Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models

Transcription

1 Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models José E. Figueroa-López 1 1 Department of Statistics Purdue University Financial Statistics Stevanovich Center The University of Chicago September 27, 2014 (Joint work with Jeff Nisen)

2 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

3 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

4 Set-up 1 Finite-Jump Activity (FJA) Itô Semimartingales: Continuous-time stochastic process t X t with dynamics where 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 is a piece-wise constant process of finite jump activity; t γ t and t σ t are adapted processes; 2 FJA Lévy Model: where N t X t = γt + σw t + 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 +; j=1 the triplet ({W t}, {N t}, {ζ j }) are mutually independent.

5 Statistical Problems Given a discrete record of observations, X t0, X t1,..., X tn, 0 = t 0 < t 1 < < t n = T, the following problems are of common interest in 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: {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.

6 Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 QV (X) π ( T := Xti+1 X ) 2 t i, (π : 0 = t0 < < t n = T ). i=0 QV (X) π T mesh(π) 0 σ 2 T + N T j=1 ζ2 j. 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard (2004)): n 1 BPV (X) T := Xti+1 X ti Xti+2 X ti+1, MPV (X) (r 1,...,r k ) T := i=0 n k Xti+1 X ti r 1... Xti+k X ti+k 1 r k. i=0 2 Threshold Realized Variations (Mancini (2003), Jacod(2007)): n 1 ( TRV (X)[B] π T := Xti+1 X ) 2 t i 1{ Xti+1 Xti B}, (B (0, )). i=0

7 Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 QV (X) π ( T := Xti+1 X ) 2 t i, (π : 0 = t0 < < t n = T ). i=0 QV (X) π T mesh(π) 0 σ 2 T + N T j=1 ζ2 j. 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard (2004)): n 1 BPV (X) T := Xti+1 X ti Xti+2 X ti+1, MPV (X) (r 1,...,r k ) T := i=0 n k Xti+1 X ti r 1... Xti+k X ti+k 1 r k. i=0 2 Threshold Realized Variations (Mancini (2003), Jacod(2007)): n 1 ( TRV (X)[B] π T := Xti+1 X ) 2 t i 1{ Xti+1 Xti B}, (B (0, )). i=0

8 Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 QV (X) π ( T := Xti+1 X ) 2 t i, (π : 0 = t0 < < t n = T ). i=0 QV (X) π T mesh(π) 0 σ 2 T + N T j=1 ζ2 j. 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard (2004)): n 1 BPV (X) T := Xti+1 X ti Xti+2 X ti+1, MPV (X) (r 1,...,r k ) T := i=0 n k Xti+1 X ti r 1... Xti+k X ti+k 1 r k. i=0 2 Threshold Realized Variations (Mancini (2003), Jacod(2007)): n 1 ( TRV (X)[B] π T := Xti+1 X ) 2 t i 1{ Xti+1 Xti B}, (B (0, )). i=0

9 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Pros: Easy to implement as it requires no parameter tune-up; Cons: Can exhibit high bias and variability levels in the presence of jumps; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), (i) E [ π 2 BPV (X) ] T σ 2 T 2T E ζ 1 E N (0, 1) 1 σλh 1/2 n (ii) [ ] E 1 MPV Cr (X)(r 1,...,r k ) T σ T 2 T C r h max i r i n C r = k i=1 E N (0, 1) r i r r k = 2. (iii) ( ) Var 1 MPV Cr (X)(r 1,...,r k ) T T C r h n,, r r k = 2 and r i 1.

10 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Pros: Easy to implement as it requires no parameter tune-up; Cons: Can exhibit high bias and variability levels in the presence of jumps; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), (i) E [ π 2 BPV (X) ] T σ 2 T 2T E ζ 1 E N (0, 1) 1 σλh 1/2 n (ii) [ ] E 1 MPV Cr (X)(r 1,...,r k ) T σ T 2 T C r h max i r i n C r = k i=1 E N (0, 1) r i r r k = 2. (iii) ( ) Var 1 MPV Cr (X)(r 1,...,r k ) T T C r h n,, r r k = 2 and r i 1.

11 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Pros: Easy to implement as it requires no parameter tune-up; Cons: Can exhibit high bias and variability levels in the presence of jumps; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), (i) E [ π 2 BPV (X) ] T σ 2 T 2T E ζ 1 E N (0, 1) 1 σλh 1/2 n (ii) [ ] E 1 MPV Cr (X)(r 1,...,r k ) T σ T 2 T C r h max i r i n C r = k i=1 E N (0, 1) r i r r k = 2. (iii) ( ) Var 1 MPV Cr (X)(r 1,...,r k ) T T C r h n,, r r k = 2 and r i 1.

12 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Pros: Easy to implement as it requires no parameter tune-up; Cons: Can exhibit high bias and variability levels in the presence of jumps; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), (i) E [ π 2 BPV (X) ] T σ 2 T 2T E ζ 1 E N (0, 1) 1 σλh 1/2 n (ii) [ ] E 1 MPV Cr (X)(r 1,...,r k ) T σ T 2 T C r h max i r i n C r = k i=1 E N (0, 1) r i r r k = 2. (iii) ( ) Var 1 MPV Cr (X)(r 1,...,r k ) T T C r h n,, r r k = 2 and r i 1.

13 Advantages and Drawbacks 1 Multipower Realized Variations (MPV) Pros: Easy to implement as it requires no parameter tune-up; Cons: Can exhibit high bias and variability levels in the presence of jumps; e.g., for a FJA Lévy model and regular sampling (t i = ih n with h n = T /n), (i) E [ π 2 BPV (X) ] T σ 2 T 2T E ζ 1 E N (0, 1) 1 σλh 1/2 n (ii) [ ] E 1 MPV Cr (X)(r 1,...,r k ) T σ T 2 T C r h max i r i n C r = k i=1 E N (0, 1) r i r r k = 2. (iii) ( ) Var 1 MPV Cr (X)(r 1,...,r k ) T T C r h n,, r r k = 2 and r i 1.

14 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

15 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

16 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

17 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

18 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

19 Advantages and Drawbacks 2 Threshold Realized Variations (TRV): Pros: Cons: Can be adapted for estimating other jump features: n 1 N[B] π T := n 1 1 { } >B, Ĵ[B] π Xti+1 X T := ( ) Xti+1 X ti 1 { } ti i=0 i=0 Xti+1 X ti >B Can exhibit reduced bias in the presence of jumps 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), (i) TRV (X)[B n] π T is consistent for σ2 T iff hn B n n and B n 0; (ii) [ ] E TRV (X)[B n] π T σ T 2 ( Thn γ 2 λσ 2) ) 2T σφ( Bn σ Bn hn + 2T λb3 n C(f ζ ) h n 3 The rate h n is attainable if B n 0 slow enough (rel. to h 1/2 n ) but faster than h 1/3 n (iii) Var ( TRV (X)[B n] n T ) 2T σ 4 h n T λb5 nc 0 (f )

20 Selection of the threshold parameter B Literature consists of mostly somewhat ad hoc" selection methods for B: Power Threshold: 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

21 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

22 Optimal Threshold Realized Estimators 1 Question: Can the threshold parameter be chosen in a meaningfully objective manner? 2 Aims Develop a well-posed 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. 3 Assumptions Finite-Jump Activity Lévy Model: N t i.i.d. X t = γt + σw t + ζ j, {N t} t 0 Pois(λ), ζ j f ζ, ; j=1 Regular sampling scheme with mesh h n := T n ; i.e., π : t i = it n. The jump density function f ζ takes the mixture form: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, p, q 0, f ± : [0, ) R + C 1 b(0, )

23 Optimal Threshold Realized Estimators 1 Question: Can the threshold parameter be chosen in a meaningfully objective manner? 2 Aims Develop a well-posed 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. 3 Assumptions Finite-Jump Activity Lévy Model: N t i.i.d. X t = γt + σw t + ζ j, {N t} t 0 Pois(λ), ζ j f ζ, ; j=1 Regular sampling scheme with mesh h n := T n ; i.e., π : t i = it n. The jump density function f ζ takes the mixture form: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, p, q 0, f ± : [0, ) R + C 1 b(0, )

24 Optimal Threshold Realized Estimators 1 Question: Can the threshold parameter be chosen in a meaningfully objective manner? 2 Aims Develop a well-posed 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. 3 Assumptions Finite-Jump Activity Lévy Model: N t i.i.d. X t = γt + σw t + ζ j, {N t} t 0 Pois(λ), ζ j f ζ, ; j=1 Regular sampling scheme with mesh h n := T n ; i.e., π : t i = it n. The jump density function f ζ takes the mixture form: f ζ (x) = pf +(x)1 {x 0} + qf ( x)1 {x<0} with p + q = 1, p, q 0, f ± : [0, ) R + C 1 b(0, )

25 Loss Function 1 Alternative 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 Idea Loss n (B) favors sequences that minimizes the Total Number of Jump Miss-Classifications: to flag that a jump occurred during [t i 1, t i ], when no jump occurred, fail to identify the occurrence of a jump during [t i 1, t i ].

26 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

27 Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (2013)) 1 There exists an N N such that for all n N, 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 = pf + (0) + qf (0).

28 Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (2013)) 1 There exists an N N such that for all n N, 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 = pf + (0) + qf (0).

29 Infill Asymptotics for the Mean and Variance Theorem (FL & Nisen (2014)) 1 For the optimal TRV estimator, as n, E(TRV (X)[B n]) T σ 2 T (γ 2 λσ 2 )h n, Var (TRV (X)[B n]) 2T σ 4 h n. 2 For some pre-estimates σ and ν(0) for the true σ and ν(0) := λc(f ζ ), let ( 1 B n := 3 σh n log Then, in terms of ω := σ 2 /σ 2, ( ) E TRV (X)[ B n ] T σ 2 ( ) Var TRV (X)[ B n ] 2T σ 4 h n. h n ) log( 2π σ ν(0)) σh 1/2 n, 3 log(1/hn ) h n T (γ 2 λσ 2 ), ω 2 3, h 3ω/2 n T E Z ( 2π σ ν(0)) ω, 0 < ω < 2 3 log(1/hn)ω 3,

30 Infill Asymptotics for the Mean and Variance Theorem (FL & Nisen (2014)) 1 For the optimal TRV estimator, as n, E(TRV (X)[B n]) T σ 2 T (γ 2 λσ 2 )h n, Var (TRV (X)[B n]) 2T σ 4 h n. 2 For some pre-estimates σ and ν(0) for the true σ and ν(0) := λc(f ζ ), let ( 1 B n := 3 σh n log Then, in terms of ω := σ 2 /σ 2, ( ) E TRV (X)[ B n ] T σ 2 ( ) Var TRV (X)[ B n ] 2T σ 4 h n. h n ) log( 2π σ ν(0)) σh 1/2 n, 3 log(1/hn ) h n T (γ 2 λσ 2 ), ω 2 3, h 3ω/2 n T E Z ( 2π σ ν(0)) ω, 0 < ω < 2 3 log(1/hn)ω 3,

31 Infill Asymptotics for the Mean and Variance Theorem (FL & Nisen (2014)) 1 For the optimal TRV estimator, as n, E(TRV (X)[B n]) T σ 2 T (γ 2 λσ 2 )h n, Var (TRV (X)[B n]) 2T σ 4 h n. 2 For some pre-estimates σ and ν(0) for the true σ and ν(0) := λc(f ζ ), let ( 1 B n := 3 σh n log Then, in terms of ω := σ 2 /σ 2, ( ) E TRV (X)[ B n ] T σ 2 ( ) Var TRV (X)[ B n ] 2T σ 4 h n. h n ) log( 2π σ ν(0)) σh 1/2 n, 3 log(1/hn ) h n T (γ 2 λσ 2 ), ω 2 3, h 3ω/2 n T E Z ( 2π σ ν(0)) ω, 0 < ω < 2 3 log(1/hn)ω 3,

32 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 constitute the first and second-order approx. for B n, and their resulting TRV estimators attain the same rates as the optimal TRV estimators; and Bn 2 mass concentration of f ζ near 0, as measured by C(f ζ ). 3 B 1 n account for the volatility σ, the intensity of jumps λ, and the 4 They provide blueprints" for devising a good threshold sequence.

33 A Feasible Iterative Algorithm to Find B n 1 Key Issue: The optimal threshold B 1 would allow us to find a approximately optimal estimate ˆσ 2 for σ 2 of the form but B 1 depends precisely on σ 2. ˆσ 2 := 1 T TRV (X)[B 1 (σ 2 )] n, 2 The previous issue suggests a fixed-point" type of implementation: Set σ 2 n,0 := 1 T n i=1 X t i X ti 1 2 and while σ 2 n,k 1 > σ2 n,k do σ n,k T TRV (X)[ B n,k ] and B B 1 n,0 := (3 σ 2 n,0 h n log(1/h n ) n,k+1 (3 σ 2 n,k+1 h n log(1/h n ) end while { } Let kn := inf k 1 : σ n,k+1 2 = σ2 n,k and take σ n,k 2 n as the final ) 1/2 estimate for σ 2 1 and the corresponding B n,k as an estimate for n B n. 3 The previous algorithm generates a non-increasing sequence of estimators { σ 2 n,k } k and finish in finite time. ) 1/2

34 A Feasible Iterative Algorithm to Find B n 1 Key Issue: The optimal threshold B 1 would allow us to find a approximately optimal estimate ˆσ 2 for σ 2 of the form but B 1 depends precisely on σ 2. ˆσ 2 := 1 T TRV (X)[B 1 (σ 2 )] n, 2 The previous issue suggests a fixed-point" type of implementation: Set σ 2 n,0 := 1 T n i=1 X t i X ti 1 2 and while σ 2 n,k 1 > σ2 n,k do σ n,k T TRV (X)[ B n,k ] and B B 1 n,0 := (3 σ 2 n,0 h n log(1/h n ) n,k+1 (3 σ 2 n,k+1 h n log(1/h n ) end while { } Let kn := inf k 1 : σ n,k+1 2 = σ2 n,k and take σ n,k 2 n as the final ) 1/2 estimate for σ 2 1 and the corresponding B n,k as an estimate for n B n. 3 The previous algorithm generates a non-increasing sequence of estimators { σ 2 n,k } k and finish in finite time. ) 1/2

35 A Feasible Iterative Algorithm to Find B n 1 Key Issue: The optimal threshold B 1 would allow us to find a approximately optimal estimate ˆσ 2 for σ 2 of the form but B 1 depends precisely on σ 2. ˆσ 2 := 1 T TRV (X)[B 1 (σ 2 )] n, 2 The previous issue suggests a fixed-point" type of implementation: Set σ 2 n,0 := 1 T n i=1 X t i X ti 1 2 and while σ 2 n,k 1 > σ2 n,k do σ n,k T TRV (X)[ B n,k ] and B B 1 n,0 := (3 σ 2 n,0 h n log(1/h n ) n,k+1 (3 σ 2 n,k+1 h n log(1/h n ) end while { } Let kn := inf k 1 : σ n,k+1 2 = σ2 n,k and take σ n,k 2 n as the final ) 1/2 estimate for σ 2 1 and the corresponding B n,k as an estimate for n B n. 3 The previous algorithm generates a non-increasing sequence of estimators { σ 2 n,k } k and finish in finite time. ) 1/2

36 A Feasible Iterative Algorithm to Find B n 1 Key Issue: The optimal threshold B 1 would allow us to find a approximately optimal estimate ˆσ 2 for σ 2 of the form but B 1 depends precisely on σ 2. ˆσ 2 := 1 T TRV (X)[B 1 (σ 2 )] n, 2 The previous issue suggests a fixed-point" type of implementation: Set σ 2 n,0 := 1 T n i=1 X t i X ti 1 2 and while σ 2 n,k 1 > σ2 n,k do σ n,k T TRV (X)[ B n,k ] and B B 1 n,0 := (3 σ 2 n,0 h n log(1/h n ) n,k+1 (3 σ 2 n,k+1 h n log(1/h n ) end while { } Let kn := inf k 1 : σ n,k+1 2 = σ2 n,k and take σ n,k 2 n as the final ) 1/2 estimate for σ 2 1 and the corresponding B n,k as an estimate for n B n. 3 The previous algorithm generates a non-increasing sequence of estimators { σ 2 n,k } k and finish in finite time. ) 1/2

37 A numerical illustration I Merton Model: 4-year / 1-day σ = 0.3, λ = 5, γ = 0 µ = 0, δ = 0.6 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 i.i.d. based on 5, 000 sample paths for the Merton model ζ i N (µ, δ 2 ). 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.

38 A numerical illustration II (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 for 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.

39 A numerical illustration III (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 for 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.

40 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

41 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

42 Additive Processes 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 (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 ( i X := X ti 1 +h i X ti 1, i N := N ti 1 +h i N ti 1, h i = t i t i 1 ) )

43 Additive Processes 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 (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 ( i X := X ti 1 +h i X ti 1, i N := N ti 1 +h i N ti 1, h i = t i t i 1 ) )

44 Additive Processes 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 (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 ( i X := X ti 1 +h i X ti 1, i N := N ti 1 +h i N ti 1, h i = t i t i 1 ) )

45 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 possess 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) ln(1/h)

46 Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1 (Mesh), K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm: [Kernel estimators by, e.g., Fan & Wang(2008), Kristensen(2010)] For each i {1, 2,..., n}, set l σ 0(t 2 i ) := i+j X 2 K θ (t i t i+j ) and j= l B,0 t i := [ 3 σ 2 0(t i )h i ln(1/h i ) ] 1/2 with θ chosen by the cross-validation method proposed in Kristensen(2010). while there exists i {1, 2,..., m} such that σ k 1 2 (t i) > σ k 2(t i) do σ k+1 2 (t i) l j= l i+jx 2 K θ (t i t i+j )1 [ ] and i+j X B,k t i+j B,k+1 t i [ 3 σ k+1 2 (t i)h i ln(1/h i ) ] 1/2 end while Let k := inf { k 1 : σ k+1 2 (t i) = σ k 2(t i); for all i = 1, 2,..., n } and take σ k 2 (t,k i) as the final estimate for σ(t i ) and the corresponding B t i as an estimate for Bt i. For each i, σ 2 k (t i) σ 2 k+1 (t i) and, thus, finish in finite time.

47 Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1 (Mesh), K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm: [Kernel estimators by, e.g., Fan & Wang(2008), Kristensen(2010)] For each i {1, 2,..., n}, set l σ 0(t 2 i ) := i+j X 2 K θ (t i t i+j ) and j= l B,0 t i := [ 3 σ 2 0(t i )h i ln(1/h i ) ] 1/2 with θ chosen by the cross-validation method proposed in Kristensen(2010). while there exists i {1, 2,..., m} such that σ k 1 2 (t i) > σ k 2(t i) do σ k+1 2 (t i) l j= l i+jx 2 K θ (t i t i+j )1 [ ] and i+j X B,k t i+j B,k+1 t i [ 3 σ k+1 2 (t i)h i ln(1/h i ) ] 1/2 end while Let k := inf { k 1 : σ k+1 2 (t i) = σ k 2(t i); for all i = 1, 2,..., n } and take σ k 2 (t,k i) as the final estimate for σ(t i ) and the corresponding B t i as an estimate for Bt i. For each i, σ 2 k (t i) σ 2 k+1 (t i) and, thus, finish in finite time.

48 Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1 (Mesh), K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm: [Kernel estimators by, e.g., Fan & Wang(2008), Kristensen(2010)] For each i {1, 2,..., n}, set l σ 0(t 2 i ) := i+j X 2 K θ (t i t i+j ) and j= l B,0 t i := [ 3 σ 2 0(t i )h i ln(1/h i ) ] 1/2 with θ chosen by the cross-validation method proposed in Kristensen(2010). while there exists i {1, 2,..., m} such that σ k 1 2 (t i) > σ k 2(t i) do σ k+1 2 (t i) l j= l i+jx 2 K θ (t i t i+j )1 [ ] and i+j X B,k t i+j B,k+1 t i [ 3 σ k+1 2 (t i)h i ln(1/h i ) ] 1/2 end while Let k := inf { k 1 : σ k+1 2 (t i) = σ k 2(t i); for all i = 1, 2,..., n } and take σ k 2 (t,k i) as the final estimate for σ(t i ) and the corresponding B t i as an estimate for Bt i. For each i, σ 2 k (t i) σ 2 k+1 (t i) and, thus, finish in finite time.

49 Spot Volatility Estimation via Kernel Methods Notation: h i = t i t i 1 (Mesh), K θ (t) = 1 θ K ( t θ ) (Kernel), θ = Bandwidth Algorithm: [Kernel estimators by, e.g., Fan & Wang(2008), Kristensen(2010)] For each i {1, 2,..., n}, set l σ 0(t 2 i ) := i+j X 2 K θ (t i t i+j ) and j= l B,0 t i := [ 3 σ 2 0(t i )h i ln(1/h i ) ] 1/2 with θ chosen by the cross-validation method proposed in Kristensen(2010). while there exists i {1, 2,..., m} such that σ k 1 2 (t i) > σ k 2(t i) do σ k+1 2 (t i) l j= l i+jx 2 K θ (t i t i+j )1 [ ] and i+j X B,k t i+j B,k+1 t i [ 3 σ k+1 2 (t i)h i ln(1/h i ) ] 1/2 end while Let k := inf { k 1 : σ k+1 2 (t i) = σ k 2(t i); for all i = 1, 2,..., n } and take σ k 2 (t,k i) as the final estimate for σ(t i ) and the corresponding B t i as an estimate for Bt i. For each i, σ 2 k (t i) σ 2 k+1 (t i) and, thus, finish in finite time.

50 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).

51 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).

52 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

53 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 is itself stochastic. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σ 2 t ) dt + βσt dw (σ) t, (2κα β 2 > 0, W (σ) W ) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received relatively limited attention: e.g., Fan & Wang(2008), Kristensen(2010), Reno & Mancini(2011) considered Kernel Estimators; Mykland & Lan(2008), Alvarez et al.(2010), and others estimate [X, X] t = σ t via a finite-difference approx. of the QV of X; Available kernel estimation methods for stochastic volatilities, {σt 2 } t 0, are quite sensitive to the bandwidth. Few Bandwidth Selection Methods (e.g., Kristensen(2010))

54 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 is itself stochastic. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σ 2 t ) dt + βσt dw (σ) t, (2κα β 2 > 0, W (σ) W ) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received relatively limited attention: e.g., Fan & Wang(2008), Kristensen(2010), Reno & Mancini(2011) considered Kernel Estimators; Mykland & Lan(2008), Alvarez et al.(2010), and others estimate [X, X] t = σ t via a finite-difference approx. of the QV of X; Available kernel estimation methods for stochastic volatilities, {σt 2 } t 0, are quite sensitive to the bandwidth. Few Bandwidth Selection Methods (e.g., Kristensen(2010))

55 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 is itself stochastic. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σ 2 t ) dt + βσt dw (σ) t, (2κα β 2 > 0, W (σ) W ) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received relatively limited attention: e.g., Fan & Wang(2008), Kristensen(2010), Reno & Mancini(2011) considered Kernel Estimators; Mykland & Lan(2008), Alvarez et al.(2010), and others estimate [X, X] t = σ t via a finite-difference approx. of the QV of X; Available kernel estimation methods for stochastic volatilities, {σt 2 } t 0, are quite sensitive to the bandwidth. Few Bandwidth Selection Methods (e.g., Kristensen(2010))

56 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 is itself stochastic. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σ 2 t ) dt + βσt dw (σ) t, (2κα β 2 > 0, W (σ) W ) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received relatively limited attention: e.g., Fan & Wang(2008), Kristensen(2010), Reno & Mancini(2011) considered Kernel Estimators; Mykland & Lan(2008), Alvarez et al.(2010), and others estimate [X, X] t = σ t via a finite-difference approx. of the QV of X; Available kernel estimation methods for stochastic volatilities, {σt 2 } t 0, are quite sensitive to the bandwidth. Few Bandwidth Selection Methods (e.g., Kristensen(2010))

57 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 is itself stochastic. 2 Prototypical Example: Mean-reverting square-root process (CIR Model): dσ 2 t = κ ( α σ 2 t ) dt + βσt dw (σ) t, (2κα β 2 > 0, W (σ) W ) 3 Pitfalls of the Estimation Methods for σ: Spot volatility estimation has received relatively limited attention: e.g., Fan & Wang(2008), Kristensen(2010), Reno & Mancini(2011) considered Kernel Estimators; Mykland & Lan(2008), Alvarez et al.(2010), and others estimate [X, X] t = σ t via a finite-difference approx. of the QV of X; Available kernel estimation methods for stochastic volatilities, {σt 2 } t 0, are quite sensitive to the bandwidth. Few Bandwidth Selection Methods (e.g., Kristensen(2010))

58 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 ), 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 optimal" bandwidths: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 ( ) 1 2 2υ+1 2, bw glb υl 2 t (0) opt = n 1 T 2υ+1 0 σ4 t dt K 2 2υ+1 2 υ T 0 L2 t (0)dt ( ) 3 Pitfall: In general, it is hard to check ( ) with explicit (nonzero) L t (0) and υ. 4 A (heuristic) alternative method: Suppose that [ (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. Then, use ( ).

59 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 ), 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 optimal" bandwidths: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 ( ) 1 2 2υ+1 2, bw glb υl 2 t (0) opt = n 1 T 2υ+1 0 σ4 t dt K 2 2υ+1 2 υ T 0 L2 t (0)dt ( ) 3 Pitfall: In general, it is hard to check ( ) with explicit (nonzero) L t (0) and υ. 4 A (heuristic) alternative method: Suppose that [ (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. Then, use ( ).

60 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 ), 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 optimal" bandwidths: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 ( ) 1 2 2υ+1 2, bw glb υl 2 t (0) opt = n 1 T 2υ+1 0 σ4 t dt K 2 2υ+1 2 υ T 0 L2 t (0)dt ( ) 3 Pitfall: In general, it is hard to check ( ) with explicit (nonzero) L t (0) and υ. 4 A (heuristic) alternative method: Suppose that [ (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. Then, use ( ).

61 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 ), 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 optimal" bandwidths: ( ) bw loc opt,t = n 1 σ 4 2υ+1 t K 1 ( ) 1 2 2υ+1 2, bw glb υl 2 t (0) opt = n 1 T 2υ+1 0 σ4 t dt K 2 2υ+1 2 υ T 0 L2 t (0)dt ( ) 3 Pitfall: In general, it is hard to check ( ) with explicit (nonzero) L t (0) and υ. 4 A (heuristic) alternative method: Suppose that [ (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. Then, use ( ).

62 Implementation for the CIR model 1 Recall σ 2 t+δ σ 2 t = t+δ t κ ( α σu 2 ) t+δ du + β t σ u dw (σ) u, 2 The leading term of ( σ 2 t+δ σt 2 2 ) is β 2 t+δ 2, σ t u dw (σ) u which is such that ( ) 2 t+δ ( ) t+δ E β 2 σ u dw (σ) u t F t = β 2 E σudu 2 t F t = β 2 σt 2 δ + o(δ). 3 The previous heuristics suggest that ( σ 2 E t+δ σt 2 2 ) Ft = β 2 σt 2 δ + o(δ). 4 Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. 5 This suggests the following local optimal estimator: ( 2σ bw loc 2 opt,t = t K 2 ) 1/2 2 n 1/2. β 2

63 Implementation for the CIR model 1 Recall σ 2 t+δ σ 2 t = t+δ t κ ( α σu 2 ) t+δ du + β t σ u dw (σ) u, 2 The leading term of ( σ 2 t+δ σt 2 2 ) is β 2 t+δ 2, σ t u dw (σ) u which is such that ( ) 2 t+δ ( ) t+δ E β 2 σ u dw (σ) u t F t = β 2 E σudu 2 t F t = β 2 σt 2 δ + o(δ). 3 The previous heuristics suggest that ( σ 2 E t+δ σt 2 2 ) Ft = β 2 σt 2 δ + o(δ). 4 Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. 5 This suggests the following local optimal estimator: ( 2σ bw loc 2 opt,t = t K 2 ) 1/2 2 n 1/2. β 2

64 Implementation for the CIR model 1 Recall σ 2 t+δ σ 2 t = t+δ t κ ( α σu 2 ) t+δ du + β t σ u dw (σ) u, 2 The leading term of ( σ 2 t+δ σt 2 2 ) is β 2 t+δ 2, σ t u dw (σ) u which is such that ( ) 2 t+δ ( ) t+δ E β 2 σ u dw (σ) u t F t = β 2 E σudu 2 t F t = β 2 σt 2 δ + o(δ). 3 The previous heuristics suggest that ( σ 2 E t+δ σt 2 2 ) Ft = β 2 σt 2 δ + o(δ). 4 Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. 5 This suggests the following local optimal estimator: ( 2σ bw loc 2 opt,t = t K 2 ) 1/2 2 n 1/2. β 2

65 Implementation for the CIR model 1 Recall σ 2 t+δ σ 2 t = t+δ t κ ( α σu 2 ) t+δ du + β t σ u dw (σ) u, 2 The leading term of ( σ 2 t+δ σt 2 2 ) is β 2 t+δ 2, σ t u dw (σ) u which is such that ( ) 2 t+δ ( ) t+δ E β 2 σ u dw (σ) u t F t = β 2 E σudu 2 t F t = β 2 σt 2 δ + o(δ). 3 The previous heuristics suggest that ( σ 2 E t+δ σt 2 2 ) Ft = β 2 σt 2 δ + o(δ). 4 Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. 5 This suggests the following local optimal estimator: ( 2σ bw loc 2 opt,t = t K 2 ) 1/2 2 n 1/2. β 2

66 Implementation for the CIR model 1 Recall σ 2 t+δ σ 2 t = t+δ t κ ( α σu 2 ) t+δ du + β t σ u dw (σ) u, 2 The leading term of ( σ 2 t+δ σt 2 2 ) is β 2 t+δ 2, σ t u dw (σ) u which is such that ( ) 2 t+δ ( ) t+δ E β 2 σ u dw (σ) u t F t = β 2 E σudu 2 t F t = β 2 σt 2 δ + o(δ). 3 The previous heuristics suggest that ( σ 2 E t+δ σt 2 2 ) Ft = β 2 σt 2 δ + o(δ). 4 Hence, ( ) holds with υ = 1 2 and L2 t (0) = β2 σ 2 t. 5 This suggests the following local optimal estimator: ( 2σ bw loc 2 opt,t = t K 2 ) 1/2 2 n 1/2. β 2

67 Outlined of the Estimation Method (No Jumps) 1 Get a rough" estimate of t σt 2 ; e.g., using Alvarez et al. (2010), ˆσ 2 0(t i ) = QV t i + h i QV ti hi = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n, since σ, σ t = β2 4 ; denote β such an estimate; 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 ) := 5 Iterate steps 3 and 4. l i+j X 2 K ˆθ0 (t i ) (t i t i+j ) j= l

68 Outlined of the Estimation Method (No Jumps) 1 Get a rough" estimate of t σt 2 ; e.g., using Alvarez et al. (2010), ˆσ 2 0(t i ) = QV t i + h i QV ti hi = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n, since σ, σ t = β2 4 ; denote β such an estimate; 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 ) := 5 Iterate steps 3 and 4. l i+j X 2 K ˆθ0 (t i ) (t i t i+j ) j= l

69 Outlined of the Estimation Method (No Jumps) 1 Get a rough" estimate of t σt 2 ; e.g., using Alvarez et al. (2010), ˆσ 2 0(t i ) = QV t i + h i QV ti hi = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n, since σ, σ t = β2 4 ; denote β such an estimate; 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 ) := 5 Iterate steps 3 and 4. l i+j X 2 K ˆθ0 (t i ) (t i t i+j ) j= l

70 Outlined of the Estimation Method (No Jumps) 1 Get a rough" estimate of t σt 2 ; e.g., using Alvarez et al. (2010), ˆσ 2 0(t i ) = QV t i + h i QV ti hi = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n, since σ, σ t = β2 4 ; denote β such an estimate; 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 ) := 5 Iterate steps 3 and 4. l i+j X 2 K ˆθ0 (t i ) (t i t i+j ) j= l

71 Outlined of the Estimation Method (No Jumps) 1 Get a rough" estimate of t σt 2 ; e.g., using Alvarez et al. (2010), ˆσ 2 0(t i ) = QV t i + h i QV ti hi = 1 hi j:t j (t i,t i + h i] ( n j X ) 2. 2 Estimate β using the realized variation of {ˆσ 0 (t i )} i=1,...,n, since σ, σ t = β2 4 ; denote β such an estimate; 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 ) := 5 Iterate steps 3 and 4. l i+j X 2 K ˆθ0 (t i ) (t i t i+j ) j= l

72 Numerical Illustration (no jumps): Uniform Kernel Model: Normal jump sizes and CIR stochastic volatility. 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 Kernel Est. with opt. loc. bw Oracle 1 Kernel Est. with opt. loc. bw Using true parameters values for γ and σ

73 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ 1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

74 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ 1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

75 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ 1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

76 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

77 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

78 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

79 Outlined of Estimation Method (With Jumps) 1 Get a rough" estimate of t σt 2 by σ 2 (t i ) = 1 ( ) 2. hi j:t j (t i,t i + h i] n j X 2 Using σ 2, estimate the optimal threshold B ti := [ 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 ti ] 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 1 (t i ) := 3ˆσ 0(t 2 i )h i ln 1 ) 1/2 ( 2ˆσ, ˆθ1 (t i ) := n 1/2 2 0 (t i ) K 2 2 h i β 2 6 Based on ˆθ 1 (t i ) and B ti, refine our estimate ˆσ 2 0 : σ 2 1(t i ) := l i+j X 2 K ˆθ1 (t i ) (t i t i+j )1 [ j X B 1 (t i )] j= l 7 Iterate steps 5 and 6. ) 1/2

80 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 loc. bw selection Twice Kernel Est. with thresholding and loc. bw selection Oracle 2 Kernel Est. with thresholding and loc. bw selection Using true parameters values for γ and σ

81 Outline 1 The Statistical Problems and the Main Estimators 2 Optimally Thresholded Estimators for Finite-Jump Activity Processes 3 Main Results 4 Extensions Additive Processes Stochastic Volatility Processes 5 Conclusions

82 Conclusions 1 Introduce an objective threshold selection procedure based on statistical optimality reasoning via a well-posed optimization problem. 2 Characterize precisely the infill asymptotic behavior of the optimal threshold sequence. 3 Proposed an iterative algorithm to find the optimal threshold sequence. 4 Extend the approach to more general stochastic models, which allows time-varying volatility and jump intensity.

83 For Further Reading I Figueroa-López & Nisen. Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models Stochastic Processes and their Applications 123(7), , Available at figueroa. Figueroa-López & Nisen. Optimality properties of thresholded multi power variation estimators. In preparation, 2014.

84 Numerical illustration I 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.