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

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

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

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

Market microstructure features

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Financial transaction data

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Autocorrelation of the returns (returns = first differences)

Main features

• discreteness of prices

• many zero returns

• bid-ask bounce

Assumption: unobserved ecient price process

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45.8045.8245.8445.86

Simulated data

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Autocorrelation of the returns (returns = first differences)

A nonlinear state-space model in transaction time

Notation:

• tj, j = 1, 2, . . ., transaction times (strictly increasing)

• Yt_j observed transaction prices

• Xt_j unobserved ecient log-prices Nonlinear state-space model:

Yt_j = gt_j;Y_t1:j−1 exp[Xt_j]

(observation equation) Xt_j = Xt_j−1+ Zt_j (state equation)

where Zt_j ∼ N (0, σ²_t

j)with constant or time-varying volatility σt_j. Key quantity: p(xt_j|yt_1:j)ltering distribution¹

1yt_1:j = {yt₁, . . . , yt_j}

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

Ytj= gt_j;Y_t1:j−1 exp[Xtj]

Model assumptions:

1 The distribution of Yt_j = gt_j;Y_t1:j−1 exp[Xt_j]

is discrete.

2 The conditional distribution of Yt_j given Xt_j and previous observations is of the form p yt_j

yt_1:j−1, exp[xt_j]
∝ 1_Atj exp[xt_j]

(a.s.) where Atj depends on ytj and on the past observations yt1, . . . , ytj−1. Remarks:

• _{If gtj} = gt_j;Y_t1:j−1 is deterministic then At_j := g⁻¹_t

j (yt_j) = {z : gt_j(z) = yt_j}is the inverse image of yt_j under gt_j.

• Concrete specications of the set At_j give quite dierent models

Examples

1 Rounding to the nearest cent

• Deterministic: A^tj = [yt_j− 0.5, yt_j+ 0.5)and y^tj =round(exp[x^tj])

• Stochastic: e.g. rounding up/down with probability 1/2, Atj= (ytj− 1, y_tj+ 1)

2 Rounding to the nearest order book level

• At_j= {z ∈ R : argmin_γ∈M

tj-|z − γ| = y^tj} and gt_j;y_t1:j−1(z) =argminγ∈M_tj-|z − γ|

3 Rounding to the nearest market maker quote

4 A general rounding for transaction data

• At_j= [yt_j− ∆_tj, yt_j+ ∆t_j)with ∆t_j = (

0.5 |ytj− y_tj−1| if ytj 6= y_tj−1,

∆tj−1 else

Example: order book data

Economic intuition: The ecient price should be closer to the observed price than to any other order book level.

t1 t2 t3t4 t5 t6t7 t8 t9 t10t11

10.00 10.01 10.02 10.03 10.04

yt₁ At₁

exp[xt1]

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)

Example: order book data

Economic intuition: The ecient price should be closer to the observed price than to any other order book level.

t1 t2 t3t4 t5 t6t7 t8 t9 t10t11

10.00 10.01 10.02 10.03 10.04

yt₁ At₁

exp[xt1]

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)

Example: order book data

Economic intuition: The ecient price should be closer to the observed price than to any other order book level.

t1 t2 t3t4 t5 t6t7 t8 t9 t10t11

10.00 10.01 10.02 10.03 10.04

yt₁ At₁

exp[xt1]

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)

Example: order book data

Economic intuition: The ecient price should be closer to the observed price than to any other order book level.

t1 t2 t3t4 t5 t6t7 t8 t9 t10t11

10.00 10.01 10.02 10.03 10.04

yt₁ At₁

exp[xt1]

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)

Example: order book data

Economic intuition: The ecient price should be closer to the observed price than to any other order book level.

t1 t2 t3t4 t5 t6t7 t8 t9 t10t11

10.00 10.01 10.02 10.03 10.04

yt₁ At₁

exp[xt1]

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)

Real data vs. deterministic rounding vs. stochastic rounding

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45.7545.85

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45.7545.85

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45.8045.8245.8445.86

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45.8045.8245.8445.86

0 50 100 150 200

45.8045.8245.8445.86

0 5 10 15 20 25 30 35

−0.20.20.61.0

0 5 10 15 20 25 30 35

−0.20.20.61.0

0 5 10 15 20 25 30 35

−0.50.00.51.0

0 5 10 15 20 25 30 35

−0.30−0.150.00

0 5 10 15 20 25 30 35

−0.30−0.150.00

0 5 10 15 20 25 30 35

−0.4−0.20.0

⇒Deterministic rounding reproduces the major microstructure features

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

The synchronous trading case

• Consider S securities: ecient log-prices X^t,sand transaction prices Y^t,s, s = 1, . . . , S

Yt_j = gt_j exp[Xt_j]
, Xt_j = Xt_j−1+ Zt_j, where

Zt_j ∼ N (0, Σt_j),

gt_j(exp[Xt_j]) = (g

t⁽¹⁾_j (exp[Xt_j,1]), . . . , g

t^(S)_j (exp[Xt_j,S]))^T, X^t= (Xt,1, . . . , Xt,S)^T.

• _Aim:On-line estimation of covariance matrix Σ^t(spot cross-volatilities)

• _Remarks:

- Assumption: Σ^tis slowly varying (or constant) - Spot volatility estimation: S = 1 and Σ^t= σ²_t

An ecient particle lter

Assume weighted particles {xⁱt_1:j−1, ωⁱ_t

j−1}^N_i=1approximating p(xt_1:j−1|yt_1:j−1)are given.

1 For i = 1, . . . , N:

• Sample xⁱt_j ∼ p(xt_j|yt_1:j, xⁱ_t

j−1) ∝ N (xt_j|xⁱ_t

j−1,Σt_j)

log A_{tj ,1}×···×log A_{tj ,S}

• Compute importance weights ˘ωtⁱ_j ∝ ωⁱ_t

j−1

R

log A_tjN (xt_j|xⁱ_t

j−1,Σt_j)dxt_j 2 For i = 1, . . . , N: Normalize importance weights ωtⁱ_j= ˘ωⁱ_t

j/(PN k=1ω˘^k_t

j) 3 Resample particles

Result: Particles {xⁱt_1:j, ωⁱ_t

j}^N_i=1approximating 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|yt_1:j, xt_j−1).

• In practice,Σt_j is replaced by an estimateΣˆ^pf_t

j.

EM algorithms

• The standard EM algorithm:

- Maximization of

Q(Σ| ˆΣ^(m)) =const +

T

X

j=2

EΣˆ^(m)[log pΣ(Xt_j|X_tj−1)|yt_1:T]

with respect to Σ

- Depends on smoothing distributions (⇒ o-line algorithm)

• Our sequential EM algorithm:

- Recursive maximization of

Qt_j(Σ| ˆΣt_1:j−1) = {1−λj} Qt_j−1(Σ| ˆΣt_1:j−2)+λjEΣˆ_t1:j−1[log pΣ(Xt_j|Xt_j−1)|yt_1:j] with respect to Σ

- Depends on ltering distributions (⇒ on-line algorithm)

Our sequential EM-type algorithm

• E-step: (based on particles {xⁱt_j−1:j, ω_tⁱ

j}^N_i=1) Qˆ_tj(Σ| ˆΣt_1:j−1) = {1 − λj} ˆQ_tj−1(Σ| ˆΣt_1:j−2)

− λj

1 2

N

X

i=1

ω_tⁱ_jh

S log 2π + log |Σ| +trn

Σ⁻¹(xⁱ_tj− xⁱ_tj−1)(xⁱ_tj− xⁱ_tj−1)^Toi

• _M-step:

Σˆt_j = {1 − λj} ˆΣt_j−1+ λjΣ˘t_j(ωt_j) with

Σ˘t_j(ωt_j) =

N

X

i=1

ωⁱ_tj(xⁱ_tj − xⁱ_t

j−1)(xⁱ_tj− xⁱ_t

j−1)^T

Step size selection:

• Simple approach: constant step size λj= λ

• Advanced approach: adaptive time-varying step size

e.g. spatially aggregated exponential smoothing (SAGES) developed by Chen and Spokoiny (2009)

Back and forth between particle lter and seq. EM algorithm

Time Particle Filter Sequential EM algorithm

tj−1

tj

tj+1

Σˆ_tj−1

- Simulate transition xⁱ_tj−1 xⁱ_tj using ytj and ˆΣ^pf

tj= ˆΣ_tj−1 - Obtain particles {xⁱ_tj−1:j, ωⁱ_tj}

{xⁱ_tj−1:j, ωⁱ_tj}

- Compute update ˘Σ_tj(ω_tj) using {xⁱ_tj−1:j, ωⁱ

tj} - Obtain estimate ˆΣ_tj Σˆ_tj

Clock time estimation

Until now we considered the estimation of Σt_j = Σ(tj)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}

Σˆ^c_tj = (1 − λj) ˆΣ^c_tj−1+ λj N

X

i=1

ω^ci_tj

x^ci_tj− x^ci_t

j−1

x^ci_tj− x^ci_t

j−1

T

|t_j− t_j−1| with modied particles {x^cit_j−1:j, ω_t^ci_j}^N_i=1.

An alternative clock time estimator: see our paper

Non-synchronous trading case

Let Zt_j = Xt_j− Xt_j−1 be the returns of the ecient log-prices.

Security 1

Security 2

t⁽¹⁾₁ t₁⁽²⁾ t₂⁽¹⁾ t⁽¹⁾₃ t⁽²⁾₂ t₃⁽²⁾ t₄⁽²⁾ t⁽¹⁾₄ = t₅⁽²⁾ t₅⁽¹⁾t⁽²⁾₆ t⁽²⁾₇

overlapping time of z_t(1) 4

and z_t(2) 2

Facts:

• Trading times are non-synchronous

• Each security s evolves in an individual transaction time {t^(s)j }^T_j=1^s

• Overlapping time is the key quantity for cross-volatility estimation

Our model: a non-standard state-space model

Our model:

Yt^(s)_j = g

t^(s)_j exp[X

t^(s)_j ]

(observation equation) Xt^(s)_j = X

t^(s)_j−1+ Z

t^(s)_j (state equation) for s = 1, . . . , S and Z_t(s)

j

∼ N (0, (Σ

t^(s)_j )ss), Cov(Z_t(s1)

j

, Zt^(s2)_k ) = ^|[t

(s1)

j−1,t^(s1)_j )∩[t^(s2)_k−1,t^(s2)_k )|

|t^(s1)_j −t^(s1)_j−1|^1/2|t^(s2)_k −t^(s2)_k−1|^1/2(Σ

min{t^(s1)_j ,t^(s2)_k })s₁s₂. Properties:

• Nonlinear observation equation

• Components evolve non-synchronously in dierent times (non-standard state-space model)

• Standard particle lters cannot be applied

• Synchronous trading is a special case

The recursive covariance estimator

Let tvdenote the joint transaction time. Our EM-type estimator is given, componentwise, by

( ˆΣtv)s₁s₂= (1 − λv,s₁,s₂)( ˆΣt_v−1)s₁s₂+ λv,s₁,s₂( ˘Σtv)s₁s₂, where

( ˘Σtv)s1s2=













 PN

i=1ωⁱ

max{t^(s1)_hv 1

,t^(s2)_hv 2

}zⁱ

t^(s1)_hv 1

zⁱ

t^(s2)_hv 2

|t^(s1) hv1

−t^(s1)

hv1−1|^1/2|t^(s2) hv2

−t^(s2) hv2−1|^1/2

|[t^(s1) hv1−1,t^(s1)_hv

1 )∩[t^(s2)_hv

2−1,t^(s2)_hv 2

)| if t^(sh^v₁¹⁾= tv

or t^(sh^v₂²⁾= tv

( ˆΣt_v−1)s₁s₂ else

with h^vs = min{hs: t^(s)_h

s ≥ tv}. Remark:

• The number of updates for each component is dierent

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

Real data: volatility estimation in transaction time

46.647.047.4

10:00 11:00 12:00 13:00 14:00 15:00 16:00

1e−043e−045e−04

10:00 11:00 12:00 13:00 14:00 15:00 16:00

1e−043e−045e−04

09:40 09:50

Figure: The upper plot shows transaction data of Citigroup for the 3rd September 2007. The middle and the lower plot give the volatility estimators ˆΣt_j (black line), ˆΣ^St_j (red line), and the benchmark estimator ˆΣ^Btj (gray line).

Real data: volatility estimation in clock time

0.00000.0015

10:00 11:00 12:00 13:00 14:00 15:00 16:00

1e−043e−045e−04

13:00 14:00 15:00

0123456

13:00 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 2007. Simple clock time estimator ˆΣ^cSt_j (black line) and alternative clock time estimator ˆΣ^cS_alt(tj) = ˆΣ^S_tj/¯δj(red line). Lower plot: averaged duration times ¯δj(y-axis shows seconds).

Real data: cross-volatility estimation

−0.20.00.20.40.6

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0.000050.000150.00025

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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).

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

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

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?