Particle Filter-Based On-Line Estimation of Spot (Cross-)Volatility with Nonlinear Market Microstructure Noise Models. Jan C.

Transcription

1 Particle Filter-Based On-Line Estimation of Spot (Cross-)Volatility with Nonlinear Market Microstructure Noise Models Jan C. Neddermeyer (joint work with Rainer Dahlhaus, University of Heidelberg) DZ BANK AG, Frankfurt Vienna, 17th June 2011

2 Contents 1 Modeling Aspects Features of Ultra High-Frequency Financial Data A Nonlinear State-Space Model for Tick-by-Tick Data A Class of Nonlinear Market Microstructure Noise Models 2 Spot (Cross-)Volatility Estimation Synchronous Trading Case Non-Synchronous Trading Case 3 Applications Results for Real Data 4 Conclusion

3 Outline 1 Modeling Aspects Features of Ultra High-Frequency Financial Data A Nonlinear State-Space Model for Tick-by-Tick Data A Class of Nonlinear Market Microstructure Noise Models 2 Spot (Cross-)Volatility Estimation Synchronous Trading Case Non-Synchronous Trading Case 3 Applications Results for Real Data 4 Conclusion

4 Market microstructure features Financial transaction data Main features discreteness of prices many zero returns bid-ask bounce Autocorrelation of the returns (returns = first differences)

5 Assumption: unobserved ecient price process Simulated data Autocorrelation of the returns (returns = first differences)

6 A nonlinear state-space model in transaction time Notation: t j, j = 1, 2,..., transaction times (strictly increasing) Y tj observed transaction prices X tj unobserved ecient log-prices Nonlinear state-space model: Y tj = g tj ;Y t1:j 1 ( exp[xtj ] ) (observation equation) X tj = X tj 1 + Z tj (state equation) where Z tj N (0, σ 2 t j ) with constant or time-varying volatility σ tj. Key quantity: p(x tj y t1:j ) ltering distribution 1 1 y t1:j = {y t1,..., y tj }

7 A class of nonlinear market microstructure noise models Aim: A model for the nonlinear time-varying (possibly stochastic) dependence of the observed transaction prices and the latent ecient prices Y tj = g tj ;Y t1:j 1 ( exp[xtj ] ) Model assumptions: 1 The distribution of Y tj = g tj ;Y t1:j 1 ( exp[xtj ] ) is discrete. 2 The conditional distribution of Y tj given X tj and previous observations is of the form p ( y tj y t1:j 1, exp[x tj ] ) 1 Atj ( exp[xtj ] ) (a.s.) where A tj depends on y tj and on the past observations y t1,..., y tj 1. Remarks: If g tj = g tj ;Y t1:j 1 is deterministic then A tj := gt 1 j (y tj ) = {z : g tj (z) = y tj } is the inverse image of y tj under g tj. Concrete specications of the set A tj give quite dierent models

8 Examples 1 Rounding to the nearest cent Deterministic: A tj = [y tj 0.5, y tj + 0.5) and y tj = round(exp[x tj ]) Stochastic: e.g. rounding up/down with probability 1/2, A tj = (y tj 1, y tj + 1) 2 Rounding to the nearest order book level A tj = {z R : argmin γ Mtj - z γ = yt j } and g tj ;y t1:j 1 (z) = argmin γ Mtj - z γ 3 Rounding to the nearest market maker quote 4 A general rounding for transaction data A tj = [y tj tj, y tj + tj ) with tj = { 0.5 y tj y tj 1 if y tj y tj 1, tj 1 else

9 Example: order book data Economic intuition: The ecient price should be closer to the observed price than to any other order book level y t1 A t exp[x t1 ] t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10t 11 Figure: Bid and ask levels of the order book (green and blue lines), supports of the ltering distributions (gray vertical lines), transaction prices (red circles), ecient prices in transaction time (diamonds), the ecient price process in clock time (black line)

10 Example: order book data Economic intuition: The ecient price should be closer to the observed price than to any other order book level y t1 A t exp[x t1 ] t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10t 11 Figure: Bid and ask levels of the order book (green and blue lines), supports of the ltering distributions (gray vertical lines), transaction prices (red circles), ecient prices in transaction time (diamonds), the ecient price process in clock time (black line)

11 Example: order book data Economic intuition: The ecient price should be closer to the observed price than to any other order book level y t1 A t exp[x t1 ] t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10t 11 Figure: Bid and ask levels of the order book (green and blue lines), supports of the ltering distributions (gray vertical lines), transaction prices (red circles), ecient prices in transaction time (diamonds), the ecient price process in clock time (black line)

12 Example: order book data Economic intuition: The ecient price should be closer to the observed price than to any other order book level y t1 A t exp[x t1 ] t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10t 11 Figure: Bid and ask levels of the order book (green and blue lines), supports of the ltering distributions (gray vertical lines), transaction prices (red circles), ecient prices in transaction time (diamonds), the ecient price process in clock time (black line)

13 Example: order book data Economic intuition: The ecient price should be closer to the observed price than to any other order book level y t1 A t exp[x t1 ] t 1 t 2 t 3 t 4 t 5 t 6 t 7 t 8 t 9 t 10t 11 Figure: Bid and ask levels of the order book (green and blue lines), supports of the ltering distributions (gray vertical lines), transaction prices (red circles), ecient prices in transaction time (diamonds), the ecient price process in clock time (black line)

14 Real data vs. deterministic rounding vs. stochastic rounding Deterministic rounding reproduces the major microstructure features

15 Outline 1 Modeling Aspects Features of Ultra High-Frequency Financial Data A Nonlinear State-Space Model for Tick-by-Tick Data A Class of Nonlinear Market Microstructure Noise Models 2 Spot (Cross-)Volatility Estimation Synchronous Trading Case Non-Synchronous Trading Case 3 Applications Results for Real Data 4 Conclusion

16 The synchronous trading case Consider S securities: ecient log-prices X t,s and transaction prices Y t,s, s = 1,..., S where Y tj = g tj ( exp[xtj ] ), X tj = X tj 1 + Z tj, Z tj N (0, Σ tj ), g tj (exp[x tj ]) = (g t (1) j (exp[x tj,1]),..., g (S)(exp[X t tj,s])) T, X t = (X t,1,..., X t,s ) T. j Aim: On-line estimation of covariance matrix Σ t (spot cross-volatilities) Remarks: - Assumption: Σ t is slowly varying (or constant) - Spot volatility estimation: S = 1 and Σ t = σ 2 t

17 An ecient particle lter Assume weighted particles {x i t 1:j 1, ωt i j 1 } N i=1 approximating p(xt 1:j 1 yt 1:j 1 ) are given. 1 For i = 1,..., N: Sample x i t j p(x tj y t1:j, x i t j 1 ) N (x tj x i t j 1, Σ tj ) log Atj,1 log A tj,s Compute importance weights ω i t j ω i t j 1 log A tj N (x tj x i t j 1, Σ tj )dx tj 2 For i = 1,..., N: Normalize importance weights ω i t j = ω i t j /( N k=1 ωk t j ) 3 Resample particles Result: Particles {x i t 1:j, ω i t j } N i=1 approximating p(xt 1:j yt 1:j ). Remarks: The particle lter is very ecient because the particles are sampled from the optimal proposal p(x tj y t1:j, x tj 1 ). In practice, Σ tj is replaced by an estimate ˆΣ pf t j.

18 EM algorithms The standard EM algorithm: - Maximization of T Q(Σ ˆΣ (m) ) = const + E ˆΣ(m)[log p Σ (X tj X tj 1 ) y t1:t ] j=2 with respect to Σ - Depends on smoothing distributions ( o-line algorithm) Our sequential EM algorithm: - Recursive maximization of Q tj (Σ ˆΣ t1:j 1 ) = {1 λ j } Q tj 1 (Σ ˆΣ t1:j 2 )+λ j E ˆΣt1:j 1 [log p Σ (X tj X tj 1 ) y t1:j ] with respect to Σ - Depends on ltering distributions ( on-line algorithm)

19 Our sequential EM-type algorithm E-step: (based on particles {x i t j 1:j, ω i t j } N i=1 ) ˆQ tj (Σ ˆΣ t1:j 1 ) = {1 λ j } ˆQ tj 1 (Σ ˆΣ t1:j 2 ) λ j 1 2 N i=1 ωt i j [S log 2π + log Σ + tr {Σ }] 1 (x i tj x i tj 1 )(x i tj x i tj 1 ) T M-step: with ˆΣ tj = {1 λ j } ˆΣ tj 1 + λ j Σtj (ω tj ) N Σ tj (ω tj ) = ωt i j (x i t j x i t j 1 )(x i t j x i t j 1 ) T i=1 Step size selection: Simple approach: constant step size λ j = λ Advanced approach: adaptive time-varying step size e.g. spatially aggregated exponential smoothing (SAGES) developed by Chen and Spokoiny (2009)

20 Back and forth between particle lter and seq. EM algorithm Time Particle Filter Sequential EM algorithm t j 1 ˆΣ tj 1 - Simulate transition x i t x i j 1 t j pf using y tj and ˆΣ t = ˆΣ tj 1 j - Obtain particles {x i t j 1:j, ω i t j } t j {x i t j 1:j, ω i t j } - Compute update Σ tj (ω tj ) using {x i t j 1:j, ω i t j } ˆΣ tj - Obtain estimate ˆΣ tj t j+1

21 Clock time estimation Until now we considered the estimation of Σ tj = Σ(t j ) in a transaction time model. A simple clock time estimator: Consider dx(t) = Γ(t) dw(t) where Γ(t) Γ T (t) = Σ c (t). W(t) is a multivariate Brownian motion and Σ c (t) denotes the volatility per time unit (while Σ(t) denotes volatility per transaction) We obtain ˆΣ N c t j = (1 λ j )ˆΣ c t j 1 + λ j i=1 with modied particles {x ci t j 1:j, ω ci t j } N i=1. ω ci t j ( x ci t j x ci t j 1 )( x ci t j x ci t j 1 ) T t j t j 1 An alternative clock time estimator: see our paper

22 Non-synchronous trading case Let Z tj = X tj X tj 1 be the returns of the ecient log-prices. Security 1 overlapping time of z t (1) 4 and z t (2) 2 Security 2 t (1) 1 t (2) t (1) 2 t (1) 3 t (1) 4 = t (2) 5 t (1) 1 t (2) 2 t (2) 3 t (2) 4 5 t (2) 6 t (2) 7 Facts: Trading times are non-synchronous Each security s evolves in an individual transaction time {t (s) j } Ts j=1 Overlapping time is the key quantity for cross-volatility estimation

23 Our model: a non-standard state-space model Our model: Y t (s) j X t (s) j = g t (s) j ( exp[xt (s)] ) (observation equation) j = X (s) + Z (s) t j 1 t j (state equation) for s = 1,..., S and Z t (s) j Cov(Z t (s 1 ) j Properties:, Z t (s 2 ) k ) = [t(s 1 ) N (0, (Σ (s)) t ss), j Nonlinear observation equation j 1,t(s 1 ) j ) [t (s 2 ) k 1,t(s 2 ) ) k (Σ t (s 1 ) j t (s 1 ) j 1 1/2 t (s 2 ) t (s 2 ) k k 1 1/2 min{t (s 1 ) j,t (s 2 ) } )s 1s 2. k Components evolve non-synchronously in dierent times (non-standard state-space model) Standard particle lters cannot be applied Synchronous trading is a special case

24 The recursive covariance estimator Let t v denote the joint transaction time. Our EM-type estimator is given, componentwise, by where (ˆΣ tv ) s1 s 2 = (1 λ v,s1,s 2 )(ˆΣ tv 1 ) s1 s 2 + λ v,s1,s 2 ( Σ tv ) s1 s 2, ( Σ tv ) s1 s 2 = N i=1 ωi max{t (s 1 ) (ˆΣ tv 1 ) s1 s 2 h v 1,t (s 2 ) h v 2 } zi z i t (s 1 ) h v t (s 2 ) 1 h v 2 t (s 1 ) h v t (s 1 ) 1 h v 1 1 1/2 t (s 2 ) h v 2 [t (s 1 ) ) [t (s 2 ) h v 1 ) 1 1,t(s h v 1 t (s 2 ) h v 2 1 1/2 h v 2 ) 2 1,t(s h v ) 2 if t (s 1) h v 1 or t (s 2) h v 2 else = t v = t v with h v s = min{hs : t(s) h s t v}. Remark: The number of updates for each component is dierent

25 Outline 1 Modeling Aspects Features of Ultra High-Frequency Financial Data A Nonlinear State-Space Model for Tick-by-Tick Data A Class of Nonlinear Market Microstructure Noise Models 2 Spot (Cross-)Volatility Estimation Synchronous Trading Case Non-Synchronous Trading Case 3 Applications Results for Real Data 4 Conclusion

26 Real data: volatility estimation in transaction time e 04 3e 04 5e 04 10:00 11:00 12:00 13:00 14:00 15:00 16:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 1e 04 3e 04 5e 04 09:40 09:50 Figure: The upper plot shows transaction data of Citigroup for the 3rd September The middle and the lower plot give the volatility estimators ˆΣ tj (black line), ˆΣ S t j (red line), and the benchmark estimator ˆΣ B t j (gray line).

27 Real data: volatility estimation in clock time e 04 3e 04 5e 04 10:00 11:00 12:00 13:00 14:00 15:00 16:00 13:00 14:00 15: :00 14:00 15:00 Figure: Estimation of time-varying spot volatility in clock time based on the transactions data of Citigroup for the 3rd September Simple clock time estimator ˆΣ cs t j (black line) and alternative clock time estimator ˆΣ cs (t alt j) = ˆΣ S t j / δ j (red line). Lower plot: averaged duration times δ j (y-axis shows seconds).

28 Real data: cross-volatility estimation :00 11:00 12:00 13:00 14:00 15: :00 11:00 12:00 13:00 14:00 15: :00 11:00 12:00 13:00 14:00 15:00 Figure: Upper plot: Transaction data of BAC (black line), C (red line), and JPM (green line) plotted with osets. Middle plot: Volatility estimates. Lower plot: Correlation estimates for BAC/C (black line), BAC/JPM (green line), and C/JPM (red line).

29 Outline 1 Modeling Aspects Features of Ultra High-Frequency Financial Data A Nonlinear State-Space Model for Tick-by-Tick Data A Class of Nonlinear Market Microstructure Noise Models 2 Spot (Cross-)Volatility Estimation Synchronous Trading Case Non-Synchronous Trading Case 3 Applications Results for Real Data 4 Conclusion

30 Conclusion New modeling and estimation aspects: A class of nonlinear market microstructure noise models A method for spot volatility estimation based on a particle lter and a new sequential EM-type algorithm Estimation in transaction time and clock time Our method works on-line and is computationally very ecient Empirical results: Deterministic rounding schemes reproduce the major microstructure features The volatility in transaction time is often rougly constant (in contrast to volatility in clock time) The correlations of real stock returns vary signicantly over the trading day

31 Thank you for your attention! Dahlhaus, R. and Neddermeyer, J.C. (2010), Particle Filter-Based On-Line Estimation of Spot Volatility with Nonlinear Market Microstructure Noise Models, under revision. Neddermeyer, J.C. (2010), Importance Sampling-Based Monte Carlo Methods with Applications to Quantitative Finance, PhD dissertation, University of Heidelberg. Questions?