Estimation of Stochastic Volatility Models Using Optimized Filtering Algorithms

In this paper, we describe and implement two recursive ﬁltering algorithms, the optimized particle ﬁlter, and the Viterbi algorithm, which allow the joint estimation of states and parameters of continuous-time stochastic volatility models, such as the Cox Ingersoll Ross and Heston model. In practice, good parameter estimates are required so that the models are able to generate accurate forecasts. To achieve the objectives the proposed algorithms were implemented using daily empirical data from the time series of the S & P 500 returns of the stock exchange index. The proposed methodology facilitates computational calculations of the marginal likelihood of states and allows the reconstruction of unknown states in a suitable way, and reliable estimation of the parameters. To measure the quality of estimation of the algorithms, we used the square root of the mean square error and relative deviation standard as measures of goodness of ﬁt. The estimated errors are insigniﬁcant for the analyzed data and the two models considered. We also calculated the execution times of the algorithms, demonstrating that the Viterbi algorithm has less execution time than the optimized particle ﬁlter.


Introduction
The Black-Scholes model described by Black and Scholes (1973) is an equation derived from the financial mathematics used to determine the prices of certain financial assets.The assumptions of this model are given in terms of an ideal scenario where it is assumed that it is possible to conduct continuous trading, the markets are perfect, the interest rate is risk-free and fixed and the price of the underlying asset behaves as a random variable that is modeled as a stochastic process.However, it has been shown by empirical studies that these assumptions are not realistic because the model does not explain the true impact of change on financial markets, such as changes in volatility.Currently, there are more sophisticated models that incorporate volatility as a random variable related to risk factors.Two most representative models are the model of Cox Ingersoll Ross (CIR) (see Cox, Ingersoll, and Ross (1985)) and the model of Heston (see Heston (1993)).The problem with these models lies in the estimation of the parameters.When the maximum likelihood estimation method is used, there is a first drawback related to obtaining a solution in closed form for the transition density when the solution of the transition density between prices and volatility is not known in closed form, see Ait-Sahalia (2002) for detail.A second problem arises when the time series of prices are only partially observed, and volatility is unknown when integrating the joint density to obtain the marginal density, the process results in the handling of analytically intractable integrals, which need to be solved by numerical or technical Monte Carlo methods.In recent years some solutions have been proposed to solve the above problems.For example, Ait-Sahalia and Kimmel (2007) and Ait-Sahalia and Kimmel (2010), proposed closed-form approximations to the log-likelihood function.MCMC methods in Eraker (2001), Eraker (2004), Eraker, Johannes, and N. (2003), Jacquier, Polson, and P. (2004) have been used for such purposes.In Bates (1996), Johannes, Polson, and Stroud (2009) and Christoffersen, Jacobs, and Mimouni (2010), proposed computational algorithms based on filter theory.Hurn, Lindsay, and McClelland (2005) described a maximum likelihood method for estimating the parameters of the Heston volatility model using market index data and the prices of the options inscribed in that index using a particle filter algorithm and the study shows the efficiency of the filter on simulated and real data taken from the S&P 500 index.Kleppe, Jun, and Skaug (2009) implemented a method for maximum likelihood estimation in stochastic volatility models, which does not require observations of either the price option or the volatility.In order to integrate the volatility of the latent process from the joint density of profitability and volatility, the use of a technique of modified importance sampling for the approximation of the continuous time model using the Euler-Maruyama scheme was proposed.Kleppe, Jun, and Skaug (2014) develop a maximum likelihood method to estimate partially observed diffusion models based on data sampled at discrete times.The method combines two estimation techniques, the first, proposed by Ait-Sahalia (2008), and which was used to obtain an exact approximation of the joint transition probability density of latent states and observed measurements.The second was proposed in Richard and Zhang (2007) and used an importance sampling technique to integrate the latent states and obtain an approximation of the likelihood function.Javaheri, Lautier, and Galli (2003) introduced the use of filtering algorithms in the world of the quantitative finances.In particular, the Kalman filter, the extended Kalman filter, the unscented Kalman filter, the nonlinear Kushner's filter and the particle filter are implemented for stochastic volatility models and structure models of commodity price terms.Spall (2003) proposed a dual estimation algorithm based on Markov chain Monte Carlo to estimate states and parameters in a state space model.Recently, in Javaheri (2015) established a methodology to estimate parameters in volatility models of time series of market price.In particular, problems related to stochastic volatility models, statistical inference techniques, filtering algorithm and optimization are considered, the consistency of the parameters is demonstrated, and recommends the use of observations of the markets of better quality options.The main contribution of this article is based on the use of the filtering theory for the joint estimation of states and parameters (Haykin (2001), Liu and West (2001)) in the stochastic volatility models of CIR and Heston.In particular, the optimized particle filter algorithm proposed by Yang and Xing (2011) was implemented in this context.Assuming the known parameters, the Viterbi algorithm, Viterbi (1967), was implemented to estimate the maximum a posteriori states.For both models, we obtain the estimated likelihood and show how the optimized algorithms work in an application with real data.In summary, the proposed in this article: • Two stochastic volatility models are formulated, these models are defined on terms of stochastic processes of continuous times with continuous space state.
• The dynamic system in continuous time is approximate by a discrete system, i.e., the derivatives in conitnuous time are approximated by difference equations in discrete time, which can be expressed in the form of space state model.
• Once that the models are in the form space state, it is necessary to estimate the states and parameters for which two algorithms are proposed.
The rest of article is summarized as follows: in section 2, Cox Ingersoll Ross model is defined, this model describes the evolution of interest rates (i.e specifies that the instantaneous interest rate follows a stochastic differential equation) and can be used in the valuation of interest rate derivatives; the section 3 contains the Heston model, this is a mathematical model that describing the evolution of the volatility of an underlying asset, assumes that the price of the asset is determined by a stochastic process; in section 4, optimized particle filter is using for generate new observations into sampling process and also optimizes it, through optimized particle filter, particles are moved towards regions where they have larger values of posterior density function; in section 5 the Viterbi algorithm are developed, this algorithm is a maximum a posteriori (MAP) estimation method that rely on a particle cloud representation of the filtering distribution which evolves through time using importance sampling and resampling ideas.The MAP estimation is then performed using a classical dynamic programming technique applied to the discretised version of the state space model; in section 6 the results obtained for two different models are shown, in section 7 contains a final discussion and conclusions and lastly, section 8 contains the acknowledgements.

Cox Ingersoll Ross model
The Cox Ingersoll Ross (CIR) model, describes the evolution of the interest rate by the following stochastic differential equation where r t is the interest rate process, B t is a Brownian motion (which models the random risk factor of the market), β, µ and σ are parameters.The parameter β corresponds to the speed of adjustment, µ to the mean and σ to volatility.The drift factor β(µ − r t ) is exactly the same as in the Vasicek model.It ensures mean reversion of the interest rate towards the long run value µ, with speed of adjustment governed by the strictly positive parameter β.The CIR model ensures that the process has mean reversion and avoids the possibility of negative interest rates.A process satisfying the equation ( 1) is called the CIR process, the transition density of the CIR model is not Gaussian.Stochastic volatility models are generally constructed based on mean reversion of volatility parameter, which reflect through observations that periods of low volatility tend to be followed by a reversion to a more moderate long-term level.
In order to solve analytically the equation given in (1), we need to introduce the chain rule for stochastic differentials, called Itô's Lemma, which is defined as follows: Theorem 1 (Itô's Lemma).Let B t be a Brownian motion and x t be an Itô drift-diffusion process which satisfies the stochastic differential equation dx t = µ(x t , t)dt + σ(x t , t)dB t .Suppose that y t = u(x t , t), u : R × [0, T ] → R, is continuous, ∂u ∂t , ∂u ∂x , ∂ 2 u ∂x 2 , exist and are continuous then The term (dx t ) 2 is interpreted using the following identities: we have where taking exponential and solving, we obtain In order to illustrate the methodology of estimation of parameters and states proposed in this article, we use the reparametrized version of the CIR model in continuous time, which is discussed in Chib, Pitt, and Shephard (2006) and Poyiadjis, Doucet, and Singh (2011), where the volatility follows a square root process given by and where ζ t+1 and ξ t are two identically distributed independent random variables, ζ t+1 ∼ N (0, σ 2 ζ ) and ξ t ∼ N (0, σ 2 ξ ).We are interested in estimating the states v 0:t = (v 0 , . . ., v t ) and parameters Θ = (µ, β, σ ζ ) of system placed in (10), and (11), given the observation y 1:t = (y 1 , ..., y t ).

The Heston model
The Heston model ( 1993) is a mathematical model used to describe the behavior of a bivariate stochastic process between stock prices s t and its variance v t .It initially emerged as a generalization of the Black and Scholes option pricing model, but assuming that volatility is a stochastic process.The model is governed by the following system of stochastic differential equations and If y t = √ v t 2 and Itô's Lemma is applied, then developing, we obtain If we make k = 2β; θ = δ 2 2β and σ = 2δ, we get: to obtain this last result, the following properties were used dtdt = 0, dtdB t = dB t dt = 0 and dB t dB t = dt.Then the given system in ( 12) and ( 18) becomes and dB 1,t dB 2,t = ρdt (20) where the system variables are defined as follows: • s t : is the asset price, • v t : is the volatility of the asset, • µ : is the expected return on the asset, • θ : is the long-term price change, • k : is the rate at which volatility tends towards its long-term average, • ρ : is the correlation of Brownian motions, • dt = t k − t k−1 : is a small increase over time, • dB 1,t : is a standard one dimensional Brownian motion, • dB 2,t : is a standard one dimensional Brownian motion.
The equations ( 12) and ( 13) is a system of stochastic differential equations, where the solutions represent a set of random variables x t and v t indexed by real numbers t > 0 is called a continuous-time stochastic process.Each instance, or realization of the stochastic process is a choice from the random variables x t and v t for each t, and is therefore a function of t.The equations ( 12) and ( 13) are, by defnition, an integral equation (integrating from t to t + dt) and where the meaning of the last integral in ( 21) and ( 22), are called an Ito integral.
The simplest efective computational method for the approximation of ordinary differential equations is Euler's method.The Euler-Maruyama method is the analogue of the Euler method for ordinary differential equations.To develop an approximate solution on the interval [c; d], assign a grid of points will be determined at the respective t points, given the SDE initial value problem.
The simplest way to discretize the process in the equations ( 21) and ( 22) is to use Euler discretization, this discretization is necessary to find the solution of the equations ( 18) and ( 19).In the following, is shown the process of discretization of s t , v t and ln(s t ).
• Discretization of the process s t The stochastic differential equation given in ( 12) and ( 13) in integral form can be expressed as follows Using a discretization of Euler we obtain that and so the stochastic differential equation of the stock price in its discretized version of Euler is given by • Discretization of the process v t With a relation similar to the discretization of the stochastic difference equation of s t , a version for v t is obtained, which is written as follows Using a discretization of Euler we obtain that where Corr(z u , z s ) = ρ.Finally, the equation of the volatility in the discrete version of Euler is given by To avoid obtaining a negative variance, v t is replaced by • Discretization of the process ln(s t ) If we now consider y t = ln (s t ), using Itô's Lemma, we obtain Integrating we obtain then Euler's discretization for ln (s t ) is given by or equivalently • The model in the state space The Heston's stochastic volatility model can be written as state space models, these models are useful for describing data in many different areas, such as financial time series, environmental data, and clinical trial, among other applications; and are used to estimate latent state processes (unknown processes), based on measurements from the observation process.Also, allows to write equations ( 18) and ( 19) in discretized form in time, facilitating the computational implementation of the estimation algorithms, this representation as follows where dt = 1 and y t+1 = lns t+1 are assumed.The equation given in ( 35) is the state equation and the equation given in (36) is the observation equation.A way to generate z u and z s with correlation ρ, consists of generating identically distributed independent normal random variables z 1 ∼ N (0, 1) and z 2 ∼ N (0, 1), such that z u = z 1 and z s = ρz 1 + 1 − ρ 2 z 2 .The equations given in ( 35) and ( 36) can be rewritten as The model parameters are Θ = (κ, θ, µ, σ, ρ).The joint distribution of (v t+1 , y t+1 ) T is given by where and The bivariate transition density of a correlated normal distribution is given by Where and Therefore, marginal distributions p(v t+1 |v t ) and p(y t+1 |y t ) are given by The conditional distribution p(y t+1 |v t+1 ), is given by Finally, the joint transition density can be rewritten as where One way to eliminate the correlation between the errors associated with the state and observation equations in the state space model is to subtract on both sides of the equation ( 37), a multiple of the quantity This procedure would eliminate the correlation (Javaheri (2015)); that is to say then, the state space model with uncorrelated noises is as follows or equivalently: where z 1 and z 2 are independent and identically distributed standard Gaussian random variables.Using this system of equations the calculations are faster and the filters are implemented more efficiently.

Optimized particle filter (FPO)
In the previous sections, two models of stochastic volatility, the CIR model and the Heston model, were proposed, then a discretization of both was performed, and then represented in the state space form.Now, it is proposed to define the optimized particulate filter to be used to estimate the states, solutions of the SDE, and parameters of the proposed models.Consider a dynamic system where v t is unobserved state of the system, y t are observations at time t ∈ {0, . . ., T }, φ ∈ R m is parameters vector, f (.|.) is a known density given by the evolution of the states, h(.|.) is also a known density by the observations, which may be linear or non-linear, t and ζ t are estimation errors of the state and observation equations which can be Gaussian or non Gaussian.It is assumed that {v t } is a Markov process generated according to the evolution of the previous state, where the observation process {y t } is conditional independent of the process {v t }.
When φ is known the inference falls on the posterior distribution p(v 0:t |y 1:t ), where the joint distribution is given by p(v 0:t , y f (v 0 ) is an initial state, v 0:t = (v 0 , . . ., v t ) and y 1:t = (y 1 , . . ., y t ), denote the states and observations from a time 0 to t.If φ is unknown parameter, an a priori distribution p(φ) is assigned and the Bayesian inference falls on the estimation of the posterior distribution of parameters and states, given by p(v 0:t , φ t |y 1:t ) ∝ p(v 0:t , y 1: In particular, we are interested in estimating the filtering distribution p(v t |y 1:t ), the predictive distribution p(v t+1 |y 1:t ), a posteriori mean µ t|t = E(v t |y 1:t ) and a posteriori covariance T , which are calculated as follows: • Prediction of the filtering distribution: • The predictive distribution: • The a posteriori mean: • The a posteriori covariance: In models with nonlinear structures and with non-Gaussian errors, a posteriori distributions p(v 0:t |y 1:t ) or p(φ, v 0:t |y 1:t ) does not have closed form expression, so making inference is complicated; in practice, it is difficult to calculate the integrals given in equations ( 54) and ( 55).
It is therefore necessary to use approximation Markov Chain Monte Carlo (MCMC) and Sequential Monte Carlo methods (SMC).The MCMC and SMC provide efficient computational tools for inference in state space models (See Andrieu, Doucet, and Holenstein ( 2010)).
In this work we propose a method that allows to simultaneously estimate the states and parameters of the posterior distribution, using a sequence of discrete particles generated by a particle filter algorithm.The approximation is given by where v are the particles of states and parameters and δ is the Dirac delta, which represents the generated distribution of the N particles.Taking into account the approximation given in ( 59), the problem now focuses on how to extract the samples sequentially from the posterior distribution.This step is complicated because the propagation of states depends on the parameters, and vice versa.To solve this situation, to use a density of importance based on the sampling technique of importance is suggested (Carvalho, Johannes, Lopez, and Nicholas (2010)).The proposed methodology combines the maximum likelihood method with an optimized stochastic particle filter algorithm.The particle filter algorithm works properly for dynamic models with Gaussian and non-Gaussian distributions, with linear and non-linear structures.On the other hand, the classical method of estimation by maximum likelihood implies the optimization of the parameters estimated in such way that the observed data are most likely to be chosen and the likelihood function is defined as where: ) is given by the equation ( 50), h(y i+1 |v i+1 , φ) is given by the equation ( 51), φ = (β, µ, σ), denotes the parameters of the CIR model and φ = (κ, θ, µ, σ, ρ) are the Heston model parameters.In practice, one uses the log-likelihood which is numerically better behaved and satisfies The maximum likelihood method is most commonly done by taking the partial derivative of the equation given in (61) for each parameter and setting it equal to zero and the system depending on the parameters is solved.
On the other hand, if we consider the joint density between observations y 1:t , states v 0:t and parameters φ, we have Furthermore; p(v 0:t , y t , y 1:t−1 , φ) = p(y t |y 1:t−1 , v 0:t , φ)p(y 1:t−1 , v 0:t , φ) = p(y 1 , v 0:t , φ) where: and The equation given in ( 63) represents the likelihood of the data at time t.Taking logarithm in the equation ( 63), we get To simplify the computation, the cost function is chosen as the predicted likelihood at time t, that is: However, except in a few simple cases, it is impossible to compute the optimal filter and the likelihood in closed form, the numerical approximation methods are required; this is The problem of maximizing the cost function is equivalent to finding the zeros of the gradient ∇C(φ).A recursive procedure to estimate the parameter vector φ such that ∇C(φ) = 0 proceeds as follows where ∇C(φ) is the estimation of gradient at the point φ t−1 and {γ t > 0} denotes a sequence of decreasing step-size.One selects a step-size sequence satisfying γ t → 0, ∞ t=1 γ t = ∞.Under appropriate conditions, the iteration defined in (68) will converge to the true value of φ in some stochastic sense.The essential part of the equation ( 68) is how to obtain the gradient estimate, however, in most cases, it is impossible to compute the closed-form gradient and we must resort to the numerical approximation.On the other hand, the particulate filters (Gordon, Salmond, and Smith (1993), Doucet, De Freitas, and Gordon (2001)), are based on importance sampling where v is simulated sequentially from some importance density function q(v t |y 1:t ) and all particles v t are selected according to a weight of importance ω t , defined by To develop the particle filter algorithm in detail consider that {v t } N i=1 is a random sample that characterizes the filtered density function p(v t |y 1:t ), where {v i=1 is a set of particles obtained with associated weights {ω Then a posteriori distribution also called In this article, we intend to estimate the unknown states of the stochastic CIR and Heston models, assumingthe known parameters through the MAP estimation.The method consists of sample particles v according to the methodology used to approximate the given filtered density in (99) and then selecting vMAP If the support of q(.) includes the support of p(v (i) 0:t |y 1:t ) then the estimator converges asymptotically when N → ∞ to vMAP 0:t (t) (Godsill et al. (2001)).The choice of q(.) has a great influence on the performance of the algorithm.Sampling directly from p(v 0:t |y 1:t ) is usually impossible, but an approximation can be used by the particle filter, then the following approximation can be obtained vMAP This is an easy method to implement, but has the degenerative problem that affects particle filters, the quality of the estimators decrease as t is increased.To solve this problem, an alternative is to use a dynamic programming algorithm.The method assumes that the filtered distribution given in the equation ( 99) has been calculated and stored for each time t.Then the approximation vMAP To evaluate the function given in the equation ( 110) to gross force would imply an exhaustive search of all possible trajectories of the model given by the equation ( 50).However, the maximizing function is given by the equation ( 61), which is additive.Then the maximizing function is vMAP This property allows the use of the dynamic programming technique called Viterbi algorithm (Viterbi (1967)), which allows estimation of vMAP 0:t (t) as follows vMAP The Viterbi algorithm is a widely used technique for estimation in discrete state space models of hidden Markov processes as given by equations ( 50) and ( 51), they are widely used for recognition voice and decoding codes in information theory (Godsill et al. (2001)), among other applications.The algorithm proceeds as follows: Algorithm: Viterbi • Step 2. for 2 ≤ k ≤ t and 2 ≤ i ≤ N, do:

Results
In order to show the proposed methodology, the joint parameter and state values of the CIR and Heston stochastic models are estimated using 2610 daily data from the time series of the S&P 500 index.
where: R t denotes the net return on an investment of a S&P 500 asset at the end of the month between the time t − 1 and the time t; s t denotes the S&P 500 index at the end of the month in time t and s t−1 denotes the S&P 500 index at the end of the month at time t − 1.In the Figure 1 a graph is displayed where the daily S&P 500 is observed for the raw data and for the transformed data in the figure 2, observing a rather volatile behavior of the time series.
To initialize the FPO algorithm in the CIR model, a priori the following values were chosen  In the Table 2 the estimations of the parameters of the Heston model are shown.In the The Root Mean Squared Error (RMSE) was also calculated as a measure of goodness of fit for the comparison between the estimated states vi and the observed true values v i .The formula provides a quantitative measurement for the comparison between two models, the smaller the RMSE value, the closer the estimated values to the observed values will be.The

RMSE is defined as
A measure of relative dispersion of the data, which takes into account its magnitude, is given by the relative standard deviation (RSD), where it is a measure of the relative dispersion of a data set estimated vi , which is obtained by dividing the standard deviation of the set by its mean arithmetic and is usually expressed in percentage terms.

RSD
In the Table 4, we show the errors estimated by the two optimized filtering algorithms proposed for the two models considered, observing little variability between the real states and the estimated states.In addition, the relative standard deviation is observed showing low variability in the estimated states.

Discussion and conclusions
There are economic models used to determine the prices of certain financial assets.However, it has been shown that the assumptions are not realistic because the model does not explain the true impact of the change in financial markets, such as changes in volatility.Volatility is what helps us to know the status of investors, that is, the feeling of levels of complacency or confidence as well as fear and extreme panic, these two opposite levels are very interesting when it comes to making a decision about a certain value.Currently, financial models are estimated with the maximum likelihood estimation, presenting a drawback related to obtaining the solution in closed form when the density of transition between prices and volatility is not known in a closed manner, also when prices are partially and partially observed.volatile is unknown, the process in the maximum likelihood estimation results in the management of analytically intractable integrals.In this sense, this research presents a Bayesian estimation methodology that estimates the states and market parameters of the S&P 500 index considering volatile models and incomplete observations on line with low execution time and immediate response of the algorithms, observing a good estimate of the real data, describing the market of the S&P 500 index when the market is presented with confidence or complacency because the volatility is very low and the market rises slowly without scarcely scaring, so investors feel safe and this feeling is the prelude to the falls, also, describing the market in the points presented with fear or panic because the volatility is very high and the market falls sharply scare the great part of the investors, therefore the investors feel insecure and this feeling is the prelude to the rises.In this article, two algorithms for estimation of states and parameters in stochastic volatility models were proposed.The methodology used is a useful and elegant tool that serves to approximate in a recursive way the parameters and the filtered distribution of the states not observed from a process observed with errors using weighted random samples.The FPO algorithm is based on a particle filter structure and a gradient stochastic approximation algorithm is used to optimize the cost function.The estimation of states and parameters is performed simultaneously.On the other hand, the Viterbi algorithm uses the properties of the hidden Markov chains in conjunction with dynamic programming techniques to obtain an optimal sequence of states that maximizes the joint posterior distribution of all states.To estimate the parameters and to reconstruct the unknown states in the two models, both algorithms were implemented, observing little variability with respect to the real states.To measure the relative success of the estimation algorithms, the RMSE and RSD was calculated showing that the estimates produced by the two models have small errors.Finally, we compared the execution times of the algorithms and showed that the Viterbi algorithm has shorter execution time than the FPO.Some of the works under investigation currently related to this methodology, we can point out the study of the stochastic behavior of the prices of a commodity taking into account the reversal of the average in terms of futures contract prices and real prices.The implementation of algorithms type Metropolis Hastings, Monte Carlo Sequential, and Particle Independent Metropolis Hastings to make simultaneous estimation of the parameters and states solutions in models of stochastic differential equations in situations where the diffusion process is partially observed.Finally it is developing a mixed effects model defined through a stochastic deferential equation, these models arise in many fields of research such as: clinical trials, growth studies in agriculture, dispersion processes of epidemiological diseases, series of financial time, population dynamics (including predatory dam systems), and intracellular processes, among many other applications.

Acknowledgments
We are infinitely grateful to the Editor Matthias Templ, for their valuable suggestions and contributions to improving this manuscript.

Figure 1 :
Figure 1: Representation of raw data of the Dow Jones Index.

Figure 2 :
Figure 2: Representation of transformed data of the Dow Jones Index.

Figure 3 :
Figure 3: Real and estimated state by the FPO algorithm, CIR model.

Figure 4 :
Figure 4: Real and estimated state by the Viterbi algorithm, CIR model.

Figure 5 :
Figure 5: Real and estimated state by the FPO algorithm, Heston model.

Figure 6 :
Figure 6: Real and estimated state by the Viterbi algorithm, Heston model.
The time series analyzed correspond to the period from 29/09/2006 up to 30/09/2016 and it is available at the following address: http://us.spindices.com/indices/equity/sp-500.The algorithms was programmed in the Octave GNU programming environment, in an Intel CPU Core i7 3.6GHz with 16GB RAM running 64Bit Windows.To analyze the data series, the following transformation was performed R t = s t − s t−1 s t−1

Table 1 :
Parameters estimated by the CIR model using the FPO.Figures5 and 6the states estimated by the FPO (red color) and by the Viterbi algorithm (red color), together with the observed data (blue color) are shown.As with the CIR model, states estimated by the Heston model show erratic behavior when calculated by the FPO and Viterbi algorithms with respect to the actual data.In the Table3the execution times of

Table 2 :
Estimated parameters of the Heston model using the FPO.
the algorithms used are shown, speed and efficiency in the calculations for a high number of data in the cases analyzed in this work are observed.

Table 3 :
Elapsed time of the algorithms for the CIR and Heston models.

Table 4 :
RSME and RSD estimated for the CIR and Heston models.