Stochastic Volatility Modeling withComputational Intelligence Particle Filters

Ayub HanifIntelligent Systems GroupUniversity College London

Gower Street, London, WC1E 6BTayub.hanif@ucl.ac.uk

Robert E. SmithIntelligent Systems GroupUniversity College London

Gower Street, London, WC1E 6BTrobert.elliott.smith@gmail.com

ABSTRACTStochastic volatility estimation is an important task for cor-rectly pricing derivatives in mathematical finance. Suchderivatives are used by varying types of market participantas either hedging tools or for bespoke market exposure. Weevaluate our adaptive path particle filter, a recombinatoryevolutionary algorithm based on the generation gap conceptfrom evolutionary computation, for stochastic volatility esti-mation of three real financial asset time series. We calibratethe Heston stochastic volatility model employing a Markov-chain Monte Carlo, enabling us to understand the latentstochastic volatility process and parameters. In our experi-ments we find the adaptive path particle filter to be superiorto the standard sequential importance resampling particlefilter, the Markov-chain Monte Carlo particle filter and theparticle learning particle filter. We present a detailed analy-sis of the results and suggest directions for future research.

Categories and Subject DescriptorsG.3 [Probability and Statistics]: Probabilistic algorithms- sequential Monte Carlo; J.4. [Social and BehavioralSciences]: Economics

General TermsAlgorithms, economics

KeywordsNonlinear filters, sequential Monte Carlos, particle filters,recursive Bayesian estimation, evolutionary computation,stochastic volatility estimation, Heston model.

1. INTRODUCTIONCommon derivative securities pricing involves assumptions

of constant volatility, either as a known constant or a knowndeterministic function. Empirically, such assumptions are in-correct and lead to mispricing which in turn offers arbitrage

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opportunities to sophisticated market participants. Variousmeasures of volatility show it to be unstable and variableleading to stochastic volatility models. These models ad-dress and allay the inconsistencies of common pricing mea-sures. As the volatility measure of an underlying security islatent, estimation is a non-trivial undertaking. In this paperwe are aiming to correctly model the volatility process ofan underlying security through the calibration of a commonstochastic volatility model.Sequential Monte Carlo methods (commonly known as

particle filters) are extensively applied to the stochastic vol-atility estimation problem. Particle filters are populationbased metaheuristics. The synergy between their statisti-cal foundations and computational intelligence (evolutionarycomputation (EC) and genetic algorithms) has led to a num-ber of ever-powerful computational intelligence particle fil-ters. Kwok, et al. [23] propose a hybrid particle filter whichincludes an evolutionary stochastic universal sampling step.Simulated on the gambler’s ruin problem they find their par-ticle filter to outperform traditional particle filters. Han, etal. [18] embed an immune genetic algorithm into the tra-ditional particle filter which effectively controls diversity astested on the univariate non-stationary growth model fromeconomics. Uosaki, et al. [34] combine evolutionary compu-tation into their evolutionary strategies particle filter, whichis used for fault detection. Duan & Cai [11] build an evo-lutionary particle filter for robust simultaneous localizationand map building tasks of autonomous mobile robots. Hanif& Smith [20] employed an evolutionary computation particlefilter to simulated univariate log-stochastic volatility series,where it excelled and outperformed contemporary and state-of-the-art particle filters.In this paper, we apply sequential Monte Carlo methods

to the stochastic volatility estimation problem. We compareour computational intelligence particle filter to contempo-rary filters across a number of real asset time series and findour method provides enhanced estimation accuracy.

2. BAYESIAN REASONINGTo facilitate probabilistic reasoning over time we will be

adopting state space representations under a Bayesian frame-work. Stochastic models overcome shortcomings of deter-ministic models and enable postulation of ideas encompass-ing uncertainty. A state space model of a time series {yt : t =1, 2, ...} is composed of two equations: the state equation andthe observation equation. The observation equation relatesthe observed data {yt} to the latent states {xt : t = 1, 2, ...}.

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Consider the discrete time estimation problem with the sys-tem model

yt = ht(xt,vt) (1)

xt+1 = ft(xt,wt) (2)

in which we represent the observation vector at time t byyt ∈ R

p, which satisfies (1) where ht : Rn × R

r → Rp is the

observation function and vt ∈ Rr is the state error term of

which the known distribution is independent of both systemnoise and time. Similarly, we represent the state vector attime t by xt ∈ R

n, which satisfies (2) where ft : Rn ×R

m →R

n is the system transition function and wt is an error termwhose known distribution is temporally independent.

There are very few system models for which an analyticsolution is available. Consider the linear Gaussian model

Yt = HtXt +Wt (3)

Xt+1 = FtXt +Vt (4)

where (3) is the measurement equation with {Wt} ∼N (0, Rt) observation errors; and (4) is the state equationwith {Vt} ∼ N (0, Qt) state description errors. We assumethat the observation errors and state errors are uncorrelated.That is ∀s,∀t : E[WsVt] = 0. Finally, the initial state X0 isassumed to be uncorrelated with all errors {Vt} and {Wt}.

The principal foundation of stochastic filtering lies in re-cursive Bayesian estimation where we are essentially tryingto compute the joint posterior. This is an inverse statisticalproblem: you want to find inputs as you are given outputs[8]. There are two key assumptions in deriving the recursiveBayesian filter: (i) that the state process follows a first-orderMarkov process

(xn|x0:n−1,y0:n−1) = p(xn|xn−1)

and (ii) that the observations and states are independent

(yn|x0:n−1,y0:n−1) = p(yn|xn).

For simplicity, we shall denote Yn as the set of observationsy0:n := {y0, ...,yn} and p(xn|Yn) as the conditional pdf ofxn. From Bayes rule we have

p(xn|Yn) =p(yn|xn)p(xn|Yn−1)

p(yn|Yn−1), (5)

the calculation and or approximation of which is the base ofBayesian filtering and inference.

3. SEQUENTIAL MONTE CARLOMETHODS

There are two approaches to obtain the posterior distri-bution of concern defined in (5): parametric Gaussian ap-proximation or non-parametric approximation using MonteCarlo techniques [32]. Though they are computationally ef-ficient, parametric techniques such as the Kalman filter andits extensions do not reliably compute states and cannot beused to learn stochastic problems where the underlying pro-cess is non-linear and non-Gaussian. Some applications ofKalman filtering techniques are highlighted by [2] though,conversely, they highlight difficulties in calculating closed-form solutions to distribution approximations and proposeapplication of numerical methods to overcome these difficul-ties.

Sequential Monte Carlo approximation, a numerical met-hod, of optimal estimation problems in non-linear non-Gaus-

sian settings is commonly performed using particle methods[8, 17, 22, 27, 6, 5, 9, 29]. The main advantage of thesemethods is that they do not rely on any local linearization orabstract functional approximation. This is at the cost of in-creased computational expense though given breakthroughsin computing technology and the related decline in process-ing costs, this is not considered a barrier except in extremecircumstances.Monte Carlo approximation using particle methods cal-

culates the expectation of the pdf by importance sampling(IS) [8]. The state space is partitioned, into which particlesare filled with respect to some probability measure. Thehigher this measure the denser the particle concentration.The state space evolves temporally and the particle systemevolves around this. Specifically, from (5) and [32]

p(xt|y0:t) =p(yt|xt)p(xt|y0:t−1)

p(yt|y0:t−1)

where p(xt|y0:t) is the state posterior (filtering distribution),p(yt|xt) is the likelihood, p(xt|y0:t−1) is the state prior (pre-dictive distribution) and the denominator p(yt|y0:t−1) is theevidence. The state prior is defined by

p(xt|y0:t−1) =

∫p(xt|xt−1,y0:t−1)p(xt−1|y0:t−1)dxt−1

where p(xt|xt−1,y0:t−1) is the transition density andp(xt−1|y0:t−1) is the previous filtering distribution.We approximate the state posterior by f(xt) with i sam-

ples of x(i)t . To find the mean E[f(xt)] of the state pos-

terior p(xt|y0:t) at t, we generate the state samples x(i)t ∼

p(xt|y0:t). Though theoretically plausible, empirically weare unable to observe and sample directly from the stateposterior. We replace the state posterior by the proposalstate distribution (importance distribution) π which is pro-portional to the true posterior at every point π(xt|y0:t) ∝p(xt|y0:t). We are thus able to sample sequentially i.i.d.draws from π(xt|y0:t) giving us

E[f(xt)] =

∫f(xt)

p(xt|y0:t)

π(xt|y0:t)π(xt|y0:t)dxt

≈∑N

i=1 f(x(i)t )w

(i)t∑N

i=1 w(i)t

.

When increasing the number of draws N this average con-verges asymptotically to the expectation of the true posterioraccording to the central limit theorem [14]. This convergenceis the primary advantage of sequential Monte Carlo meth-ods as they provide asymptotically consistent estimates ofthe true distribution p(xt|y0:t) [10].IS allows us to sample from complex highly-dimensional

distributions though exhibits linear increases in complexityupon each subsequent draw [10]. To admit fixed computa-tional complexity we can use sequential importance sampling(SIS). The SIS approach is sensitive to the choice for the pro-posal density, with typical choices being the transition prob-ability of states [22, 13]. This choice minimizes the varianceof weights critical to the asymptotic convergence of the al-gorithm. However, SIS suffers from a debilitating problem:the variance of the estimates increases exponentially with nleading to fewer and fewer non-zero importance weights in aproblem known as weight degeneracy. To alleviate the issue,states are resampled to retain the most pertinent contribu-tors in a method known as sequential importance resampling

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x̂36 x̌36

W(i)36 = max

[p(y36|x̂(i)

36 ), p(y36|x̌(i)36 )

]1 2 3 4 5 6 7 8 9 1011121314151617181920

Figure 1: APPF weight update fitness assessment through inter-generational competition for N = 20 at t = 36

(SIR - also referred to as the particle filter (PF)). We providethe SIR algorithm using the transition prior as the proposaldistribution [8] in Table 4, in Appendix A.

SIR attempts to address weight degeneracy by replacingparticles with high weight with many particles with highinter-particle correlation [17, 8]. Frequent resampling leadsto the marginal distribution p̂(x0:t|y0:n) collapsing onto afew or single unique particle(s) in a problem similar to weightdegeneracy. The samples are impoverished and resamplinghas not helped.

There are a number of schemes proposed to address weightdegeneracy and sample impoverishment simultaneously. TheMarkov-chain Monte Carlo particle filter (MCMC-PF) at-tempts to reduce degeneracy by jittering particle locations,using Metropolis-Hastings to accept moves, whilst the par-ticle learning particle filter (PLA) performs an MCMC afterevery 50 iterations. Details of the MCMC step embeddedinto the MCMC-PF and applied in the PLA every 50 itera-tions can be found in Table 5, in Appendix B..

The adaptive path particle filter (APPF) uses techniquesfrom computational intelligence to alleviate weight degener-acy and sample impoverishment together [20]. It maintainsdiversity through embedding a generation based adaptivepath switching step into the particle filter weight update,making use of previously discarded particles if their discrim-inatory power is higher than the current particle set. Thecomplete APPF algorithm is detailed in Table 1.

Upon closer inspection we can see that the APPF is es-sentially a recombinatory evolutionary algorithm. In theexample given in Figure 1 we have N = 20 particles: att = 36 upon evaluation of the fitness function select 11 par-ticles from the current importance samples x̂36 (marked asthe blue particles) and 9 particles from importance samplesbased on the previously resampled out set x̌36 (marked as

the red particles). For example we can see W(1)36 is assigned

as x̂(1)36 , similarly W

(8)36 is assigned as x̌

(8)36 . As can be seen

in the resultant particle set W36 we are leveraging the fittestparticles amongst and across the two generations.

At the level of the representation of the probability dis-tribution (the primary output of a sequential Monte Carlomethod), APPF is using a fitness-based recombination of apast representation with a current representation. In thissense, we believe the algorithm draws upon the body of ECexperience and theory. Admittedly, this is in essence a ECalgorithm with a population of size 2. A natural extension(which would make APPF appear to be a more traditionalEC algorithm) would be the maintenance and fitness-basedrecombination of several past iterations particles (and thusa larger population). However, note that due to the natureof the PF process itself - that of the state process being a

Table 1: APPF Algorithm

1. Initialization - forming the initial population:for i = 1, ..., Np, sample

x(i)0 ∼ p(x0)

ψ(i)0 ∼ p(x0)

with weights W(i)0 = 1

Np.

For t ≥ 1

2. Importance sampling - selection:for i = 1, ..., Np, draw samples

x̂(i)t ∼ p(xt|x(i)

t−1)

set x̂(i)0:t = {x(i)

0:t−1, x̂(i)t }

and draw x̌(i)t ∼ p(xt|ψ(i)

t−1)

set x̌(i)0:t = {x(i)

0:t−1, x̌(i)t }

3. Weight update: assess fitness

W(i)t = max

[p(yt|x̂(i)

t ), p(yt|x̌(i)t )

]evaluate - recombination through inter-generationalcompetition:

if p(yt|x̌(i)t ) > p(yt|x̂(i)

t ) then

x̂(i)t = ψ

(i)t

end if

4. Normalize weights:

W̃(i)t =

W(i)t∑Np

j=1 W(j)t

5. Commit (generation gap) pre-resample set of parti-cles to memory:

{ψ(i)t } = {x̂(i)

t }

6. Resampling: Generate Np new particles x(i)t from

the set {x̂(i)t } according to importance weights W̃

(i)t .

7. Repeat from importance sampling step 2.

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Markov process by assumption (i) in Section 2, the imme-diate previous particle set contains information about thoseprevious representations explicitly. In this sense, the recom-bination of only these two sets of particles in APPF containsthe implicit schema average fitness information upon whichmuch of EC theory is based [16]. Thus, we feel that thedramatic improvements in performance the APPF offers (asshown in the following sections, and in other applications[19]) is directly related to EC theory and practice, and canbe built on further using EC ideas.

4. STOCHASTIC VOLATILITYESTIMATION PROBLEM

Understanding the dynamics of the volatility process intandem with the dynamics of the underlying asset in thesame timescale enable us to measure the stochastic volatil-ity process. The most popular stochastic volatility model isthe Heston model which uses both square-root and Ornstein-Uhlenbeck dynamics. We shall proceed to describe and es-timate this model. Recent reformulations of the originalFourier integrals of the Heston stochastic volatility model[21] have led to numerically stable and efficient computationsof derivative prices [1, 25, 26, 7, 24]. The Heston model toestimate stochastic volatility is defined by the coupled two-dimensional stochastic differential equation:

dX(t)/X(t) =√

V (t)dWX(t) (6)

dV (t) = κ(θ − V (t))dt+ ε√

V (t)dWV (t) (7)

where κ, θ, ε are strictly positive constants, and where WX

and WV are scalar Brownian motions in some probabilitymeasure; we assume that dWX(t) · dWV (t) = ρdt, wherethe correlation measure ρ is some constant in [−1, 1]. X(t)represents an asset price process and is assumed to be a mar-tingale in the chosen probability measure. V (t) representsthe instantaneous variance of relative changes to X(t) - thestochastic volatility - and is modeled as a mean-revertingsquare-root diffusion, with Ornstein-Uhlenbeck dynamics (acontinuous-time analogue of the discrete-time first-order au-toregressive process).

Recognizing that the asset price process X(t) is relativelyclose to geometric Brownian motion, it is sensible to workwith logarithms of X(t). By Ito’s lemma we have:

d lnX(t) = −1

2V (t)dt+

√V (t)dWX(t) (8)

dV (t) = κ(θ − V (t))dt+ ε√

V (t)dWV (t) (9)

Euler discretization of the stochastic differential equation(8)-(9) takes the form:

ln X̂(t+Δ) = ln X̂(t)− 1

2V̂ (t)Δ +

√V̂ (t)ZX

√Δ (10)

V̂ (t+Δ) = V (t) + κ(θ − V̂ (t))Δ + ε

√V̂ (t)ZV

√Δ (11)

where X̂ and V̂ are discrete-time approximation to X andV , respectively, and where ZX and ZV are Gaussian ran-dom variables with correlation ρ. ZX and ZV are drawn asindependent uniform samples from the inverse cumulativeGaussian distribution function.

A critical problem with the naive Euler discretization abo-ve enables the discrete process for V to become negative with

non-zero probability, which makes the computation of√

V̂

impossible. A full truncation scheme produces the smallestdiscretization bias [28], leading to the dynamics:

ln X̂(t+Δ) = ln X̂(t)− 1

2V̂ (t)+Δ+

√V̂ (t)+ZX

√Δ (12)

V̂ (t+Δ) = V (t) + κ(θ − V̂ (t)+)Δ + ε

√V̂ (t)+ZV

√Δ

(13)

where the operator x+ = max(x, 0) enables the process forV to go below zero thereafter becoming deterministic withan upward drift κθ, and where X̂(t) is the observed price

process and V̂ (t) is the stochastic volatility process to beestimated. In addition we need to calibrate the parametersκ, θ, ε to run the recursive Bayesian estimators.

5. EXPERIMENTAL RESULTSWe ran state estimation experiments on three assets’ daily

closing prices from the beginning of 2010 till the end of 2012:two common stocks, General Electric (GE) and Citigroup(C), and one index, the S&P500 (SPX). Prior to runningthe filters we ran a 10,000 iteration MCMC calibration tobuild an understanding of the prices process (observationequation) and volatility process (state equation). The priceprocess is the evolution of log returns which for SPX canbe seen in Figure 2. Our Heston model stochastic volatilitycalibration for SPX can be seen in Figure 3, where we cansee the full truncation scheme forcing the SV process to bepositive, and the associated parameter evolution can be seenin Figure 4. This process was repeated for the remainingsecurities.Of note, we can see ε is a small constant (this is the same

throughout all the securities). This is attributable to the factε represents the volatility of volatility. If it were large andor varying across the securities we would not observe thecoupling between and amongst securities in these marketsas we do. This coupling, referred to as trend/momentum infinance, can be seen as the measure of similarity between thereturn processes. There are large periods of activity whichare similar across the securities indicative of an underlyingtrend within the market.Given the price process we estimate the latent stochas-

tic volatility process using the SIR, MCMC-PF, PLA andAPPF particle filters run with 1,000 particles and system-atic resampling. Furthermore, we shall explore the effectsof increasing the number of particles. We take the jointmaximum a posteriori (MAP) estimates of κ and θ from ourMCMC calibration and calculate the root-mean-square error(RMSE) between the estimated volatility and actual stochas-tic volatility process for each particle filter. Furthermore, weexplore the effects of increasing the number of particles. Ourexperiments were run on an Intel Core i7-2600k @ 3.4GHzprocessor with 16GB of DDR3 memory. Our results can beseen in Table 2, noting the RMSE and execution time inseconds, and visually in Figures 5 - 7.We can clearly see the APPF providing more accurate

estimates of volatility compared to the other particle filters.In estimation of SPX SV in Figure 5 we observe noticeablyworse estimation by the MCMC-PF RMSE = 0.05393 andPLA RMSE = 0.05317 compared to the SIR PF RMSE =0.05282 with the APPF RMSE = 0.04961 outperforming allthese filters by a significant margin, a 7% improvement onSIR with moderate computational expense in reference tothe other filters.

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0 100 200 300 400 500 600 700 8001000

1050

1100

1150

1200

1250

1300

1350

1400

1450

1500

0 100 200 300 400 500 600 700 800−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06PriceReturn Series

Time

Price

Retu

rn

Figure 2: SPX daily closing price process [04-Jan-2010 – 28-Dec-2012]

0 100 200 300 400 500 600 700 800−0.08

−0.06

−0.04

−0.02

0

0.02

0.04

0.06

0 100 200 300 400 500 600 700 8000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5Return SeriesSV Process

Time

Return V

Figure 3: Heston model SPX daily closing stochasticvolatility calibration [04-Jan-2010 – 28-Dec-2012]

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−0.2

−0.1

0

0.1

0.2Kappa Evolution

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−4

−2

0

2

4Theta Evolution

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−1

0

1

2Epsilon Evolution

Figure 4: Heston model SPX daily closing stochasticvolatility calibration, parameter estimates and evo-lution [04-Jan-2010 – 28-Dec-2012]

0 100 200 300 400 500 600 700 800−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6True xSIR estimateMCMC−PF estimatePLA estimateAPPF estimate

Time

V

Figure 5: Heston model estimates for SPX - filterestimates (posterior means) vs. true state

0 100 200 300 400 500 600 700 800−0.2

0

0.2

0.4

0.6

0.8

1

1.2True xSIR estimateMCMC−PF estimatePLA estimateAPPF estimate

Time

V

Figure 6: Heston model estimates for GE - filterestimates (posterior means) vs. true state

0 100 200 300 400 500 600 700 800−0.2

0

0.2

0.4

0.6

0.8

1

1.2True xSIR estimateMCMC−PF estimatePLA estimateAPPF estimate

Time

V

Figure 7: Heston model estimates for C - filter esti-mates (posterior means) vs. true state

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Table 2: Heston model experimental results. RMSEmean and execution time in seconds using 1,000 par-ticles and systematic resampling

SPX

RMSE Exec. (s)

PF 0.05282 3.79MCMC-PF 0.05393 59.37PLA 0.05317 21.30APPF 0.04961 39.33

GE

RMSE Exec. (s)

PF 0.06121 3.79MCMC-PF 0.06166 58.34PLA 0.06095 20.91APPF 0.05108 38.90

C

RMSE Exec. (s)

PF 0.06162 3.88MCMC-PF 0.06233 59.32PLA 0.06175 21.33APPF 0.05887 39.65

This trend continues upon estimation of the two commonstocks. Estimating GE SV in Figure 6 the MCMC-PF RMSE= 0.06166 and PLA RMSE= 0.06095 perform comparably tothe SIR PF RMSE = 0.06121, however the APPF RMSE =0.05108 markedly outperforms all three of these displaying a16% improvement in performance. In estimation of C SV inFigure 7, the MCMC-PF RMSE = 0.06233 and PLA RMSE= 0.06175 perform worse than the SIR PF RMSE=0.06162,with the APPF RMSE = 0.05887 outperforming all thesefilters once again with a 4.5% improvement in estimationaccuracy compared to SIR with a similar computational ex-pense gain as before. Such impairments across the board byfilters other than the APPF can lead to both problematictrading and mispricing compared to derivatives factoring inthe APPF SV estimates. The APPF provides a marked andstatistically significant increase in performance.

Estimating the stochastic volatility for GE using 5,000particles we find similar results to the above. Please seeTable 3 noting the RMSE and execution time in seconds.By taking the MAP estimate we reduced the parameter es-timation problem to a 2-dimensional, deterministic controlfunction and as such this enables us to focus our efforts onstate estimation. We observe similar results as in the lit-erature with only the PF showing any discernible increasein estimation performance. It is evident that increasing thenumber of particles does not increase APPF performanceand as such we can say it provides robust performance witha limited particle set in comparison to the PF, MCMC-PFand PLA.

Table 3: Heston model experimental results - par-ticle size experiment. RMSE mean and executiontime in seconds using 5,000 particles and systematicresampling

GE

RMSE Exec. (s)

PF 0.06001 3.84MCMC-PF 0.06154 58.80PLA 0.06110 21.22APPF 0.05108 39.22

Building on [20] the evolutionary computation step thatis embedded in the APPF is successfully addressing sampleimpoverishment and weight degeneracy of sequential MonteCarlos. Maintaining diversity amongst the particle systemallows the APPF to provide more accurate estimates com-pared to the the other filters. The execution time is slightlyincreased for the APPF however given compute resourceswill not prove a hindrance in the real-world.

6. CONCLUSIONSWe have proposed an evolutionary computation based par-

ticle filter for the estimation of stochastic volatility of indicesand stocks. We have compared our method, the APPF withthe traditional SIR particle filter, and the more advancedMCMC-PF and PLA. We find our filter to perform better inall state estimation experiments compared to these other fil-ters, with marked and statistically significant improvementsin places. Additionally, performance does not increase whenincreasing the number of particles indicating we can get op-timal results with a small number of particles.These results go some one way in showing that selective

pressure from our generation-gap and distribution-recomb-ination method does not lead to premature convergence.However, the inclusion of various evolutionary computationschemes to mitigate this should be considered in the future.Such schemes include fitness sharing [3, 15], crowding [30,12], niche specialization [4] and triggered hypermutation [31]which aim to preserve diversity in populations and acrossgenerations providing further avenues of potential enhance-ment of the APPF. In addition, we hope to look into incor-porating recombination at the state (particle) level as in theon-going, parallel effort [33], as well as at the distributionlevel, to yield even more enhanced performance through theexploitation of evolutionary computation ideas in sequentialMonte Carlo methods.

7. ACKNOWLEDGMENTSThis work is supported by an EPSRC studentship under

the RCUK Digital Economy program.

8. REFERENCES[1] L. Andersen. Simple and efficient simulation of the

heston stochastic volatility model. Journal ofComputational Finance, 11(3):42, 2008.

[2] B. Anderson and J. Moore. Optimal filtering.Prentice-Hall Information and System Sciences Series,Englewood Cliffs: Prentice-Hall, 1979, 1, 1979.

[3] M. Arulampalam, S. Maskell, N. Gordon, andT. Clapp. A tutorial on particle filters for onlinenonlinear/non-gaussian bayesian tracking. SignalProcessing, IEEE Transactions on, 50(2):174–188,2002.

[4] D. Ashlock. Evolutionary computation for modelingand optimization, volume 200. Springer-Verlag NewYork Inc, 2006.

[5] L. Bauwens, M. Lubrano, and J. Richard. Bayesianinference in dynamic econometric models. OxfordUniversity Press, USA, 1999.

[6] J. Carpenter, P. Clifford, and P. Fearnhead. Improvedparticle filter for nonlinear problems. In Radar, Sonar

1274

and Navigation, IEE Proceedings-, volume 146, pages2–7. IET, 1999.

[7] P. Carr and D. Madan. Option valuation using the fastfourier transform. Journal of Computational Finance,2(4):61–73, 1999.

[8] Z. Chen. Bayesian filtering: From Kalman filters toparticle filters. Technical report, and beyond.Technical report, Adaptive Systems Lab, McMasterUniversity, 2003.

[9] A. Doucet, N. De Freitas, and N. Gordon. SequentialMonte Carlo methods in practice. Springer Verlag,2001.

[10] A. Doucet and A. Johansen. A tutorial on particlefiltering and smoothing: Fifteen years later, 2008.

[11] Z. Duan and Z. Cai. Evolutionary particle filter forrobust simultaneous localization and map buildingwith laser range finder. In Natural Computation, 2008.ICNC’08. Fourth International Conference on,volume 1, pages 443–447. IEEE, 2008.

[12] A. Eiben and J. Smith. Introduction to evolutionarycomputing (natural computing series). 2008.

[13] J. Freitas, M. Niranjan, A. Gee, and A. Doucet.Sequential monte carlo methods to train neuralnetwork models. Neural computation, 12(4):955–993,2000.

[14] J. Geweke. Bayesian inference in econometric modelsusing monte carlo integration. Econometrica: Journalof the Econometric Society, pages 1317–1339, 1989.

[15] D. Goldberg and J. Richardson. Genetic algorithmswith sharing for multimodal function optimization. InProceedings of the Second International Conference onGenetic Algorithms on Genetic algorithms and theirapplication, pages 41–49. L. Erlbaum Associates Inc.,1987.

[16] D. E. Goldberg and J. H. Holland. Genetic algorithmsand machine learning. Machine Learning, 3(2):95–99,1988.

[17] N. Gordon, D. Salmond, and A. Smith. Novelapproach to nonlinear/non-Gaussian Bayesian stateestimation. In Radar and Signal Processing, IEEProceedings F, volume 140, pages 107–113. IET, 1993.

[18] H. Han, Y. Ding, K. Hao, and X. Liang. Anevolutionary particle filter with the immune geneticalgorithm for intelligent video target tracking.Computers & Mathematics with Applications,62(7):2685–2695, 2011.

[19] A. Hanif and P. Protopapas. Recursive bayesianestimation of regularized and irregular astrophysicaltime series. Monthly Notices of the Royal AstronomicalSociety, 2013.

[20] A. Hanif and R. Smith. Generation path-switching insequential monte-carlo methods. In Evolutionary

Computation (CEC), 2012 IEEE Congress on, pages1–7. IEEE, 2012.

[21] S. Heston. A closed-form solution for options withstochastic volatility with applications to bond andcurrency options. Review of Financial Studies,6(2):327–343, 1993.

[22] G. Kitagawa. Monte carlo filter and smoother fornon-gaussian nonlinear state space models. Journal ofcomputational and graphical statistics, 5(1):1–25, 1996.

[23] N. Kwok, G. Fang, and W. Zhou. Evolutionaryparticle filter: re-sampling from the genetic algorithmperspective. In Intelligent Robots and Systems,2005.(IROS 2005). 2005 IEEE/RSJ InternationalConference on, pages 2935–2940. IEEE, 2005.

[24] R. Lee. Option pricing by transform methods:extensions, unification and error control. Journal ofComputational Finance, 7(3):51–86, 2004.

[25] A. Lewis. Option valuation under stochastic volatility.Option Valuation under Stochastic Volatility, 2000.

[26] A. Lipton. The vol smile problem. Risk, 15(2):61–66,2002.

[27] J. Liu and R. Chen. Sequential monte carlo methodsfor dynamic systems. Journal of the Americanstatistical association, 93(443):1032–1044, 1998.

[28] R. Lord, R. Koekkoek, and D. Van Dijk. A comparisonof biased simulation schemes for stochastic volatilitymodels. Quantitative Finance, 10(2):177–194, 2006.

[29] S. Maskell. An introduction to particle filters. StateSpace and Unobserved Component Models, Theory andApplications, editors Harvey, AC and Koopman, SJand Shephard, N, 2004.

[30] O. Mengshoel and D. Goldberg. The crowdingapproach to niching in genetic algorithms.Evolutionary computation, 16(3):315–354, 2008.

[31] R. Morrison and K. De Jong. Triggered hypermutationrevisited. In Evolutionary Computation, 2000.Proceedings of the 2000 Congress on, volume 2, pages1025–1032. IEEE, 2000.

[32] N. Nikolaev and E. Smirnov. Stochastic volatilityinference with monte carlo filters. Wilmott Magazine,John Wiley and Sons, July, pages 72–81, 2007.

[33] R. Smith and M. Hussain. Hybrid metaheuristicparticle filters for stochastic volatility estimation. InProceedings of the fourteenth international conferenceon Genetic and evolutionary computation conference,pages 1167–1174. ACM, 2012.

[34] K. Uosaki, Y. Kimura, and T. Hatanaka. Nonlinearstate estimation by evolution strategies based particlefilters. In Evolutionary Computation, 2003. CEC’03.The 2003 Congress on, volume 3, pages 2102–2109.IEEE, 2005.

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APPENDIX

A. PARTICLE FILTER ALGORITHM

Table 4: SIR Algorithm

1. Initialization: for i = 1, ..., Np, sample

x(i)0 ∼ p(x0)

with weights W(i)0 = 1

Np.

For t ≥ 1

2. Importance sampling: for i = 1, ..., Np, draw samples

x̂(i)t ∼ p(xt|x(i)

t−1)

set x̂(i)0:t = {x(i)

0:t−1, x̂(i)t }

3. Weight update: calculate the importance weights

W(i)t = p(yt|x̂(i)

t )

4. Normalize weights:

W̃(i)t =

W(i)t∑Np

j=1 W(j)t

5. Resampling: Generate Np new particles x(i)t from the

set {x̂(i)t } according to importance weights W̃

(i)t .

6. Repeat from importance sampling step.

B. MARKOV-CHAIN MONTE CARLOALGORITHM STEP

Table 5: Metropolis-Hastings Step

.

.

.

6. Smoothing Metropolis-Hastings Step

i) Sample:

v ∼ U(0, 1)ii) Sample proposal candidate:

x�(i)t ∼ p(xt|x(i)

t−1)

iii) If

v ≤ min

[p(yt|x�(i)

t )

p(yt|x̃(i)t )

, 1

]

accept move: x(i)0:t = (x̃

(i)0:t−1,x

�(i)t )

else reject move: x(i)0:t = x̃

(i)0:t

.

.

.

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