 log 2πσλC(fζ ) σh 1/2 ( ) Bn = 3σ 2 n h h n log n + o, h n 3 log(1/hn ) log(1/h n ) 1 ε where C(f ζ ) = lim ε 0 2ε ε f ζ(x)dx. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
16 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Remarks 1 The leading term of the optimal sequence is proportional to Lévy s modulus of continuity of the Brownian motion: lim sup h 0 1 2h log(1/h) sup W t W s = 1, a.s. t s <h,s,t [0,1] Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
17 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Remarks 1 The leading term of the optimal sequence is proportional to Lévy s modulus of continuity of the Brownian motion: lim sup h 0 1 2h log(1/h) 2 The threshold sequences B 1 n := ( 1 3σ 2 h n log h n sup W t W s = 1, a.s. t s <h,s,t [0,1] ), Bn 2 := Bn 1 ( ) log 2πσλC(fζ ) 3 log(1/hn ) σh 1/2 n are the first and second-order approximations for B n, and it can be shown that the biases of their corresponding TRV estimators attain the optimal" rate of O(h n ) as n., Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
18 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process Remarks 1 The leading term of the optimal sequence is proportional to Lévy s modulus of continuity of the Brownian motion: lim sup h 0 1 2h log(1/h) 2 The threshold sequences B 1 n := ( 1 3σ 2 h n log h n sup W t W s = 1, a.s. t s <h,s,t [0,1] ), Bn 2 := Bn 1 ( ) log 2πσλC(fζ ) 3 log(1/hn ) σh 1/2 n are the first and second-order approximations for B n, and it can be shown that the biases of their corresponding TRV estimators attain the optimal" rate of O(h n ) as n. 3 They both provide blueprints" for devising good threshold sequences!, Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
19 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A Feasible Implementation Algorithm Based on B 1 n 1 Key Issue: The threshold B 1 would allow us to find an (approximately) optimal" estimate ˆσ 2 for σ 2 of the form ˆσ 2 := 1 T TRV (X)[B 1 ] n Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
20 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A Feasible Implementation Algorithm Based on B 1 n 1 Key Issue: The threshold B 1 would allow us to find an (approximately) optimal" estimate ˆσ 2 for σ 2 of the form but B 1 n ˆσ 2 := 1 T TRV (X)[B 1 ] n := B 1 n (σ 2 ) depends precisely on σ 2. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
21 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A Feasible Implementation Algorithm Based on B 1 n 1 Key Issue: The threshold B 1 would allow us to find an (approximately) optimal" estimate ˆσ 2 for σ 2 of the form ˆσ 2 := 1 T TRV (X)[B 1 ] n but Bn 1 := Bn 1 (σ 2 ) depends precisely on σ 2. 2 The previous issue suggests a fixed-point" type of implementation: Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
22 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A Feasible Implementation Algorithm Based on B 1 n 1 Key Issue: The threshold B 1 would allow us to find an (approximately) optimal" estimate ˆσ 2 for σ 2 of the form ˆσ 2 := 1 T TRV (X)[B 1 ] n but Bn 1 := Bn 1 (σ 2 ) depends precisely on σ 2. 2 The previous issue suggests a fixed-point" type of implementation: (i) Get a rough" estimate of σ 2 via, e.g., the QV: σ n,0 2 := 1 T QV T := 1 n X ti X ti 1 2 T (ii) Use σ n,0 2 1 to estimate the optimal threshold B n,0 := ( 3 σ n,0h 2 n log(1/h ) 1/2 n) (iii) Refine σ n,0 2 using thresholding, ˆσ n,1 2 = 1 n X ti X ti T [ ] X ti X ti 1 B n,0 1 (iv) Iterate Steps (ii) and (iii): i=1 σ 2 n,0 i=1 1 B n,0 ˆσ n,1 2 1 B n,1 ˆσ n,2 2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
23 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A numerical illustration I (S2) Kou Model: 1-week / 5-minute σ = 0.5, λ = 50, γ = 0 p = 0.45, α + = 0.05, α = 0.1 Method TRV S TRV Loss S Loss B n,k 1 n Pow ω=0.495;α= BF Table: Finite-sample performance of the threshold realized variation (TRV) estimators based on 5, 000 sample paths of the Kou model: f Kou (x) = p α + e x/α+ 1 [x 0] + (1 p) α e x /α 1 [x<0]. Loss represents the total number of Jump Misclassification Errors, while TRV, Loss, S TRV, and S Loss denote the corresponding sample means and standard deviations, respectively. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
24 Optimally Thresholded Estimators Finite-Jump Activity Lévy Process A numerical illustration II (S3) Kou Model: 1-year / 5-minute σ = 0.4, λ = 1000, γ = 0 p = 0.5, α + = α = 0.1 Method TRV S TRV Loss S Loss B n,k 1 n Pow ω=0.495;α= BF Table: Finite-sample performance of the threshold realized variation (TRV) estimators based on K = 5, 000 sample paths of the Kou model: f ζ (x) = p α + e x/α+ 1 [x 0] + q α e x /α 1 [x<0]. Loss represents the total number of Jump Misclassification Errors, while TRV, Loss, S TRV, and S Loss denote the corresponding sample means and standard deviations, respectively. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
25 Additive Processes Additive Processes and general sampling schemes 1 The model: X s := s 0 γ(u)du + s 0 N s σ(u)dw u + ζ j =: Xs c + J s, where ζ j i.i.d. f ζ and (N s ) s 0 Poiss ({λ(s)} s 0 ), independent of W, and deterministic smooth functions σ, λ : [0, ) R + and γ : [0, ) R with σ and λ bounded away from 0. j=1 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
26 Additive Processes Additive Processes and general sampling schemes 1 The model: X s := s 0 γ(u)du + s 0 N s σ(u)dw u + ζ j =: Xs c + J s, where ζ j i.i.d. f ζ and (N s ) s 0 Poiss ({λ(s)} s 0 ), independent of W, and deterministic smooth functions σ, λ : [0, ) R + and γ : [0, ) R with σ and λ bounded away from 0. 2 Optimal Threshold Problem Given a sampling scheme π : 0 = t 0 < < t n = T, determine the vector B π, = (B π, t 1 inf E B=(Bt1,...,B tn ) R m + j=1,..., B π, t n ) that minimizes the problem n i=1 ( 1 [ Xti X ti 1 >B ti,n ti N ti 1 =0] + 1 [ Xti X ti 1 B ti,n ti N ti 1 0] ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
27 Additive Processes Additive Processes and general sampling schemes 1 The model: X s := s 0 γ(u)du + s 0 N s σ(u)dw u + ζ j =: Xs c + J s, where ζ j i.i.d. f ζ and (N s ) s 0 Poiss ({λ(s)} s 0 ), independent of W, and deterministic smooth functions σ, λ : [0, ) R + and γ : [0, ) R with σ and λ bounded away from 0. 2 Optimal Threshold Problem Given a sampling scheme π : 0 = t 0 < < t n = T, determine the vector B π, = (B π, t 1 inf E B=(Bt1,...,B tn ) R m + n = i=1 j=1,..., B π, t n ) that minimizes the problem n i=1 ( 1 [ Xti X ti 1 >B ti,n ti N ti 1 =0] + 1 [ Xti X ti 1 B ti,n ti N ti 1 0] inf {P ( i X > B ti, i N = 0) + P ( i X B ti, i N 0)}, B ti Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31 )
28 Additive Processes Well-posedness and Asymptotic Characterization Theorem (FL & Nisen, 2014) For any fixed T > 0, there exists h 0 := h 0 (T ) > 0 such that, for each t [0, T ] and h (0, h 0 ], the function L t,h (B) := P( X t+h X t > B, N t+h N t = 0) + P( X t+h X t B, N t+h N t 0), is quasi-convex and possesses a unique global minimum, Bt,h, such that, as h 0, ( ) ( ) 1 Bt,h = 3σ 2 (t)h log log( 2πσ(t)λ t C(f ζ ))σ(t)h 1/2 h + o. h 3 log(1/h) log(1/h) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
29 Additive Processes Well-posedness and Asymptotic Characterization Theorem (FL & Nisen, 2014) For any fixed T > 0, there exists h 0 := h 0 (T ) > 0 such that, for each t [0, T ] and h (0, h 0 ], the function L t,h (B) := P( X t+h X t > B, N t+h N t = 0) + P( X t+h X t B, N t+h N t 0), is quasi-convex and possesses a unique global minimum, Bt,h, such that, as h 0, ( ) ( ) 1 Bt,h = 3σ 2 (t)h log log( 2πσ(t)λ t C(f ζ ))σ(t)h 1/2 h + o. h 3 log(1/h) log(1/h) Again, the leading term Bt,h 1 = 3σ 2 (t)h log ( ) 1 h provides a blueprint to devise a good threshold parameter. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
30 Additive Processes Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1, K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm based on Kernel estimation (Fan & Wang(2008), Kristensen(2010)) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
31 Additive Processes Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1, K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm based on Kernel estimation (Fan & Wang(2008), Kristensen(2010)) 1 Get a rough" estimate for t σ 2 t via the Kernel estimator: σ 2 0(t i ) := l i+j X 2 K θ (t i t i+j ), j= l with θ chosen by a cross-validation type method (cf. Kristensen(2010)). Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
32 Additive Processes Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1, K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm based on Kernel estimation (Fan & Wang(2008), Kristensen(2010)) 1 Get a rough" estimate for t σ 2 t via the Kernel estimator: σ 2 0(t i ) := l i+j X 2 K θ (t i t i+j ), j= l with θ chosen by a cross-validation type method (cf. Kristensen(2010)). 2 Get an initial estimate for the leading term of the optimal threshold: B 0 1 (t i ) := 3 σ 0 2(t i)h i log (1/h i ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
33 Additive Processes Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1, K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm based on Kernel estimation (Fan & Wang(2008), Kristensen(2010)) 1 Get a rough" estimate for t σ 2 t via the Kernel estimator: σ 2 0(t i ) := l i+j X 2 K θ (t i t i+j ), j= l with θ chosen by a cross-validation type method (cf. Kristensen(2010)). 2 Get an initial estimate for the leading term of the optimal threshold: B 0 1 (t i ) := 3 σ 0 2(t i)h i log (1/h i ) 3 Refine the estimate σ 2 0 (t i) using thresholding: σ 2 1(t i ) := l j= l i+j X 2 K θ (t i t i+j ) 1 [ i+j X B 1 0 (t i+j )] Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
34 Additive Processes Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1, K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm based on Kernel estimation (Fan & Wang(2008), Kristensen(2010)) 1 Get a rough" estimate for t σ 2 t via the Kernel estimator: σ 2 0(t i ) := l i+j X 2 K θ (t i t i+j ), j= l with θ chosen by a cross-validation type method (cf. Kristensen(2010)). 2 Get an initial estimate for the leading term of the optimal threshold: B 0 1 (t i ) := 3 σ 0 2(t i)h i log (1/h i ) 3 Refine the estimate σ 2 0 (t i) using thresholding: σ 2 1(t i ) := l j= l i+j X 2 K θ (t i t i+j ) 1 [ i+j X B 1 0 (t i+j )] 4 Iterate Steps 2 and 3 : σ ( ) B 0 ( ) ˆσ2 1 1 ( ) B 1 ( ) ˆσ2 2 ( ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
35 Additive Processes Illustration of Opt. Thresh. Spot Vol. Estimation Alg. (A) Initial Estimates (B) Intermediate Estimates Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Time Horizon Time Horizon Figure: Estimation of Spot Volatility using Adaptive Kernel Weighted Realized Volatility. (A) The initial estimates. (B) Intermediate estimates. Parameters: γ(t) = 0.1t, σ(t) = 4.5t sin(2πe t2 ) , λ(t) = 25(e 3t 1), ζ i i.i.d. = D N (µ = 0.025, δ = 0.025). Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
36 Additive Processes Illustration of Opt. Thresh. Spot Vol. Estimation Alg. (C) Terminal Estimates Actual Spot Volatility Est. Spot Vol. (Uniform) Est. Spot Vol. (Quad) Sample Increments Time Horizon Figure: Estimation of Spot Volatility using Adaptive Kernel Weighted Realized Volatility. (C) The terminal estimates. (D) Estimation variability, based on 100 generated sample paths, for the Quadratic Kernel based estimator. Parameters: γ(t) = 0.1t, σ(t) = 4.5t sin(2πe t2 ) , λ(t) = 25(e 3t 1), ζ i i.i.d. = D N (µ = 0.025, δ = 0.025). Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
37 Stochastic Volatility Processes Stochastic Volatility Models with FJA 1 Motivation: In financial application, one usually encounters models like X s := s γ u du + s 0 0 j=1 N s σ u dw u + ζ j =: Xs c + J s, where {σ t } t 0 itself is erratic and not smooth. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σt 2 ) dt+βσt dw (σ) t, (2κα β 2 > 0, Cov(dW (σ) t, dw t ) = ρdt) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
38 Stochastic Volatility Processes Stochastic Volatility Models with FJA 1 Motivation: In financial application, one usually encounters models like X s := s γ u du + s 0 0 j=1 N s σ u dw u + ζ j =: Xs c + J s, where {σ t } t 0 itself is erratic and not smooth. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σt 2 ) dt+βσt dw (σ) t, (2κα β 2 > 0, Cov(dW (σ) t, dw t ) = ρdt) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received comparatively much less attention than integrated variance estimation; Few bandwidth selection methods (e.g., Kristensen(2010)) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
39 Stochastic Volatility Processes A Proposed Bandwidth Selection Method 1 Technical Assumption (Kristensen, 2010): σ 2 t+δ σ 2 t = L t (δ)δ υ + o P (δ υ ), t, a.s., (δ 0), ( ) where υ (0, 1] and δ L t (δ) is slowly varying (random) function at 0 and t L t (0) := lim δ 0 + L t (δ) is continuous. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
40 Stochastic Volatility Processes A Proposed Bandwidth Selection Method 1 Technical Assumption (Kristensen, 2010): σ 2 t+δ σ 2 t = L t (δ)δ υ + o P (δ υ ), t, a.s., (δ 0), ( ) where υ (0, 1] and δ L t (δ) is slowly varying (random) function at 0 and t L t (0) := lim δ 0 + L t (δ) is continuous. 2 Under ( ), Kristensen(2010) proposes the following (asymptotically) optimal" bandwidth: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 2 2υ+1 2 υl 2 t (0) ( ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
41 Stochastic Volatility Processes A Proposed Bandwidth Selection Method 1 Technical Assumption (Kristensen, 2010): σ 2 t+δ σ 2 t = L t (δ)δ υ + o P (δ υ ), t, a.s., (δ 0), ( ) where υ (0, 1] and δ L t (δ) is slowly varying (random) function at 0 and t L t (0) := lim δ 0 + L t (δ) is continuous. 2 Under ( ), Kristensen(2010) proposes the following (asymptotically) optimal" bandwidth: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 2 2υ+1 2 υl 2 t (0) ( ) 3 Pitfall: In general, it is hard to check ( ) with explicit constants υ and L t (0) 0. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
42 Stochastic Volatility Processes Heuristic alternative approach 1 Idea: Replace the path-wise holder continuity assumption" with: [ (σ 2 E t+δ σt 2 ) 2 ] Ft = L 2 t (0)δ 2υ + o P (δ υ ), a.s. (δ 0), ( ) for some positive adapted {L t (0)} t 0. 2 Then, use ( ): ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 2 2υ+1 2 υl 2 t (0) ( ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
43 Stochastic Volatility Processes Heuristic alternative approach 1 Idea: Replace the path-wise holder continuity assumption" with: [ (σ 2 E t+δ σt 2 ) 2 ] Ft = L 2 t (0)δ 2υ + o P (δ υ ), a.s. (δ 0), ( ) for some positive adapted {L t (0)} t 0. 2 Then, use ( ): ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 2 2υ+1 2 υl 2 t (0) 3 Example: For the CIR model dσt 2 = κ ( α σt 2 that ( (σ 2 E t+δ σt 2 ) 2 ) Ft = β 2 σt 2 δ + o(δ). Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. ( ) ) dt + βσt dw (σ) t, it turns out 4 This in turn suggests the following local bandwidth selection method: bw loc opt,t = n 1/2 ( 2σ 2 t K 2 2 β 2 ) 1/2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
44 Stochastic Volatility Processes Estimation Method for the CIR model (No Jumps) 1 Get a rough" estimate of σt 2 ; e.g., using Alvarez et al. (2010): ˆσ 0(t 2 i ) = QV t i + h i QV ti hi = 1 ( n j X ) 2. hi j:t j (t i,t i + h i] Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
45 Stochastic Volatility Processes Estimation Method for the CIR model (No Jumps) 1 Get a rough" estimate of σt 2 ; e.g., using Alvarez et al. (2010): ˆσ 0(t 2 i ) = QV t i + h i QV ti hi = 1 ( n j X ) 2. hi j:t j (t i,t i + h i] 2 Estimate the vol vol β using the realized variation of {ˆσ 0 (t i )} i=1,...,n since, for the CIR model, σ, σ t = β2 4 ; denote such an estimate by β; Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
46 Stochastic Volatility Processes Estimation Method for the CIR model (No Jumps) 1 Get a rough" estimate of σt 2 ; e.g., using Alvarez et al. (2010): ˆσ 0(t 2 i ) = QV t i + h i QV ti hi = 1 ( n j X ) 2. hi j:t j (t i,t i + h i] 2 Estimate the vol vol β using the realized variation of {ˆσ 0 (t i )} i=1,...,n since, for the CIR model, σ, σ t = β2 4 ; denote such an estimate by β; 3 Estimate the optimal local bandwidth based on ˆσ 0 2 and β: ( 2ˆσ ˆθ 0 (t i ) := n 1/2 2 0 (t i ) K 2 ) 1/2 2 β 2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
47 Stochastic Volatility Processes Estimation Method for the CIR model (No Jumps) 1 Get a rough" estimate of σt 2 ; e.g., using Alvarez et al. (2010): ˆσ 0(t 2 i ) = QV t i + h i QV ti hi = 1 ( n j X ) 2. hi j:t j (t i,t i + h i] 2 Estimate the vol vol β using the realized variation of {ˆσ 0 (t i )} i=1,...,n since, for the CIR model, σ, σ t = β2 4 ; denote such an estimate by β; 3 Estimate the optimal local bandwidth based on ˆσ 0 2 and β: ( 2ˆσ ˆθ 0 (t i ) := n 1/2 2 0 (t i ) K 2 ) 1/2 2 β 2 4 Apply kernel estimation with the estimated optimal local bandwidth: σ 2 1(t i ) := l i+j X 2 K ˆθ 0 (t i ) (t i t i+j ) j= l Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
48 Stochastic Volatility Processes Estimation Method for the CIR model (No Jumps) 1 Get a rough" estimate of σt 2 ; e.g., using Alvarez et al. (2010): ˆσ 0(t 2 i ) = QV t i + h i QV ti hi = 1 ( n j X ) 2. hi j:t j (t i,t i + h i] 2 Estimate the vol vol β using the realized variation of {ˆσ 0 (t i )} i=1,...,n since, for the CIR model, σ, σ t = β2 4 ; denote such an estimate by β; 3 Estimate the optimal local bandwidth based on ˆσ 0 2 and β: ( 2ˆσ ˆθ 0 (t i ) := n 1/2 2 0 (t i ) K 2 ) 1/2 2 β 2 4 Apply kernel estimation with the estimated optimal local bandwidth: σ 2 1(t i ) := l i+j X 2 K ˆθ 0 (t i ) (t i t i+j ) j= l 5 Iterate steps 3 and 4 : ˆσ 2 0 ( ) ˆθ 0 ( ) ˆσ 2 1 ( ) ˆθ 1 ( ) ˆσ 2 2 ( ) Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
49 Stochastic Volatility Processes Numerical Illustration (no jumps): Uniform Kernel Model: CIR stochastic volatility dσt 2 = κ ( α σt 2 ) dt + βσt dw (σ) t. Parameters: γ t 0.05, λ = 0, ζ i N (0, 0.3), κ = 5, α = 0.04, β = 0.5, Cov(dW t, dw (σ) t ) = ρdt Regular Sampling Scheme: T=1/12 (one month) and h n = 5 min Monte Carlo results of MSE = n i=1 (ˆσ 2 (t i ) σ 2 (t i ) ) 2 based on 500 runs Method ρ = 0.5 ρ = 0 ρ = 0.5 Alvarez et al. Method Iterated Kernel Est. with opt. loc. bw Oracle 1 Kernel Est. with opt. loc. bw Using true parameters values for γ and σ Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
50 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
51 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Using σ 2, estimate the optimal threshold B(t i ) := [ 3 σ 2 (t i )h i ln(1/h i ) ] 1/2 ; Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
52 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Using σ 2, estimate the optimal threshold B(t i ) := [ 3 σ 2 (t i )h i ln(1/h i ) ] 1/2 ; 3 Refine σ 2 (t i ) using thresholding, ˆσ 0(t 2 i ) = 1 ( n j X ) 2 1[ j hi j:t i (t i,t i + X B(t i )] h i] Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
53 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Using σ 2, estimate the optimal threshold B(t i ) := [ 3 σ 2 (t i )h i ln(1/h i ) ] 1/2 ; 3 Refine σ 2 (t i ) using thresholding, ˆσ 0(t 2 i ) = 1 ( n j X ) 2 1[ j hi j:t i (t i,t i + X B(t i )] h i] 4 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n = β. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
54 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Using σ 2, estimate the optimal threshold B(t i ) := [ 3 σ 2 (t i )h i ln(1/h i ) ] 1/2 ; 3 Refine σ 2 (t i ) using thresholding, ˆσ 0(t 2 i ) = 1 ( n j X ) 2 1[ j hi j:t i (t i,t i + X B(t i )] h i] 4 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n = β. 5 Refine the estimates of the optimal threshold and local bandwidth: ( B 0 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ 0 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 ) 1/2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
55 Stochastic Volatility Processes Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Using σ 2, estimate the optimal threshold B(t i ) := [ 3 σ 2 (t i )h i ln(1/h i ) ] 1/2 ; 3 Refine σ 2 (t i ) using thresholding, ˆσ 0(t 2 i ) = 1 ( n j X ) 2 1[ j hi j:t i (t i,t i + X B(t i )] h i] 4 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n = β. 5 Refine the estimates of the optimal threshold and local bandwidth: ( B 0 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ 0 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 0 (t i ) and B 0 (t i ), refine the estimate ˆσ 2 0 (t i): σ 2 1(t i ) := l i+j X 2 K ˆθ0 (t i ) (t i t i+j )1 [ j X B 0 (t i )] j= l ) 1/2 Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
56 Stochastic Volatility Processes Numerical Illustration (with jumps): Uniform Kernel Model: Normal jump sizes and CIR stochastic volatility. Parameters: γ t 0.05, λ = 120, ζ i N (0, 0.3), κ = 5, α = 0.04, β = 0.5. Regular Sampling Scheme: T=1/12 (one month) and h n = 5 min Monte Carlo results of MSE = n i=1 (ˆσ 2 (t i ) σ 2 (t i ) ) 2 based on 500 runs Method MSE Alvarez et al. Method Alvarez et al. Method with thresholding Kernel Est. with thresholding and opt. loc. bw selection Twice Kernel Est. with thresholding and opt. loc. bw selection Oracle 2 Kernel Est. with thresholding and loc. bw selection Using true parameters values for γ and σ Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
57 Conclusions Main Contributions 1 Introduced an objective threshold selection procedure based on statistically optimal criteria. 2 Developed the infill asymptotic characterization of the optimal threshold. 3 Proposed an iterative algorithm to find the optimal threshold sequence. 4 Proposed extensions to more general stochastic models, which allows time-varying stochastic volatility and jump intensity. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
58 Conclusions Ongoing and Future Work 1 Connection between the optimal threshold B 1 and the data-based threshold ˆB 1 obtained by the iterative method 2 Implementation based on the second-order approximation B 2 of B 3 Extensions to infinite-jump activity processes 4 Incorporation of a microstructure noise component 5 Extensions to other loss functions 6 Extensions to other estimation problems where parameter tuning is needed. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
59 Appendix Bibliography For Further Reading I Figueroa-López & Nisen. Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models Stochastic Processes and their Applications 123(7), , Figueroa-López & Nisen. Optimality properties of thresholded multi power variation estimators. In preparation, Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
60 Classical Path of a FJA Lévy Model Back Back 2 Figure: Classical Path of X t = γt + σw t + N t i=1 ζ i; times and sizes of consecutive jumps are denoted by τ 1 < < τ n and ζ 1,..., ζ n, respectively. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
61 Numerical illustration I Back Back Diffusion Volatility Parameter Estimates TRV(GE) TRV(Hist) TRV(BF) TRV(Pow(0.35)) TRV(Pow(0.4)) TRV(Pow(0.45)) TRV(Pow(0.495)) MPV(1, 1) MPV(2 3,, 2 3) MPV(1 2,, 1 2) MPV(2 5,, 2 5) MPV(1 3,, 1 3) MinRV MedRV Merton Model: Diffusion Volatility Parameter Estimates Calibrated TRV Estimators Uncalibrated TRV Estimators Power Variation Estimators T = 6 months Freq. 5 min. σ = 0.35 λ = δ = Figure: Boxplots for Volatility Estimation: Based on 1,000 sample paths. Parameters: σ = 0.35, λ = , ζ = D N (0, ), T = 6-months, sampling frequency = 5-min. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
62 Numerical illustration II Back Diffusion Volatility Parameter Estimates TRV(B 1 ) TRV(B 2 ) TRV(GE) TRV(Hist) TRV(BF) TRV(Pow) MPV(1, 1) MPV(2 3,, 2 3) MPV(1 2,, 1 2) MPV(2 5,, 2 5) MPV(1 3,, 1 3) MinRV MedRV Student's t Model: Diffusion Volatility Parameter Estimates Oracle Optimal TRV Estimators Estimated Optimal TRV Estimators Alternative TRV Estimators Multi Power Variation Estimators Local Order Statistics Based Estimators σ = 0.45 Sparsity = Signal to Noise Ratio = 30 T = 1 year Freq. = 15 min. Figure: Boxplots for Volatility Estimators: Based on 1,000 sample paths. Parameters: σ = 0.45, λ = 32.84, ζ = D t-student(3 d.f.), T = 1-year, h n = 15-min. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
63 Numerical illustration III Back Jump Rate Parameter Estimates Student's t Model: Jump Rate Parameter Estimation Sparsity = Oracle Optimal TRV Estimators Estimated Optimal TRV Estimators Alternative TRV Estimators λ = Signal to Noise Ratio = 30 T = 1 year Freq. = 15 min. B 1 B 2 GE Hist BF Pow Figure: Boxplots for rate estimators λ := N[B] T : Based on 1,000 sample paths. T Parameters: σ = 0.45, λ = 32.84, ζ = D t-student(3 d.f.), T = 1-year, h n = 15-min. Figueroa-López, José (Purdue, Statistics) Optimally Thresholded Realized Variations ISI / 31
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