Nonparametric Minimax Nonparametric Minimax Estimation of the Estimation of the Volatility in High- Volatility in High-Frequency Models Corrupted Frequency Models Corrupted by by Noise Noise Schmidt- Hieber J. Schmidt-Hieber Models Joint work with Axel Munk and Tony Cai Estimation Numerical Results Institut f¨ ur Mathematische Stochastik, G¨ ottingen Summary/ Outlook www.stochastik.math.uni-goettingen.de schmidth@math.uni-goettingen.de
Introduction Nonparametric Minimax Estimation of the Volatility in High- Frequency 1 Models Models Corrupted by Noise Schmidt- Hieber 2 Estimation Models Estimation 3 Numerical Results Numerical Results Summary/ Outlook 4 Summary/ Outlook
Statistical Inverse Problems Nonparametric Minimax Statistical Inverse Problems: Given observations of the Estimation of the Volatility model in High- Frequency Models Y = Kf + ǫ Corrupted by Noise Schmidt- where Hieber K is an operator Models ǫ measurement noise Estimation Numerical estimate (reconstruct) the function f . Results Summary/ Outlook If K is linear, we say that this is a linear inverse problem. If the errors are not identically distributed, than the noise process is called heteroscedastic.
The models Nonparametric Minimax We consider two models. Estimation of the Volatility in High- For i = 1 , . . . , n Frequency Models � i � i / n Corrupted by � Noise Y i , n = σ ( s ) dW s + τ ǫ i , n . (1) Schmidt- n 0 Hieber � i � i � � ˜ Y i , n = σ W i / n + τ ǫ i , n , (2) Models n n Estimation Numerical σ, τ are deterministic, unknown, positive functions. Results � � � � ǫ 2 ǫ 4 Summary/ ǫ i , n , i.i.d., E ( ǫ i , n ) = 0, E = 1, E < ∞ . Outlook i , n i , n ( W t ) t ≥ 0 is a Brownian motion. ( ǫ 1 , n , . . . , ǫ n , n ) and ( W t ) t ≥ 0 are considered to be independent for i = 1 , . . . , n .
Connections to inverse problems The models can be seen as linear statistical inverse problems Nonparametric Minimax with random operator. For i = 1 , . . . , n Estimation of the Volatility � i in High- � Frequency Y i , n = ( K σ ) + heteroscedastic noise , Models n Corrupted by � � i Noise � � ˜ ˜ Schmidt- Y i , n = K σ + heteroscedastic noise , Hieber n Models where Estimation � t Numerical ( K σ ) ( t ) = σ ( s ) dW s , Results 0 � � Summary/ ˜ Outlook K σ ( t ) = σ ( t ) W t . Statistical Problem Estimation of the functions σ 2 and τ 2 , pointwise.
Differences � t σ ( t ) W t 0 σ ( s ) dW s σ Nonparametric Minimax Estimation of the Volatility in High- Frequency Models Corrupted by Noise Schmidt- Hieber Models Estimation Numerical Results Summary/ Outlook
Nonparametric Minimax Estimation of the Volatility in High- Frequency Models Corrupted by Noise Schmidt- Hieber From now on we will only consider Model (1). Estimation and theoretical results are similar for the second model. Models Estimation Numerical Results Summary/ Outlook
Origin Nonparametric Minimax Estimation of The model stems from high-frequency modeling of stock the Volatility in High- returns (Fan et al. ’03, Barndorff-Nielsen et al. ’06). Frequency Models Corrupted by These models are however much more general. Economic Noise theory indicates that σ is stochastic itself. Schmidt- Hieber Theory focusses so far only on estimation of the integrated Models moments of volatility (Ait-Sahalia et al. ’05, Podolskij and Estimation Vetter ’09), i.e. Numerical Results � t σ 2 p ( s ) ds p = 1 , 2 , . . . Summary/ Outlook 0 Remarkable exceptions are Malliavin and Mancino ’05 and Hoffmann ’99.
Estimation of σ 2 and τ 2 Nonparametric Minimax Estimation of the Volatility in High- Frequency Models Corrupted by Noise Non-parametric approach necessary. Schmidt- Hieber We do not have independent observations. Transformation that diagonalizes the process depends on Models the unkown quantities σ ( t ) and τ ( t ) and can not be Estimation Numerical computed explicitly in general. Results Summary/ Outlook
Estimation Step 1(4): Nonparametric Minimax Estimation of the Volatility Consider the weighted increments in High- Frequency ∆ i Y ( g ) := g ( i / n ) ( Y i +1 , n − Y i , n ) and observe that Models � i � i Corrupted by � � Noise ∆ i Y ( g ) ≈ n − 1 / 2 ( g σ ) η i , n + ( g τ ) ( ǫ i +1 , n − ǫ i , n ) , Schmidt- n n Hieber � �� � MA(1) Models where η i , n ∼ N (0 , 1) , iid. Estimation Numerical Results Summary/ Outlook
Estimation Step 1(4): Nonparametric Minimax Estimation of the Volatility Consider the weighted increments in High- Frequency ∆ i Y ( g ) := g ( i / n ) ( Y i +1 , n − Y i , n ) and observe that Models � i � i Corrupted by � � Noise ∆ i Y ( g ) ≈ n − 1 / 2 ( g σ ) η i , n + ( g τ ) ( ǫ i +1 , n − ǫ i , n ) , Schmidt- n n Hieber � �� � MA(1) Models where η i , n ∼ N (0 , 1) , iid. Estimation The noise term “dominates”. Numerical Results Important: The variance of the first term is of order 1 / n Summary/ Outlook whereas the eigenvalues of the covariance of the MA (1)-process behave like i 2 / n 2 . In spectral domain, we can use the first √ n observations. This can be viewed as the degree of ill-posedness of the problem.
Estimation Nonparametric Step 2(4): Minimax Estimation of the Volatility in High- Frequency The increment process Models Corrupted by ∆ Y ( g ) := (∆ 1 Y ( g ) , . . . , ∆ n − 1 Y ( g )) is stationary, if Noise σ, τ, g are constants. Schmidt- Hieber Stationary processes are “almost” diagonalized by Discrete Fourier Transforms. Models Estimation Therefore, we transform by DST, i.e. we consider Numerical Results Z := D n (∆ Y ( g )) , Summary/ Outlook where �� �� � ij π 2 D n := n sin . n i , j =1 ,..., n − 1
Estimation Nonparametric Minimax Estimation of Step 3(4): the Volatility in High- Frequency Models Corrupted by Let Z := D n (∆ Y ( g )) and define the estimator Noise Schmidt- n − 1 Hieber 1 � � λ − 1 Z 2 � τ 2 , g 2 � := i , i n − m Models i = m +1 Estimation where λ i = 4 sin 2 ( i π/ (2 n )) and m = m n , s.t. Numerical Results m / √ n → ∞ and m / n → 0. Summary/ Outlook Under smoothness assumptions on σ, τ , � � τ 2 , g 2 � estimates � 1 0 τ 2 ( s ) g 2 ( s ) ds at a convergence rate n − 1 / 2 .
Estimation Nonparametric Minimax Estimation of the Volatility in High- Frequency � Models � σ 2 , g 2 � Similar for Corrupted by Noise 2 [ n 1 / 2 ] Schmidt- � σ 2 , g 2 � := √ n Hieber � � i − λ i � Z 2 � τ 2 , g 2 � . Models � �� � i = [ n 1 / 2 ] +1 bias correction Estimation Numerical � Results � σ 2 , g 2 � estimates Under smoothness assumptions on σ, τ , � 1 0 σ 2 ( s ) g 2 ( s ) ds at a convergence rate n − 1 / 4 . Summary/ Outlook
Nonparametric Minimax Estimation of the Volatility in High- Frequency Models Corrupted by Step 4(4): Noise Schmidt- Hieber � σ 2 , g 2 � We are able to estimate for all sufficiently smooth Models g . Estimation We use this to construct a series estimator. Numerical Results Summary/ Outlook
Basis Functions Nonparametric Minimax Estimation of the Volatility Consider the L 2 [0 , 1] ONS, in High- √ Frequency � � { ψ k } = 1 , 2 cos ( k π t ) , k = 1 , . . . . Models Corrupted by Noise Further we introduce f k : [0 , 1] → R , Schmidt- Hieber f k ( x ) := ψ k ( x / 2) , k = 0 , 1 , . . . Models Note Estimation Numerical f 2 k ( x ) = 1 + cos( k π x ) =: ψ 0 ( x ) + 2 − 1 / 2 ψ k ( x ) , k ≥ 1 . Results Summary/ Outlook ψ i ψ j = 2 − 1 / 2 ( ψ i − j + ψ i + j ) sin( 2 i − 1 2 π ) sin( 2 j − 1 2 π ) = 2 − 3 / 2 ( ψ i − j + ψ i + j +1 )
The pointwise estimator Nonparametric Minimax Estimation of the Volatility in High- Frequency Models Corrupted by The estimator of σ 2 ( t ) is then given by Noise Schmidt- Hieber � � N �� � � − � � � � � � σ 2 σ 2 , f 2 σ 2 , f 2 σ 2 , f 2 ˆ N ( t ) = + 2 cos ( i π t ) , Models 0 0 i i =1 Estimation Numerical where N is some threshold parameter. Results Summary/ Outlook
Assumptions Nonparametric Minimax Function space: (Truncated) Sobolev s-ellipsoid Estimation of the Volatility in High- Θ b Θ b = s ( α, C , [ l , u ]) Frequency s Models � f ∈ L 2 [0 , 1] : l ≤ f ≤ u , Corrupted by ∃ ( θ n ) n , = Noise � ∞ ∞ Schmidt- � � Hieber i 2 α θ 2 i ≤ C s. t. f ( x ) = θ 0 + 2 θ i cos ( i π x ) , Models i =1 i =1 Estimation We always assum that l > 0 , u < ∞ . Numerical Results Characterisation: For any q odd, q < α ∈ N , f ( q ) (0) = f ( q ) (1) = 0 and Summary/ Outlook � 1 ( f ( α ) ) 2 ( x ) dx ≤ ˜ C 0
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