Bayesian Filtering on Realised, Bipower and Option Implied Volatility

Transcription

1 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

2 1 Declaration I hereby declare that this submission is my own work and to the best of my knowledge it contains no material previously written by another person, or material which to a substantive extent has been accepted for the award of any other degree or diploma of a university or other institute of higher learning, except where referenced in the text. I also declare that the intellectual content of this thesis is the product of my own work, and any assistance that I have received in preparing the project, writing the program as well as presenting the thesis, has been duly acknowledged. Nelson Qu 1

3 Contents 1 Declaration 1 2 Acknowledgements 5 3 Abbreviations 5 4 Abstract 6 5 Introduction 7 6 Literature Review Stochastic Volatility Models Volatility Stylised Facts Kalman Filters Markov Chain Monte Carlo Methods(MCMC) Sequential Monte Carlo Methods/ Particle Filtering Models Stochastic Volatility Model (SVM) Realized Volatility Bipower Volatility VIX Data Data transformation Method MCMC Method Outline of MCMC Method Particle Filter Outline of Particle Filter Methods Liu and West Filter Particle Learning Filter Models How observation errors are distributed Results Marginal Log Likelihoods Sequential Bayes Factor Parameter Estimation

4 10.4 Forecast Robustness and Analysis Prior Sensitivity Prior distribution difficulty for Option Implied Volatility Number of iterations Models Conclusion Appendix Prior Specifications Particle Learning Filter Liu and West Filter List of Figures 1 Difference in Realised and Bipower volatility of S&P500 Index 27 2 Graph of Data QQ plots of Data and Log Data Sequential Bayes Factor Sequential Bayes Factor Sequential bayes factor Liu and West: Latent State of Realised Volatility Model B Particle Learning: Latent State of Realised Volatility Model B 49 9 MCMC: Latent State of Realised Volatility Model B Liu and West: Latent State of Bipower Volatility Model B Particle Learning: Latent State of Realised Volatility Model B MCMC: Latent State of Bipower Volatility Model B Liu and West: Latent State of Option Volatility Model B Particle Learning: Latent State of Realised Volatility Model B MCMC: Latent State of Option Volatility Model B Probabilistic Forecast of Realised Volatility Probabilistic Forecast of Bipower Volatility Probabilistic Forecast of Option Implied Volatility Particle Learning: Parameter history of Realised Volatility Model B Particle Learning: Parameter history of Bipower Volatility Model B Particle Learning: Parameter history of Option Volatility Model B

5 22 Liu and West: Parameter history of Realised Volatility Model B Liu and West: Parameter history of Bipower Volatility Model B Liu and West: Parameter history of Option Volatility Model B 66 List of Tables 1 Summary of S&P 500 data Transformed data Marginal Log Likelihood for Realised Volatility Marginal Log Likelihood for Bipower Volatility Marginal Log Likelihood for Option Implied Volatility Realised volatility: Posterior estimates of parameters Bipower volatility: Posterior estimates of parameters Option Implied volatility: Posterior estimates of parameters Forecast deviation measures Marginal Likelihood sensitivity to changes in Prior

6 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 Python T M. 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 observation errors LW-B = Liu and West filter for model B with normally distributed observation errors 5

7 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 different 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 volatility 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

8 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 nonparametric measures of volatility like the standard variance formula or the realised variance formula. The motivation for the study of latent volatility 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

9 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 models, 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 literature 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 modelling literature field however this can become a complicated process if we are unsure which models to apply in certain situations. However model selection 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

10 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 statistical 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 dynamic 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 system. 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

11 2007 Global financial Crisis. We will consider three different variance measures 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 interesting 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 volatility 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

12 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

13 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 approach 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 approximate the sampling distribution of the likelihood was developed and these include 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

14 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 volatility, 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 returns 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 periods 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 between 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 Taylor(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: y t = σ x ɛ t σ t log(σ 2 t ) = φlog(σ 2 t 1) + η t, t=1,2,...,t-1 13

15 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 volatility 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 volatility 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

16 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 nonlinearity 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

17 Monte Carlo (sampling) methods. 6.4 Markov Chain Monte Carlo Methods(MCMC) The concept of applying a Monte Carlo approximation method to solve a difficult integral is suppose f(x)dp (x), if we can draw size N of identical and x independently distributed random samples of {x 1, x 2,...x N } from the probability distribution P(x) then we can approximate f(x) by f ˆ N N = 1 f(x i ). N From the concept of Strong Law of Large Numbers we can say that our estimated value of the function ˆ f N 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 distribution 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 i=1 16

18 Markov Chain Monte Carlo filters represents a general family of filtering algorithms that uses the fundamental idea that if there is a mulitvariate distribution that is difficult to integrate we are able to draw random samples 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 versions 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 accurate 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 Gibbssampling 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

19 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 Filtering 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 discrete 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 sampling 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 implement 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 approximate 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

20 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 distribution 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 generally 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 described 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. Another 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

21 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 algorithms 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 conditional 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 parameter 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 multivariate 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

22 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

23 7 Models In this section we will define latent volatility, so first lets define spot asset price as P t at time t, if we assume a continuous diffusion process then similar to what is shown in Harvey et al(1998) and Taylor (1986); ds t = µ t dt + V t db p t (1) d(v t ) = κ(θ V t )dt + τdb V t (2) where the parameters (µ t, κ, θ, τ) are evolving with volatility and B p t, B V t 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]. ds t = µ t dt + V t db p t + dj p t (3) The jump component dj p t is described as a random jump process such that dj p t = Z p t dn p t where the jump size is Z p t follows an exponential distribution with probability of jump dn p t happening with a bernoulli distribution. To incorporate volatility jumps that Eraker et al(2003) [13] suggests improves model fit we would need to modify equation (2) by incorporating a jump component. d log(v t ) = κ(θ log(v t ))dt + τdb V t + dj V t (4) 22

24 Similar to the definition for the jump component for asset price, volatility jumps is defined as djt V = Zt V dnt V where Zt V is exponential distribution for size of the volatility jump and dnt V 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). S t = S t t + µ t t + V t t ξ 1t (5) V t = V t t + κθ t κv t t t + τξ 2t (6) where (ξ 1t, ξ 2t ) N(0 2, Σ), Σ = [ 1 ρ ρ if we set t = 1 the above equation will explain daily change in latent variance. Equation (6) will become V t = V t 1 + κθ κv t 1 + τξ 2t = κθ + (1 κ)v t 1 + τξ 2t. If we let α v = κθ and β v = 1 κ then equation (6) can be transformed into a more simplified version: σ 2 v ] V t = α v + β v V t 1 + τξ 2t (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 generally models without jumps in both asset price and volatility are misspecified. Euler discretization were applied to equation (3) and (4), 23

25 S t = S t t + µ t t + V t t ξ 1t + Z S t N S t (8) V t = V t t + κθ t κv t t t + τξ 2t + Z V t N V t (9) N S t where Zt V iid exp(µ v ), Zt S iid Bernoulli(δ S t). iid exp(µ s ), N V t iid Bernoulli(δ v t) and If we let t = 1 and reparameterize for α v = κθ and β v = 1 κ equation (9) can be rewritten in as: V t = α v + β v V t 1 + τξ 2t + Z V t N V 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 volatility measures by a generalized autoregressive conditional heteroskedasticity model (GARCH) or by latent stochastic volatility models (SV-M). Survey papers 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 integrated variance RV t IVt, where IV t = σ 2 (t + s 1)ds. It is p s i said that realised variation can be a consistent estimator of integrated variation. s i 1 24

26 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 (RV t IV t ) d N(0, 1) where IQ t = 1 0 σ 4 (t + s 1)ds is integrated quarticity. In terms of modelling realised volatility many different papers offer different models to try to model realised volatility. Christoffersen et al (2007) [10] defined realised variance to have the form of RV t+1 = E[RV t+1 V t ] + u t, using the results found from Ait-Sahalia (2007) [1] we can create the observation equation to have the form: RV t+1 = θ + [ exp( κ/252) 1 ](V t θ) + u t κ/252 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). Another model that is found in the literature on modelling realised volatility is RV t+1 = α 1 β β 1 exp( ) 1 (V t θ) + u t (11) β RV t+1 = V t + σ rv u t (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

27 and three halves stochastic volatility model to fit realised volatility. Below is a simplified example of the three stochastic volatility model forms: V t+1 = α + βv t + τ rv Vt u t (13) V t+1 = α + βv t + τ rv V t u t (14) V t+1 = α + βv t + τ rv V t 3/2 u t (15) 7.3 Bipower Volatility Bipower volatility is described by Anderson et al(2007) as the variation definition when there is no jump in volatility. The bipower volatility has a mathematical form of BV t = π 2 M r ti r ti 1 t 1<t i t 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 otherwise 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 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

28 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 λ rn V t : dv t = (κ(θ V t ) + λ rn V t )dt + τdb t dv t = (κ λ rn )(κθ/(κ λ rn ) V t )dt + τdb t dv t = κ (θ V t )dt + τdb t (16) 27

29 where db t is a brownian motion for the risk neutral process, κ = (κ λ rn ) and θ = κθ/κ. If we took the Euler discretization of equation (16) V t+1 = V t + κ θ κ V t + τξt V t+1 = α + β V t + τξt (17) now we let α = κ θ = κθ and β = (1 κ ) = (1 + λ rn κ) to be risk neutral parameters for the stochastic volatility model. The λ rn 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

30 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 It covers many significant financial 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 (RV t = M rt 2 i ) 2. Bipower variation (BV t = π 2 t 1<t t M t 1<t i t r ti r ti 1 ) 3. Option variation denoted as MF t ( MF t = ( 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

31 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 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

32 Realised Volatility Bipower Volatility Volatility Index Minimum st Quartile Median Mean rd Quartile Maximum 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 Volatility Minimum st quartile Median Mean rd quartile Maximum Table 2: Transformed data 31

33 Figure 2: Graph of Data 32

34 RV Q Q Plot Theoretical Quantiles Sample Quantiles BV Q Q Plot Theoretical Quantiles Sample Quantiles VIX Q Q Plot Theoretical Quantiles Sample Quantiles Log RV Q Q Plot Theoretical Quantiles Sample Quantiles Log BV Q Q Plot Theoretical Quantiles Sample Quantiles Log VIX Q Q Plot Theoretical Quantiles Sample Quantiles Figure 3: QQ plots of Data and Log Data 33

35 9 Method 9.1 MCMC Method 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 represent 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 follows a markov chain we can say that the distribution of the state variables for the future, present and past is P (x t 1, x t, x t+1 Y t ) = P (x t 1, x t Y t )P (x t+1 x t, x t 1, Y t ) = P (x t 1, x t Y t )P (x t+1 x t ). Since on the right hand side we have P (x t 1, x t Y t ) which is the distribution we should know from our previous step then we can marginalise out x t 1 so the distribution of the state variables can be represented in the form of: P (x t, x t+1 Y t ) = P (x t Y t )P (x t+1 x t ) 34

36 Update step: Applying bayes theorem of the above equation such that P (x t, x t+1 Y t+1 ) P (x t+1, x t, Y t+1 Y t ) = P (x t, x t+1 Y t )P (Y t+1 x t, x t+1 ) P (x t, x t+1 Y t )P (Y t+1 can be found in the previous evolution step and the second term P (Y t+1 x t, x t+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 1 P (x 1, x 2,..., x t Y t ) = P (x t, x t 1 Y t ) p(x i x i+1, x i+2,..., x t, Y t ) i=t Particle Filter 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