Least squared Itô's drift-diffusion Kernel function for fat tailed distribution simulation of realtime online stock price

Abstract

Real time online data truly are high frequency data as part of the Big Data problem from the Internet Of Things. High frequency data, particularly in finance, tend to be more fat-tailed in distribution than lower frequency data. Fat tailed distribution simulation of random variables like high frequency data has been brought into attention as a better phenomenon in in-sample fit and out-of-sample prediction. The existing approaches include parameter estimation by Least Square Regression (LSR) and Maximum Likelihood Estimation (MLE), and exponential generalized autoregressive conditional heteroscedastic (EGARCH) model simulation based on Stochastic Volatility Jump-Diffusion process. However, there is still room for accuracy of fat tailed distribution simulation. We propose using Ito's drift-diffusion equation as the kernel function for a univariate stochastic process without introducing extra jump coefficients in EGARCH or performing LSR or MLE between x t and x t−1 . The mean reverting rate and stochastic volatility are conditional variables and automatically become correlated. Their estimation can be solved by applying Least Square Estimation on the kernel function given the mean value as unconditional variable. The proposed method is proved to be a fat tailed distribution simulation of original data with smaller mean squared error. The paper compares the simulation of the stochastic process of Google finance's online real time stock price by the proposed method to that by LSR, MLE and EGARCH. The proposed method simulates a fat tailed distribution of original data with lower mean squared error and yet being outlier-prone in a more reasonable way.