Joshin Murai

Stock price process and long memory in trade signs

Empirical study on tick by tick data in stock markets shows us that there exists a long memory in trade signs and signed trade volumes. This means that an order flow is a highly autocorrelated long memory process. We present a mathematical model of trade signs and trade volumes in which traders decompose their orders into small pieces. We prove that fractional Brownian motions are obtained as a scaling limit of the signed volume process induced by the model.

Stock price process and the long-range percolation

Using a Gibbs distribution developed in the theory of statistical physics and a long-range percolation theory, we present a new model of a stock price process for explaining the fat tail in the distribution of stock returns.

Signs of Market Orders and Human Dynamics

A time series of signs of market orders was found to exhibit long memory. There are several proposed explanations for the origin of this phenomenon. A cogent one is that investors tend to strategically split their large hidden orders into small pieces before execution to prevent the increase in the trading costs. Several mathematical models have been proposed under this explanation.In this paper, taking the bursty nature of the human activity patterns into account, we present a new mathematical model of order signs that have a long memory property. In addition, the power law exponent of distribution of a time interval between order executions is supposed to depend on the size of hidden order. More precisely, we introduce a discrete time stochastic process for polymer model, and show it’s scaled process converges to a superposition of a Brownian motion and countably infinite number of fractional Brownian motions with Hurst exponents greater than one-half.

Long Memory in Finance and Fractional Brownian Motion

We present a mathematical model of the trade signs and trade volumes, and derive a fractional Brownian motion as a scaling limit of the signed volume process which describes a super-diffusive nature. In our model, we assume that traders place a market order at a single time or divide their order into two chunks and place orders at different times. When they divide their order into two chunks, the probability distribution of the time lag t of divided orders is assumed to decay as an inverse power law of t with exponent α. We obtain three types of scaling limit of the signed volume process according to the three cases of the value of α ,( i) α 1. (See Theorem 4.1.) We prove that a fractional Brownian motion having a super diffusive nature is obtained in a scaling limit of a signed volume process if and only if α< 1.