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University of New South Wales
Bayesian Filtering on Realised,Bipower and Option Implied
Volatility
Honours Student:Nelson Qu
Supervisors:Dr Chris Carter
Dr Valentyn Panchenko
1 Declaration
I hereby declare that this submission is my own work and to the best ofmy knowledge it contains no material previously written by another person,or material which to a substantive extent has been accepted for the awardof any other degree or diploma of a university or other institute of higherlearning, except where referenced in the text.
I also declare that the intellectual content of this thesis is the productof my own work, and any assistance that I have received in preparing theproject, writing the program as well as presenting the thesis, has been dulyacknowledged.
−−−−−−−−−−−Nelson Qu
1
Contents
1 Declaration 1
2 Acknowledgements 5
3 Abbreviations 5
4 Abstract 6
5 Introduction 7
6 Literature Review 126.1 Stochastic Volatility Models . . . . . . . . . . . . . . . . . . . 126.2 Volatility Stylised Facts . . . . . . . . . . . . . . . . . . . . . 146.3 Kalman Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 156.4 Markov Chain Monte Carlo Methods(MCMC) . . . . . . . . . 166.5 Sequential Monte Carlo Methods/ Particle Filtering . . . . . . 18
7 Models 227.1 Stochastic Volatility Model (SVM) . . . . . . . . . . . . . . . 237.2 Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . 247.3 Bipower Volatility . . . . . . . . . . . . . . . . . . . . . . . . . 267.4 VIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
8 Data 298.1 Data transformation . . . . . . . . . . . . . . . . . . . . . . . 31
9 Method 349.1 MCMC Method . . . . . . . . . . . . . . . . . . . . . . . . . . 34
9.1.1 Outline of MCMC Method . . . . . . . . . . . . . . . . 349.2 Particle Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9.2.1 Outline of Particle Filter Methods . . . . . . . . . . . . 359.2.2 Liu and West Filter . . . . . . . . . . . . . . . . . . . . 369.2.3 Particle Learning Filter . . . . . . . . . . . . . . . . . . 37
9.3 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 389.3.1 How observation errors are distributed . . . . . . . . . 39
10 Results 4110.1 Marginal Log Likelihoods . . . . . . . . . . . . . . . . . . . . . 4110.2 Sequential Bayes Factor . . . . . . . . . . . . . . . . . . . . . 4610.3 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 48
2
10.4 Forecast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11 Robustness and Analysis 5611.1 Prior Sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 5611.2 Prior distribution difficulty for Option Implied Volatility . . . 5711.3 Number of iterations . . . . . . . . . . . . . . . . . . . . . . . 5811.4 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
12 Conclusion 60
13 Appendix 6213.1 Prior Specifications . . . . . . . . . . . . . . . . . . . . . . . . 6213.2 Particle Learning Filter . . . . . . . . . . . . . . . . . . . . . . 6313.3 Liu and West Filter . . . . . . . . . . . . . . . . . . . . . . . . 65
List of Figures
1 Difference in Realised and Bipower volatility of S&P500 Index 272 Graph of Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 QQ plots of Data and Log Data . . . . . . . . . . . . . . . . . 334 Sequential Bayes Factor . . . . . . . . . . . . . . . . . . . . . 475 Sequential Bayes Factor . . . . . . . . . . . . . . . . . . . . . 476 Sequential bayes factor . . . . . . . . . . . . . . . . . . . . . . 487 Liu and West: Latent State of Realised Volatility Model B . . 498 Particle Learning: Latent State of Realised Volatility Model B 499 MCMC: Latent State of Realised Volatility Model B . . . . . . 5010 Liu and West: Latent State of Bipower Volatility Model B . . 5011 Particle Learning: Latent State of Realised Volatility Model B 5112 MCMC: Latent State of Bipower Volatility Model B . . . . . . 5113 Liu and West: Latent State of Option Volatility Model B . . . 5214 Particle Learning: Latent State of Realised Volatility Model B 5215 MCMC: Latent State of Option Volatility Model B . . . . . . 5216 Probabilistic Forecast of Realised Volatility . . . . . . . . . . . 5317 Probabilistic Forecast of Bipower Volatility . . . . . . . . . . . 5418 Probabilistic Forecast of Option Implied Volatility . . . . . . . 5419 Particle Learning: Parameter history of Realised Volatility
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6320 Particle Learning: Parameter history of Bipower Volatility
Model B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6321 Particle Learning: Parameter history of Option Volatility Model
B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
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22 Liu and West: Parameter history of Realised Volatility Model B 6523 Liu and West: Parameter history of Bipower Volatility Model B 6524 Liu and West: Parameter history of Option Volatility Model B 66
List of Tables
1 Summary of S&P 500 data . . . . . . . . . . . . . . . . . . . . 312 Transformed data . . . . . . . . . . . . . . . . . . . . . . . . . 313 Marginal Log Likelihood for Realised Volatility . . . . . . . . 434 Marginal Log Likelihood for Bipower Volatility . . . . . . . . . 445 Marginal Log Likelihood for Option Implied Volatility . . . . . 456 Realised volatility: Posterior estimates of parameters . . . . . 487 Bipower volatility: Posterior estimates of parameters . . . . . 508 Option Implied volatility: Posterior estimates of parameters . 519 Forecast deviation measures . . . . . . . . . . . . . . . . . . . 5310 Marginal Likelihood sensitivity to changes in Prior . . . . . . 57
4
2 Acknowledgements
My sincere gratitude goes to both Dr Valentyn Panchenko and Dr Chris
Carter for their supervision and assistance in helping my understanding in
the bayesian filtering literature field, without their guidance I do not think
I would have been able to finish my honours year. Dr Gael Martin and
Worapree Maneesoonthorn from Monash University for their willingness to
share their dataset and their helpful suggestions in modelling volatility. My
gratitude towards Dr Christopher Strickland from Queensland University of
Technology for his help on PythonTM . I would also like to thank my parents
for their continued support.
3 Abbreviations
LEGENDS:
LW= Liu and west filter
PL=Particle Learning Filter
MCMC=Markov chain Monte Carlo
t = student t-distributed observation errors
A, B, C, D, E and F = models A, B, C, D, E and F respectively
Example:
PL-C t=Particle learning filter for model c with student t-dstributed obser-
vation errors
LW-B = Liu and West filter for model B with normally distributed observa-
tion errors
5
4 Abstract
Volatility plays an important role specifically in the finance sector it is used
for pricing securities and managing portfolio risk. We will explore three dif-
ferent Bayesian filter methods developed over the last two decades which
are the forward filtering backward sampling (MCMC) algorithm (1994), the
Liu and West filter (2001) and the particle learning algorithm (2010). Using
these three filtering methods we will apply these methods on three variance
measures of the S&P 500 index the realised, bipower and option implied
volatility to see how to Bayesian filters fare in filtering real world variance
measures to determine the integrated variation (the unobserved/true volatil-
ity state of the S&P 500). However stochastic volatility is only one important
topic in Economics, another is model selection and forecasting so at the same
time we will explore stochastic volatility selection and how the particle filter
performs in forecasting in a latest financial crisis(2007).
6
5 Introduction
Bayesian filtering is a recursive method to solve a dynamic model by filtering
out observation and state transition noises to determine an unobserved latent
state variable, these unobserved latent state variables can be the true state of
volatility or the Non-Accelerating Inflation Rate of Unemployment (NAIRU).
The current Bayesian filtering methods are in the category of the Kalman
filter, Markov chain Monte Carlo method and the Particle filter. There are
many problems where the Bayesian filtering is applied to solve real world
problems such as tracking, GPS, robotics and stochastic volatility modelling.
The last example which is stochastic volatility modelling will be explored in
this paper particular on the intention of model selection and the effectiveness
and efficiency of the Markov chain Monte Carlo compared to the particle
filter. Another difference in this paper is unlike many previous papers such as
Kim et al (1998)[24], Johannes and Polson(2002) [21] and Nakajima(2010) we
will investigate filtering three different variance measures realised volatility,
bipower volatility and the option implied volatility of the S&P 500 index to
determine the integrated variance of the S&P 500 index.
Why are we interested in the latent state when we already have non-
parametric measures of volatility like the standard variance formula or the
realised variance formula. The motivation for the study of latent volatil-
ity state is that there are observation errors in real data collection as such
it transfers to the non-parametric measures of variation so we can either
underestimate or overestimate volatility in a time period. This becomes a
problem particularly in the business sector where volatility is used to account
for portfolio risk, business forecasting and to price financial derivatives such
as futures and options. As such the latent volatility state can be described as
the true volatility state of the variation measure of the time series when we
filtered out the observation and state transition errors via a Bayesian filtering
method.
The literature in stochastic volatility modelling began as early as 1986
7
by Taylor[31], Taylor in 1986 first suggested to solve the stochastic volatility
problem by adjusting the dynamic model to be linear as such via the Kalman
filter we can approximately filter the latent volatility state from the returns of
the security prices. Since the model was adjusted to be linear the estimated
results were not the best maximised likelihood estimate. The Kalman filter
required both the dynamic model to be linear and the observation and state
transition errors to be normally distributed to find the best linear estimator
as such it was restrictive method for data filtering.
Jacquier, Polson and Rossi (2004) [19] and Kim, Shephard and Chib
(1998)[24] are one of first papers to apply Markov chain Monte Carlo (MCMC)
to recursively solve for the latent volatility state for stochastic volatility mod-
els, in both papers they showed that the MCMC method is able to solve more
general dynamic models that are non-linear and are not normally distributed.
With the ability to model more general models, development in the litera-
ture occurred to try to filter volatility jumps and leverage. Shephard and
Pitt(1999) later applied stochastic volatility modelling through the particle
filter developed by Gordon, Salmond and Smith (1993). The particle filter is
the same as the MCMC method as it is possible to adjust the problem for the
particle filter to solve. Over the last two decade most papers in stochastic
volatility filtering looked at the observation errors as normally distributed
models however only recently such as Nakajima (2010) looked at fat-tailed
distributed models like the student t-distribution for the observation errors
using the Markov chain Monte Carlo method.
There are a variety of models available in the stochastic volatility mod-
elling literature field however this can become a complicated process if we
are unsure which models to apply in certain situations. However model se-
lection is not explored thoroughly in this area of research as the literature
in filtering has only been recent. Therefore it is important to look at model
selection as current literature in the field of stochastic volatility modelling
do not look at model selection and hence usually papers do not justify their
selected model so in this paper we will explore the reason of the different
8
volatility models and the method to select different volatility models. Since
we are applying the a Bayesian filter to determine the latent volatility state
in the S&P 500 index we would have the benefit of many Bayesian statisti-
cal tools for model comparison such as the marginal log-likelihood and the
sequential Bayes factor.
The Markov Chain Monte Carlo method is a recursive method to solve dy-
namic model problems, Carter and Kohn (1994) [8] and Furthwirth-Schnatter
(1994) [14] developed the forward filtering and Backward sampling algorithm.
The forward filtering backward sampling algorithm can be described as we
start with the fixed parameters which we will use to recursively solve a sys-
tem. This creates one path of the behaviour of the latent state but the
parameters could be wrong so then the parameters are resampled according
to either Gibbs-Sampling algorithm or the Metropolis-Hastings algorithm to
get new parameters and then it is repeated to solve for the latent state.
The sequential monte carlo/particle filter as stated before was developed by
Gordon, Salmond and Smith (1993) [16] it is similar to the MCMC method
however instead of performing one path of estimation of the latent volatility
state we perform multiple paths of the latent state but for only one time
step. We later resample the each path and keep the good paths and drop
the bad paths. This is where the name sequential Monte Carlo comes from
as the states are learnt over each time step, we will sample the states and
parameters for the next time step then resample them when we have new
observations. In most cases the particle filter will perform much faster than
the MCMC method however the particle filter may requires a large size of
particles/paths to be able to perform consistent and efficient estimates of the
state and parameters. Therefore to sample the posterior distribution of the
system via a particle filter it may require a highly intensive memory machine
to get very consistent posterior estimates.
Investigating the S&P 500 from 2 July 1996 to 31st December 2008 we
will have modelling the latent state over multiple Financial crises such as the
1997 Asian Financial Crisis, the 2000 Dot-Com bubble burst and the recent
9
2007 Global financial Crisis. We will consider three different variance mea-
sures of the S&P 500 which are Realised volatility, Bipower volatility and
the Option implied volatility. The realised volatility is describe as historical
returns of a financial security for a period of time. The bipower volatility is
defined as the adjacent intraday returns of a financial security and an inter-
esting property between bipower volatility and realised volatility is that the
difference between the distribution of the two volatility measures represents
volatility jumps. The option implied volatility is from the spot based volatil-
ity index (VIX) from the Chicago Board of Option Exchange (CBOE). We
should expect that the latent volatility state from both realised volatility,
bipower volatility and the option implied volatility to be the same since they
are all measuring the same financial security. However it is not the case,
both realised and Bipower volatility do share the same parameter estimates
and latent state movement however the Option implied volatility behaves
differently to the other two variance measures.
After modelling for our three volatility measurements we can say that
a parsimonious model such as a local level model can be as effective as a
more complicated non-linear Heston stochastic volatility model. Through
the sequential bayes factor and the marginal log-likelihood the best model
according to these results suggests that model B which is an autoregressive
one process to be able to capture the behaviour of the variance measures.
This implies that behaviour between the volatility measures are similar even
though their numbers are different we can also say autoregressive of order one
process volatility measures are very similar to the true integrated variation,
therefore proving that Anderson et al [2] studies are true.
Using the particle filter we generated the predictive forecast of the three
variance measures over the volatile period in the 2008 if you look in section
10. It is fair to say that the predictive probabilistic forecast covers almost
all the real world variance movement.
With the limited amount of comparison tool available for model selection
10
for particle filters we made further checks to see if our results are consistent.
Our robustness check illustrates that the priors of the parameter distributions
can play a big part in determining model selection but at the same time the
priors can affect the model’s fit. Though it did cause a lose of fit in the model
during our robustness check but it did not affect the rank of the preferred
models so we believe our results are consistent.
11
6 Literature Review
6.1 Stochastic Volatility Models
In finance volatility plays an important role in managing portfolio risk,
derivative security pricing and analysing asset risks. As such development in
volatility modelling over the last three decades has been rich, with models
modelling stochastic/changing variances by methods of looking at conditional
variance and stochastic variance.
Two main methods for volatility modelling; the first popular method was
the autoregressive conditional heteroskedasticity(ARCH) model and general
autoregressive conditional heteroskedasticity(GARCH) model developed by
Engle [12], it is used to model for conditional variance. Conditional variance
is the variance calculated given that we have information of past returns. The
ARCH/GARCH models was successful in modelling time varying volatility
clustering a stylised fact of volatility. However volatility has a few more
stylised facts that were not successfully captured in ARCH/GARCH models
so further extensions of the ARCH/GARCH models were created to take
into account of leverage effects, long memory, persistency and Engle’s other
volatility stylised facts.
The other method for volatility modelling is via a Bayesian approach
which is explored in this paper. A Bayesian approach would be to evaluate
the likelihood/fit for unknown parameters, it may sound simple but this ap-
proach could be difficult as there are no exact forms of the likelihood function
and we would be required to integrate a t-dimensional integral (t periods of
observations). To be able to evaluate a difficult integral a method to approx-
imate the sampling distribution of the likelihood was developed and these in-
clude the sequential Monte Carlo and Markov Chain Monte Carlo methods.
Another method is application of a Quasi-Maximum likelihood estimation
via a Kalman filter to solve the stochastic volatility model however, since
12
the Kalman filter only can solve for linear and gaussian distributed errors
the results that can be found in Harvey et al(1994) [17] says is that it only
yields a minimum mean square linear estimator(MMSLE) of the underlying
volatility not the minimum mean square estimators (MMSE). As a result,
the Kalman filter would work well if the series filtered was linear otherwise
it is not the best method to use for non-linear series.
In volatility modelling, we generally use terminologies like realised volatil-
ity, bipower volatility and quadratic variance. Realised volatility is defined as
the actual historical volatility of the security price and is usually represented
over a period of time. It is generally calculated as the intraday squared re-
turns over short periods such as 5 or 15 minutes. The bipower volatility is
defined as the actual historical volatility that is calculated as the product of
absolute value of returns over two intraday periods. Anderson et al(2007)
[2] was able to solve that to see if there is a large jump between two pe-
riods of returns, such as if returns were small for two periods the bipower
volatility would be much smaller but if returns over two periods were large
the bipower volatility value would be larger. An interesting property be-
tween realised volatility and bipower volatility is that if there is a presence of
volatility jumps, Barndorff and Shephard (2004)[4] says that the difference
between the two measures would represent the jump component.
Earliest proposal adjusting the stochastic volatility model was by Tay-
lor(1986) where the natural logarithm of volatility was modelled as a linear
autoregressive process of order one, in this form it was known as the ARSV
(Autoregressive Stochastic Volatility) model. An example of an ARSV(1)
model is shown below:
yt = σxεtσt
log(σ2t ) = φlog(σ2
t−1) + ηt, t=1,2,...,T-1
13
6.2 Volatility Stylised Facts
In Engle and Patton (2001) [12] paper they explained that a good volatility
model is able to capture the stylised facts of volatility, the first fact is volatil-
ity persistence (i.e. generally volatility cluster together and move in the same
direction for extended periods so large movements in volatility will later lead
to more large movements in the future). The second stylised fact for volatil-
ity is mean reverting suggest that after some periods large movements in
volatility, the volatility state would always go back towards its usual(mean)
levels. The third fact, is the leverage effect that is when returns are negative
volatility tends to be larger but when returns are positive volatility tends to
be smaller.
Over the years development in the stochastic volatility models suggests
inclusion of a jump parameter as empirical evidence suggests there were
periods when there were positive jumps in volatility. Eraker et al (2003)
[13] says that there were evidence to show that there were volatility jumps
and jumps in returns. Return jumps can cause large movements resulting to
a market crash however volatility jumps can remain persistent which does
affect future returns. Eraker et al (2003) [13] suggests incorporating jumps
in the stochastic volatility model it allows us to better model rapid volatility
increases such as in market crashes and provides evidence to support other
literatures such as Bates(2000) and Pan (2002) that including jumps removes
model misspecification .
Recent papers such as Nakajima(2011)[29] and Malik and Pitt(2011) [26]
both explored GARCH models and filter methods to capture the leverage
effects, jumps and volatility persistence. They were able to show that on
average for most stochastic models they used the MCMC method performed
better than the GARCH counterpart, Nakajima(2011) also found that the
stochastic models that incorporated jumps, fat tails and leverage tend to do
much better than any other specification.
14
6.3 Kalman Filters
Earlier methods to solve problems such as (i) prediction of random signals;
(ii) separation of random signals from random noise (Kalman R., 1960) [22]
were pioneered by Wiener (1949). However there were limitations as the
filter was determined by the impulse response functions consequently if our
problem is complex (large amount of observations or non-stationary series)
then it will become difficult to derive the impulse response function. Another
limitation was we cannot filter for non-stationary series as it was created with
the assumption that the noise and signal of the series are stationary.
In 1960, R. Kalman published his first extension to the Wiener filter for
solving dynamic models in a discrete time series allowing for non-stationary
time series and in the next year Kalman and Bucy (1961) published another
version which accounted for a continuous time series. The Kalman filter was a
recursive method for solving linear data problems that has either observation,
measurement noise or both. Welch (1995)[32] states that Ho and Lee (1964)
was able to determine that the Kalman filter was optimal Bayesian filter
under 3 assumptions for the time series i.e. linear, quadratic and Gaussian.
The restrictive assumptions for the time series to be able to obtain an
optimal Kalman filter is important as in this paper we would be filtering time
series for the underlying volatility and we know that stochastic volatility for
large data series can be multi-dimensional series. As a result it took many
years before a viable adaptation of a likelihood function to use as the filter
equation and in 1994, Harvey et al suggested the use of Quasi-Maximum
likelihood functions.
Further extensions of the Kalman filters were developed for either non-
linearity or non-gaussian data series such as the extended Kalman filter,
Gaussian sum filter and the unscented Kalman filter. However each new
method would still suffer problems if there were severe non-linearity and/or
non-Gaussianity, as such a different class of filters were developed using
15
Monte Carlo (sampling) methods.
6.4 Markov Chain Monte Carlo Methods(MCMC)
The concept of applying a Monte Carlo approximation method to solve a dif-
ficult integral is suppose∫x
f(x)dP (x), if we can draw size N of identical and
independently distributed random samples of {x1, x2, ...xN} from the proba-
bility distribution P(x) then we can approximate f(x) by fN = 1N
N∑i=1
f(xi) .
From the concept of Strong Law of Large Numbers we can say that our esti-
mated value of the function fN would almost surely converge to the expected
value of the function E[f(x)]. However there arises two main problems; one
is in what method do we draw random samples from the probability distri-
bution and the second is what method are we going to take to determine the
expectation of the function.
Markov chains can be described as a type of probability process where the
outcome of the only the current state would affect the outcome of a future
state. Markov chains have a few properties that must be satisfied for MCMC
to be a viable option for filtering and that is the distribution we want to
sample must be homogeneous, reversible and ergodic. For Markov chains to
be homogeneous it requires the solution to depend on the elapsed time of the
process but not the absolute time, so if we want calculate Markov chains for
the probability of changing state i to state j over one day the probability of
the state transition will not change. Reversible requires if it is possible to
change from state i to state j it is also possible for there to be a transition
from state j to i. For Markov chains to be ergodic that means it is possible
to transition from one state to any other possible state but not necessarily in
one jump. What this means for Monte Carlo methods is that if our problem
does exhibit a Markov process then we can use Monte Carlo sampling to draw
samples and these samples will also be a Markov process and it is possible to
16
Markov Chain Monte Carlo filters represents a general family of filter-
ing algorithms that uses the fundamental idea that if there is a mulitvariate
distribution that is difficult to integrate we are able to draw random sam-
ples that are homogeneous, reversible and ergodic Markov Chain with an
invariant distribution that is similar to our target density(problem). MCMC
filters are also known as offline/batch sampling method since for most ver-
sions of the MCMC method given our initial draw for the initial parameters
and state variables we can draw for next run of the MCMC filter by using
the parameters and state variables given the previous draw. Jacquier et al
(1994)[20] provided one of the earlier implementations of the MCMC method
on stochastic volatility models which they reported that it provided very ac-
curate results by using Gibbs-sampling and Metropolis-Hastings algorithm.
In Geweke and Tanizaki (2001) [15] they explained the benefits of using
a Metropolis-Hastings algorithm and the Gibbs-sampling method together
but first we need to define what are these two algorithms. The Metropolis-
Hastings algorithm is a version of MCMC algorithm where assuming if it is
difficult to use the exact distribution to draw samples from we can use draws
from a proposal distribution and later accept/reject the draws according to
an acceptance/rejection criterion if we reject we will redraw the samples.
Gibbs-sampling can be described as a special case of Metropolis-Hastings
algorithm, where under certain conditions the algorithm to solve a MCMC
problem becomes easier to solve. Cassella and George (1992) [9] explains that
normally for a metropolis-hastings algorithm we would need to approximate
probability density function by making random draws. However with Gibbs-
sampling we do not need to approximate probability density function, we
can sample from the conditional distributions which we would know from our
problem and under special conditions if we generate a very large sample from
the conditional distributions (e.g. f(x|y) and f(y|x) the results will converge
to the true marginal density f(x). Carlin et al (1992) [7], Carter and Kohn
(1994) [8] illustrated an approach that Gibbs-sampling can be used for solving
non-linear and non-gaussian state space models which were important for
17
future literatures that applied MCMC methods to solve stochastic volatility
models since we able to simplify problems more by not requiring to draw
from the true marginal density or a proposal density.
6.5 Sequential Monte Carlo Methods/ Particle Filter-
ing
Recent development in bayesian estimation of stochastic volatility lead to
the use of particle filters. First published paper by Gordon, Salmond and
Smith(1993) [16] the method illustrated in the paper is commonly called the
bootstrap filter, however the it was first mentioned in 1970s however due to
the restraint in computer memory and power MCMC methods still remained
the popular method for nonlinear recursive filtering methods. The particle
filter is a recursive approximation method for filtering random variables by
particles. We do this by approximating a continuous variable by using dis-
crete sampling points; this method has been popular recently due to increase
in computing power.
Bootstrap filter (Gordon et al(1993)[16])is a sequential importance sam-
pling method however we eliminate low importance weights and multiply
particles that have high importance to avoid the weight degeneracy problem
that happens for sequential importance sampling. There are many benefits
of using the bootstrap filter is that it is generally quick and easy to imple-
ment for large variety of problems as we only need to change the importance
weight distribution to modify the bootstrap filter for a new problem.
The auxiliary particle filter was developed by Pitt and Shephard (1999)
[30] it was an improvement over the bootstrap filter as it allows us to ap-
proximate better the tails of distribution. It works by selecting a proposal
distribution which has a fatter tail than the true posterior distribution, by
resampling from the proposal distribution we are able to get a larger variety
of particles compared to resampling with the posterior distribution. Using
18
this method helps reduce the effect of the convergence rate to get the true
posterior estimate but we can get a better picture of the posterior distribu-
tion and therefore it helps reduce the effects of the problem of particle decay
in the boostrap filter.
Particle filters have the advantage over MCMC methods is that it gen-
erally is much faster to apply and obtain results than the MCMC methods.
However one disadvantage is that particle filters with unknown parameters
have difficulties to obtain optimal estimations. There have been multiple
methods found in literature on particle filters; one class of methods is the
maximum likelihood method. The maximum likelihood method can be de-
scribed as we start off with initial estimates of the parameters to perform the
a version of the particle filter methods such as bootstrap/auxiliary filter then
parameters are changed slightly until we obtain the largest likelihood value
available in the search. The problem with the maximum likelihood method
is that it can be slow if the number of parameters needed to be estimate is
large and if the data series is large, so it can take a long time for the method
to converge to the maximum likelihood value for a complicated problem. An-
other problem would be the initial estimates used could become a problem
if it is far off from the true values of the parameters so it may take longer
to determine the results or it will never find the true parameters. Malik and
Pitt(2011)[26] has developed a method to improve the speed of finding the
parameters by smoothing the importance weights and sorting the particle in
order for each time step which in the long run improves the efficiency of the
algorithm to determine the unknown parameter.
Another method to determine the unknown parameters is the MCMC
particle filter where the Metropolis-Hastings or Gibbs-sampling algorithm is
used to sample the unknown parameters then we apply these parameters via
the auxiliary particle filter to obtain the latent state and the two steps are
performed for many iterations to obtain convergence in the parameters. The
MCMC particle filter that is explained in depth in Andrieu et al [3] and
discussed in Kantas et al (2009) [23] can also be a slow method to determine
19
unknown parameters as there is a requirement to perform the particle filter
until there is convergence in the parameters. Though the MCMC particle
filter can be effective method to determine unknown parameters however it
does not take advantage of idea of the particle filter that is the latent states
are learnt sequentially over time so it must also be possible for the parameters
to be learnt sequentially over time.
The last class of particle filter methods are the sequential parameter
learning particle filters which will appear in this paper is a class of algo-
rithms where the parameters and latent states are sequentially learnt over
each time step in the algorithm. Development in sequential estimation of
parameters with particle filter usage includes papers by Liu and West(2001),
Storvik(2002) and Carvalho, Johannes, Lopes and Polson(2010). There have
been difficulties with sequential parameter learning methods as that there
was no successful solution since the development of the bootstrap filter to
sample for parameter evolution as using the parameter distribution condi-
tional on the observations will result in particle decay. With no parameter
evolution we could be left with particle decay in the parameter particles (in
general the particle filter will include a importance weights step to keep only
the good particles as a result there will be a decrease in different param-
eter particles until there is only one particle left) within a few time steps
in the particle filter algorithm which does not help in estimating unknown
parameters.
One of the first method successful sequential parameter learning methods
is the Liu and West filter (2001) which suggests the parameter distribution
is based on the mixture of multivariate normal. Using the mixtures of mul-
tivariate normal to generate samples for the parameters we will avoid the
particle decay problem in the parameters as the mixture distribution keep
dispersion in the parameter particles preventing particle decay. In a later
section we will discuss the steps in the Liu and West filter. The second
popular sequential parameter learning method is the Storvik filter (2002)
where the parameter distribution is dependent on a recursively updated suf-
20
ficient statistics. The updated sufficient statistics is an independent process
which solves the particle decay problem as the parameter distribution will
not shrink during the resample step to choose good particles. The latest
sequential parameter learning method is known as the particle learning filter
developed by Carvalho et al (2010) which is similar to the Storvik filter as the
sufficient statistic is used to sample from the parameter distribution but it
improves the efficiency of the Storvik filter by incorporating a state sufficient
statistic which allows for faster convergence by minimising the variance of
the importance weights.
21
7 Models
In this section we will define latent volatility, so first lets define spot asset
price as Pt at time t, if we assume a continuous diffusion process then similar
to what is shown in Harvey et al(1998) and Taylor (1986);
dSt = µtdt+√VtdB
pt (1)
d(Vt) = κ(θ − Vt)dt+ τdBVt (2)
where the parameters (µt, κ, θ, τ) are evolving with volatility and Bpt , B
Vt
are described as Brownian motions that can correlate.
If there are jumps in the model we can add in another component into
equation (1) that would capture jumps in asset prices, resulting in the same
equation found in Maneesoonthorn et al (2012) [27] and Eraker et al(2003)
[13].
dSt = µtdt+√VtdB
pt + dJpt (3)
The jump component dJpt is described as a random jump process such that
dJpt = Zpt dN
pt where the jump size is Zp
t follows an exponential distribution
with probability of jump dNpt happening with a bernoulli distribution.
To incorporate volatility jumps that Eraker et al(2003) [13] suggests im-
proves model fit we would need to modify equation (2) by incorporating a
jump component.
d log(Vt) = κ(θ − log(Vt))dt+ τdBVt + dJVt (4)
22
Similar to the definition for the jump component for asset price, volatility
jumps is defined as dJVt = ZVt dN
Vt where ZV
t is exponential distribution for
size of the volatility jump and dNVt is a bernoulli distribution to model
volatility jump happening.
7.1 Stochastic Volatility Model (SVM)
Data collection is discrete however our model that was described in equation
(1) (observation equation) and (2) (state equation) are in continuous cases,
as Kim et al(1998), Maneesoonthorn et al (2012), and an application of Euler
discretization would be required on equation (1) and (2).
St = St−∆t + µt∆t+√Vt−∆tξ1t (5)
Vt = Vt−∆t + κθ∆t− κVt−∆t∆t+ τξ2t (6)
where (ξ1t, ξ2t) ∼ N(02,Σ), Σ =
[1 ρ
ρ σ2v
]
if we set ∆t = 1 the above equation will explain daily change in latent
variance. Equation (6) will become Vt = Vt−1 + κθ − κVt−1 + τξ2t = κθ +
(1 − κ)Vt−1 + τξ2t. If we let αv = κθ and βv = 1 − κ then equation (6) can
be transformed into a more simplified version:
Vt = αv + βvVt−1 + τξ2t (7)
The model allows for depiction of leverage effects between the stock price
and the volatility effect however Eraker et al (2003) [13] suggests that gener-
ally models without jumps in both asset price and volatility are misspecified.
Euler discretization were applied to equation (3) and (4),
23
St = St−∆t + µt∆t+√Vt−∆tξ1t + ZS
t ∆NSt (8)
Vt = Vt−∆t + κθ∆t− κVt−∆t∆t+ τξ2t + ZVt ∆NV
t (9)
where ZVt
iid∼ exp(µv), ZSt
iid∼ exp(µs), ∆NVt
iid∼ Bernoulli(δv∆t) and
∆NStiid∼ Bernoulli(δS∆t).
If we let ∆t = 1 and reparameterize for αv = κθ and βv = 1− κ equation
(9) can be rewritten in as:
Vt = αv + βvVt−1 + τξ2t + ZVt ∆NV
t (10)
7.2 Realized Volatility
Extensive studies have been undertaken to look at the behaviour of realised
variance since the mid 1990s where many papers have modelled volatil-
ity measures by a generalized autoregressive conditional heteroskedasticity
model (GARCH) or by latent stochastic volatility models (SV-M). Survey pa-
pers of the research topic on realised volatility include McAleer and Medeiros
(2008)[28] and Corsi et al (2008)[11] where in McAleer and Medeiros it was
a review paper of the last decade of work on realised volatility and in Corsi
et al a proposed a different model (HAR-GARCH and ARFIMA)to forecast
and model realised volatility. Realized variance is formally defined as the
sum of squared returns and in this paper it will defined as the daily sum of
squared returns over five minute intervals.
Anderson et al (2003) found that if there were no microstructure noise
almost always the realised variance will converge in probability to the inte-
grated variance RVtp→ IVt, where IVt =
si∫si−1
σ2(t + s − 1)ds . It is said
that realised variation can be a consistent estimator of integrated variation.
24
Now why are we interested inintegrated variation as it is stated as the true
underlying volatility state. Barndorff-Nielsen and Shephard (2002) was able
to show that if no microstructure noise assumption was applied then realised
variation is asymptotic normally distributed of the form:
√nt√
2IQt
(RVt − IVt)d→ N(0, 1)
where IQt =1∫0
σ4(t+ s− 1)ds is integrated quarticity.
In terms of modelling realised volatility many different papers offer differ-
ent models to try to model realised volatility. Christoffersen et al (2007) [10]
defined realised variance to have the form of RVt+1 = E[RVt+1|Vt]+ut, using
the results found from Ait-Sahalia (2007) [1] we can create the observation
equation to have the form:
RVt+1 = θ + [exp(−κ/252)− 1
−κ/252](Vt − θ) + ut
By making substitutions of β = 1 − κ and α = κθ we can rewrite the
above equation with the same parameters like equation (7) and (10).
RVt+1 =α
1− β+
exp(β−1252
)− 1β−1252
(Vt − θ) + ut (11)
Another model that is found in the literature on modelling realised volatil-
ity is
RVt+1 = Vt + σrvut (12)
this form is shown in Lopes and Tsay (2011) [25] where they had an example
of modelling realised volatility of Alcoa.
Many papers applied the various Heston’s stochastic volatility models
such as the Heston’s square root model, standard stochastic volatiliy model
25
and three halves stochastic volatility model to fit realised volatility. Below
is a simplified example of the three stochastic volatility model forms:
Vt+1 = α + βVt + τrv√Vtut (13)
Vt+1 = α + βVt + τrvVtut (14)
Vt+1 = α + βVt + τrvVt3/2ut (15)
7.3 Bipower Volatility
Bipower volatility is described by Anderson et al(2007) as the variation def-
inition when there is no jump in volatility. The bipower volatility has a
mathematical form of
BVt =π
2
M∑t−1<ti≤t
|rti ||rti−1|
Where the√
2/√π is the expected value of an absolute value of a standard
normal distribution.
The graph of the difference in the realised variation and bipower variation
is shown in figure 1. In Anderson et al (2007) they suggested that only when
difference between realised volatility and bipower volatility is significantly
positive then it suggests that there is a volatility jump in that period oth-
erwise small jumps in the volatility indicates possibly noise in collection of
data. We can see from the figure 1 there is a large spike on 11 January 2001
which coincides with the U.S. Federal Trade Commision approving a merger
between AOL and Time Warner. Another spike is on 18 September 2007,
this coincides with the Bank of England injecting 4.4 billion pounds into
the U.K. Financial system and U.S. Federal Reserve cutting interest rates
As such Eraker et al (2003) suggests that if there is volatility jumps in the
26
dataset then to avoid model misspecification it is better to take into account
both price and volatility jumps.
Figure 1: Difference in Realised and Bipower volatility of S&P500 Index
7.4 VIX
To be able to work out how to price options we need to specify a volatility
risk premium to the latent volatility equation to show a relationship of a
risk neutral stochastic volatility process. Using equation (2) to show the risk
neutral dynamics, we express volatility risk premium by λrnVt:
dVt = (κ(θ − Vt) + λrnVt)dt+ τdB∗t
dVt = (κ− λrn)(κθ/(κ− λrn)− Vt)dt+ τdB∗t
dVt = κ∗(θ∗ − Vt)dt+ τdB∗t (16)
27
where dB∗t is a brownian motion for the risk neutral process, κ∗ = (κ− λrn)
and θ∗ = κθ/κ∗. If we took the Euler discretization of equation (16)
Vt+1 = Vt + κ∗θ∗ − κ∗Vt + τξ∗t
Vt+1 = α∗ + β∗Vt + τξ∗t (17)
now we let α∗ = κ∗θ∗ = κθ and β∗ = (1 − κ∗) = (1 + λrn − κ) to be risk
neutral parameters for the stochastic volatility model. The λrn parameter
represents the size of the risk premium that is placed when pricing options,
so the volatility risk premium will adjust to the size of the latent volatility.
Similar to modelling realised volatility different papers utilised different
models so in this paper we will explore the different VIX models to see which
performs better.
28
8 Data
To model realised volatility, bipower volatility and option volatility we took
data from S&P 500 spot and option index. The observation range of the
collected data is July 1997 to August 2007. It covers many significant finan-
cial events such as the 1997 Asian Financial Crisis, the 2001 dot-com bubble
burst and the start of the recent 2007 Global Financial Crisis.
The data consists of three measures of variance:
1. Realised Variance (RVt =M∑
t−1<t≤tr2ti
)
2. Bipower variation (BVt = π2
M∑t−1<ti≤t
|rti ||rti−1|)
3. Option variation denoted as MFt ( MFt = (V IXt
100)2
Variance measure constructed by the prices of options denoted as , VIX is
the volatility index that is available on Chicago Board of Option Exchange).
Microstructure noise can be inherent in high frequency data which can
be problem in modelling high frequency data. Microstructure noise are any
types of trading activities that can cause bias results in the collection of
data and modelling phases these noises arises from infrequent trading, price
discreteness (rounding of prices) and bounces in bid/ask prices. The data
has been cleaned by a method mentioned in Maneesoonthorn et al (2012)
[27] and Brownless and Gallo (2006) [6] which should remove most problems
of microstructure noise. The data cleaning approach is each daily time series
collection of data is filtered such that outliers are trimmed down according
to a trimming parameter to preserve the behaviour of the series of the day.
Realised Variance was calculated as the annualised 5 minute intraday
returns for each trading day. From figure (1), we see that for the first 1500
observations oscillate around within 0 and 0.2 where once in a while there
is a large spike to 0.3 at around 1996, 1997, 1998 and then 2000, which
29
first well to show that during financial crises the volatility tends to be more
unstable and would spike up higher than normal levels. After the first 1500
observations (around 2003) the realised variance fluctuates less until August
2007 (the start of the defaults of sub-prime mortgages began affect financial
institutions) where volatility began fluctuating more with a large spike to
1.03 in October 2008.
The bipower variance was calculated as the annualised 5 minute apart
returns which is calculated to represent return movements. From figure 2 we
notice that the bipower variance closely follows the shape and size of realised
variance.
The VIX which will now be described as the Option implied volatility
is the . From figure 2, the annualised variance over stable financial periods
is always higher than both than realised and bipower volatility. There is a
lag in its movement compared to realised and bipower volatility as it takes a
few days before a movement in realised/bipower volatility causes the option
implied volatility to move which makes sense as the option volatility is used
to price securities over 3 or multiple months so one time volatility shocks will
not enter into the option volatility until a few days later.
Figure 3, shows the quantile-quantile plots of the data and the natural
logarithm of the data. We see that for the non-transformed variation measure
time series it is not normally distributed as it does not fit well with the
straight line of the normal distributed quantile line. However interestingly
if we transformed the variation measures by a natural logarithm we are able
to see that the if fits well with the straight line so its possible to make
an assumption that the natural logarithm of the high frequency variation
measures are normally distributed or student t distributed.
30
Realised Volatility Bipower Volatility Volatility IndexMinimum 0.000667 0.000383 0.0097811st Quartile 0.006112 0.005402 0.024930Median 0.011660 0.010440 0.042560Mean 0.022290 0.020770 0.0540503rd Quartile 0.021750 0.019760 0.062250Maximum 1.036000 1.033000 0.653800
Table 1: Summary of S&P 500 data
8.1 Data transformation
Due to the size of the raw data series if we took the natural logarithm of the
data series majority of the observations would become negative. As a result
can affect the estimation of the parameters and latent states since the latent
state must always remain positive, therefore we multiplied the raw series
by ten thousand. The table below is the summary statistic of the natural
logarithm of ten thousand multiplied by the raw data series.
Realised Volatility Bipower Volatility Option Implied VolatilityMinimum 6.672 3.826 97.81
1st quartile 59.898 53.092 243.20Median 113.163 100.739 419.43Mean 173.717 161.059 474.85
3rd quartile 203.928 189.164 601.72Maximum 2993.111 3459.148 2092.15
Table 2: Transformed data
31
Figure 2: Graph of Data
32
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−3 −1 1 3
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2025
3035
BV Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
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−3 −1 1 3
−2
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01
23
Log RV Q−Q Plot
Theoretical Quantiles
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ple
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ntile
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−2
−1
01
23
Log BV Q−Q Plot
Theoretical Quantiles
Sam
ple
Qua
ntile
s
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−3 −1 1 3
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Log VIX Q−Q Plot
Theoretical Quantiles
Sam
ple
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ntile
s
Figure 3: QQ plots of Data and Log Data
33
9 Method
9.1 MCMC Method
9.1.1 Outline of MCMC Method
Markov chain monte carlo methods are a family of algorithms that uses
the concept of markov chains to propagate the future samples of the latent
variables. For all models we applied a 2000 burn in and 5000 draws.
The Forward Filtering Backward Sampling method was applied to repre-
sent the markov chain monte carlo filter methods. As the name of the method
implies there are two steps in this application, the first is the forward filter
which is described in Hore et al (2010) [18] as an evolution and update step.
The next section of the method is the backward sampling where we draw
samples from a joint distribution of the states.
Forward Filter
Evolution step: Using bayes theorem and assuming the distribution fol-
lows a markov chain we can say that the distribution of the state variables for
the future, present and past is P (xt−1, xt, xt+1 | Yt) = P (xt−1, xt | Yt)P (xt+1 |xt, xt−1, Yt) = P (xt−1, xt | Yt)P (xt+1 | xt). Since on the right hand side we
have P (xt−1, xt | Yt) which is the distribution we should know from our pre-
vious step then we can marginalise out xt−1 so the distribution of the state
variables can be represented in the form of:
P (xt, xt+1 | Yt) = P (xt | Yt)P (xt+1 | xt)
34
Update step: Applying bayes theorem of the above equation such that
P (xt, xt+1 | Yt+1) ∝ P (xt+1, xt, Yt+1 | Yt)
= P (xt, xt+1 | Yt)P (Yt+1 | xt, xt+1)
P (xt, xt+1 | Yt)P (Yt+1 can be found in the previous evolution step and
the second term P (Yt+1 | xt, xt+1) is determined by the observation equation.
Backward sampling
As the name of the step implies we sample from a distribution determined
by our previous states so the distribution would have the form
P (x1, x2, ..., xt | Yt) = P (xt, xt−1 | Yt)1∏
i=t−2
p(xi | xi+1, xi+2, ..., xt, Yt)
9.2 Particle Filter
9.2.1 Outline of Particle Filter Methods
Two particle filter methods called Liu and West (LW) filter and the particle
learning (PL) filter were used to model the realised variance, bipower variance
and the option variance. These two methods sequential parameter estimation
allows the user to model for both state and parameters evolve over time.
Previous particle filter methods before the LW and PL filters that could
estimate both parameters and states were methods where a particle filter
was applied to model and using maximum likelihood estimation we adjust the
initial parameters for the best fit or they could first apply an MCMC method
to obtain draws from the parameter distribution then using the parameters
we apply it to a particle filter algorithm. These two general methods did not
35
take advantage of the sequential nature of the particle filter (such as MCMC
then particle filter) or are very computational difficult and expensive to do
(particle filter and apply MLE to obtain parameter estimates). N=5000
particles were used for both the Liu and West and Particle Learning filter,
this allows us to be able to compare the effectiveness of the particle and
MCMC filter as the number of samples drawn would be equal.
9.2.2 Liu and West Filter
Liu and West particle filter (LW) is an extension of auxiliary particle filter
where the method was adjusted to account for sequential parameter esti-
mation. The auxiliary particle filter was developed to account for better
modelling of the tails of a distribution. The general algorithm for the Liu
and West filter is:
1. Resample the latent state variables and parameters using weights (at
the first time step the importance weights has the expression and for t > 1
weights are generated from step 4)
2. Propagate the parameter from sampling from a normal distribution
with mean m(θ(i)t ) and variance V
3. Propagate the latent state variables by sampling from a state transition
distribution with the form of p(xt+1 | x(i)t , θ
(i)t+1)
4. Compute the weights to be used for step 1, these weights are generated
by w(i)t+1 ∝ w
(i)t
P (yt+1|x(i)t+1,θ(i)t+1)
P (yt+1|g(xi),m(˜θ(i)t ))
5. Repeat step 1
36
9.2.3 Particle Learning Filter
Particle Learning(PL) uses the concept of the kalman filter where a state
sufficient statistic and parameter statistics are updated for each time step
of the particle filter, this allows us to sample from an updated parameter
distribution and we can resample the parameter samples from a distribution
determined by both the state sufficient parameters and the previous latent
states. Lopes and Tsay (2011) were able to show that for general cases
the particle learning filter can converge faster than the Liu and West filter
however the particle learning method becomes more difficult to use for non-
general state space models.
The general particle learning algorithm is:
1. Resample the parameters(θ), parameter statistics (st) and state-sufficient
statistics (sxt ) using weights generated from the distribution weights ∝ P (yt |st−1, θ) which is the
2. Sample the latent state variables (xt) from a distribution P (xt |sxt−1, θ, yt)
3. Update the parameter statistics statistics st = (st−1, xt, yt)
4. Sample the parameters from p(θ | st)
5. Update the state sufficient statistics sxt = (sxt−1, θ, yt)
6. Repeat from step 1
37
9.3 Models
Integrated variation is the unobserved state(Vt) that we would be looking for
using the following models below:
Model A: Local level model
Yt+1 = Vt+1 + σyεt (18)
Vt+1 = Vt + τvηt (19)
The local level model is able to show that the variation measures are
the approximately the true measures of integrated variation. The local level
model is shown in Carvalho et al(2010) for realised volatility filtering.
Model B -AR(1) model:
Yt+1 = Vt+1 + σyεt (20)
Vt+1 = α + βVt + τvηt (21)
The autoregressive model is one of the earlier models in stochastic volatil-
ity modelling that was used to determine the latent volatility state as such
it will also appear in this paper to determine which model is better.
Model C -SV-AR(1) model:
Yt+1 = Vt+1 + σyVt+1εt (22)
Vt+1 = α + βVt + τvηt (23)
The stochastic volatility autoregressive model is an extension of the au-
toregressive model since in the observation equation there is an extra variable
for the observation noise that is affected by the size of the latent state. It
allows us to take account for the leverage effect of volatility that is the noise
38
in the observation will get larger during times when volatility is greater and
the noise will become smaller when the latent volatility is smaller.
Model D- Heston’s Square Root Stochastic Volatility Model
Yt+1 = Vt+1 + σyVt+1εt (24)
Vt+1 = α + βVt + τv√Vtηt (25)
Model E- Heston’s Stochastic Volatility Model
Yt+1 = Vt+1 + σyVt+1εt (26)
Vt+1 = α + βVt + τvVtηt (27)
Model F- Heston’s 3/2 Stochastic Volatility Model
Yt+1 = Vt+1 + σyVt+1εt (28)
Vt+1 = α + βVt + τvV3/2t ηt (29)
Model D, E and F Models D, E and F are stochastic volatility models
that appear in Christoffersen et al (2010)[10]. The latent state equation
uses the Heston stochastic volatility model which Steven Heston was able to
show that these models were able to represent the underlying volatility of
a model very well in his 1993 paper about bond and currency options. By
incorporating these models we would have a larger variety of models to do
perform model selection from.
9.3.1 How observation errors are distributed
According to our analysis of the QQ-plot the natural logarithm of the vari-
ance measures are either normally distributed or student-t distributed. As
39
such we will assume that observation errors are either normally distributed
or student t-distributed, for simplicity we will assume the state transition
equations to be normally distributed.
An example of model D which is now normally distributed
P (Yt+1 | Vt+1, θ) ∼ N(Vt+1, σ2V 2
t+1)
P (Xt+1 | Vt, θ) ∼ N(α + βVt, τ2Vt)
An example of model E which has observation errors student t-distributed
P (Yt+1 | Vt+1, θ) ∼ T (Vt+1, σ2V 2
t+1)
P (Xt+1 | Vt, θ) ∼ N(α + βVt, τ2V 2
t )
We will assume that the student-t distribution has 10 degrees of freedom.
Therefore the importance weights calculated for the particle filter would now
be student t-distributed.
40
10 Results
10.1 Marginal Log Likelihoods
From table 3, we can see that for all models the marginal log likelihood values
were have very close implying that the model’s selected were good models to
use for comparison of the various filter techniques but still only models A to
D can be effective models to use to model realised volatility as the marginal
log-likelihood are close and fall within a range of -2500 to -2600. Further
comparison of the marginal log-likelihood suggests that model B and model
C for the case where both observation and state transition are normally
distributed performs best as they had the largest marginal log-likelihood
values. The student t-distributed observation errors do not perform as well
as the normally distributed errors hence the distribution of realised volatility
is not significantly affected by the tails of the distribution.
Henceforth model B and model C for the normally distributed observa-
tion errors suggests that the realised volatility follows an autoregressive of
order one process so realised volatility has dependence on the lag of inte-
grated variation and possibly measurement noise is a problem that we need
to account for in modelling realised volatility to further reduce the amount
of models we consider the sequential bayes factor of the two models this will
be found in the next section.
According to the dataset of bipower volatility of the S&P 500 we expect
should expect the results from realised variation to also hold here since they
behave similarly in the time series. Similar to the realised volatility it seems
all models can be effective models to predict the latent volatility state as
the marginal log likelihood value are relatively close but closer examination
says that the preferred models are models C and B as they have the highest
marginal log-likelihood values.
41
When the model is non-linear such as for models C, D, E and F for filtering
both realised volatility and bipower volatility the Markov chain Monte Carlo
method performs not as well compared to the particle filter. For the same
number of iterations/particles we see the benefit of the importance weights
42
Model Marginal Log-Likelihood RankingLW-A -2523.520 7PL-A -2474.048 5
MCMC-A -2648.940 12LW-A t -5042.359 30PL-A t -3741.110 27LW-B -2473.803 4PL-B -2456.685 2
MCMC-B -2439.831 1LW-B t -3653.631 26PL-B t -3585.185 25LW-C -2485.466 6PL-C -2468.032 3
MCMC-C -2962.528 23LW-C t -4015.245 29PL-C t -3987.007 28LW-D -2547.847 9PL-D -2546.481 8
MCMC-D -2877.018 16LW-D t -2948.965 21PL-D t -2950.044 22LW-E -2604.841 10PL-E -2611.793 11
MCMC-E -2977.741 24LW-E t -2905.482 18PL-E t -2921.141 19LW-F -2728.858 14PL-F -2713.701 13
MCMC-F -2939.352 20LW-F t -2899.811 17PL-F t -2870.488 15
Table 3: Marginal Log Likelihood for Realised Volatility
43
Model Marginal Log-Likelihood RankingLW-A -2592.729 7PL-A -2538.630 5
MCMC-A -2638.574 10LW-A t -5033.072 30PL-A t -3634.138 28LW-B -2520.915 4PL-B -2512.838 3
MCMC-B -2502.731 1LW-B t -3719.315 29PL-B t -3591.816 27LW-C -2560.902 6PL-C -2509.616 2
MCMC-C -2640.777 11LW-C t -3005.172 24PL-C t -3116.110 14LW-D -2601.530 9PL-D -2594.679 8
MCMC-D -2703.430 12LW-D t -2964.297 23PL-D t -2866.141 18LW-E -2767.562 15PL-E -2733.110 13
MCMC-E -2844.013 17LW-E t -2931.104 21PL-E t -2900.114 20LW-F -2835.598 16PL-F -2933.983 22
MCMC-F -3102.103 24LW-F t -2889.599 19PL-F t -2744.111 14
Table 4: Marginal Log Likelihood for Bipower Volatility
44
Model Marginal Log-Likelihood RankingLW-A -3182.191 12PL-A -2612.673 5
MCMC-A -2574.111 4LW-A t -3274.474 16PL-A t -2848.225 6LW-B -2398.484 3PL-B -1167.031 2
MCMC-B -899.107 1LW-B t -3338.770 18PL-B t -3294.516 17LW-C -4015.245 21PL-C -3353.274 19
MCMC-C -3221.017 14LW-C t -3273.411 15PL-C t -3073.110 8LW-D -6896.688 24PL-D -5033.670 22
MCMC-D -5433.774 23LW-D t -3122.523 11PL-D t -3014.507 7LW-E -8009.858 27PL-E -7539.411 25
MCMC-E -7644.110 26LW-E t -3415.092 20PL-E t -3211.144 13LW-F -9517.800 30PL-F -8752.774 28
MCMC-F -9258.114 29LW-F t -3081.448 9PL-F t -3101.152 10
Table 5: Marginal Log Likelihood for Option Implied Volatility
45
10.2 Sequential Bayes Factor
The sequential bayes factors allows us to compare two different models for
each time step by looking at the ratio of fit/likelihood between the two differ-
ent models, it is an analysis tool for looking at predictive power of a particular
model over another. Since we have eliminated most of the models from the
marginal log-likelihoods however the likelihood values are close together so
then we can make the best judgment on the forecast model. It is one of the
few benefits of the sequential monte carlo/particle filter method as we can
see the fit of model over the observation periods hence we can determine
if one model performs better in different periods of the observation dataset
as such the results from the particle learning method would be used for the
sequential bayes factor.
To obtain the sequential bayes factor we first need the marginal likelihood
of each model in
p(yn+1 | yn,model1) =1
N
N∑i=1
p(yn+1 | θ(i), x(i)n )
then we need to create the ratio to obtain the results:
Sequential Bayes Factor =p(yn+1 | model1)
p(yn+1 | model2)
To determine which model is better at how the graph behaves, so when
the graph is rising or greater than one it suggests that during the time period
model one is performing better than model two.
Realised Volatility
Figure 3, contains the sequential bayes factor for model B over model C,
since the line is greater than one it suggests that model b predictive ability
46
Time
SB
F
0 500 1000 1500 2000 2500 3000
1.0
21
.08
Sequential Bayes Factor for Realised Volatility
Figure 4: Sequential Bayes Factor
is better than model C for almost all time periods.
Bipower Volatility
Figure 4, contains the sequential bayes factor for model B over C, we see an
Time
SB
F
0 500 1000 1500 2000 2500 3000
0.9
81
.02
1.0
6
Sequential Bayes Factor for Bipower Volatility
Figure 5: Sequential Bayes Factor
upward sloping line which suggests that model B is slowly performing better
than model C. After 1000 observations we see that the the sequential bayes
factor value is always greater than one hence we can say that model B has
better predictive power over model C.
Option Implied Volatility
Figure 5 is the sequential bayes factor of model B over model A, it shows an
upward sloping line which is always greater than one implying that model B
47
Time
SB
F
0 500 1000 1500 2000 2500 3000
0.5
1.5
2.5
Sequential Bayes Factor for Option Implied Volatility
Figure 6: Sequential bayes factor
has better predictive power than model A.
10.3 Parameter Estimation
The models shown below are the posterior estimates for preferred models
which is model B for the three difference variance measures and the three
different filter methods. The first row is the posterior median and the second
row is the confidence interval (2.5th quantile and the 97.5th quantile).
Liu and West Particle Learning MCMCα 0.161 0.265 0.203
(0.124,0.213) (0.220,0.320) (0.144,0.263)β 0.972 0.943 0.957
(0.967,0.979) (0.929,0.953) (0.945,0.970)τ 0.048 0.071 0.053
(0.045,0.051) (0.068,0.075) (0.044,0.064)σ 0.201 0.187 0.182
(0.188,0.214) (0.176,0.197) (0.168,0.196)
Table 6: Realised volatility: Posterior estimates of parameters
Both parameter estimation of the particle learning and MCMC posterior
parameter results are closer than the Liu and West filter. At the same time
48
Time
Vt
0 500 1000 1500 2000 2500 3000
35
7
RVb−V
Figure 7: Liu and West: Latent State of Realised Volatility Model B
Time
Vt
0 500 1000 1500 2000 2500 3000
35
7
RVb−V
Figure 8: Particle Learning: Latent State of Realised Volatility Model B
the confidence interval of the Liu and west filter is much larger than both
the particle learning and the markov chain monte carlo methods for model
B.
However for both the bipower and option implied volatility, the Liu and
West filter obtains parameter estimates closer to the Markov chain Monte
Carlo than the particle learning filter. As a result it is inconclusive of which
method produces the most consistent results but for most parameter posterior
estimates the confidence intervals over laps on another or are close to one
another.
49
Time
Vt
0 500 1000 1500 2000 2500 3000
35
7
RVb−V
Figure 9: MCMC: Latent State of Realised Volatility Model B
Liu and West Particle Learning MCMCα 0.239 0.297 0.214
(0.200,0.277) (0.231,0.345) (0.153,0.276)β 0.949 0.927 0.954
(0.941,0.956) (0.914,0.942) (0.941,0.967)τ 0.059 0.09 0.06
(0.056,0.063) (0.081,0.098) (0.050,0.071)σ 0.171 0.182 0.184
(0.162,0.179) (0.172,0.194) (0.170,0.200)
Table 7: Bipower volatility: Posterior estimates of parameters
Time
Vt
0 500 1000 1500 2000 2500 3000
24
68
BVb−V
Figure 10: Liu and West: Latent State of Bipower Volatility Model B
50
Time
Vt
0 500 1000 1500 2000 2500 3000
24
68
BVb−V
Figure 11: Particle Learning: Latent State of Realised Volatility Model B
Time
Vt
0 500 1000 1500 2000 2500 3000
35
7
BVb−V
Figure 12: MCMC: Latent State of Bipower Volatility Model B
Liu and West Particle Learning MCMCα 0.231 0.312 0.245
(0.150,0.327) (0.231,0.387) (0.162,0.323)β 0.945 0.942 0.959
(0.832,1.07) (0.929,0.954) (0.946,0.973)τ 0.066 0.055 0.058
(0.053,0.075) (0.053,0.058) (0.055,0.061)σ 0.04 0.083 0.052
(0.031,0.051) (0.081,0.087) (0.0486,0.055)
Table 8: Option Implied volatility: Posterior estimates of parameters
51
Time
Vt
0 500 1000 1500 2000 2500 3000
46
8
VIXb−V
Figure 13: Liu and West: Latent State of Option Volatility Model B
Time
Vt
0 500 1000 1500 2000 2500 3000
56
78
VIXb−V
Figure 14: Particle Learning: Latent State of Realised Volatility Model B
Time
Vt
0 500 1000 1500 2000 2500 3000
4.5
6.0
7.5
VIXb−V
Figure 15: MCMC: Latent State of Option Volatility Model B
52
10.4 Forecast
According to our results from the marginal log likelihood comparisons the
models we would use to forecast . The priors for the forecast are the pos-
terior distribution estimated by the first 3000 observations of the S&P500
variance measures. In the table below are the average absolute deviations
and squared deviations of the predicted volatility measures between the pe-
riod 13th August 2008 to 31st December 2008. The forecasted values are
generated from particle learning draws for the next business days as a result
it is a probabilistic forecast of realised, bipower and option implied volatility.
Models Average Absolute Deviation Average Squared DeviationRealised volatility 0.1140 0.0372Bipower volatility 0.1109 0.0363
Option implied volatility 0.1237 0.0354
Table 9: Forecast deviation measures
Time
Ann
ualis
ed V
aria
nce
0 20 40 60 80 100
0.0
0.4
0.8
1.2
Probabilistic Forecast of RV
Figure 16: Probabilistic Forecast of Realised Volatility
A probabilistic forecast is a better method to forecast real world events
as suggested by Gneiting (2007) as nothing is for certain. The probabilistic
forecast where the upper solid line is the 97.5th percentile, the lower solid
line is the 2.5th percentile and the middle solid line is the median of the
predicted of the predicted observation states. During the volatile period of
the 2007 Financial crisis, we can see that the forecast range is large over this
period where there was a range from 1.2 to 0.2 at the 45th period which
53
Time
Ann
ualis
ed V
aria
nce
0 20 40 60 80 100
0.0
0.4
0.8
1.2
Probabilistic Forecast of BV
Figure 17: Probabilistic Forecast of Bipower Volatility
Time
Ann
ualis
ed V
aria
nce
0 20 40 60 80 100
0.0
0.4
0.8
1.2
Probabilistic Forecast of VIX
Figure 18: Probabilistic Forecast of Option Implied Volatility
is two days after the 10th October 2008 when the news of a market crash
caused panic securities selling causing the realised and bipower volatility to
rise to 1.03 on the 43th period .
The probabilistic forecast of the option volatility covers for all the real
observations, there are no cases where the option volatility jumps out of the
forecast range which implies we do have a good model. We can see that
from the first 30 observations the market was still stable however leading up
to more bad news of the global market the range of forecast grew and by
October the 35 th observation the market continued to stay volatile which is
depicted in the predicated forecast range.
54
However all our forecast contained a large probabilistic range which is
not a desirable attribute as it suggest inaccuracy in the forecast model used.
As a result there is a possibility of a better model that can improve forecast
accuracy (discussion in Robustness section).
55
11 Robustness and Analysis
To determine the best model to forecast from we applied three Bayesian
filter methods to recursively solve for integrated variation (latent state). By
making our conclusion from multiple Bayesian filter methods we would be
able to pick the best model that fits our data to perform a consistent and
efficient forecast of volatility, however there can be multiple parts in the
filtering process which can affect our conclusions such as the priors, number
of iterations or models.
11.1 Prior Sensitivity
As stated in previous section our model selection analysis tools were the
marginal log-likelihoods and the sequential bayes factor however it does not
necessary suggest that our results are valid. Since we are comparing the
marginal log-likelihoods to determine the best model the results are highly
dependent on the priors distributions used for the initial estimated latent
state and parameter distribution, as a result the priors can affect our judg-
ment on the best model to use. Henceforth, the table below is the marginal
log-likelihoods for the our two most preferred models for each variance mea-
sure when we double the variance of the priors and the parameter posterior
results.
From table 10, it shows that by doubling the variance or scale parameters
for the both the parameter and initial latent state priors we lose some fit
according to the marginal log-likelihood values as for all cases they have
fallen. Though not all models are shown in the table above the ranking for
the top two preferred models with the original priors and new priors were the
same. However it still suggests that model B is the preferred model as it has
the highest marginal log-likelihood. Both the realised volatility and bipower
volatility did not change much in the marginal log-likelihood compared to
56
Particle Learning Marginal Log-Likelihood valuesModel Original Priors Doubling variance/scale prior parameter
Realised volatilityModel B -2456.685 -2513.594Model C -2468.032 -2673.339
Bipower volatilityModel B -2512.838 -2578.327Model C -2509.616 -2710.641
Option Implied VolatilityModel A -2612.673 -3642.057Model B -1167.031 -2060.145
Table 10: Marginal Likelihood sensitivity to changes in Prior
the option implied volatility as the values for marginal log-likelihoods almost
doubled by doubling variance/scale prior distribution.
11.2 Prior distribution difficulty for Option Implied
Volatility
The option implied volatility as shown in table 10 can be affected significantly
by the prior distributions used unlike the realised and bipower volatility
which are relatively stable compared to option implied volatility. Similar to
the exercise performed in the Robustness: Multiple filter method section,
if we reduced the variance and scale priors for the option volatility by half
we would get positive marginal log-likelihood values which suggests model
misspecification as the model is very sensitive to our prior selections.
From the S&P 500 data series we see that both realised and bipower
volatility are identical however the option implied volatility has a lagged
reaction to changes in realised and bipower volatility. We also noted in the
data discussion section that the option volatility is relatively more smooth
compared to the realised and bipower volatility this can imply that lags of the
latent states have an important impact to the option volatility. The desired
57
model could have been a p-th order autoregressive model or a GARCH model
as they can take into account for heteroskedasticity in the model and as well
as take into account for lags of the latent state. This could be used for future
research into the modelling of option implied volatility.
11.3 Number of iterations
Time efficiency is an advantage of particle filters over markov chain monte
carlo methods as by performing the particle filter method we used 5000 par-
ticles (though more particles would lead to more consistent results memory
limitations prevented usage of over 5000 particles) for the particle learning
models and the Liu and West filter method for 3000 observations and both
particle filter methods take approximately 2-3 minutes to complete. How-
ever with the markov chain monte carlo method for 8000 iterations (3000
iterations are used as burn-in) for the same observations it will take approxi-
mately an hour to complete. All simulations were performed using a 3.1Ghz
Intel Core i3 CPU with 4gb of RAM for memory. We believe that to be able
to compare the consistency of our results we applied multiple filter methods
and by keeping the number of iterations and particles the same it will be a
reliable approach for comparison between all filter methods, as by comparing
the marginal log-likelihoods we can see how the same number of iterations
affects the convergence of the filter methods. We can conclude that for non-
linear models the Markov chain Monte Carlo method converges slower than
the particle filter methods but for linear models all three methods.
11.4 Models
In this section we would be discussing about the realised and bipower volatil-
ity models, there are a variety of models investigated however there can lim-
itations since the models we looked at do not have long memory properties
58
that Engle suggested. Though our model did not account for some of the
stylised facts of volatility we can see from our results that more complicated
non-linear models did not perform as well compared to the simple local level
model and the autoregressive of order one model perform slightly better than
the local level model. This is helpful as it implies that the two variance mea-
sures are good non-parametric estimates of the true volatility (integrated
variance) which is consistent to Anderson et al(2007) results.
From the S&P500 index we observe that the series is very volatile however
from our marginal log-likelihood for the first 3000 observations are relatively
low at around -2500 for majority of models and methods modelling realised
volatility and bipower volatility. Consequently these two arguments implies
that the models investigated are a valid selection of models however further
improvements can be looked at such as the SV-GARCH models found in
Pitt(2010) which captures most of the stylised facts of volatility presented
by Engle.
59
12 Conclusion
In this paper we have reviewed three bayesian filter methods: the Liu and
West filter, the particle learning filter and the forward filtering backward
sampling algorithm. The particle filter performs more efficiently than the
markov chain monte carlo method to recursively solve a dynamic model as
reported in robustness section the particle filter is more time efficient than the
markov chain monte carlo method. The Markov chain Monte Carlo method
has a faster convergence (in terms of iterations) than the particle filter(s) if
the model is relatively simple and linear on the other hand if the model is
non-linear and begins to become more complicated the particle filter has its
benefits of speed and general practabiliity to be adapted to another problem.
Current research in bayesian filtering with stochastic volatility models
suggest to investigate stochastic volatility model via particle filter approach
is to determine fat tailed distributions in stochastic volatility, where it is
possible to at the same time to determine the degrees of freedom of the t-
dstribution, estimate the parameters and solve the system. A few other aca-
demics such as Hedibert Lopes and Carlos Carvalho (University of Chicago)
are researching on multivariate stochastic volatility models and how to in-
corporate the markov switching process into particle filtering. Literature in
the particle filter field is mainly on parameter estimation or a new particle
filter algorithm that solves a problem with the previous particle filter such
as particle degeneracy problems was fixed with the auxiliary particle filter
(resample-sample filter). Currently the authors of the particle learning al-
gorithm (Lopes, Carvalho, Johannes and Polson) are working on ways to
extend and reduce the Monte Carlo errors and sampling improverishments
that particle learning has.
Particle filters had been developed in 1993 by Gordon, Salmond and Smith
primary for solving mathematical problems as a result has become extremely
popular in engineering and machine learning. It has taken many years for
academics in the economics field to start applying the particle filter The
60
literature field in bayesian filtering in econometrics is vast with academics
looking into incorporating particle filters in solving macroeconomic problems
via dynamic stochastic general equilibrium models such as Flury and Shep-
hard (2008) and Fernandez-Villaverde and Rubio-Ramirez (2007) or solving
stochastic volatility models in finance. There is a bright future in researching
in Bayesian filtering as there are still new particle filter algorithms to develop
and new applications of the particle filter.
61
13 Appendix
13.1 Prior Specifications
The priors used were not determined by previous sampling of the series, they
are all just guesses that are approximately the size and magnitude the param-
eters should be. Note that both the alpha and beta are normal distributed
and both measurement and observation errors are inverse gamma distributed
as it keeps all errors to be positive. Since we stated that the in the case of
normally distributed observation errors we assumed that the initial latent
variable is also normally distributed.
Realised Volatility (RV)
Alpha∼ Normal(0.1,0.05)
Beta∼ Normal(1,0.1)
Tau∼ Inverse Gamma(20,2.2)
Sigma∼ Inverse Gamma(20,1.1)
x0 ∼ Normal(4,2)
Bipower Volatility (BV)
Alpha∼ Normal(0.1,0.05)
Beta∼ Normal(1,0.1)
Tau∼ Inverse Gamma(20,2.2)
Sigma∼ Inverse Gamma(20,1.1)
x0 ∼ Normal(4,2)
Option Implied Volatility (VIX)
Alpha∼ Normal (0.1,0.05)
Beta∼ Normal(1,0.1)
Tau ∼ Inverse Gamma(400,220)
Sigma ∼Inverse Gamma(400,220)
x0 ∼ Normal(4,2)
62
13.2 Particle Learning Filter
Time
α
0 500 1500 2500
−0
.50
.00
.51
.0
RVb−alpha
Time
β
0 500 1500 2500
0.7
0.8
0.9
1.0
1.1
1.2
RVb−beta
Timeτ
0 500 1500 2500
0.1
00
.15
0.2
00
.25
RVb−tau
Time
σ
0 500 1500 2500
0.2
0.3
0.4
0.5
0.6
0.7
RVb−sigma
Figure 19: Particle Learning: Parameter history of Realised Volatility ModelB
Time
α
0 500 1500 2500
−0
.50
.00
.51
.0
BVb−alpha
Time
β
0 500 1500 2500
0.6
0.7
0.8
0.9
1.0
1.1
1.2
BVb−beta
Time
τ
0 500 1500 2500
0.1
00
.15
0.2
00
.25
0.3
0
BVb−tau
Time
σ
0 500 1500 2500
0.2
0.3
0.4
0.5
0.6
0.7
BVb−sigma
Figure 20: Particle Learning: Parameter history of Bipower Volatility ModelB
63
Time
α
0 500 1500 2500
−0
.50
.00
.51
.01
.5
VIXb−alpha
Time
β
0 500 1500 2500
0.8
0.9
1.0
1.1
1.2
1.3
VIXb−beta
Time
τ
0 500 1500 2500
0.0
60
.08
0.1
0
VIXb−tau
Time
σ
0 500 1500 2500
0.0
1.0
2.0
3.0
VIXb−sigma
Figure 21: Particle Learning: Parameter history of Option Volatility ModelB
64
13.3 Liu and West Filter
Time
α
0 500 1500 2500
0.0
00
.05
0.1
00
.15
0.2
0
RVb−alpha
Time
β
0 500 1500 2500
0.8
0.9
1.0
1.1
RVb−beta
Timeτ
0 500 1500 2500
0.0
30
.05
0.0
70
.09
RVb−tau
Time
σ
0 500 1500 2500
0.1
00
.20
0.3
0
RVb−sigma
Figure 22: Liu and West: Parameter history of Realised Volatility Model B
Time
α
0 500 1500 2500
0.0
0.1
0.2
0.3
BVb−alpha
Time
β
0 500 1500 2500
0.8
0.9
1.0
1.1
BVb−beta
Time
τ
0 500 1500 2500
0.0
40
.08
0.1
2
BVb−tau
Time
σ
0 500 1500 2500
0.1
00
.15
0.2
00
.25
BVb−sigma
Figure 23: Liu and West: Parameter history of Bipower Volatility Model B
65
Time
α
0 500 1500 2500
−0
.6−
0.4
−0
.20
.00
.2
VIXb−alpha
Time
β
0 500 1500 2500
0.8
0.9
1.0
1.1
VIXb−beta
Time
τ
0 500 1500 2500
0.4
0.6
0.8
1.0
VIXb−tau
Time
σ
0 500 1500 2500
0.4
0.6
0.8
1.0
VIXb−sigma
Figure 24: Liu and West: Parameter history of Option Volatility Model B
66
References
[1] Ait Sahalia, Y., and Kimmel, R. Maximum likelihood estimationof stochastic volatility models. Journal of Financial Econometrics .
[2] Anderson, T., Bollerslev, T., and Diebold, F. Roughing itup: Including jump components in the measurement, modelling andforecasting of return volatility.
[3] Andrieu, C., Doucet, A., and Holenstein, R. Particle markovchain monte carlo methods. Journal of the Royal Statistical Society:Series B (Statistical Methodology) 72, 3 (2010), 269–342.
[4] Barndorff-Nielsen, O., and Shephard, N. Power and bipowervariation with stochastic volatility and jumps. Journal of FinancialEconometrics 2, 1 (2004), 1–37.
[5] BBC News. Fear grips global stock market, Oct. 2008.
[6] Brownlees, C., and Gallo, G. Financial econometric analysis atulta-high frequency; data handling concerns. Computational statisticsand Data analysis 51 (2006).
[7] Carlin, B., Polson, N., and Stoffer, D. A monte carlo approachto nonnormal and nonlinear state-space modeling. Journal of the Amer-ican Statistical Association (1992), 493–500.
[8] Carter, C., and Kohn, R. On gibbs sampling for state space models.Biometrika 81, 3 (1994), 541–553.
[9] Casella, G., and George, E. Explaining the gibbs sampler. Amer-ican Statistician (1992), 167–174.
[10] Christoffersen, P., Jacobs, K., and Mimouni, K. Volatility dy-namics for the s&p500: Evidence from realized volatility, daily returns,and option prices. Review of Financial Studies 23, 8 (2010), 3141–3189.
[11] Corsi, F., Mittnik, S., Pigorsch, C., and Pigorsch, U. Thevolatility of realized volatility. Econometric Reviews 27 (2008), 46–78.
[12] Engle, R., and Patton, A. What good is a volatility model? NYUworking paper (2001).
67
[13] Eraker, B., Johannes, M., and Polson, N. The impact of jumpsin volatility and returns. The Journal of Finance 58, 3 (2003), 1269–1300.
[14] Fruhwirth-Schnatter, S. Applied state space modelling of non-gaussian time series using integration-based kalman filtering. Statisticsand Computing 4, 4 (1994), 259–269.
[15] Geweke, J., and Tanizaki, H. Bayesian estimation of state-spacemodels using the metropolis–hastings algorithm within gibbs sampling.Computational statistics & data analysis 37, 2 (2001), 151–170.
[16] Gordon, N.J, S. D., and Smith, A. Novel approach tononlinear/non-gaussian bayesian state estimation. In Radar and Sig-nal Processing, IEE Proceedings F (1993), vol. 140, IET, pp. 107–113.
[17] Harvey, A., Ruiz, E., and Shephard, N. The Review of EconomicStudies .
[18] Hore, S., Johannes, M., Lopes, H., McCulloch, R., and Pol-son, N. Bayesian computation in finance. Frontiers of Statistical Deci-sion Making and Bayesian Analysis (2010).
[19] Jacquier, E., Polson, N., and Rossi, P. Bayesian analysis ofstochastic volatility models with fat-tails and correlated errors. Journalof Econometrics 122, 1 (2004), 185–212.
[20] Jacquier, S., Polson, N., and Rossi, P. Bayesian analysis ofstochastic volatility models. Journal of Business and Economic Statis-tics 12, 4.
[21] Johannes, M., and Polson, N. Mcmc methods for financial econo-metrics. working paper (2002).
[22] Kalman, R. A new approach to linear filtering and prediction prob-lems. Journal of basic Engineering 82, 1 (1960), 35–45.
[23] Kantas, N., Doucet, A., Singh, S., and Maciejowski, J. Anoverview of sequential monte carlo methods for parameter estimationin general state-space models. In Proc. IFAC Symposium on SystemIdentification (SYSID) (2009).
[24] Kim, S., Shephard, N., and Chib, S. Stochastic volatility: likeli-hood inference and comparison with arch models. The Review of Eco-nomic Studies 65, 3 (1998), 361–393.
68
[25] Lopes, H., and Tsay, R. Particle filters and bayesian inference infinancial econometrics. Journal of Forecasting 30 (2011), 168–209.
[26] Malik, S., and Pitt, M. Modelling stochastic volatility with lever-age and jumps: A simulated maximum likelihood approach via particlefiltering. Available at SSRN 1763783 (2011).
[27] Maneesoonthorn, W., Martin, G., Forbes, C., and Grose, S.Probabilistic forecast of volatility and its risk premia. Working Paper:forthcoming Journal of Econometrics (2012).
[28] McAleer, M., and Medeiros, M. Realized volatility: A review.Econometric Reviews 27:1-3 (2008), 10–45.
[29] Nakajima, J. Bayesian analysis of generalized autoregressive condi-tional heteroskedasticity and stochastic volatility: Modeling leverage,jumps and heavy-tails for financial time series*. Japanese EconomicReview 63, 1 (2011), 81–103.
[30] Pitt, M., and Shephard, N. Filtering via simulation: Auxiliaryparticle filters. Journal of the American statistical association 94, 446(1999), 590–599.
[31] Taylor, S. Modeling stochastic volatility: A review and comparativestudy. Mathematical Finance 4, 2 (1994), 183–204.
[32] Welch, G., and Bishop, G. An introduction to the kalman filter.University of North Carolina at Chapel Hill, Chapel Hill, NC 7, 1 (1995).
69