Optimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error

Transcription

1 Optimum Thresholding for Semimartingales with Lévy Jumps under the mean-square error José E. Figueroa-López Department of Mathematics Washington University in St. Louis Spring Central Sectional Meeting Indiana University, Bloomington, IN April 2, 2017 (Joint work with Cecilia Mancini from University of Florence) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 1 / 35

2 Outline 1 Motivational Example 2 The Problem Multipower Variations and Truncated Realized Variations The problem of Parameter Tuning 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Ongoing and Future Research J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 2 / 35

3 Motivational Example Outline 1 Motivational Example 2 The Problem Multipower Variations and Truncated Realized Variations The problem of Parameter Tuning 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Ongoing and Future Research J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 3 / 35

4 Motivational Example Merton s Log-Normal Model Consider the following model for the log-return process X t = log St S 0 of a financial asset: X t = at + σw t + ζ i, i.i.d. ζ i N(µ jmp, σjmp), 2 {τ i } i 1 Poisson(λ) i:τ i t Goal: Estimate the volatility σ based on a discrete record X t1,..., X tn J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 4 / 35

5 Motivational Example Merton s Log-Normal Model Consider the following model for the log-return process X t = log St S 0 of a financial asset: X t = at + σw t + ζ i, i.i.d. ζ i N(µ jmp, σjmp), 2 {τ i } i 1 Poisson(λ) i:τ i t Goal: Estimate the volatility σ based on a discrete record X t1,..., X tn (σ = 0.4, λ = 200, σ jmp = 0.02, µ jmp = 0, t i t i 1 = 5 min, T = t n = 1/12) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 4 / 35

6 Motivational Example Model Simulation 1 Fix the time horizon T and the number of sampling points n. Set the time mesh and the sampling times as 2 Generate Z i i.i.d. N(0, 1), h = h n = T n, t i = t i,n = ih n, i = 0,..., n. γ i i.i.d. N(µ jmp, σ 2 jmp ), 3 Iteratively generate the process starting from X 0 : I i i.i.d. Bernoulli(λh) X ti = X ti 1 + ah + σ hz i + I i γ i J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 5 / 35

7 The Problem Outline 1 Motivational Example 2 The Problem Multipower Variations and Truncated Realized Variations The problem of Parameter Tuning 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Ongoing and Future Research J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 6 / 35

8 The Problem Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 ( RQV [X] n := Xti+1 X ) 2 t i i=0 RQV [X] n n T σ 2 + j:τ j T ζ2 j J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 7 / 35

9 The Problem Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 ( RQV [X] n := Xti+1 X ) 2 t i RQV [X] n i=0 n T σ 2 + j:τ j T ζ2 j 1 Bipower Realized Variations (Barndorff-Nielsen and Shephard): n 1 BPV [X] n := Xti+1 X ti Xti+2 X ti+1, i=0 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 7 / 35

10 The Problem Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: n 1 ( RQV [X] n := Xti+1 X ) 2 t i RQV [X] n i=0 n T σ 2 + j:τ j T ζ2 j 1 Bipower Realized Variations (Barndorff-Nielsen and Shephard): n 1 BPV [X] n := Xti+1 X ti Xti+2 X ti+1, i=0 2 Truncated Realized Variations (Mancini): n 1 ( TRV n [X](ε) := Xti+1 X ) 2 t i 1 { Xti+1 X ti i=0 ε }, (ε [0, )). J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 7 / 35

11 The Problem The problem of Parameter Tuning Calibration or Tuning of the Estimator Problem: How do you choose the threshold parameter ε? J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 8 / 35

12 The Problem The problem of Parameter Tuning Calibration or Tuning of the Estimator Problem: How do you choose the threshold parameter ε? Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price Price process Continuous process Times of jumps Truncated Realized Variation (TRV) TRV True Volatility (0.4) Realized Variation (0.51) Bipower Variation (0.42) Time in years (252 days) Truncation level (epsilon) Figure: (left) 5-min Merton observations with σ = 0.4, σ jmp = 3 h, µ jmp = 0, λ = 200; (right) TRV performance wrt the truncation level J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 8 / 35

13 The Problem The problem of Parameter Tuning Calibration or Tuning of the Estimator Problem: How do you choose the threshold parameter ε? Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price Price process Continuous process Times of jumps Truncated Realized Variation (TRV) TRV True Volatility (0.2) Realized Variation (0.4) Bipower Variation (0.26) Cont. Realized Variation (0.19) Time in years (252 days) Truncation level (epsilon) Figure: (left) 5 minute Merton observations with σ = 0.2, σ jmp = 1.5 h, µ jmp = 0, λ = 1000; (right) TRV performance wrt the truncation level J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 9 / 35

14 The Problem The problem of Parameter Tuning Truncation Selection Literature consists of mostly ad hoc" selection methods for ε, aimed to satisfy sufficient conditions for the consistency and asymptotic normality of the associated estimators. The most popular is the so-called Power Threshold: ε Pwr α,ω := α h ω, for α > 0 and ω (0, 1/2). The rule of thumb says to use a value of ω close to 1/2 (typically, 0.495). J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 10 / 35

15 Outline 1 Motivational Example 2 The Problem Multipower Variations and Truncated Realized Variations The problem of Parameter Tuning 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Ongoing and Future Research J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 11 / 35

16 Classical Approach 1 Fix a suitable and sensible metric of the estimation error; say, [ ( ) ] 1 2 MSE(ε) = E T TRV n(ε) σ 2 2 Show the existence of the optimal threshold ε n minimizing the error function; 3 Analyze the asymptotic behavior ε n (when n ) to infer qualitative information such as rate of convergence on n and dependence on the underlying parameters of the model (σ, σ J, λ) Devise a plug-in type calibration of ε by estimating those parameters (if possible). J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 12 / 35

17 via Expected number of jump misclassifications Via Expected number of jump misclassifications J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 13 / 35

18 via Expected number of jump misclassifications Via Expected number of jump misclassifications 1 Estimation Error: (F-L & Nisen, SPA 2013) n ) Loss n (ε) := E (1 [ ni X >ε, ni N=0] + 1 [ ni X ε, ni N 0]. 2 Notation: i=1 N t := # of jumps by time t n i X := X t i X ti 1 n i N := N t i N ti 1 = # of jumps during (t i 1, t i ] J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 13 / 35

19 via Expected number of jump misclassifications Via Expected number of jump misclassifications 1 Estimation Error: (F-L & Nisen, SPA 2013) n ) Loss n (ε) := E (1 [ ni X >ε, ni N=0] + 1 [ ni X ε, ni N 0]. 2 Notation: 3 Idea i=1 N t := # of jumps by time t n i X := X t i X ti 1 n i N := N t i N ti 1 = # of jumps during (t i 1, t i ] Intuitively, the estimation error should be directly linked to the ability of the threshold to detect jumps. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 13 / 35

20 via Expected number of jump misclassifications Existence and Infill Asymptotic Characterization Theorem (FL & Nisen, SPA 2013) a J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 14 / 35

21 via Expected number of jump misclassifications Existence and Infill Asymptotic Characterization Theorem (FL & Nisen, SPA 2013) 1 For n large enough, the loss function Loss n (ε) is quasi-convex and, moreover, possesses a unique global minimum ε n. a J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 14 / 35

22 via Expected number of jump misclassifications Existence and Infill Asymptotic Characterization Theorem (FL & Nisen, SPA 2013) 1 For n large enough, the loss function Loss n (ε) is quasi-convex and, moreover, possesses a unique global minimum ε n. 2 As n, the optimal threshold sequence (ε n) n is such that ( ) 1 ε n = 3σ 2 h n log + h.o.t., 1 δ where C(f ζ ) = lim δ 0 2δ δ f ζ(x)dx > 0 and f ζ is the density of the jumps ζ i. a a (throughout, h.o.t. refers to higher order terms ) h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 14 / 35

23 via Expected number of jump misclassifications Remarks 1 The threshold sequence ε 1 n := ( 1 3σ 2 h n log is the first-order approximations for ε n. h n ), J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 15 / 35

24 via Expected number of jump misclassifications Remarks 1 The threshold sequence ε 1 n := ( 1 3σ 2 h n log is the first-order approximations for ε n. 2 Why h log(1/h)? h n ), J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 15 / 35

25 via Expected number of jump misclassifications Remarks 1 The threshold sequence ε 1 n := ( 1 3σ 2 h n log is the first-order approximations for ε n. 2 Why h log(1/h)? This is proportional to modulus of continuity of the B.M.: lim sup h 0 1 2h log(1/h) h n ), sup W t W s = 1. s,t [0,1]: t s <h J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 15 / 35

26 via Expected number of jump misclassifications Remarks 1 The threshold sequence ε 1 n := ( 1 3σ 2 h n log is the first-order approximations for ε n. 2 Why h log(1/h)? This is proportional to modulus of continuity of the B.M.: lim sup h 0 1 2h log(1/h) h n ), sup W t W s = 1. s,t [0,1]: t s <h 3 Practically, ε 1 n provides us with a blueprint" for devising good threshold sequences! J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 15 / 35

27 via Expected number of jump misclassifications A Feasible Implementation based on ε 1 n (i) Get a rough" estimate of σ 2 via, e.g., the realized QV: ˆσ n,0 2 := 1 n X ti X ti 1 2 T i=1 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 16 / 35

28 via Expected number of jump misclassifications A Feasible Implementation based on ε 1 n (i) Get a rough" estimate of σ 2 via, e.g., the realized QV: ˆσ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use ˆσ n,0 2 to estimate the optimal threshold ˆε n,0 := (3 ˆσ 2 n,0 h n log(1/h n ) ) 1/2 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 16 / 35

29 via Expected number of jump misclassifications A Feasible Implementation based on ε 1 n (i) Get a rough" estimate of σ 2 via, e.g., the realized QV: ˆσ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use ˆσ n,0 2 to estimate the optimal threshold ˆε n,0 := (3 ˆσ 2 n,0 h n log(1/h n ) ) 1/2 (iii) Refine ˆσ n,0 2 using thresholding, ˆσ n,1 2 = 1 n X ti X ti T [ X ti X ti 1 ˆε n,0 i=1 ] J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 16 / 35

30 via Expected number of jump misclassifications A Feasible Implementation based on ε 1 n (i) Get a rough" estimate of σ 2 via, e.g., the realized QV: ˆσ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use ˆσ n,0 2 to estimate the optimal threshold ˆε n,0 := (3 ˆσ 2 n,0 h n log(1/h n ) ) 1/2 (iii) Refine ˆσ n,0 2 using thresholding, ˆσ n,1 2 = 1 n X ti X ti T [ X ti X ti 1 ˆε n,0 i=1 (iv) Iterate Steps (ii) and (iii): ˆσ 2 n,0 ˆε n,0 ˆσ2 n,1 ˆε n,1 ˆσ2 n,2 ˆσ2 n, ] J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 16 / 35

31 via Expected number of jump misclassifications Illustration I Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price Truncated Realized Variation (TRV) TRV True Volatility (0.4) Realized Variation (0.46) Bipower Variation (0.41) Cont. Realized Variation (0.402) Time in years (252 days) Truncation level (epsilon) Figure: (left) Merton Model with σ = 0.4, σ jmp = 3 h, µ jmp = 0, λ = 200; (right) TRV performance wrt the truncation level. Red dot is ˆσ n,1 = 0.409, while purple dot is the limiting estimator ˆσ n, = J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 17 / 35

32 via Expected number of jump misclassifications Illustration II Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price Truncated Realized Variation (TRV) TRV True Volatility (0.2) Realized Variation (0.33) Bipower Variation (0.24) Cont. Realized Variation (0.195) Time in years (252 days) Truncation level (epsilon) Figure: (left) Merton Model with σ = 0.2, σ jmp = 1.5 h, µ jmp = 0, λ = 1000; (right) TRV performance wrt the truncation level. Red dot is ˆσ n,1 = 0.336, while purple dot is the limiting estimator ˆσ n, = J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 18 / 35

33 via conditional Mean Square Error (cmse) Via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 19 / 35

34 via conditional Mean Square Error (cmse) Via conditional Mean Square Error (cmse) 1 We now propose a second approach in which we aim to control ( T 2 MSE c (ε) := E TRV n (ε) σsds) 2 0 σ, J. This is in the more general semimartingale setting: where J is a pure-jump process. t X t = X 0 + σ s dw s + J t 0 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 19 / 35

35 via conditional Mean Square Error (cmse) Via conditional Mean Square Error (cmse) 1 We now propose a second approach in which we aim to control ( T 2 MSE c (ε) := E TRV n (ε) σsds) 2 0 σ, J. This is in the more general semimartingale setting: where J is a pure-jump process. 2 Assumptions: t X t = X 0 + σ s dw s + J t 0 σ t > 0, t, and σ and J are independent of W. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 19 / 35

36 via conditional Mean Square Error (cmse) Key Relationships 1 Clearly, under the previous non-leverage assumption, i X σ, J independent N ( m i, σi 2 ), where m i := n i J = J t i J ti 1 and σ 2 i := t i t i 1 σ 2 s ds. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 20 / 35

37 via conditional Mean Square Error (cmse) Key Relationships 1 Clearly, under the previous non-leverage assumption, i X σ, J independent N ( m i, σi 2 ), where m i := n i J = J t i J ti 1 and σi 2 := t i t i 1 σs 2 ds. 2 Then, b i (ε) := E [ ( i X) 2 ] 1 { i X ε} σ, J J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 20 / 35

38 via conditional Mean Square Error (cmse) Key Relationships 1 Clearly, under the previous non-leverage assumption, i X σ, J independent N ( m i, σi 2 ), where m i := n i J = J t i J ti 1 and σi 2 := t i t i 1 σs 2 ds. 2 Then, b i (ε) := E [ ( i X) 2 ] 1 { i X ε} σ, J is such that b i (ε) = σ ( i 2π 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) e (ε mi ) 2 ε+m i + (mi 2 + σi 2 σ i ) ε m i σ i e x 2 /2 dx 2π J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 20 / 35

39 via conditional Mean Square Error (cmse) Key Relationships 1 Clearly, under the previous non-leverage assumption, i X σ, J independent N ( m i, σi 2 ), where m i := n i J = J t i J ti 1 and σi 2 := t i t i 1 σs 2 ds. 2 Then, b i (ε) := E [ ( i X) 2 ] 1 { i X ε} σ, J is such that b i (ε) = σ ( i 2π 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) e (ε mi ) 2 ε+m i + (mi 2 + σi 2 σ i ) ε m i σ i e x 2 /2 dx 2π db i (ε) dε = J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 20 / 35

40 via conditional Mean Square Error (cmse) Key Relationships 1 Clearly, under the previous non-leverage assumption, i X σ, J independent N ( m i, σi 2 ), where m i := n i J = J t i J ti 1 and σi 2 := t i t i 1 σs 2 ds. 2 Then, b i (ε) := E [ ( i X) 2 ] 1 { i X ε} σ, J is such that b i (ε) = σ ( i 2π 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) e (ε mi ) 2 ε+m i + (mi 2 + σi 2 σ i ) ε m i σ i e x 2 /2 dx 2π (ε m i ) 2 db i (ε) = ε 2 a i (ε), with a i (ε) := e 2σ i 2 + e (ε+m i ) 2σ i 2 dε σ i 2π J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 20 / 35 2

41 via conditional Mean Square Error (cmse) Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini (2017)) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 21 / 35

42 via conditional Mean Square Error (cmse) Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini (2017)) 1 Let IV := T 0 σ2 sds be the integrated variance of X and let MSE c (ε) := E [(TRV n (ε) IV ) 2 ] σ, J J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 21 / 35

43 via conditional Mean Square Error (cmse) Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini (2017)) 1 Let IV := T 0 σ2 sds be the integrated variance of X and let MSE c (ε) := E [(TRV n (ε) IV ) 2 ] σ, J 2 Then, MSE c (ε) is differential in (0, ) and, furthermore, d MSE c (ε) = ε 2 G(ε), dε where G(ε) := ( i a i(ε) ε ) j i b j(ε) 2IV. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 21 / 35

44 via conditional Mean Square Error (cmse) Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini (2017)) 1 Let IV := T 0 σ2 sds be the integrated variance of X and let MSE c (ε) := E [(TRV n (ε) IV ) 2 ] σ, J 2 Then, MSE c (ε) is differential in (0, ) and, furthermore, d MSE c (ε) = ε 2 G(ε), dε where G(ε) := ( i a i(ε) ε ) j i b j(ε) 2IV. 2 Furthermore, there exists an optimal threshold ε n MSE c (ε) and is such that G(ε n ) = 0. that minimizes J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 21 / 35

45 via conditional Mean Square Error (cmse) Asymptotics: FA jumps with constant variance Theorem (F-L & Mancini (2017)) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 22 / 35

46 via conditional Mean Square Error (cmse) Asymptotics: FA jumps with constant variance Theorem (F-L & Mancini (2017)) 1 Suppose that σ t σ is constant and J is a finite jump activity process (with or without drift; not necessarily Lévy): N t X t = σw t + j=1 ζ j J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 22 / 35

47 via conditional Mean Square Error (cmse) Asymptotics: FA jumps with constant variance Theorem (F-L & Mancini (2017)) 1 Suppose that σ t σ is constant and J is a finite jump activity process (with or without drift; not necessarily Lévy): N t X t = σw t + 2 Then, as n, the optimal threshold ε n is such that ( ) 1 2σ 2 h n log ε n j=1 ζ j h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 22 / 35

48 via conditional Mean Square Error (cmse) A Feasible Implementation of ε n (i) Get a rough" estimate of σ 2 via the realized QV: σ n,0 2 := 1 n X ti X ti 1 2 T i=1 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 23 / 35

49 via conditional Mean Square Error (cmse) A Feasible Implementation of ε n (i) Get a rough" estimate of σ 2 via the realized QV: σ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use σ n,0 2 to estimate the optimal threshold ˆε n,0 := (2 σ 2 n,0 h n log(1/h n ) ) 1/2 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 23 / 35

50 via conditional Mean Square Error (cmse) A Feasible Implementation of ε n (i) Get a rough" estimate of σ 2 via the realized QV: σ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use σ n,0 2 to estimate the optimal threshold ˆε n,0 := (2 σ 2 n,0 h n log(1/h n ) ) 1/2 (iii) Refine σ n,0 2 using thresholding, σ n,1 2 = 1 n X ti X ti T [ ] X ti X ti 1 ˆε n,0 i=1 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 23 / 35

51 via conditional Mean Square Error (cmse) A Feasible Implementation of ε n (i) Get a rough" estimate of σ 2 via the realized QV: σ n,0 2 := 1 n X ti X ti 1 2 T i=1 (ii) Use σ n,0 2 to estimate the optimal threshold ˆε n,0 := (2 σ 2 n,0 h n log(1/h n ) ) 1/2 (iii) Refine σ n,0 2 using thresholding, σ n,1 2 = 1 n X ti X ti T [ ] X ti X ti 1 ˆε n,0 i=1 (iv) Iterate Steps (ii) and (iii): σ 2 n,0 ˆε n,0 σ2 n,1 ˆε n,1 σ2 n,2 σ2 n, J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 23 / 35

52 via conditional Mean Square Error (cmse) Illustration II. Continued... Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price Truncated Realized Variation (TRV) TRV True Volatility (0.2) Realized Variation (0.33) Bipower Variation (0.24) Cont. Realized Variation (0.195) Time in years (252 days) Truncation level (epsilon) Figure: (left) Merton Model with λ = Red dot is ˆσ n,1 = 0.336, while purple dot is the limiting ˆσ n,k = Orange square is σ n,1 = 0.225, while brown square is the limiting estimator σ n, = J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 24 / 35

53 via conditional Mean Square Error (cmse) Monte Carlo Simulations Estimator ˆσ std(ˆσ) Loss ε N RQV ˆσ n, ˆσ n, σ n, σ n, BPV Table: Estimation of the volatility σ = 0.2 for a log-normal Merton model based on simulations of 5-minute observations over a 1 month time horizon. The jump parameters are λ = 1000, σ Jmp = 1.5 h and µ Jmp = 0. Loss is the number of jump misclassifications and N is the number of iterations. bar is used to denote average. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 25 / 35

54 via conditional Mean Square Error (cmse) Heuristics of Proof I: FA jumps with constant variance Recall that the optimum threshold ε n G(ε) := i ( a i (ε) ε j i is a solution of the equation: ) b j (ε) 2IV } {{ } g i (ε) = 0 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 26 / 35

55 via conditional Mean Square Error (cmse) Heuristics of Proof I: FA jumps with constant variance Recall that the optimum threshold ε n G(ε) := i ( a i (ε) ε j i is a solution of the equation: ) b j (ε) 2IV } {{ } g i (ε) = 0 It can be shown that, as n, σ i 2 2σ i εe ε 2 2σ i 2 + h.o.t., if m 2π i = 0, b i (ε) = σ i m i 2π ε2 e ( m i ε) 2 2σ i 2 + h.o.t., if m i 0. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 26 / 35

56 via conditional Mean Square Error (cmse) Heuristics of Proof II: FA jumps with constant variance These in turn allows to show that g i (ε) := ε n b j (ε) 2 σj 2 j i j=1 = ε [ ] b j (ε) σj = ε 2 4ε = ε 2 4ε j i:m j =0 j i:m j =0 σ j e ε 2 2σ j 2 2π n σ j e ε 2π j=1 2 2σ 2 j + h.o.t. + h.o.t. j i:m j 0 b j (ε) 2 j:m j 0 σ 2 j J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 27 / 35

57 via conditional Mean Square Error (cmse) Heuristics of Proof III: FA jumps with constant variance Then, up to higher-order terms, ε n ε 4 is a solution of the equation: n σ j e ε 2 2σ j 2 = 0. 2π j=1 In the case of constant σ, we have: ε n 4n σ h n e ε n 2 2σ 2 hn + h.o.t. = 0, 2π which implies ε n 2σ 2 h n ln(1/h n ), n. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 28 / 35

58 via conditional Mean Square Error (cmse) Generalization I What if t σ t is not constant? J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 29 / 35

59 via conditional Mean Square Error (cmse) Generalization I What if t σ t is not constant? By the mean value theorem, σj 2 ε n = ε n is such that := t j t j 1 σ 2 sds = σ 2 s j h n and, thus, J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 29 / 35

60 via conditional Mean Square Error (cmse) Generalization I What if t σ t is not constant? By the mean value theorem, σj 2 ε n = ε n is such that := t j t j 1 σ 2 sds = σ 2 s j h n and, thus, where w n := ε 2 n/h n. 0 = ε n 4 n σ j e ε 2π j=1 = ε n 4 1 hn n j=1 = ε n 4 hn T 0 2 n 2σ j 2 σ sj 2π e + h.o.t. ε2 n 2σ 2 s j hn h n + h.o.t. σ s 2π e w n 2σ 2 s ds + h.o.t., J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 29 / 35

61 via conditional Mean Square Error (cmse) Generalization II It can be shown that w n = ε 2 n/h n. Then, by the Laplace method, J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 30 / 35

62 via conditional Mean Square Error (cmse) Generalization II It can be shown that w n = ε 2 n/h n. Then, by the Laplace method, T where σ s0 := max s [0,T ] σ s. 0 σ s e w n 2σs 2 ds σ5/2 s σ 0 2π s 0 1 e wn wn 2σ 2 s 0 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 30 / 35

63 via conditional Mean Square Error (cmse) Generalization II It can be shown that w n = ε 2 n/h n. Then, by the Laplace method, T where σ s0 := max s [0,T ] σ s. 0 σ s e w n 2σs 2 ds σ5/2 s σ 0 2π s 0 1 e wn wn Then, for some constant K and σ := max s [0,T ] σ s, which again implies ε n 0 = ε n K 1 e ε 2 n 2 σ 2 hn ε n + h.o.t., 2 σ 2 h n ln(1/h n ), n. 2σ 2 s 0 J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 30 / 35

64 Ongoing and Future Research Outline 1 Motivational Example 2 The Problem Multipower Variations and Truncated Realized Variations The problem of Parameter Tuning 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Ongoing and Future Research J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 31 / 35

65 Ongoing and Future Research Ongoing and Future Research 1 In principle, we can apply the constant-volatility method for varying volatility t σ t by localization; i.e., applying it to periods where σ is approximately constant. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 32 / 35

66 Ongoing and Future Research Ongoing and Future Research 1 In principle, we can apply the constant-volatility method for varying volatility t σ t by localization; i.e., applying it to periods where σ is approximately constant. 2 This could also help us to estimate σ = max s [0,T ] σ s and, then, apply the asymptotic equivalence ε n 2 σ 2 h n ln(1/h n ). J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 32 / 35

67 Ongoing and Future Research Ongoing and Future Research 3 As it turns, for a Lévy process J and constant σ, the expected [ mean square error, MSE(ε) := E (TRV n (ε) IV ) 2], is such that d dε MSE(ε) = nε2 E[a 1 (ε)] (ε 2 + 2(n 1)E[b 1 (ε)] 2nh n σ 2) J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 33 / 35

68 Ongoing and Future Research Ongoing and Future Research 3 As it turns, for a Lévy process J and constant σ, the expected [ mean square error, MSE(ε) := E (TRV n (ε) IV ) 2], is such that d dε MSE(ε) = nε2 E[a 1 (ε)] (ε 2 + 2(n 1)E[b 1 (ε)] 2nh n σ 2) Therefore, there exists a unique minimum point ε n which is the solution of the equation ε 2 + 2(n 1)E[b 1 (ε)] 2nh n σ 2 = 0. J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 33 / 35

69 Ongoing and Future Research Ongoing and Future Research 3 Furthermore, J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 34 / 35

70 Ongoing and Future Research Ongoing and Future Research 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 ε n 2σ 2 h n log h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 34 / 35

71 Ongoing and Future Research Ongoing and Future Research 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 ε n 2σ 2 h n log But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 ε n (2 Y )σ 2 h n log Thus the higher the jump activity is, the lower the optimal threshold has to be to discard the higher noise represented by the jumps. h n h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 34 / 35

72 Ongoing and Future Research Ongoing and Future Research 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 ε n 2σ 2 h n log But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 ε n (2 Y )σ 2 h n log Thus the higher the jump activity is, the lower the optimal threshold has to be to discard the higher noise represented by the jumps. Does this phenomenon holds for the minimizer ε n of the cmse? h n h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 34 / 35

73 Ongoing and Future Research Ongoing and Future Research 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 ε n 2σ 2 h n log But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 ε n (2 Y )σ 2 h n log Thus the higher the jump activity is, the lower the optimal threshold has to be to discard the higher noise represented by the jumps. Does this phenomenon holds for the minimizer ε n of the cmse? Can we generalize it to Lévy processes with stable like jumps? h n h n J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 34 / 35

74 Ongoing and Future Research Further Reading J.E. Figueroa-López & C. Mancini. Optimum thresholding using mean and conditional mean square error. Available at J.E. Figueroa-López & J. Nisen. Optimally Thresholded Realized Power Variations for Lévy Jump Diffusion Models. Stochastic Processes and their Applications 123(7), , J.E. Figueroa-López, C. Li, & J. Nisen. Optimal iterative threshold-kernel estimation of jump diffusion processes. In preparation, J.E. Figueroa-López (WUSTL) Optimum Thresholding AMS Sectional Meeting 35 / 35