UNSTABLE VOLATILITY: THE BREAK PRESERVING LOCAL LINEAR ESTIMATOR - PowerPoint PPT Presentation

Motivation Volatility estimation Simulations Summary UNSTABLE VOLATILITY: THE BREAK PRESERVING LOCAL LINEAR ESTIMATOR Isabel Casas – CREATES, Aarhus University joint work with Irene Gijbels – K.U. Leuven 6 th Bachelier Finance Society World Congress Toronto, 2010

Motivation Volatility estimation Simulations Summary What is our work about? Aim?: estimation of discontinuous volatility functions. Discontinuities?: abrupt structural changes. Method?: nonparametric kernel estimation. Contribution?: Break preserving local linear.

Motivation Volatility estimation Simulations Summary Model Y i = m ( X i ) + σ ( X i ) ǫ i ǫ ∼ i.i.d (0 , 1) Fixed design or random design. E ( ǫ | X ) = 0 , E ( ǫ 2 | X ) = 1 and E ( ǫ 4 | X ) < ∞ E ( Y | X = x ) = m ( x ) E (( Y − m ( X )) 2 | X = x ) = σ 2 ( x )

Motivation Volatility estimation Simulations Summary Previous work: drift estimator

Motivation Volatility estimation Simulations Summary Drift estimation Y i = m ( X i ) + 0 . 4 ǫ i with ǫ ∼ IID (0 , 1) Given a point x in the continuous part, estimator of m ( x ) ? 3.0 2.5 2.0 Y 1.5 1.0 point to estimate 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 X

Motivation Volatility estimation Simulations Summary Drift estimation: centred estimator The centred estimator, ˆ m c ( x ) , is obtained as a regression using the points in a neighbourhood of x , Fan and Gijbels (1997). 3.0 2.5 2.0 Y 1.5 1.0 0.5 0.0 0.0 0.2 0.4 0.6 0.8 1.0 X

Motivation Volatility estimation Simulations Summary Drift estimation: centred estimator 1.5 LL (centred) estimator 1.0 Y point to estimate 0.5 0.0 0.36 0.38 0.40 0.42 0.44 0.46 X

Motivation Volatility estimation Simulations Summary Drift estimation: What happens at discontinuities? 3.0 point to estimate is a discontinuity 2.5 2.0 Y 1.5 1.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 X We expect the centred estimator to fall in the middle of the jump.

Motivation Volatility estimation Simulations Summary Drift estimation: What happens at discontinuities? The asymmetric estimator: find two estimators, left and right, and choose appropriately, Qiu (2003). 3.0 2.5 2.0 Y 1.5 1.0 0.5 0.0 0.2 0.4 0.6 0.8 1.0 X

Motivation Volatility estimation Simulations Summary Drift estimation: What happens at discontinuities? We have three estimator, which is the best choice? 3.0 right estimator 2.5 point to estimate 2.0 Y 1.5 centred estimator 1.0 left estimator 0.5 0.26 0.28 0.30 0.32 0.34 X

Motivation Volatility estimation Simulations Summary Contribution: volatility estimator

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility Y i = m ( X i ) + σ ( X i ) ǫ i ǫ ∼ i.i.d (0 , 1) r 2 | X = x ) = ˆ σ 2 ( x ) . Define ˆ r i = ( Y i − ˆ m ( X i )) . Then, E (ˆ Fan and Yao (1998): m itself is of order O ( h 2 “While the bias of ˆ 1 ) , its contribution to σ 2 ( · ) is only of o ( h 2 ˆ 1 ) ”. So, we expect to get a good estimate of the volatility even if the drift function is unknown.

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility Do you think that the centred estimator (Fan and Yao, 1998) is a good choice to estimate a discontinuous volatility function?

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility Do you think that the centred estimator (Fan and Yao, 1998) is a good choice to estimate a discontinuous volatility function? No, because it is not consistent at discontinuities. Solution : the break preserving local linear (BPLL) estimator.

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility ˆ σ 2 σ 2 ˆ k ( x ) = ˆ a 0 ,k ( x ) and ˙ k = ˆ a 1 ,k � X i − x n � � 2 K k � r 2 � (ˆ a 0 ,k ( x ) , ˆ a 1 ,k ( x )) = min ˆ i − a 0 , k − a 1 ,k ( X i − x ) h 2 ( a 0 ,a 1 ) i =1 K l ( x ) K c ( x ) K r ( x ) 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0.0 0.0 0.0 x−h/2 x x+h/2 x−h/2 x x+h/2 x−h/2 x x+h/2 left (k=l) centred (k=c) right (k=r)

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility The expression of the three volatility estimators: � s k, 2 − s k, 1 ( X i − x ) n � X i − x σ 2 � r 2 ˆ k ( x ) = ˆ k = c, l, r i K k s k, 0 s k, 2 − s 2 h 2 k, 1 i =1 where � X i − x � � ( X i − x ) j K k s k,j = h 2 . Easy to compute. No numerical minimisation.

Motivation Volatility estimation Simulations Summary Estimation of a discontinuous volatility How well are the estimators fitted to the data set? Weighted Residuals Mean Square . � X i − x � � 2 K k � n r 2 � ˆ i − ˆ a 0 , c − ˆ a 1 ,c ( X i − x ) i =1 h 2 WRMS k ( x ) = � X i − x � � n i =1 K k h 2

Motivation Volatility estimation Simulations Summary Break preserving local linear The break preserving local linear estimator: σ 2 ˆ c ( x ) diff ( x ) < u       σ 2 ˆ l ( x ) diff ( x ) ≥ u and WRMS l ( x ) < WRMS r ( x )      σ 2 ˆ BP LL ( x ) = σ 2 ˆ r ( x ) diff ( x ) ≥ u and WRMS l ( x ) > WRMS r ( x )       σ 2 σ 2 ˆ l ( x ) + ˆ r ( x )    diff(x) ≥ u and WRMS l ( x ) = WRMS r ( x )  2 where diff ( x ) = max( WRMS c ( x ) − WRMS l ( x ) , WRMS c ( x ) − WRMS r ( x )) , and 0 ≤ u ≤ Q for all x and Q a constant .

Motivation Volatility estimation Simulations Summary How is the WRMS for each estimator? Let [ a, b ] be the support of X and { x q } for q = 1 , . . . , m be the finite set of points where the volatility function is discontinuous. Then, two regions can be differentiated: D 1 is the region where the volatility function is continuous, � a + h 2 2 , b − h 2 � D 1 = \ D 2 2 D 2 contains the points of discontinuity and their neighbourhoods: m � � x q − h 2 2 , x q + h 2 � D 2 = 2 q =1

Motivation Volatility estimation Simulations Summary How is the WRMS for each estimator? Under certain regularity conditions : For x ∈ D 1 , WRMS k ( x ) = σ 4 ( x )( E ( ǫ 4 | X ) − 1) + R k, 1 ( x )

Motivation Volatility estimation Simulations Summary How is the WRMS for each estimator? Under certain regularity conditions : For x ∈ D 1 , WRMS k ( x ) = σ 4 ( x )( E ( ǫ 4 | X ) − 1) + R k, 1 ( x ) For x ∈ D 2 such that x = x q + τh 2 with τ ∈ [0 , 1 2 ] and a jump of magnitude d , σ 4 ( x )( E ( ǫ 4 | X ) − 1) + d 2 C 2 WRMS l ( x ) = l,τ + R l, 2 ( x ) σ 4 ( x )( E ( ǫ 4 | X ) − 1) + R r, 2 ( x ) WRMS r ( x ) = σ 4 ( x )( E ( ǫ 4 | X ) − 1) + d 2 C 2 WRMS c ( x ) = c,τ + R c, 2 ( x )

Motivation Volatility estimation Simulations Summary How is the WRMS for each estimator? Under certain regularity conditions : For x ∈ D 1 , WRMS k ( x ) = σ 4 ( x )( E ( ǫ 4 | X ) − 1) + R k, 1 ( x ) For x ∈ D 2 such that x = x q + τh 2 with τ ∈ [ − 1 2 , 0] and a jump of magnitude d , σ 4 ( x )( E ( ǫ 4 | X ) − 1) + R l, 3 ( x ) WRMS l ( x ) = σ 4 ( x )( E ( ǫ 4 | X ) − 1) + d 2 C 2 WRMS r ( x ) = r,τ + R r, 3 ( x ) σ 4 ( x )( E ( ǫ 4 | X ) − 1) + d 2 C 2 WRMS c ( x ) = c,τ + R c, 3 ( x )

Motivation Volatility estimation Simulations Summary MSE (continuous points) Under certain regularity conditions and with � 2 � µ k, 2 − µ k, 1 u � � u j K k ( u ) du and V k = K 2 µ k,j = k ( u ) du : µ k, 0 µ k, 2 − µ 2 k, 1 For x ∈ D 1 ( Continuous points ), µ 2 k, 2 − µ k, 1 µ k, 3 k ( x )) = h 2 σ 2 ( x ) 2 ¨ 1 σ 2 + o p ( h 2 1 + h 2 Bias (ˆ 2 + nh 2 ) 2 µ k, 2 µ k, 0 − µ 2 k, 1 k ( x )) = ( E ( ǫ 4 | X ) − 1) σ 4 ( x ) � � σ 2 1 Variance (ˆ V k + o p nh 2 f X ( x ) nh 2 k ( x )) = Bias 2 + Variance σ 2 MSE (ˆ

Motivation Volatility estimation Simulations Summary MSE (continuous points) Under certain regularity conditions and with � 2 � µ k, 2 − µ k, 1 u � � u j K k ( u ) du and V k = K 2 µ k,j = k ( u ) du : µ k, 0 µ k, 2 − µ 2 k, 1 For x ∈ D 1 ( Continuous points ), µ 2 k, 2 − µ k, 1 µ k, 3 k ( x )) = h 2 σ 2 ( x ) 2 ¨ 1 σ 2 + o p ( h 2 1 + h 2 Bias (ˆ 2 + nh 2 ) 2 µ k, 2 µ k, 0 − µ 2 k, 1 k ( x )) = ( E ( ǫ 4 | X ) − 1) σ 4 ( x ) � � σ 2 1 Variance (ˆ V k + o p nh 2 f X ( x ) nh 2 If h 1 , h 2 → 0 , n → ∞ and nh 2 → ∞ k ( x )) = Bias 2 + Variance σ 2 MSE (ˆ

Motivation Volatility estimation Simulations Summary MSE (continuous points) Under certain regularity conditions and with � 2 � µ k, 2 − µ k, 1 u � � u j K k ( u ) du and V k = K 2 µ k,j = k ( u ) du : µ k, 0 µ k, 2 − µ 2 k, 1 For x ∈ D 1 ( Continuous points ), σ 2 Bias (ˆ k ( x )) = k ( x )) = ( E ( ǫ 4 | X ) − 1) σ 4 ( x ) � � 1 σ 2 Variance (ˆ V k + o p nh 2 f X ( x ) nh 2 If h 1 , h 2 → 0 , n → ∞ and nh 2 → ∞ k ( x )) = Bias 2 + Variance σ 2 MSE (ˆ

Motivation Volatility estimation Simulations Summary MSE (continuous points) Under certain regularity conditions and with � 2 � µ k, 2 − µ k, 1 u � � u j K k ( u ) du and V k = K 2 µ k,j = k ( u ) du : µ k, 0 µ k, 2 − µ 2 k, 1 For x ∈ D 1 ( Continuous points ), σ 2 Bias (ˆ k ( x )) = σ 2 Variance (ˆ k ( x )) = If h 1 , h 2 → 0 , n → ∞ and nh 2 → ∞ k ( x )) = Bias 2 + Variance σ 2 MSE (ˆ