Application of the Cluster Expansion to a Mathematical Model of the Long Memory Phenomenon in a Financial Market

Kuroda, Koji; Maskawa, Jun ichi; Murai, Joshin

doi:10.1007/s10955-013-0783-z

Cited by 4 publications

(8 citation statements)

References 12 publications

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“…This situation is also included in our model as m D 1. Although the model proposed by Kuroda et al [7] assumed that m is bounded above, that is, the maximum value of m is finite, our model does not require any restriction on the maximum value of m. More precisely, m has an upper bound log log n, and n tends to infinity in our main theorem.…”

Section: Modelmentioning

confidence: 97%

“…A method of the cluster expansion is developed in the study of the statistical physics and is applied for instance to convergence theorems of the phase separation line of the two dimensional Ising model [4,12]. Since the cluster expansion is defined in an abstract setting [5], it can be applied to a financial model [7,14].…”

Section: Modelmentioning

confidence: 99%

“…The detail of the proof is complicated. The interested reader is referred to Kuroda et al [7]. For any 0 < t 1 < : : : < t k Ä 1 and any z D .z 1 ; : : : ; z k / 2 R k , we define…”

Section: Outline Of the Proof Of Theoremmentioning

confidence: 99%

“…Taking account of the bursty nature of human dynamics [1], Kuroda et al [7] proposed another theoretical model with this explanation. They assumed that a time interval between order executions distributes as a power law, and that the power law exponent does not depend on the size of hidden order.…”

Section: Introductionmentioning

confidence: 99%

“…Remark 3 In the model of Kuroda et al [7], since they assume that the exponent of inter-event time distribution is a constant˛.m/ D˛, they need to put a limitation on the size of hidden order: m Ä m max where m max is a positive number. Then, they derive finite number of fractional Brownian motions.…”

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confidence: 99%

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Signs of Market Orders and Human Dynamics

Murai

2015

Proceedings of the International Conference on Social Modeling and Simulation, Plus Econophysics Colloquium 2014

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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. IntroductionEmpirical studies [2,6,8,11] on high frequency financial data of stock markets that employ the continuous double auction method have revealed a time series of signs of market orders has long memory property. In contrast, a time series of stock returns is known to have short memory property. A time series of order signs is defined by changing transactions at the best ask price into C1 and transactions at the best bid price into 1. The auto-correlation function of the order signs decays as a power law of the lag and the exponent of the decay is less than 1, which is equivalent to a Hurst exponent of the time series is greater than one-half.In this paper, we propose a new mathematical model which takes account of origin of the long memory in order signs. As a first step, we define a discrete time stochastic process of cumulative order signs in accordance with some explanation

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Section: Modelmentioning

confidence: 97%

Section: Modelmentioning

confidence: 99%

“…The detail of the proof is complicated. The interested reader is referred to Kuroda et al [7]. For any 0 < t 1 < : : : < t k Ä 1 and any z D .z 1 ; : : : ; z k / 2 R k , we define…”

Section: Outline Of the Proof Of Theoremmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Signs of Market Orders and Human Dynamics

Murai

2015

Proceedings of the International Conference on Social Modeling and Simulation, Plus Econophysics Colloquium 2014

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A model of transaction signs with order splitting and public information

Murai

2016

Evolut Inst Econ Rev

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Applying a method of the cluster expansion developed in a study of statistical mechanics, a new mathematical model has been structured to account for the reason that a time series of transaction signs in financial markets has a long memory property. A basic assumption for the model was that investors split their hidden orders into small pieces before execution. The effect of public information also was taken into consideration. A mathematical expression of investors' investment behavior generates a discrete time stochastic process of cumulative transaction signs. The strong law of large numbers holds for the process: it converges to a trend term almost surely. The distribution of the fluctuation around the trend term weakly converges to the distribution of a superposition of a stochastic integral with respect to a Brownian motion and stochastic integrals with respect to a fractional Brownian motions with Hurst exponents greater than one-half. Namely, increments of the derived process have a long memory property.

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Investment time horizon and multifractality of stock price process

Kuroda

2016

Evolut Inst Econ Rev

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We construct a multifractal random walk for a stock trades model with inverse power law interaction. Consider a stock in a stock market and introduce a discrete time model for pair trades [(first trade ! second (reverse) trade] transacted by various types of traders. The type of trader is characterized by the investment time horizon defined as a time difference of the pair trades. We assume that probability distributions of the investment time horizons are given by inverse power law interactions with different exponents depending on the types of traders, and define a discrete time log-volatility process of the stock. Using the method of abstract polymer expansion developed in the study of mathematical physics we obtain a continuous type log-volatility process as a scale limit, and define a logreturn process from time 0 as a stochastic integral with respect to a Brownian motion. We then find a condition of the continuous cascade equation which derives a multifractality in the log-return process. Finally, we construct a multifractal random walk using the martingale convergence theorem.

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Application of the Cluster Expansion to a Mathematical Model of the Long Memory Phenomenon in a Financial Market

Cited by 4 publications

References 12 publications

Signs of Market Orders and Human Dynamics

Signs of Market Orders and Human Dynamics

A model of transaction signs with order splitting and public information

Investment time horizon and multifractality of stock price process

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