Whitt, W., 1981, ‘The stationary distribution of a stochastic clearing process’, Operations Research 29(2), 294–308. The data series is transformed into daily log returns series so that we can obtain stationary series. MathWorks is the leading developer of mathematical computing software for engineers and scientists. 307–312. Kariya, T., Tsukuda, Y., Maru, J., Matsue, Y. The model uses two parameters, the rate of drift from previous values and volatility, to describe and predict how the continuous-time stochastic process evolves over time. https://doi.org/10.1086/258792, Marathe, R.R. However, many empirical studies have revealed that return distributions are usually not normal. contributed by introducing an extension of the random noise process in the GBM model from normal to student"s t-distribution. You can also select a web site from the following list: Select the China site (in Chinese or English) for best site performance. The descriptive statistics for the five indices of the NSE are given in Table 1. The results, based on daily stock prices, reveal that the student’s t GBM is comparable to the normal GBM model but empirically is more successful. Geometric Brownian motion (GBM), a stochastic differential equation, can be used to model phenomena that are subject to fluctuation and exhibit long-term trends, such as stock prices and the market value of goods. Others presented a more realistic stochastic process for the underlying process (e.g. A stochastic process St is said to follow a geometric Brownian motion (GBM) if it satisfies the above SDE. The experimental outcome of the simulation and real data analyses confirms the viability of the proposed method. This is seen as an empirical approximation for the density made up of point masses: Kitagawa and Sato (2001) and Kitagawa (1996) offer an algorithm for filtering as follows: For i = 1, …, N, generate a random number. One class of fat-tailed distributions with the potential to give a better approximation to the distribution of stock returns is the t-distribution. Nevertheless, one of the foremost challenges in applying this model to financial market data is the fact that the underlying asset value process is unobservable. Geometric Brownian motion has been extensively used as a model for stock prices, commodity prices, growth in demand for products and services, and real options analysis (Benninga & Tolkowsky 2002; Nembhard, Shi & Aktan 2002; Thorsen 1998). Opportunities for recent engineering grads. The significant kurtosis indicates that return distribution has sharp peaks compared to a normal distribution. Table 4 presents the observed values. Distribution performance was assessed through simulation study and real data. https://doi.org/10.1016/j.jbankfin.2003.10.017. Shimada, J. The study revealed that the student’s t GBM performed better than the normal GBM in estimating both the volatility and the parameters of the model. 0. Graphically, for a single run, the estimation results obtained from running these two models are shown in Figures 6–10. Practical/managerial implications: The proposed model which is heavier tailed than the normal does not only provide an approximate solution to non-normal estimation problem. where fθ is the probability density function, and are the log returns. Figures 6–10 show the plots of volatility estimation for each of the five firms’ stock prices. An SMC or particle filter technique based on student’s t-distribution is developed to estimate the parameters for the extended model. We compute the MLEs and the corresponding log-likelihood for each stock. The model uses two parameters, the rate of drift from previous values and volatility, to describe and predict how the continuous-time stochastic process evolves over time. Stein, E., & J Stein, J., 1991, ‘Stock price distributions with stochastic volatility’, Review of Financial Studies 4,727–752. Proceedings, pp. It can also follow GBM based on the Gaussian process. Previous approaches to the estimation of GBM model have revealed that return distributions are usually not normal. Equivalently, S is lognormally distributed. The GBM is one of the most popular stochastic processes and undoubtedly an effective instrument in modelling and predicting the random changes in stock prices that evolve over time. Such exercises are based on a stochastic process of stock price movements, called geometric Brownian motion, that underlies the derivation of the Black-Scholes option pricing model. This method applies the concept of sampling-importance-resampling (SIR) (Rubin 1987). Mandelbrot (1963) affirmed that although financial prices or its logarithm following a Brownian motion is mathematically convenient, it is difficult to fit the real financial data with this assumption. A particle filter technique based on student’s t-distribution is developed to estimate the random effects and parameters for the extended model. Geometric Brownian motion model usually assumes that the distribution of asset returns is either normal or lognormal. > Kitagawa, G. & Sato, S., 2001, ‘Monte carlo smoothing and self-organising state space model’, in A. Doucet, N. de Freitas & N. Gordon (eds. Tokyo, University of Tokyo Press. It usually expresses accuracy as a percentage and is defined as follows: The difference between Yt and Ft is divided by the actual value Yt again. Research approach/design and method: The particle filter based on the t-distribution is developed to estimate the random effects and parameters for the extended model; the mean absolute percentage error (MAPE) were calculated to compare distribution fit. was the project leader and was responsible for most of the theoretical and experimental work done.

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