Modeling The Stock Price With Jump By Voter Model And Stopping Time

In this paper, the return process of a stock in a stock market is constructed by the voter model and the theory of stopping time. And we also construct jump in the process of the stock which is more accordance with the real market. The paper contains two parts:Part I:as the study of mathematical finance, the financial situation is highly uncertain, and therefore the stochastic process theory as an important branch of probability theory, is widely used to study the financial problems. Based on the theory of random process, we establish the voter model. And then applying the voter model and the theory of stopping time, we establish the stock return process. Finally, we get the stock price process. This paper proved that the probability distribution of the stock price process converges to distribution of Black-Scholes model which is widely accepted This implies that the financial model of the present paper is somewhat useful for us to understand the statistical properties of the fluctuations for the stock prices.Part II:In part two, influenced by the following elements such as politic, economy, market impacts and enterprises themselves etc., the fluctuations of stock prices is unsystematic and frequent. Especially sometimes, the price of the stock will fluctuate in a violence way. We call it jump. Therefore based on the stock process that we build by voter model in part one, we add a jump function on it and made it more appropriate to the real market. Then we show that the probability distributions of the stock price converge to the corresponding distribution of the Black-Scholaes model.The financial model established in this article will be helpful for us to study to understand the statistical properties of the Fluctuations for the stock prices, while the stock price process can be applied to study the actual market fluctuations in the stock market.