Abstract
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 $S\&P500$ 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.
Highlights
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
For the Cox Ingersoll Ross (CIR) model, much variability in the approximate states by the Viterbi algorithm can be observed, whereas the states estimated by the FPO fit quite well to the real data
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, 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
Summary
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 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 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 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
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