Dissertation paperDissertation paper

Modelling and Forecasting Volatility Index

based on the stochastic volatility models

MSc Student: LAVINIA-ROXANA DAVID

Supervisor: Professor MOISA ALTAR, PhD.

Bucharest, 2006

Goals

to model and to forecast an index based on the stochastic volatility models

to analyse the predictive ability of these models one of them is based on implied volatility calculated from option

prices

to perform an empirical evidence for 2 indices:

S&P 100 index • analysis is performed based on volatility index VXO

BET index

Outline

Importance of volatility

Stochastic volatility modelsmodel specification and estimationvolatility forecasting methodology

Data description

Model estimates and volatility forecast results

Conclusions

Importance of volatility

volatility risk is considered as one of the prime and hidden risk factors on capital markets

volatility forecasting plays an important role in financial decision making

predicting volatility is quite difficult to be accurately performed

historical (realised) volatility and implied volatility

GARCH and Stochastic Volatility (SV) models

Stochastic volatility models

allow a stochastic element in the time series evolution of the conditional variance process

time-varying volatility breaches the constant volatility assumption underlying the Black-Scholes formula

by incorporating stochastic and implied volatility from options prices

new level of precision is reachednew level of precision is reached

o performance confirmed by Koopman and Hol (2002), Fleming (1998), Poon and Granger (2002)

Stochastic volatility models

mean eq.: ,t tt

yt =

variance eq.: 2 *2 exp( ), tt h 1 , t t th h

SV Model : 1 t t th h

SVX Model:

SIV Model: 1 t t th x

- based on historical returns

- implied volatility as explanatory variable

1 1(1 ) , t t t tL xh h

- obtained for 0

Model estimation

parameters are estimated by simulated maximum likelihood, using Monte Carlo importance sampling

SV/ SVX models - special cases of non-linear state space models:

where:

based on state vector:

1( ,...., ) Ty y y

2

1

( | , ) (0, ),

T

tt

p y N

*2 *2exp( ) exp( ) exp( ) h htt t t

*( , , ) ht t

1,0,1 t t1

( | , ) (0,exp ),

T

tt

p y N

1*

1 0 0 0

0 1 0 0

0

tt t

tx

*1(1 ) t t t tx L x x x

Model estimation

likelihood function of a SVX model is calculated via Monte Carlo technique of importance sampling:

o similar with likelihood function of an approx. Gaussian model (based on Kalman filter) multiplied by a correction term

( ) ( | ) ( , | ) ( | , ) ( | ) . L p y p y d p y p d

( , | ) ( | , ) ( , )( ) ( | ) ,

( | , ) ( | , )

gg y g y p

L g yg y g y

o computational implementation: Ox, SsfPack 2.2

given => ( , )

,

Volatility forecasting methodology

rolling window principle o one –step ahead volatility forecast:

2 *21| 1| 1|ˆ( ) exp(ln 0.5 ) T T T T T TE h p

o N-step ahead volatility forecast:

1|1, | 1|2 *2 exp 0.5 .ˆ T TT T N T T TE N h P

o measuring predictive forecasting ability:

o regression model:

2 2

1,1

N

T iT T Ni

RR

1,

221, | , T T N T T N Ta bE errorR

o error statistics: RMSE, MAE, Theil, Variance Prop, Covariance Prop

H0: a=0 si b=1

Data description

S&P 100 index VXO index

09/19/1997 - 12/30/2005

1100(ln ln ) t t tR P P

BET index

2060 daily returns

1100(ln ln ) t t tR P P2 2, / 252 IV t tVXO

2084 daily returns 2084 daily obs.

OEX

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

900.00

09/1

9/97

02/1

9/98

07/2

0/98

12/1

5/98

05/1

7/99

10/1

3/19

993/

13/2

000

08/0

9/00

01/0

8/20

0106

/07/

2001

11/0

8/20

0104

/11/

0209

/09/

0202

/06/

0307

/08/

2003

12/0

3/20

0305

/04/

2004

10/0

1/20

0403

/02/

0507

/29/

0512

/27/

2005

OEX

VXO

0

10

20

30

40

50

60

09/1

9/97

03/3

1/98

10/0

7/98

04/1

9/99

10/2

5/19

995/

3/20

0011

/08/

0005

/21/

2001

12/0

3/20

0106

/13/

2002

12/1

9/02

07/0

1/20

0301

/08/

2004

07/2

0/20

0401

/26/

0508

/04/

05

VXO

BET (09/19/97 - 12/30/05)

010002000300040005000600070008000

BET

Descriptive statistics S&P 100

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

0.1

0.2

0.3

0.4R_OEX

-8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

-5

0

5

Distribut ionR_OEX

R_OEX

-10-8-6-4-202468

01/0

1/00

07/0

1/00

01/0

1/01

07/0

1/01

01/0

1/02

07/0

1/02

01/0

1/03

07/0

1/03

01/0

1/04

07/0

1/04

01/0

1/05

07/0

1/05

R_OEX

0 5 10

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00ACF-R_OEX PACF-R_OEX

Daily return seriesHistogram and distribution of returnAutocorrelation and partial correlation of daily return

PeriodNo. of Obs. T

09/19/97 to 12/30/20052084

09/05/2000 – 12/30/20051338

Series S&P 100(OEX)

VXO S&P 100(OEX)

VXO

Rt Rt 2 Rt Rt

2

Mean 0.010478 1.6008 0.68134 -0.027518 1.4937 0.54602

Standard deviation

1.2652 3.5193 0.68765 1.2219 3.1832 0.78182

Skewness -0.046389 6.5425 -0.32503 0.19968 4.9532 0.00098878

Excess Kurtosis 2.8353 64.875 -0.39852 2.5621 31.849 -1.0014

Descriptive statistics BET R_BET

-15

-10

-5

0

5

10

15

9/19

/199

7

9/19

/199

8

9/19

/199

9

9/19

/200

0

9/19

/200

1

9/19

/200

2

9/19

/200

3

9/19

/200

4

9/19

/200

5

R_BET

0 5 10

-0.75

-0.50

-0.25

0.00

0.25

0.50

0.75

1.00ACF-R_OS_BET PACF-R_OS_BET

Daily return seriesHistogram and distribution of returnAutocorrelation and partial correlation of daily return

-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

10

20

30

40 DensityR_OS_BET

-0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06

-0.10

-0.05

0.00

0.05

DistributionR_OS_BET

PeriodNo. of Obs. T

09/19/1997 – 12/30/20052060

09/05/2000 – 12/30/20051314

Series BET BET

Rt Rt 2 Rt Rt

2

Mean 0.00091592 0.00031609 0.0020051 0.00020871

Standard deviation 0.017755 0.00088583 0.014307 0.00057780

Skewness -0.15528 7.3312 -0.33500 12.804

Excess Kurtosis 5.9173 71.519 6.0773 267.98

Empirical in-sample results

Period 09/19/97 - 12/30/2005 09/05/2000 - 12/30/2005

T 2084 1338

Model SVX

SIV

SVX

SIV

0.513901 0.52023 0.54687 0.55351

-0.54002 0 -0.77238 0

γ 1.1109 1.1115 1.1062 1.1146

0.073483

0.082911 0.010330 1.0440e-006

Period09/05/1997 – 12/30/2005

09/05/2000 – 12/30/2005

T 2060 1314

Model SV SV

1.8557 1.3967

0.93209 0.94676

γ 0.090144 0.076754

0.11798 0.077643

S&P 100 index BET index

*2*2

2 2

Volatility forecast resultsS&P 100 index

o based on the period: 09/05/2000 - 05/08/2005 (1172 obs.)o evaluation period: 05/09/2005 – 12/30/2005 ( 166 obs.) o based on the parameters of this initial sample

o roll it forward by one trading day, keeping the sample size constant at 1172 obs.

o horizon forecast of 1, 5 and 10 days

Volatility forecast resultsS&P 100 index

Forecasting Model

Forecasting Horizon

1 5 10

SVX Model R2 0.628585 0.142646 0.197848

RMSE 0.319605 1.522221 0.608028

MAE 0.261104 1.294826 0.470655

Theil 0.458094 0.319413 0.265894

Variance Prop 0.586634 0.451710 0.384273

Covariance Prop 0.413366 0.548290 0.615727

SIV Model R2 0.423336 0.235641 0.016977

RMSE 1.719710 0.319299 0.894773

MAE 1.305528 0.305976 0.767675

Theil 0.157008 0.191082 0.202357

Variance Prop 0.211649 0.346412 0.769449

Covariance Prop 0.788351 0.653588 0.230551

Volatility forecast results BET index

o based on the period 09/05/2000 - 05/04/2005 (1132 obs.)o evaluation period: 05/05/2005 – 12/21/2005 ( 158 obs.) o based on the parameters of this initial sample

o roll it forward by one trading day, keeping the sample size constant at 1132 obs.

o horizon forecast of 1, 5 and 10 days

Volatility forecast results BET index

Forecasting Model

Forecasting Horizon

1 5 10

SV Model

R2 0.972752 0.847383 0.523865

RMSE 0.164621 10.39685 8.760473

MAE 0.146685 6.951804 7.561559

Theil 0.056271 0.258109 0.159677

Variance Prop 0.006906 0.160237 0.041377

Covariance Prop 0.993094 0.839763 0.958623

Conclusions

Analysis of results SVX and SIV models are appropriate for prediction forecasting horizon N=1 provide a better prediction

than N=5 and N=10 (more relevant)

Forecast accuracy depends on: selecting periods as in-sample and out-of-sample selecting forecasting horizon, forecast evaluation measure volatilities ranges for in-sample unexpected change in future volatility is difficult to be predicted availability of data about implied volatility (volatility index)

Conclusions

Proposal and further research direction

better identification and selection of periods

intra-day data (high frequency data) instead of daily data

applying the model for a stock (instead of index)

• for which there are options on that stock (for calculating implied volatility)

Bibliography (selection)

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