Analysis of price diffusion in financial markets using PUCK model

doi:10.1016/j.physa.2007.02.049

Physica A: Statistical Mechanics and its Applications

Volume 382, Issue 1, 1 August 2007, Pages 187-192

https://doi.org/10.1016/j.physa.2007.02.049 Get rights and content

Abstract

Based on the new type of random walk process called the potentials of unbalanced complex kinetics (PUCK) model, we theoretically show that the price diffusion in large scales is amplified $2 (2 + b)^{- 1}$ times, where b is the coefficient of quadratic term of the potential. In short time scales the price diffusion depends on the size M of the super moving average. Both numerical simulations and real data analysis of Yen–Dollar rates are consistent with theoretical analysis.

Introduction

Crashes and uncontrollable hikes can often occur in financial markets. Such changes of the prices confuse the market and damage the economy because they start abruptly in many cases. Therefore, techniques to measure the probabilistic risk of sudden change in the prices have been studied using tick-by-tick data [1]. Recently, it was empirically found that change of prices can be approximated by the Fokker–Planck equation and the new type of random walk in a potential field [2], [3], [4], [5]. The potential field is approximated by a quadratic function with its center given by the moving average of past market prices. This random walk model is called the potentials of unbalanced complex kinetics (PUCK) model in which the potential slowly changes in the market [3], [4]. In this paper, we focus on the diffusion properties of this random walk process and calculate the diffusion coefficient which is helpful for estimating the market risk.

We first review an empirical derivation of the PUCK model. We next show that the statistically steady condition of price fluctuations depends on the potential field, and clarify relationships between the price diffusion and the potential field. We finally demonstrate that the price diffusion in short time scales depends on the size of moving average, however, large scale diffusion properties are independent of the moving average. In the paper, we used all the Bid record (about 20 million ticks) of the exchange rates for Yen/Dollar that were traded by the term from 1989 to 2002 to find the firm statistical laws.

Section snippets

Empirical derivation of PUCK model

Prices in financial markets always have violent fluctuation in a short time scale. We first eliminate the uncorrelated noise $η (t)$ from the price $P (t)$ in order to reduce the statistical error. We next investigate the dynamics of the price.

We can perform this noise elimination process by introducing an optimum moving average $\bar{P (t)}$ : $P (t) = \bar{P (t)} + η (t),$ $\bar{P (t)} = \sum_{k = 1}^{K} w_{k} \cdot P (t - k),$ where $P (t)$ is a price, $η (t)$ is an uncorrelated noise and $w_{k}$ gives the weight factors where the time is measured by ticks. The

Statistically steady condition of price fluctuations in the PUCK model

We focus on Eq. (4) with the case of a constant b because the coefficient $b (t)$ is known to change slowly in financial markets. Eq. (4) is transformed as follows: $\bar{P (t + 1)} - \bar{P (t)} = - \frac{b}{2} (\frac{2}{M (M - 1)} \sum_{k = 1}^{M - 1} (M - k) (\bar{P (t - k + 1)} - \bar{P (t - k)})) + f (t) .$ This is a type of AR process for price difference when b is a constant. We can estimate the conditions of b to make the AR process being statistically steady. Eq. (5) is transformed by the following determinant: $X_{t} = {AX}_{t - 1} + F_{t} = (A)^{t} X_{0} + (F_{t} + {AF}_{t - 1} + \dots + (A)^{t - 1} F_{1}),$ where $X_{t} = (\begin{matrix} \bar{P (t + 1)} - \bar{P (t)} \\ \bar{P (t)} - \end{matrix})$

Diffusion of prices in market potential field

As the potential coefficient $b (t)$ has a long autocorrelation, we can calculate the future price diffusion using Eq. (4). This prediction is crucial in order to evaluate the risks of market. We clarify statistical laws of price diffusion described by Eq. (4) using both simulations and theoretical analysis.

By simulating Eq. (4) for the case of $b (t)$ is a constant, we investigate the standard deviation on a time scale T defined by $σ_{b} (T) = \sqrt{〈 (\bar{P (t + T)} - \bar{P (t)})^{2} 〉} .$ In Fig. 2 we plot $σ_{b} (T)$ for $M = 4$ , 16, 64, 256

Diffusion of Yen–Dollar rates

From the data set of real market prices, we can estimate the value of b, and presume the best value of M by comparing the price diffusion of numerical simulations to real price diffusion. Fig. 4 shows the diffusions of Yen–Dollar rates from 3:35 to 8:35 and from 9:25 to 23:25 in 11/9/2001, the day of terrorism. The rates were stable till 8:35 and it became quite unstable after 9:25. The rates until 8:35 follow a slow diffusion in short time scales, namely, the market has an attractive potential

Discussion

We can approximate the change of market prices by the random walk in a potential field. The potential field is well approximated by a quadratic function with its center given by the moving average of past prices. The random walk process is called the PUCK model. By analyzing the model, we clarified that the statistically steady condition of price fluctuations depends on the potential coefficient b, and we also theoretically proved that the price diffusion in the long time scales is amplified $2 (2$

Acknowledgments

This work is partly supported by Research Fellowships of the Japan Society for the Promotion of Science for Young Scientists (T.M.). The authors appreciate H. Moriya of Oxford Financial Education Co Ltd. for providing the tick data.

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