Asymptotic Methods in Financial Mathematics

Asymptotic Methods in Financial Mathematics José E. Figueroa-López 1 1 Department of Mathematics Washington University in St. Louis Statistics Seminar Washington University in St. Louis February 17, 2017 (Joint work with Cecilia Mancini) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 1 / 29

Outline 1 Motivational Example 2 The Main Estimators Multipower Variations and Truncated Realized Variations 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Conclusions J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 2 / 29

Motivational Example Outline 1 Motivational Example 2 The Main Estimators Multipower Variations and Truncated Realized Variations 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Conclusions J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 3 / 29

Motivational Example Merton Log-Normal Model Consider the following model for the log-return process X t = log S t S 0 financial asset: X t = at + σw t + i:τ i t ζ i, ζ i i.i.d. N(µ jmp, σ 2 jmp ), Goal: Estimate σ based on a discrete record X t1,..., X tn. Log-Normal Merton Model of a {τ i} i 1 Poisson(λ) Stock Price 1.0 1.1 1.2 1.3 Price process Continuous process Times of jumps 0.00 0.02 0.04 0.06 0.08 Time in years (252 days) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 4 / 29

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 h = h n = T n, t i = t i,n = ih n, i = 0,..., n. 2 Generate i.i.d. i.i.d. Z i N(0, 1), γ i N(µ jmp, σjmp 2 ), and I i.i.d. i Bernoulli(λh) 3 Iteratively generate the process: X ti = X ti 1 + ah + σ hz i + I i γ i J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 5 / 29

The Main Estimators Outline 1 Motivational Example 2 The Main Estimators Multipower Variations and Truncated Realized Variations 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Conclusions J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 6 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: QV (X) n := 1 T n 1 ( Xti+1 X ) 2 t i i=0 QV (X) n n σ 2 + j:τ j T ζ2 j J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 7 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: QV (X) n := 1 T QV (X) n n 1 ( Xti+1 X ) 2 t i i=0 n σ 2 + j:τ j T ζ2 j 1 Multipower 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 7 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: QV (X) n := 1 T QV (X) n n 1 ( Xti+1 X ) 2 t i i=0 n σ 2 + j:τ j T ζ2 j 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard): n k MPV [X] (r 1,...,r k ) n := Xti+1 X ti r 1... X ti+k X ti+k 1 r k. i=0 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 7 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Two main classes of estimators Precursor. Realized Quadratic Variation: QV (X) n := 1 T QV (X) n n 1 ( Xti+1 X ) 2 t i i=0 n σ 2 + j:τ j T ζ2 j 1 Multipower Realized Variations (Barndorff-Nielsen and Shephard): n k MPV [X] (r 1,...,r k ) n := Xti+1 X ti r 1... X ti+k X ti+k 1 r k. 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 7 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations 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 0.85 0.90 0.95 1.00 Price process Continuous process Times of jumps Truncated Realized Variation (TRV) 0.0 0.1 0.2 0.3 0.4 0.5 TRV True Volatility (0.4) Realized Variation (0.51) Bipower Variation (0.42) 0.00 0.02 0.04 0.06 0.08 0.00 0.01 0.02 0.03 0.04 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 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 8 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations 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 0.80 0.85 0.90 0.95 1.00 Price process Continuous process Times of jumps Truncated Realized Variation (TRV) 0.0 0.1 0.2 0.3 0.4 TRV True Volatility (0.2) Realized Variation (0.4) Bipower Variation (0.26) Cont. Realized Variation (0.19) 0.00 0.02 0.04 0.06 0.08 0.000 0.005 0.010 0.015 0.020 0.025 0.030 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 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 9 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Popular truncations ε 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. Power Threshold (Mancini (2003)) ε Prw α,ω := α h ω, for α > 0 and ω (0, 1/2). J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 10 / 29

The Main Estimators Multipower Variations and Truncated Realized Variations Popular truncations ε 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. Power Threshold (Mancini (2003)) ε Prw α,ω := α h ω, for α > 0 and ω (0, 1/2). Bonferroni Threshold (Bollerslev et al. (2007) and Gegler & Stadtmüller (2010)) ( B σ,c BF := σh1/2 Φ 1 1 C h ), for C > 0 and σ > 0. 2 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 10 / 29

Classical Approach 1 Fix a suitable and sensible metric for estimation error; typically, the MSE: MSE(ε) = E [ ( ) ] 1 2 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 12 / 29

via Expected number of jump misclassifications Expected number of jump misclassifications 1 Notation: 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 ] 2 Estimation Error: (F-L & Nisen, 2013) n ) Loss n (ε) := E (1 [ ni X >ε, ni N=0] + 1 [ ni X ε, ni N 0]. 3 Motivation i=1 The underlying principle is that the estimation error is directly link to the ability of the threshold to detect jumps. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 13 / 29

via Expected number of jump misclassifications Existence and Infill Asymptotic Characterization Theorem (FL & Nisen (SPA, 2013)) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 14 / 29

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 convex and, moreover, possesses a unique global minimum ε n. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 14 / 29

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 convex and, moreover, possesses a unique global minimum ε n. 2 As n, the optimal threshold sequence (ε n) n is such that ( ) ( ) 1 log 2πσλC(fζ ) σh 1/2 ε n = 3σ 2 n h n log + h.o.t., h n 3 log(1/hn ) 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 (h.o.t. refers to high order terms). J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 14 / 29

via Expected number of jump misclassifications Remarks 1 The threshold sequences ε 1 n := ( 1 3σ 2 h n log h n ), ε 2 n := ε 1 n ( ) log 2πσλC(fζ ) 3 log(1/hn ) σh 1/2 n are the first and second-order approximations for B n, and the biases of their corresponding TRV estimators attain the optimal rate of O(h n ) as n., J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 15 / 29

via Expected number of jump misclassifications Remarks 1 The threshold sequences ε 1 n := ( 1 3σ 2 h n log h n ), ε 2 n := ε 1 n ( ) log 2πσλC(fζ ) 3 log(1/hn ) σh 1/2 n are the first and second-order approximations for B n, and 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!, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 15 / 29

via Expected number of jump misclassifications A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., realized QV: ˆσ n,0 2 := 1 n X ti X ti 1 2 T i=1 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 16 / 29

via Expected number of jump misclassifications A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 16 / 29

via Expected number of jump misclassifications A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., 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 1 2 1 T [ X ti X ti 1 ˆε n,0 i=1 ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 16 / 29

via Expected number of jump misclassifications A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., 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 1 2 1 T [ X ti X ti 1 ˆε n,0 i=1 (iv) Iterate Steps (ii) and (iii): ˆσ n,0 2 ˆε n,0 ˆσ2 n,1 ˆε n,1 ˆσ2 n,2 ˆσ2 n, ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 16 / 29

via Expected number of jump misclassifications Illustration I Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price 0.85 0.90 0.95 1.00 1.05 Truncated Realized Variation (TRV) 0.0 0.1 0.2 0.3 0.4 TRV True Volatility (0.4) Realized Variation (0.46) Bipower Variation (0.41) Cont. Realized Variation (0.402) 0.00 0.02 0.04 0.06 0.08 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 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, = 0.405 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 17 / 29

via Expected number of jump misclassifications Illustration II Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price 0.96 1.00 1.04 Truncated Realized Variation (TRV) 0.00 0.10 0.20 0.30 TRV True Volatility (0.2) Realized Variation (0.33) Bipower Variation (0.24) Cont. Realized Variation (0.195) 0.00 0.02 0.04 0.06 0.08 Time in years (252 days) 0.00 0.01 0.02 0.03 0.04 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, = 0.215 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 18 / 29

via conditional Mean Square Error (cmse) Setting 1 Framework: t X t = X 0 + σ s dw s + J t 0 where J is a pure-jump process. 2 Key Assumptions: σ t > 0 for all t, and σ and J are independent of W. 3 Notation: m i = n i J = J t i J ti 1, σ 2 i = ti t i 1 σ 2 sds 4 Key Observation: Conditioning on σ and J, the increments n i X are i.i.d. normal with mean m i and variance σ 2 i J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 19 / 29

via conditional Mean Square Error (cmse) Key Relationships Let b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 20 / 29

via conditional Mean Square Error (cmse) Key Relationships Let b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] 1 As it turns out, b i (ε) = (e (ε m i ) 2 2σ i 2 (ε + m i ) + e (ε+m i ) 2 2σ i 2 σi (ε m i )) 2π + m2 i + σi 2 π ( ε mi 2σi 0 e t 2 ε+m i 2σi ) dt + e t 2 dt 0 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 20 / 29

via conditional Mean Square Error (cmse) Key Relationships Let b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] 1 As it turns out, b i (ε) = (e (ε m i ) 2 2σ i 2 (ε + m i ) + e (ε+m i ) 2 2σ i 2 σi (ε m i )) 2π + m2 i + σi 2 ( ε mi 2σi e t 2 ε+m i 2σi ) dt + e t 2 dt π 0 0 2 Furthermore, db i (ε) dε = ε 2 a i (ε), with a i (ε) := e (ε m i ) 2 2σ 2 i + e (ε+m i ) σ i 2π 2σ 2 i 2 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 20 / 29

via conditional Mean Square Error (cmse) Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini (2017)) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 21 / 29

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 2 + 2 ) j i b j(ε) 2IV. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 21 / 29

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 2 + 2 ) 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 21 / 29

via conditional Mean Square Error (cmse) Asymptotics: FA process with constant variance Theorem (F-L & Mancini (2017)) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 22 / 29

via conditional Mean Square Error (cmse) Asymptotics: FA process 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 22 / 29

via conditional Mean Square Error (cmse) Asymptotics: FA process 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 22 / 29

via conditional Mean Square Error (cmse) A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., the QV: σ n,0 2 := 1 n X ti X ti 1 2 T i=1 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 23 / 29

via conditional Mean Square Error (cmse) A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., the 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 23 / 29

via conditional Mean Square Error (cmse) A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., the 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 1 2 1 T [ ] X ti X ti 1 ˆε n,0 i=1 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 23 / 29

via conditional Mean Square Error (cmse) A Feasible Implementation based on ε n (i) Get a rough" estimate of σ 2 via, e.g., the 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 1 2 1 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 23 / 29

via conditional Mean Square Error (cmse) Illustration II. Continued... Log-Normal Merton Model Performace of Truncated Realized Variations Stock Price 0.96 1.00 1.04 Truncated Realized Variation (TRV) 0.00 0.10 0.20 0.30 TRV True Volatility (0.2) Realized Variation (0.33) Bipower Variation (0.24) Cont. Realized Variation (0.195) 0.00 0.02 0.04 0.06 0.08 Time in years (252 days) 0.00 0.01 0.02 0.03 0.04 Truncation level (epsilon) Figure: (left) Merton Model with λ = 1000. Red dot is ˆσ n,1 = 0.336, while purple dot is the limiting ˆσ n,k = 0.215. Orange square is σ n,1 = 0.225, while brown square is the limiting estimator σ n, = 0.199 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 24 / 29

via conditional Mean Square Error (cmse) Monte Carlo Simulations Estimator ˆσ std(ˆσ) Loss ε N RQV 0.3921 0.0279 ˆσ n,1 0.29618 0.02148 70.1 0.01523 1 ˆσ n, 0.23 0.0108 49.8 0.00892 5.86 σ n,1 0.265 0.0163 62.6 0.0124 1 σ n, 0.211 0.00588 39.1 0.00671 5.10 BPV 0.2664 0.0129 Table: Estimation of the volatility σ = 0.2 for a log-normal Merton model based on 10000 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 25 / 29

Conclusions Outline 1 Motivational Example 2 The Main Estimators Multipower Variations and Truncated Realized Variations 3 Optimal Threshold Selection via Expected number of jump misclassifications via conditional Mean Square Error (cmse) 4 Conclusions J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 26 / 29

Conclusions Ongoing and Future Research 1 In principle, we can apply the proposed method for varying volatility t σ t by localization; i.e., applying to periods where σ is approximately constant. 2 However, is it possible to analyze the asymptotic behavior of MSE c = ε 2 G(ε) in terms of certain estimable summary measures? Say, G(ε) G 0 (ε, m 1,..., m n, σ, σ), where σ := inf t T σ t and σ := sup t T σ t J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 27 / 29

Conclusions Ongoing and Future Research 3 As it turns, for a Lévy process J and constant σ, the expected [ 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 28 / 29

Conclusions Ongoing and Future Research 3 As it turns, for a Lévy process J and constant σ, the expected [ 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 a solution of the equation ε 2 + 2(n 1)E[b 1 (ε)] 2nh n σ 2 = 0. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 28 / 29

Conclusions Ongoing and Future Research 3 Furthermore, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 29 / 29

Conclusions 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) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 29 / 29

Conclusions 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, in order to catch information about IV. Does this phenomenon holds for the minimizer ε n of the cmse? h n h n J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics WUSTL Statistics Seminar 29 / 29