Statistical methods for financial models driven by Lévy processes

Transcription

1 Statistical methods for financial models driven by Lévy processes Part IV: Some Non-parametric Methods for Lévy Models José Enrique Figueroa-López 1 1 Department of Statistics Purdue University PASI CIMAT, Gto. Mex May 31 - June 5, 2010

2 Outline 1 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Statistical Methodology Numerical and empirical examples 2 Non-parametric estimation in time-changed Lévy models Model Formulation of the statistical problems Recovery of the random clock An empirical example 3 Conclusions

3 Non-parametric estimation of the Lévy density in Geometric Lévy Models Outline 1 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Statistical Methodology Numerical and empirical examples 2 Non-parametric estimation in time-changed Lévy models Model Formulation of the statistical problems Recovery of the random clock An empirical example 3 Conclusions

4 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Set-up 1 The price process S t of an asset is assumed to follow a Geometric Lévy model: S t = S 0 e Xt, where X t is a Lévy Process. 2 Data consists of discrete-time observations of the process: S t0, S t1,..., S tn X t0, X t1,..., X tn 3 We assume a non-parametric model for X. Concretely, we assume that the "Lévy measure" ν of X has a density p(x): ν(dx) = p(x)dx. 4 Goal: To use the data to estimate (or predict) the function p.

5 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Tools 1 The function p can be approximately recovered from integrals of the form β(ϕ) := ϕ(x)p(x)dx, by taking test functions ϕ on certain classes; e.g. indicator functions, splines, wavelet basis, etc. 2 p controls the jump behavior of the process: s T 1 [a,b]( X s ) = 1 [a,b] (x) p(x)dx E 1 T E 1 T s T ϕ ( X s) = ϕ(x)p(x)dx 3 p determines also the short-term behavior of X: 1 t E [ϕ (X t 0 t)] ϕ(x)p(x)dx.

6 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Tools 1 The function p can be approximately recovered from integrals of the form β(ϕ) := ϕ(x)p(x)dx, by taking test functions ϕ on certain classes; e.g. indicator functions, splines, wavelet basis, etc. 2 p controls the jump behavior of the process: s T 1 [a,b]( X s ) = 1 [a,b] (x) p(x)dx E 1 T E 1 T s T ϕ ( X s) = ϕ(x)p(x)dx 3 p determines also the short-term behavior of X: 1 t E [ϕ (X t 0 t)] ϕ(x)p(x)dx.

7 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Tools 1 The function p can be approximately recovered from integrals of the form β(ϕ) := ϕ(x)p(x)dx, by taking test functions ϕ on certain classes; e.g. indicator functions, splines, wavelet basis, etc. 2 p controls the jump behavior of the process: s T 1 [a,b]( X s ) = 1 [a,b] (x) p(x)dx E 1 T E 1 T s T ϕ ( X s) = ϕ(x)p(x)dx 3 p determines also the short-term behavior of X: 1 t E [ϕ (X t 0 t)] ϕ(x)p(x)dx.

8 Non-parametric estimation of the Lévy density in Geometric Lévy Models Statistical Methodology Idea of the Estimation Method 1 Consider 2 The SLLN implies that ˆβ T (ϕ) := 1 T ϕ( X s ) s T ˆβ T (ϕ) T β(ϕ) := ϕ(x)p(x)dx. 3 Drawback: Neither the size of the jumps X t nor the times are observable from discrete observations X t0,..., X tn of the process.

9 Non-parametric estimation of the Lévy density in Geometric Lévy Models Statistical Methodology Idea of the Estimation Method 1 Consider 2 The SLLN implies that ˆβ T (ϕ) := 1 T ϕ( X s ) s T ˆβ T (ϕ) T β(ϕ) := ϕ(x)p(x)dx. 3 Drawback: Neither the size of the jumps X t nor the times are observable from discrete observations X t0,..., X tn of the process.

10 Non-parametric estimation of the Lévy density in Geometric Lévy Models Statistical Methodology Idea of the Estimation Method 1 Consider 2 The SLLN implies that ˆβ T (ϕ) := 1 T ϕ( X s ) s T ˆβ T (ϕ) T β(ϕ) := ϕ(x)p(x)dx. 3 Drawback: Neither the size of the jumps X t nor the times are observable from discrete observations X t0,..., X tn of the process.

11 Non-parametric estimation of the Lévy density in Geometric Lévy Models Statistical Methodology Idea of the Estimation Method 1 Consider 2 The SLLN implies that ˆβ T (ϕ) := 1 T ϕ( X s ) s T ˆβ T (ϕ) T β(ϕ) := ϕ(x)p(x)dx. 3 Drawback: Neither the size of the jumps X t nor the times are observable from discrete observations X t0,..., X tn of the process.

12 Non-parametric estimation of the Lévy density in Geometric Lévy Models Statistical Methodology The estimation method Woerner (2003) and FL (2004) Estimators: ˆβ n (ϕ) := 1 t n n ϕ(x ti X ti 1 ); i=1 Properties: [Woerner 03, FL & Houdré 06; FL 07] If ϕ satisfies some regularity conditions ( ), then as t n and δ n := max{t i t i 1 } 0, { 2 1 E ˆβn (ϕ) β(ϕ)} 0 2 ˆβ n (ϕ) P β(ϕ) 3 ( ) D Tn ˆβ n (ϕ) β(ϕ) N (0, β(ϕ 2 )), provided that δ n tn 0.

13 Non-parametric estimation of the Lévy density in Geometric Lévy Models Numerical and empirical examples The Gamma Lévy process 1 Model: Pure-jump Lévy process with Lévy density p(x) = α x e x/β 1 {x>0}. 2 Histogram like estimators: Outside the origin.

14 Non-parametric estimation of the Lévy density in Geometric Lévy Models Numerical and empirical examples Performance Least-square fit: Fit the model α x e x/β (using least-squares) to the histogram estimator: ˆα LSE = 0.93 and ˆβ LSE = (vs. ˆα MLE = 1.01 and ˆβ MLE = 0.94) Sampling distribution Means and standard errors of ˆα LSE and ˆβ LSE based on 1000 repetitions t PPE-LSF MLE (0.06) 1.40 (0.50) (0.01) 0.99 (0.05) (0.08) 1.12 (0.31) (0.07) 0.99 (0.08) (0.08) 1.13 (0.34) (0.07) 0.99 (0.08)

15 Non-parametric estimation of the Lévy density in Geometric Lévy Models Numerical and empirical examples Recovery of the Lévy measure ν( x a) in time Estimation based on 5-sec. sampling observations of INTEL from Jan to Dec. 2005

16 Non-parametric estimation in time-changed Lévy models Outline 1 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Statistical Methodology Numerical and empirical examples 2 Non-parametric estimation in time-changed Lévy models Model Formulation of the statistical problems Recovery of the random clock An empirical example 3 Conclusions

17 Non-parametric estimation in time-changed Lévy models Model Set-up 1 The model: S t = S 0 e Xt, X t = bt + W τ(t) + J τ(t), {W t } t 0 is a Wiener process {J t } t 0 is an indep. pure-jump Lévy process with Lévy measure ν. τ(t) = t 0 r s ds, (Random Clock) dr t = α(m r t )dt + γ r t dw t, (Speed of the random clock) 2 Observations: Sampling of the log return process X t = log(s t /S 0 ): X 0, X δ,..., X nδ with nδ T n (Regular sampling);

18 Non-parametric estimation in time-changed Lévy models Model Set-up 1 The model: S t = S 0 e Xt, X t = bt + W τ(t) + J τ(t), {W t } t 0 is a Wiener process {J t } t 0 is an indep. pure-jump Lévy process with Lévy measure ν. τ(t) = t 0 r s ds, (Random Clock) dr t = α(m r t )dt + γ r t dw t, (Speed of the random clock) 2 Observations: Sampling of the log return process X t = log(s t /S 0 ): X 0, X δ,..., X nδ with nδ T n (Regular sampling);

19 Non-parametric estimation in time-changed Lévy models Model Set-up 1 The model: S t = S 0 e Xt, X t = bt + W τ(t) + J τ(t), {W t } t 0 is a Wiener process {J t } t 0 is an indep. pure-jump Lévy process with Lévy measure ν. τ(t) = t 0 r s ds, (Random Clock) dr t = α(m r t )dt + γ r t dw t, (Speed of the random clock) 2 Observations: Sampling of the log return process X t = log(s t /S 0 ): X 0, X δ,..., X nδ with nδ T n (Regular sampling);

20 Non-parametric estimation in time-changed Lévy models Model Set-up 1 The model: S t = S 0 e Xt, X t = bt + W τ(t) + J τ(t), {W t } t 0 is a Wiener process {J t } t 0 is an indep. pure-jump Lévy process with Lévy measure ν. τ(t) = t 0 r s ds, (Random Clock) dr t = α(m r t )dt + γ r t dw t, (Speed of the random clock) 2 Observations: Sampling of the log return process X t = log(s t /S 0 ): X 0, X δ,..., X nδ with nδ T n (Regular sampling); X t n 0,..., X t n n with 0 = t n 0 < < t n n = T n (Irregular sampling);

21 Non-parametric estimation in time-changed Lévy models Formulation of the statistical problems Statistical Problems 1 Non-parametric estimation based on high-frequency data: Recovery of the random clock τ(t) = R t 0 r(u)du (e.g. Winkel (2001), Woerner (2007), FL. (2009), Aït-Sahalia & Jacod (2009)). Recovery of the jump-acivity index". Estimation of the Lévy measure ν of X (FL. (2009)).

22 Non-parametric estimation in time-changed Lévy models Formulation of the statistical problems Statistical Problems 1 Non-parametric estimation based on high-frequency data: Recovery of the random clock τ(t) = R t 0 r(u)du (e.g. Winkel (2001), Woerner (2007), FL. (2009), Aït-Sahalia & Jacod (2009)). Recovery of the jump-acivity index". Estimation of the Lévy measure ν of X (FL. (2009)).

23 Non-parametric estimation in time-changed Lévy models Formulation of the statistical problems Statistical Problems 1 Non-parametric estimation based on high-frequency data: Recovery of the random clock τ(t) = R t 0 r(u)du (e.g. Winkel (2001), Woerner (2007), FL. (2009), Aït-Sahalia & Jacod (2009)). Recovery of the jump-acivity index". Estimation of the Lévy measure ν of X (FL. (2009)).

24 Non-parametric estimation in time-changed Lévy models Formulation of the statistical problems Statistical Problems 1 Non-parametric estimation based on high-frequency data: Recovery of the random clock τ(t) = R t 0 r(u)du (e.g. Winkel (2001), Woerner (2007), FL. (2009), Aït-Sahalia & Jacod (2009)). Recovery of the jump-acivity index". Estimation of the Lévy measure ν of X (FL. (2009)).

25 Non-parametric estimation in time-changed Lévy models Formulation of the statistical problems Statistical Problems 1 Non-parametric estimation based on high-frequency data: Recovery of the random clock τ(t) = R t 0 r(u)du (e.g. Winkel (2001), Woerner (2007), FL. (2009), Aït-Sahalia & Jacod (2009)). Recovery of the jump-acivity index". Estimation of the Lévy measure ν of X (FL. (2009)).

26 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α}, }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn 3 Under x α s(x)dx c 0α β, as α 0,

27 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α}, }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn 3 Under x α s(x)dx c 0α β, as α 0,

28 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α} α 0 τ(t ), }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn 3 Under x α s(x)dx c 0α β, as α 0,

29 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α} α 0 τ(t ), }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn 3 Under x α s(x)dx c 0α β, as α 0,

30 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α} α 0 τ(t ), }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn τ(t ) provided that α n 0 and δ n := max k {t n k t n k 1 } 0 (High-frequency). 3 Under x α s(x)dx c 0α β, as α 0,

31 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α} α 0 τ(t ), }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn τ(t ) provided that α n 0 and δ n := max k {t n k t n k 1 } 0 (High-frequency). 3 Under x α s(x)dx c 0α β, as α 0,

32 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the clock based on high-frequency data 1 Winkel (2001): X infinite jump-activity Lévy and τ(t) independent continuous increasing: ˆτ α (T ) := 1 x α s(x)dx #{t T : X t α} α 0 τ(t ), }{{} Jump 2 Figueroa (2009) & Aït-Sahalia & Jacod (2009): 1 { } ˆτ n (T ) := x α n s(x)dx # tk n T : X t n k X t n k 1 αn τ(t ) provided that α n 0 and δ n := max k {t n k t n k 1 } 0 (High-frequency). 3 Under x α s(x)dx c 0α β, as α 0, { } ˆτ n,α(t ) := αn β # tk n T : X t n k X t n n k 1 αn c 0 τ(t ),

33 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the jump activity index β 1 Suppose that x α s(x)dx c 0α β, as α 0, for some β (0, 2). 2 Then, { } ˆτ n,α(t ) := αn β # tk n T : X t n k X t n n k 1 αn c 0 τ(t ), 3 Aït-Sahalia and Jacod (2009) & F. (2010): ˆβ α,α,ω n := ˆβ n := log P k 1( X ) t n X t n αδn ω k k 1 P k 1( X t n k X t n k 1 α αδ ω n log(α ) ) P β. where t n k = kδ n with δ n = T /n and T is a given fixed time horizon.

34 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the jump activity index β 1 Suppose that x α s(x)dx c 0α β, as α 0, for some β (0, 2). 2 Then, { } ˆτ n,α(t ) := αn β # tk n T : X t n k X t n n k 1 αn c 0 τ(t ), 3 Aït-Sahalia and Jacod (2009) & F. (2010): ˆβ α,α,ω n := ˆβ n := log P k 1( X ) t n X t n αδn ω k k 1 P k 1( X t n k X t n k 1 α αδ ω n log(α ) ) P β. where t n k = kδ n with δ n = T /n and T is a given fixed time horizon.

35 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the jump activity index β 1 Suppose that x α s(x)dx c 0α β, as α 0, for some β (0, 2). 2 Then, { } ˆτ n,α(t ) := αn β # tk n T : X t n k X t n n k 1 αn c 0 τ(t ), 3 Aït-Sahalia and Jacod (2009) & F. (2010): ˆβ α,α,ω n := ˆβ n := log P k 1( X ) t n X t n αδn ω k k 1 P k 1( X t n k X t n k 1 α αδ ω n log(α ) ) P β. where t n k = kδ n with δ n = T /n and T is a given fixed time horizon.

36 Non-parametric estimation in time-changed Lévy models Recovery of the random clock Recovery of the jump activity index β 1 Suppose that x α s(x)dx c 0α β, as α 0, for some β (0, 2). 2 Then, { } ˆτ n,α(t ) := αn β # tk n T : X t n k X t n n k 1 αn c 0 τ(t ), 3 Aït-Sahalia and Jacod (2009) & F. (2010): ˆβ α,α,ω n := ˆβ n := log P k 1( X ) t n X t n αδn ω k k 1 P k 1( X t n k X t n k 1 α αδ ω n log(α ) ) P β. where t n k = kδ n with δ n = T /n and T is a given fixed time horizon.

37 Non-parametric estimation in time-changed Lévy models An empirical example Data observations INTC 5 Sec. Prices (Jan. 2, 2003 Dec. 30, 2005) Stock Prices Time in years

38 Non-parametric estimation in time-changed Lévy models An empirical example Recovery of the jump activity index β

39 Non-parametric estimation in time-changed Lévy models An empirical example Recovery of the jump activity index β

40 Non-parametric estimation in time-changed Lévy models An empirical example Recovery of the intensity {v t } t 0 of random clock

41 Conclusions Outline 1 Non-parametric estimation of the Lévy density in Geometric Lévy Models Setting Statistical Methodology Numerical and empirical examples 2 Non-parametric estimation in time-changed Lévy models Model Formulation of the statistical problems Recovery of the random clock An empirical example 3 Conclusions

42 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

43 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

44 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

45 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

46 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

47 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

48 Conclusions Conclusions: In the context of a Time-changed Lévy model, 1 we develop a consistent estimation scheme for the Lévy measure ν, the random clock τ, and the index of activity β using high-frequency data. 2 We obtain central limit theorems for the estimators. Important open problems: 1 Data-driven calibration of the estimator parameters for small samples? 2 Estimators that are robust against (high-frequency) microstructure noise?

49 Conclusions Bibliography For Further Reading I Figueroa-López. Nonparametric estimation of time-changed Levy models under high-frequency data Advances in Applied Probability, Figueroa-López. Central limit theorems for the estimation of time-changed Lévy models Submitted, Figueroa-López. Statistical estimation of Levy-type stochastic volatility models To appear in Annals of Finance, 2010.

50 Conclusions Bibliography For Further Reading II Figueroa-López. Model selection for Lévy processes based on discrete-sampling. IMS volume of the 3rd Erich L. Lehmann Symposium, Figueroa-López. Small-time moment asymptotics for Lévy processes. Statistics and Probability Letters, 78, , Figueroa-Lopez & Houdré. Risk bounds for the non-parametric estimation of Lévy processes. IMS Lecture Notes - Monograph Series. High Dimensional Probability, 51:96 116, 2006.