Short-Time Asymptotic Methods in Financial Mathematics

Transcription

1 Short-Time Asymptotic Methods in Financial Mathematics José E. Figueroa-López Department of Mathematics Washington University in St. Louis Probability and Mathematical Finance Seminar Department of Mathematical Sciences Carnegie Mellon University February 20, 2017 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 1 / 39

2 Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 2 / 39

3 Short-time asymptotics of Option Prices Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 3 / 39

4 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 4 / 39

5 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics Short-time ATM Skew Asymptotics Theorem (F-L & Ólafsson, F&S 2016) For Y (1, 2), the implied volatility ˆσ (κ, t) of a vanilla option with log-moneyness κ and time-to-maturity t is such that d 1 t 1 2, if pure jump model with C + C, ˆσ (κ, t) d 1 t Y, if pure jump model with C + = C, κ κ=0 d 1 t 1 Y 2, if mixed model with C + C, d 1, if mixed model with C + = C. Furthermore, in the asymmetric cases, the sign of d 1 is the same as that of C + C. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 5 / 39

6 Short-time asymptotics of Option Prices A Calibration Case-Study Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 6 / 39

7 Short-time asymptotics of Option Prices A Calibration Case-Study Calibration of S&P 500 Option Prices Objectives: To assess the plausibility of the divergence of the skew to as a power of time-to-maturity. Calibrate the parameter Y, which measures the activity of small jumps in the price process. Data: Daily closing bid and ask prices for S&P 500 index options, across all strikes K, and maturities t from Jan. 2 to Jun. 30, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 7 / 39

8 Short-time asymptotics of Option Prices A Calibration Case-Study Implied Volatility Smile and ATM Skew (a) 0.14 (b) ATM skew -6-8 Implied vol Time-to-maturity (t) log-moneyness (κ) Figure: (a) The graphs of the ATM implied volatility skew are consistent with divergence to as a power law. (b) The implied volatility smiles on Jan. 15, 2014, corresponding to maturities ranging from (8 days) to 0.25 (3 months), show greater steepness as time-to-maturity decreases, which is consistent with the presence of jumps. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 8 / 39

9 Short-time asymptotics of Option Prices A Calibration Case-Study Calibration of the Index Of Jump Activity Y (a) 1-day regression 21-day regression (b) Y pj 1.5 Y cc day regression 21-day regression 1 Jan Feb Mar Apr May Jun Day 1 Jan Feb Mar Apr May Jun Day Figure: The Y -estimates implied by daily and monthly regressions of on log t assuming a pure-jump model (panel (a)), and a mixed κ=0 log ˆσ(κ,t) κ model (panel (b)). The first and third quantiles of the regressions R 2 are and 0.995, respectively. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 9 / 39

10 Short-time asymptotics of Option Prices A Calibration Case-Study ATM Implied Volatility (a) (b) VIX ATM vol. ATM implied vol Volatility Time-to-maturity (t) 0.06 Jan Feb Mar Apr May Jun Day Figure: (a) ATM implied volatility as a function of time-to-maturity in years, for each business day in Jan Jun, (b) ATM implied volatility for the shortest outstanding maturity compared to the VIX index. These suggest a mixed model rather than a pure jump model since ˆσ(t, 0) σ 0 > 0. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 10 / 39

11 Short-time asymptotics of Option Prices Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 11 / 39

12 Short-time asymptotics of Option Prices 1 Besides the skew, the convexity of the implied volatility, κ=0, is very important in trading and, thus, its asymptotic 2ˆσ(κ,t) κ 2 behavior is of great interest. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 12 / 39

13 Short-time asymptotics of Option Prices 1 Besides the skew, the convexity of the implied volatility, κ=0, is very important in trading and, thus, its asymptotic 2ˆσ(κ,t) κ 2 behavior is of great interest. 2 Small-time asymptotics of options written on Leveraged Exchange-Traded Fund (LETF). LETF is a managed portfolio, which seeks to multiply the instantaneous returns of a reference Exchange-Traded Fund (ETF). J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 12 / 39

14 Short-time asymptotics of Option Prices 1 Besides the skew, the convexity of the implied volatility, κ=0, is very important in trading and, thus, its asymptotic 2ˆσ(κ,t) κ 2 behavior is of great interest. 2 Small-time asymptotics of options written on Leveraged Exchange-Traded Fund (LETF). LETF is a managed portfolio, which seeks to multiply the instantaneous returns of a reference Exchange-Traded Fund (ETF). 3 There is no known results for American options. It is expected that the early exercise feature won t affect the leading order terms of the asymptotics but they should contribute to the high-order terms. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 12 / 39

15 Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 13 / 39

16 Merton 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 Log-Normal Merton Model Stock Price Price process Continuous process Times of jumps Time in years (252 days) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 14 / 39

17 Multipower Variations and Truncated Realized Variations Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 15 / 39

18 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 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 16 / 39

19 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 n t i σ 2 + i=0 j:τ j T ζ 2 j J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 16 / 39

20 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 n t i σ 2 + i=0 j:τ j T 1 Bipower Realized Variations (Barndorff-Nielsen and Shephard): n 1 BPV [X] n := Xti+1 X ti Xti+2 X ti+1, i=0 ζ 2 j J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 16 / 39

21 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 n t i σ 2 + i=0 j:τ j T 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 ζ 2 j }, (ε [0, )). J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 16 / 39

22 Multipower Variations and Truncated Realized Variations Calibration or Tuning of the Estimator Problem: How do you choose the threshold parameter ε? J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 17 / 39

23 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 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 estimates for all the truncation levels. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 17 / 39

24 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 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) Asymptotics in Financial Mathematics CCF Seminar 18 / 39

25 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. 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) Asymptotics in Financial Mathematics CCF Seminar 19 / 39

26 Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 20 / 39

27 General Approach 1 Fix a suitable and sensible metric for 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 Characterize 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 implementation of ε by estimating those parameters (if possible). J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 21 / 39

28 Via Expected number of jump misclassifications J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 22 / 39

29 Via Expected number of jump misclassifications 1 Estimation Error: (F-L & Nisen, SPA 2013) 2 Notation: Loss n (ε) := E n i=1 N t := # of jumps by time t n i X := X t i X ti 1 (1 [ ni X >ε, ni N=0] + 1 [ ni X ε, ni N 0] ). n i N := N t i N ti 1 = # of jumps during (t i 1, t i ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 22 / 39

30 Existence and Infill Asymptotic Characterization Theorem (F-L & Nisen, SPA 2013)) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 23 / 39

31 Existence and Infill Asymptotic Characterization Theorem (F-L & Nisen, SPA 2013)) 1 For n large enough, the loss function Loss n (ε) is convex and, moreover, enjoys a unique global minimum ε n. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 23 / 39

32 Existence and Infill Asymptotic Characterization Theorem (F-L & Nisen, SPA 2013)) 1 For n large enough, the loss function Loss n (ε) is convex and, moreover, enjoys 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., where h n := t i t i 1 is the time step. h n J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 23 / 39

33 Existence and Infill Asymptotic Characterization Theorem (F-L & Nisen, SPA 2013)) 1 For n large enough, the loss function Loss n (ε) is convex and, moreover, enjoys 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., where h n := t i t i 1 is the time step. Remark: The leading order term is proportional to the Lévy s modulus of continuity of the Brownian motion!. h n J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 23 / 39

34 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 CCF Seminar 24 / 39

35 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 CCF Seminar 24 / 39

36 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 T [ X ti X ti 1 ˆε n,0 i=1 ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 24 / 39

37 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 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 CCF Seminar 24 / 39

38 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) Asymptotics in Financial Mathematics CCF Seminar 25 / 39

39 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) Asymptotics in Financial Mathematics CCF Seminar 26 / 39

40 Via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 27 / 39

41 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) Asymptotics in Financial Mathematics CCF Seminar 27 / 39

42 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 Key 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) Asymptotics in Financial Mathematics CCF Seminar 27 / 39

43 Key Relationships b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 28 / 39

44 Key Relationships b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] Then, with the notation m i := n i J = J ti J ti 1, σi 2 := ti t i 1 σ 2 s ds, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 28 / 39

45 Key Relationships Then, with the notation b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] we have m i := n i J = J ti J ti 1, σ 2 i := b i (ε) = σ ( i 2π e (ε mi ) 2 ti t i 1 σ 2 s ds, 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) 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) Asymptotics in Financial Mathematics CCF Seminar 28 / 39

46 Key Relationships Then, with the notation b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] we have m i := n i J = J ti J ti 1, σ 2 i := b i (ε) = σ ( i 2π e (ε mi ) 2 ti t i 1 σ 2 s ds, 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) 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) Asymptotics in Financial Mathematics CCF Seminar 28 / 39

47 Key Relationships Then, with the notation b i (ε) := E [ ( i X) 2 1 { i X ε} σ, J ] we have m i := n i J = J ti J ti 1, σ 2 i := b i (ε) = σ ( i 2π e (ε mi ) 2 ti t i 1 σ 2 s ds, 2σ i 2 (ε + m i ) + e (ε+m i ) 2 ) 2σ i 2 (ε m i ) db i (ε) dε ε+m i + (mi 2 + σi 2 σ i ) ε m i σ i e x 2 /2 dx 2π = 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 CCF Seminar 28 / 39

48 Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini, 2017) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 29 / 39

49 Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini, 2017) 1 MSE c (ε) := E [(TRV n (ε) IV ) 2 ] σ, J is such that d MSE c (ε) = ε 2 G(ε), dε where G(ε) := ( i a i(ε) ε j i b j(ε) 2 ) j σ2 j. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 29 / 39

50 Conditional Mean Square Error (MSE c ) Theorem (F-L & Mancini, 2017) 1 MSE c (ε) := E [(TRV n (ε) IV ) 2 ] σ, J is such that d MSE c (ε) = ε 2 G(ε), dε where G(ε) := ( i a i(ε) ε j i b j(ε) 2 ) j σ2 j. 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 CCF Seminar 29 / 39

51 Asymptotics: FA process with constant variance Theorem (F-L & Mancini, 2017) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 30 / 39

52 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 CCF Seminar 30 / 39

53 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 2 Then, as n, the optimal threshold ε n cmse is such that ε n ( 1 2σ 2 h n log h n ζ j that minimizes the ), h n 0. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 30 / 39

54 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 CCF Seminar 31 / 39

55 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 CCF Seminar 31 / 39

56 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 T [ ] X ti X ti 1 ˆε n,0 i=1 J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 31 / 39

57 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 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 CCF Seminar 31 / 39

58 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) Asymptotics in Financial Mathematics CCF Seminar 32 / 39

59 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. J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 33 / 39

60 Outline 1 Short-time asymptotics of Option Prices Short-time ATM Skew Asymptotics A Calibration Case-Study 2 High-Frequency Based Estimation Methods Multipower Variations and Truncated Realized Variations via Expected number of jump misclassifications via conditional Mean Square Error (cmse) J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 34 / 39

61 1 In principle, we can apply the proposed 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) Asymptotics in Financial Mathematics CCF Seminar 35 / 39

62 1 In principle, we can apply the proposed method for varying volatility t σ t by localization; i.e., applying it 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 CCF Seminar 35 / 39

63 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 CCF Seminar 36 / 39

64 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 solution of the equation ε 2 + 2(n 1)E[b 1 (ε)] 2nh n σ 2 = 0. which is a J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 36 / 39

65 3 Furthermore, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 37 / 39

66 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 2σ 2 h n log ε n h n J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 37 / 39

67 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 2σ 2 h n log ε n But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 (2 Y )σ 2 h n log ε n 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) Asymptotics in Financial Mathematics CCF Seminar 37 / 39

68 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 2σ 2 h n log ε n But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 (2 Y )σ 2 h n log ε n 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 h n h n of the cmse? J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 37 / 39

69 3 Furthermore, As expected, in the finite jump activity case, it turns out ( ) 1 2σ 2 h n log ε n But, surprisingly, if J is a Y -stable Lévy process (IA), ( ) 1 (2 Y )σ 2 h n log ε n 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) Asymptotics in Financial Mathematics CCF Seminar 37 / 39

70 Further Reading I J.E. Figueroa-López, R. Gong, and C. Houdré, High-order short-time expansions for ATM option prices of exponential Lévy models. Mathematical Finance, 26(3), , J.E. Figueroa-López and S. Ólafsson, Short-time expansions for close-to-the-money options under a Lévy jump model with stochastic volatility. Finance & Stochastics 20(1), , J.E. Figueroa-López and S. Ólafsson, Short-time asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps. Finance & Stochastics 20(4), , J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 38 / 39

71 Further Reading II 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. Mancini. Optimum thresholding using mean and conditional mean square error. Available at J.E. Figueroa-López, C. Li, & J. Nisen. Optimal iterative threshold-kernel estimation of jump diffusion processes. In preparation, J.E. Figueroa-López (WUSTL) Asymptotics in Financial Mathematics CCF Seminar 39 / 39