Bootstrap in Finance
Esther Ruiz and Maria Rosa Nieto
(A. Rodríguez, J. Romo and L. Pascual)
Department of Statistics
UNIVERSIDAD CARLOS III DE MADRID
Workshop Modelling and Numerical
Techniques in Quantitative Finance
A Coruña
15 de octubre de 2009


Motivation
Bootstrapping to obtain prediction densities of
future returns and volatilities




GARCH
Stochastic volatility
Bootstrapping to measure risk: VaR and Expected
Shortfall
Conclusions
1. Motivation

High frequency time series of
returns are characterized by
volatility clustering:


Excess kurtosis
Significant autocorrelations of absolute
returns (not independent)
Daily Returns of Euro/Dollar from 02/09/2003
to 25/09/2009
6
4
15
Daily Returns of IBEX35 from 02/09/2003 to
25/09/2009
10
2
5
0
0
-2
-5
-10
-4
-15
700
Series: IBEX35
Sample 1 1538
Observations 1538
350
Series: EURODOLLAR
Sample 1 1582
Observations 1582
300
250
200
150
100
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.018446
0.027891
3.718798
-2.796839
0.639161
0.231583
5.464342
Jarque-Bera
Probability
414.4515
0.000000
600
500
400
300
200
100
50
0
-2.50
-1.25
0.00
1.25
2.50
3.75
0
-10
0.4
0.25
0.2
0.15
Correlogran of absolute returns Euro/Dollar
-5
0
5
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
0.031305
0.097436
10.11763
-9.585865
1.377396
-0.145046
12.45961
Jarque-Bera
Probability
5739.843
0.000000
10
Correlogram of absolute returns IBEX35
0.3
0.2
0.1
0.1
0.05
0
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
0


Financial models based on inference
on the dynamic behaviour of returns
and/or predictions of moments
associated with the density of future
returns which assume independent
and/or Gaussian observations are
inadequate.
Bootstrap methos are attractive in
this context because they do not
assume any particular distribution.

However, bootstrap procedures cannot be based on
resampling directly from observed returns as they are not
indepedent.



Assume a parametric specification of the dependence and
bootstrap from the corresponding residuals.
Generalize the bootstrap procedures to cope with
dependent observations
Li and maddala (1996) and Berkowitz and Kilian (2000)
show that the parametric approach could be preferable in
many applications.

Consequently, we focus on two of
the most popular and simpler
models to represent the dynamic
properties of financial returns:
yt   t t
GARCH(1,1)
 t2    yt21   t21
ARSV(1)
log  t2     log  t21  t
GARCH(1,1)-N: 0.5, 0.15, 0.8
4
3
SV: 0.05, 0.95, 0.05
10
2
1
5
0
0
-1
-5
-2
-10
-3
240
-4
Series: SV
Sample 1 1500
Observations 1500
-15
200
200
Series: GN
Sample 1 1500
Observations 1500
160
120
80
160
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.011948
0.024450
3.387700
-3.136800
0.899944
0.014030
3.501793
120
Jarque-Bera
Probability
15.78648
0.000373
0
80
40
40
0
-2.50
0.15
-1.25
0.00
1.25
-10.0
-7.5
-5.0
-2.5
0.0
2.5
5.0
Mean
Median
Maximum
Minimum
Std. Dev.
Skewness
Kurtosis
-0.060025
-0.007813
8.578551
-9.568826
1.830458
-0.131483
4.666630
Jarque-Bera
Probability
177.9254
0.000000
7.5
2.50
Correlogram of absolute returns
GARCH(1,1)-N
0.1
0.14
Correlogram of absolute returns SV
0.12
0.1
0.08
0.05
0.06
-0.1
0.04
0.02
0
-0.02
-0.04
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
-0.05
1
4
7
10
13
16
19
22
25
28
31
34
37
40
43
46
49
52
55
58
61
64
67
70
73
76
79
82
85
88
91
94
97
100
0

There are two main related areas of application of
bootstrap methods
 Inference: Obtaining the sample distribution of
a particular estimator or a statistic for testing,
for example, the autoregressive dynamics in
the conditional mean and variance, unit roots in
the mean, fractional integration in volatility,
inference for trading rules: Ruiz and Pascual
(2002, JES)
 Prediction: Prediction of future densities of
returns and volatilities: VaR and ES

Bootstrap methods allow to obtain
prediction densities (intervals) of future
returns without distributional assumptions
on the prediction error distribution and
incorporating the parameter uncertainty
 Thombs and Schucany (1990, JASA):
backward representation
 Cao et al. (1997, CSSC): conditional on
estimated parameters
 Pascual et al. (2004, JTSA): incorporate
parameter undertainty without backward
representation
Example AR(1)
The Minimum MSE predictor of yT+k based on the
information available at time T is given by its
conditional mean
~
yT  k    ~
yT  k 1
where
~
yT  yT
Predictions are made conditional on the available data
In practice, the parameters are substituted by consistent
estimates. Therefore, the predictions are given by
yˆ T k  ˆ  ˆyˆ T k 1
Thombs and Schucany (1990)
yˆ T*  k   *   * yˆ T*  k 1  aT*  k
*
a
where T  k are bootstrap replicates of the
standardized residuals and  * ,  * are
obtained from bootstrap replicates of the
series based on the backward
representation
yt*  ˆ  ˆyT* 1  aT*
where
yT*  yT
Should we fix yT when bootstrapping the parameters???
Pascual et al. (2004, JTSA) propose a bootstrap
procedure to obtain prediction intervals in ARIMA
models that do not requiere the backward
representation. Therefore, this procedure is
simpler and more general, as it can cope with
models for which the backward representation
does not exist as, for example, GARCH.
2. Bootstrap forecast of future returns
and volatilities
We consider the prediction of future returns
and volatilities generated by GARCH and
ARSV(1) models
2.1 GARCH
2
2
2
 t    yt 1   t 1
2.2 ARSV
log  t2     log  t21  t
Both models provide prediction intervals
which are narrow in quite times and wide
in volatile periods.
2.1 GARCH (1,1): Pascual et al. (2005, CSDA)
Consider again the GARCH(1,1) model given by
yt   t t
Therefore,
 t2    yt21   t21
yt | y1 ,..., yt 1  G(0,  t2 )
Assuming conditional Normality of returns:
· one-step ahead prediction errors are Normal.
· prediction errors for two or more steps ahead are not
Normal.
· one-step-ahead volatilities only have associated parameter
uncertainty.
· volatilities more than one-step ahead also have uncertainty
about future errors.
Bootstrap procedure
Estimate parameters and obtain standardized residuals
Obtain bootstrap replicates of the series


 t*2  ˆ  ˆyt*21  ˆ t*21
yt*   t* t*
and estimate the parameters.
 Bootstrap forecasts of future returns and volatilities
 T*2 k  ˆ *  ˆ * yT*2 k 1  ˆ * T*2 k 1
yT*  k   T*  k T*  k
yT*  yT

*2
T
*
T 2


ˆ
ˆ *

*
*
j
2
ˆ y

ˆ






T  j 1
*
*
*
* 

ˆ
ˆ
1  ˆ  
1  ˆ   
j 0

Using the bootstrap
estimates of the
parameters with the
original observations
(conditional)
One-step-ahead predictions of returns of IBEX35
Five-step-ahead predictions of returns of IBEX35
0.7
0.7
prediction returns
N(0,sigmat+1)
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-8
-6
-4
-2
0
2
prediction returns
N(0,sigmat+5)
0
-8
4
One-step-ahead predictions of volatility of IBEX35
-6
-4
-2
0
2
4
6
Five-step-ahead predictions of volatility of IBEX35
8
2.5
7
2
6
5
1.5
4
1
3
2
0.5
1
0
0.7
0.8
0.9
1
1.1
1.2
1.3
0
0
2
4
6
ˆ  0.0401, ˆ  0.1271, ˆ  0.8481
8
10
12
One-step-ahead predictions of returns of Euro/Dollar
Five-step-ahead predictions of returns of Euro/Dollar
1
0.9
prediction returns
N(0,sigmat+1)
0.9
prediction returns
N(0,sigmat+5)
0.8
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0
-3
-2
-1
0
1
2
0
-3
3
One-step-ahead predictions of volatility of Euro/Dollar
-2
-1
0
1
2
3
Five-step-ahead predictions of volatility of Euro/Dollar
20
8
18
7
16
6
14
5
12
10
4
8
3
6
2
4
1
2
0
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0
0.1
0.2
0.3
0.4
0.5
ˆ  0.0026, ˆ  0.0583, ˆ  0.9387
0.6
0.7
0.8
2.2 ARSV(1) models
The ARSV(1) model can be linearized by taking logs of
squares
yt   * t  t
log yt2  log  *  log  t2  log  t2
log  t2   log  t21  t
log  t2   log  t21   t
Bootstrap methods for unobserved component models
are much less developed. Previous procedures
cannot be implemented due to the presence of
several disturbances.
In this context, the interest is not only to construct
densities of future values of the observed variables
but also of the unobserved components.


The Kalman filter provides
 one-step-ahead (updated and smoothed)
predictions of the series together with their MSE
 estimates of the latent components and their
MSE
Bootstrap procedures can be implemented to obtain
densities of
 Estimates of the parameters
 Prediction densities of future observations: Wall
and Stoffer (2002, JTSA), Rodríguez and Ruiz
(2009, JTSA)
 Prediction densities of underlying unobserved
components: Pferfferman and Tiller (2005, JTSA),
Rodríguez and Ruiz (2009, manuscript)

Rodríguez and Ruiz (2009, JTSA)
propose a bootstrap procedure
to obtain prediction intervals of
future observations in
unobserved component models
that incorporate the parameter
uncertainty without using the
backward representation.
The proposed procedure consists on the
following steps:
1) Estimate the parameters by QML, ˆ and
obtain the standardized innovations, ˆts
2) Obtain a sequence of bootstrap replicates
of the standardized innovations, ˆt* s
3) Obtain a bootstrap replicate of the series
using the IF with the estimated
parameters ˆ Estimate the parameters,
obtaining ˆ * and aˆT*|T 1
4) Obtain the conditional bootstrap
predictions


However, as we mentioned before, when
modelling volatility, the objective is not
only to predict the density of future returns
but also to predict future volatilities.
Therefore, we need to obtain prediction
intervals for the unobserved components.
At the moment, Rodríguez and Ruiz (2009,
manuscript) propose a procedure to obtain
the MSE of the unobserved components.
Consider, for example, the random walk
plus noise model:
yt   t   t
 t   t 1   t
In this case, the prediction intervals are
given by
Estimated parameters
Normality assumption
Random walk with q=0.5
Estimates of the level and 95% confidence intervals: In red with
estimated parameters and in black with known parameters.


Our procedure is based on the following
decomposition of the MSE proposed by
Hamilton (1986)
He proposes to generate replicates of the
parameters from the asymptotic
distribution and then to estimate the MSE
by
The filter is run with original observations

We propose a non-parametric
bootstrap in which the bootstrap
replicates of the series are obtained
from the innovation form after
resampling from the innovations.
3. Var and ES
In the context of financial risk management, one of
the central issues of density forecasting is to track
certain aspects of the densities as, for example, VaR
and ES. Consider the GARCH(1,1) model
yt   t t
 t2    yt21   t21
In this context the VaR and ES are given by
VaRt   t q
ESt   t E  t |  t  VaRt 
T 1
In practice, assuming that the model is
known, both the parameters and the
distribution of the errors are unknown.
Therefore, we obtain the estimates
VaˆRt  ˆ t qˆ



ES t  ˆ t E  t |  t  VaR t 
t 1

Bootstrap procedures have been proposed to
obtain point estimates of the VaR by
computing the corresponding quantile of
the bootstrap distribution of returns (Ruiz
and Pascual, 2004, JES)


Bootstrap procedures can also be implemented to
obtain estimates of VaR and ES together with their
MSE.
Christoffersen and GonÇalves (2005, J Risk) propose
to compute bootstrap replicates of the VaR by
VaˆRt  ˆ t*qˆ 0*.01
where
q0.01 ( particular distributi on)

ˆ H

0
.
01



 ˆ(*k 1) 
( Hill estimator)


k
/
T


qˆ0*.01  
2
ˆ
ˆ

 ˆ
2
2
2



1

1
2
1
 0.01 1 
 0.01  3 
2  0.101  0.5   1  0.101  1

36
 24
 6

*
ˆ

(
0.01T )



 





 t*
ˆ  y / ˆ   *
ˆ t
*
t
*
t
*
t
*
t
Instead of using the residuals obtained in each of
the bootstrap replicates of the original series,
Nieto and Ruiz (2009, manuscript) propose to
estimate q0.01 by a second bootstrap step.
For each bootstrap replicate of the series of returns,
we obtain n random draws from the empirical
distribution of the original standardized residuals, ˆt
Then, the constant q0.01 can be estimated by any
of the three alternative estimators described
before.
In this way, we avoid the estimation error involved
*

in the residuals
ˆt*   t* t*
ˆ t
Monte Carlo experiments:
Coverages
95% VaR
d
T
N
S-8
Normal FHS
CF
B-FHS
B-CF
250
85%
88.1%
85.8%
93.6%
90.3%
500
79.6
90.3%
88.5%
92.7%
87.3%
250
84.9%
89.1%
82.8%
93.3%
87.6%
500
76.8%
87.7%
88.8%
95.2%
90.1%
95% ES
d
N
S-8
T
Normal FHS
CF
B-FHS
B-CF
250
88.5%
75.8%
92%
79.3%
90.8%
500
79.6%
81.1%
86%
82.1%
88.6%
250
84.9%
67.8%
97.2%
72.8%
99.4%
500
76.8%
72.9%
99.2%
81.7%
97.9%
Prediction bootstrap densities of VaR and ES for IBEX35
5
4.5
VaRBootstrap
VaRCGN
VaRCGH
VaRCGFHS
VaRCGGCCF
4.5
4
3.5
ESBootstrap
ESCGGN
ESCGGH
ESCGGFHS
ESCGGGCCF
4
3.5
3
3
2.5
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
-4.5
-4
-3.5
-3
-2.5
-2
0
-5
-1.5
-4.5
-4
-3.5
-3
-2.5
-2
-1.5
-1.2
-1
Prediction bootstrap densities of VaR and ES for Euro/Dollar
5
4
VaRBootstrap
VaRCGN
VaRCGH
VaRCGFHS
VaRCGGCCF
4.5
4
3.5
ESBootstrap
ESCGN
ESCGH
ESCGFHS
ESCGGCCF
3.5
3
2.5
3
2.5
2
2
1.5
1.5
1
1
0.5
0.5
0
-2.2
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
0
-3
-2.8
-2.6
-2.4
-2.2
-2
-1.8
-1.6
-1.4
-1
Conclusions and further research


Few analytical results on the statistical
properties of bootstrap procedures when
applied to heterocedastic time series
Further improvements in:



bootstrap estimation of quantiles and expectations
to compute the VaR and ES
construction of prediction intervals for unobserved
components (stochastic volatility)
Multivariate extensions: Engsted and
Tanggaard (2001, JEF)