Upload
sherman-lang
View
216
Download
0
Tags:
Embed Size (px)
Citation preview
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)
Alexander, C. (2001), “Market Models: Chapter 5: Forecasting Volatility and Correlation”
Andersen T. G., T. Bollerslev, P. F. Christoffersen and F. Diebold (2006), “Volatility and correlation forecasting”, Handbook of Economic Forecasting, Volume 1
Blair, B., S.-H. Poon and S. J. Taylor (2000), “Forecasting S&P 100 volatility: The Incremental Information Content of Implied. Volatilities and High Frequency Index Returns”
Fleming, J. (1998), “The quality of market volatility forecast implied by S&P100 index option prices”
Ghysels, E., A. Harvey and E.Renault (1995), “Stochastic Volatility” Jungbacker, B. and S. J. Koopman, “Monte Carlo likelihood estimation for three mult
Jungbacker, B. and Koopman S. J., “On Importance Sampling for State Space Models”
Bos, C. (2006), “The method of Maximum Likelihood” Koopman, S. J. and E. Hol (2002), “Forecasting the Variability of Stock Index Returns
with Stochastic Volatility Models and Implied Volatility” Koopman, S. J. and N. Shephard (2004), “Estimating the likelihood of the stochastic
volatility model: testing the assumptions behind importance sampling” Koopman, S. J. (2005), “Introduction to State Space Methods”
Bibliography (selection)
Jungbacker B. and S. J.Koopman, “On Importance Sampling for State Space Models” Holger, C., Mittnik S. (2002), “Forecasting Stock Market Volatility and the
Informational Efficiency of the DAX index Options Market” Hull, J. , “Options, futures and other derivatives” Koopman, S. J., “Modelling Volatility in Financial Time Series: Daily and Intra-daily
Data” Koopman, S. J., N. Shephard and J. Doornik (1998), “Statistical algorithms for
models in state space using SsfPack 2.2” Koopman, S. J., M. Ooms and M.-J. Boes (2006), “Case Econometrics and
Quantitative Finance: Modeling Volatility for Forecasting and Option Pricing, MSc. Econometrics”
Koopman, S. J., B. Jungbacker and E. Hol (2004), “Measuring price and volatility from high-frequency stock prices”
Koopman, S. J. and K. M. Lee, “Simulated Maximum Likeliood in Stochastic Volatility Modelling”
Lopez, J. A. (1999), “Evaluating the Predictive Accuracy of Volatility Models” Notger C. (2004), “Volatility and its Measurements: The Design of a Volatility Index
and the Execution of its Historical Time Series at the DEUTSCHE BORSE AG”, im Fach Bank-, Finanz- und Investitionswirtschaft in WS 2004/2005
Poon, S.-H. and , C. W. J Granger (2003), “Forecasting Volatility in Financial Markets: A Review”