multi-step-ahead prediction of volatility proxies · multi-step-ahead prediction of volatility...

Multi-step-ahead prediction of

volatility proxies

Jacopo De Stefani, Ir. - jdestefa@ulb.ac.beProf. Gianluca Bontempi - gbonte@ulb.ac.beOlivier Caelen, PhD - olivier.caelen@worldline.comDalila Hattab, PhD - dalila.hattab@equensworldline.com

Benelearn 2017

Eindhoven University of Technology, Eindhoven, Nethelands

Friday 9th June, 2017

Problem overview

25

30

35

40

45

First series CAC40 [2012−01−02/2013−11−04]

Last 47.255

Volume (100,000s):

345,721

0

10

20

30

40

50

Moving Average Convergence Divergence (12,26,9):

MACD: 1.335

Signal: 1.258

−3

−2

−1

0

1

2

3

Jan 022012

Mar 012012

May 022012

Jul 022012

Sep 032012

Nov 012012

Jan 022013

Mar 012013

May 022013

Jul 012013

Sep 022013

Nov 012013

What is volatility?

De�nition

Volatility is a statistical measure of the dispersion of returns for agiven security or market index.

0 20 40 60 80 1008

9

10

11

12High volatility Low volatility

t [days]

Pt

A closer look on data

0 0.2 0.4 0.6 0.8 1 1.2 1.4

9.8

10

10.2

P o0

P h0

P l0

P c0

P o1

P h1

P l1

P c1

Pre-opening

1− f f 1− f

Calendar Day 0 Calendar Day 1

t [days]

Pt

Volatility proxy

P otP htP ltP ct

σJt

Time series forecasting - Taieb [2014]

De�nition

Given a univariate time series {y1, · · · , yT } comprising Tobservations, forecast the next H observations {yT+1, · · · , yT+H}where H is the forecast horizon.

Hypotheses:

I Autoregressive model yt = f(yt−1, · · · , yt−d) + εt with lagorder d

I ε is a stochastic iid model with µε = 0 and σ2ε = σ2

Multistep ahead forecasting for volatilityState-of-the-art

m(σ)

· · · σJt−1][σJt−d

· · ·[σJt σJt+H ]

1 Input1 Output

Proposed method

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·[σJt σJt+H ]

M + 1 inputs1 output

Future work

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·· · ·· · ·

[σJt[· · ·[σXMt

σJt+H ]

· · · ]σXMt+H ]

M + 1 inputsM + 1 outputs

Multistep ahead forecasting for volatilityState-of-the-art

m(σ)

· · · σJt−1][σJt−d

· · ·[σJt σJt+H ]

1 Input1 Output

Proposed method

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·[σJt σJt+H ]

M + 1 inputs1 output

Future work

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·· · ·· · ·

[σJt[· · ·[σXMt

σJt+H ]

· · · ]σXMt+H ]

M + 1 inputsM + 1 outputs

Multistep ahead forecasting for volatilityState-of-the-art

m(σ)

· · · σJt−1][σJt−d

· · ·[σJt σJt+H ]

1 Input1 Output

Proposed method

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·[σJt σJt+H ]

M + 1 inputs1 output

Future work

m(σ)

· · ·· · ·· · ·

σXMt−1 ]

· · · ]σJt−1]

[σXMt−d

[· · ·[σJt−d

· · ·· · ·· · ·

[σJt[· · ·[σXMt

σJt+H ]

· · · ]σXMt+H ]

M + 1 inputsM + 1 outputs

Models for volatility

Volatility models

Pastvolatility

Average-based

HA

MAES

EWMA

STES

SimpleRegression

SR-AR

SR-TAR

SR-ARMA

RandomWalk

ARCH

Symmetric

ARCH(q)

GARCH(p,q)

Asymmetric

EGARCH(p,q)

GJR-GARCH(p,q)

QGARCH(p,q)

ST-GARCH(p,q)

RS-GARCH(p,q)Extended

Component-GARCH(p,q)

RGARCH(p,q)

MachineLearning

NN

k-NN SVR

GB

Models for volatility

Volatility models

Pastvolatility

Average-based

HA

MAES

EWMA

STES

SimpleRegression

SR-AR

SR-TAR

SR-ARMA

RandomWalk

ARCH

Symmetric

ARCH(q)

GARCH(p,q)

Asymmetric

EGARCH(p,q)

GJR-GARCH(p,q)

QGARCH(p,q)

ST-GARCH(p,q)

RS-GARCH(p,q)Extended

Component-GARCH(p,q)

RGARCH(p,q)

MachineLearning

NN

k-NN SVR

GB

Models for volatility

Volatility models

Pastvolatility

Average-based

HA

MAES

EWMA

STES

SimpleRegression

SR-AR

SR-TAR

SR-ARMA

RandomWalk

ARCH

Symmetric

ARCH(q)

GARCH(p,q)

Asymmetric

EGARCH(p,q)

GJR-GARCH(p,q)

QGARCH(p,q)

ST-GARCH(p,q)

RS-GARCH(p,q)Extended

Component-GARCH(p,q)

RGARCH(p,q)

MachineLearning

NN

k-NN SVR

GB

Past Research

Models for volatility

Volatility models

Pastvolatility

Average-based

HA

MAES

EWMA

STES

SimpleRegression

SR-AR

SR-TAR

SR-ARMA

RandomWalk

ARCH

Symmetric

ARCH(q)

GARCH(p,q)

Asymmetric

EGARCH(p,q)

GJR-GARCH(p,q)

QGARCH(p,q)

ST-GARCH(p,q)

RS-GARCH(p,q)Extended

Component-GARCH(p,q)

RGARCH(p,q)

MachineLearning

NN

k-NN SVR

GB

Past Research

Current Research

Proposed model

m(σ)

· · ·· · ·

σXt−1]

σJt−1]

[σXt−d

[σJt−d

· · ·[σJt σJt+H ]

2 TS Input1 TS Output

Volatility proxies σX , σJ :

I σi family - Garman and Klass[1980]

I GARCH (1,1) model - Hansenand Lunde [2005]

I Sample standard deviation

System overview

Missing values imputation

Proxy generation

Correlation analysisModel identi�cation

Model choice

Evaluation choice

Forecaster

Raw

OHLC data

Imputed

OHLC data

σit, σ

SDt , σG

t

{ANN,

KNN}

{RO,

RW}

m∗, θ∗

User choice

User choice

Data

preprocessing

System overview

Missing values imputation

Proxy generation

Correlation analysisModel identi�cation

Model choice

Evaluation choice

Forecaster

Raw

OHLC data

Imputed

OHLC data

σit, σ

SDt , σG

t

{ANN,

KNN}

{RO,

RW}

m∗, θ∗

User choice

User choice

Data

preprocessing

System overview

Missing values imputation

Proxy generation

Correlation analysisModel identi�cation

Model choice

Evaluation choice

Forecaster

Raw

OHLC data

Imputed

OHLC data

σit, σ

SDt , σG

t

{ANN,

KNN}

{RO,

RW}

m∗, θ∗

User choice

User choice

Data

preprocessing

Correlation analysis - CAC40 Time seriesMeta-analysis (cf. Field [2001]) across 40 time

series (CAC40)

?

?

?

?

?

?

?

?

?

?

?

?

?

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Volu

me

σ 1 σ 6 σ 4 σ 5 σ 2 σ 3 r t σ 0 σ SD

25

0

σ SD

10

0

σ SD

50

σ G

Volume

σ1

σ6

σ4

σ5

σ2

σ3

rt

σ0

σSD

250

σSD

100

σSD

50

σG

I Hierarchicalclusteringusing Ward Jr[1963]

I Time range:05-01-2009 to22-10-2014

I 1489 OHLCsamples perTS

I All thecorrelationsarestatisticallysigni�cant

NARX forecaster - ResultsNaive normalized MASE

σX ANN kNN ANNX kNNX GARCH(1,1)

σ6 0.07 0.08 0.06 0.11 1.34V olume 0.07 0.08 0.07 0.14 1.34σSD,5 0.07 0.08 0.07 0.09 1.34σSD,15 0.07 0.08 0.06 0.10 1.34σSD,21 0.07 0.08 0.06 0.10 1.34

Single CAC40 stock

I σJt = σG

t

I 10-step ahead

I 10-fold CV

I 05-01-2009⇒22-10-2014

Naive normalized MASE

σX ANN kNN ANNX kNNX GARCH(1,1)

σ6 0.58 0.49 0.53 0.56 1.15V olume 0.58 0.49 0.57 0.66 1.15σSD,5 0.58 0.49 0.58 0.58 1.15σSD,15 0.58 0.49 0.65 0.65 1.15σSD,21 0.58 0.49 0.56 0.65 1.15

S&P500 Index

I σJt = σG

t

I 10-step ahead

I 10-fold CV

I 01-04-2012 to30-07-2013 asin Dash andDash [2016]

Conclusions

I B Preliminary results

I CorrelationI Correlation clustering among proxies belonging to the same

family, i.e. σit and σ

SD,nt .

I ForecastingI Both machine learning methods outperform the benchmark

methods (naive and GARCH).I ANN can take advantage of the additional information

provided by the exogenous proxy better than k-NN

I Combination of proxies coming from di�erent families couldimprove forecast accuracy

I We are currently assessing the performances of the models fordi�erent forecasting horizons h and model orders d.

I Inclusion of a greater number of input TS as a future researchdirection.

Thank you for your attention! Any questions/comments?

Find the paper at:

Bibliography I

References

Tim Bollerslev. Generalized autoregressive conditionalheteroskedasticity. Journal of econometrics, 31(3):307�327,1986.

Rajashree Dash and PK Dash. An evolutionary hybrid fuzzycomputationally e�cient egarch model for volatility prediction.Applied Soft Computing, 45:40�60, 2016.

Andy P Field. Meta-analysis of correlation coe�cients: a montecarlo comparison of �xed-and random-e�ects methods.Psychological methods, 6(2):161, 2001.

Bibliography II

Mark B Garman and Michael J Klass. On the estimation of securityprice volatilities from historical data. Journal of business, pages67�78, 1980.

Peter R Hansen and Asger Lunde. A forecast comparison ofvolatility models: does anything beat a garch (1, 1)? Journal of

applied econometrics, 20(7):873�889, 2005.

Rob J Hyndman and Anne B Koehler. Another look at measures offorecast accuracy. International journal of forecasting, 22(4):679�688, 2006.

Souhaib Ben Taieb. Machine learning strategies for

multi-step-ahead time series forecasting. PhD thesis, Ph. D.Thesis, 2014.

Joe H Ward Jr. Hierarchical grouping to optimize an objectivefunction. Journal of the American statistical association, 58(301):236�244, 1963.

Appendix

Correlation analysis - Methodology

[σi(1), σSD(1), σG(1)

]

[σi(j), σSD(j), σG(j)

]

[σi(N), σSD(N), σG(N)

]

corr(·)

corr(·)

corr(·)

Meta-analysistoolkit

corr(σAGG)

corr(σ(1))

corr(σ(j))

corr(σ(N))

I 40 Time series (CAC40)I Time range: 05-01-2009 to 22-10-2014 ⇒ 1489 OHLC

samples per TS

NARX forecaster - Methodology

σJpOriginalDGP

Disturbances

d

Modelm∗(θ∗, σJ

p , σXp )

Structural

identi�cation

Parametric

identi�cation{ANN,KNN}

{RO, RW}

σXp

eσJf

σJf

m∗(·, σJp , σXp ) θ∗

Model identi�cation

Volatility proxies (1) - Garman and Klass [1980]

I Closing prices

σ0(t) =

[ln

(P

(c)t+1

P(c)t

)]2= r2t (1)

I Opening/Closing prices

σ1(t) =1

2f·

[ln

(P

(o)t+1

P(c)t

)]2︸ ︷︷ ︸

Nightly volatility

+1

2(1− f)·

[ln

(P

(c)t

P(o)t

)]2︸ ︷︷ ︸

Intraday volatility

(2)

I OHLC prices

σ2(t) =1

2 ln 4·

[ln

(P

(h)t

P(l)t

)]2(3)

σ3(t) =a

f·

[ln

(P

(o)t+1

P(c)t

)]2︸ ︷︷ ︸

Nightly volatility

+1− a1− f · σ2(t)︸ ︷︷ ︸Intraday volatility

(4)

Volatility proxies (2) - Garman and Klass [1980]

I OHLC prices

u = ln

(P

(h)t

P(o)t

)d = ln

(P

(l)t

P(o)t

)c = ln

(P

(c)t

P(o)t

)(5)

σ4(t) = 0.511(u− d)2 − 0.019[c(u+ d)− 2ud]− 0.383c2 (6)

σ5(t) = 0.511(u− d)2 − (2 ln 2− 1)c2 (7)

σ6(t) =a

f· log

(P

(o)t+1

P(c)t

)2

︸ ︷︷ ︸Nightly volatility

+1− a1− f · σ4(t)︸ ︷︷ ︸Intraday volatility

(8)

Volatility proxies (3)

I GARCH (1,1) model - Hansen and Lunde [2005]

σGt =

√√√√ω +

p∑j=1

βj(σGt−j)

2 +

q∑i=1

αiε2t−i

where εt−i ∼ N (0, 1), with the coe�cients ω, αi, βj �tted according to

Bollerslev [1986].

I Sample standard deviation

σSD,nt =

√√√√ 1

n− 1

n−1∑i=0

(rt−i − r)2

where

rt = ln

(P

(c)t

P(c)t−1

)rn =

1

n

t∑j=t−n

rj

Hyndman and Koehler [2006] - Errormeasures

Error measures

Scaleindependant

MAPE

MdAPE

RMSPE

RMdSPE

sMAPE

sMdAPE

Scaledependant

MSE

RMSE

MAE

MdAE

RelativeErrors

MRAE

MdRAE

GMRAE

MASE

RelativeMeasures

RelX

Percent-Better

Hyndman and Koehler [2006] - Scaledependant

Scale dependant

MSE

RMSE

MAE

MdAE et = yt − ytI MSE : 1

n

∑nt=0(yt − yt)2

I RMSE :√1n

∑nt=0(yt − yt)2

I MAE : 1n

∑nt=0 |yt − yt|

I MdAE :Mdt∈{1···n}(|yt − yt|)

Hyndman and Koehler [2006] - Scaleindependant

Scale independant

MAPE

MdAPE

RMSPE

RMdSPE

sMAPE

sMdAPE

I MAPE :1n

∑nt=0 | 100 ·

yt−ytyt|

I MdAPE :Mdt∈{1···n}(| 100 · yt−ytyt

|)

I RMSPE :√1n

∑nt=0(100 ·

yt−ytyt

)2

I RMdSPE :√Mdt∈{1···n}((100 · yt−ytyt

)2)

I sMAPE :1n

∑nt=0 200 ·

|yt−yt|yt+yt

I sMdAPE :Mdt∈{1···n}(200 ·

|yt−yt|yt+yt

)

Hyndman and Koehler [2006] - Relativeerrors

Relative Errors

MRAE

MdRAE

GMRAE

MASE

rt =ete∗t

I MRAE : 1n

∑nt=0 | rt |

I MdRAE : Mdt∈{1···n}(| rt |)

I GMRAE :n

√1n

∏t = 0n | rt |

I MASE :1T

∑Tt=1

(|et|

1T−1

∑Ti=2|Yi−Yi−1|

)

Hyndman and Koehler [2006] - Relativemeasures

Relative Measures

RelX

Percent-Better

I RelX : XXbench

I Percent Better :PB(X) =100 · 1n

∑forecasts I(X < Xb)

where

I X: Error measure of theanalyzed method

I Xb: Error measure of thebenchmark