# A Binary Decision Model and Fat Tails in Financial Market

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## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. An Example of a Real Trading Process

#### 2.2. The Image of a Real Trading Board

- Information is incorporated correctly into the price.
- Information is incorporated rapidly into the price.
- Arbitrage deals profit if there are errors or delays.
- Countless traders are always looking for arbitrage opportunities.
- The arbitrage opportunity is gradually lost, and the market becomes more efficient.

#### 2.3. A Simple Security Market Model

**Definition**

**1.**

**Definition**

**2.**

#### 2.4. The Kirman Process

## 3. The Simulation

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

`n=100;x=10000;e=0.003;d=0.05;i=0;j=0;m1=0;m2=0; m=0;buy=0;sell=0;`

`p1[k_]:=(1-k/n) (e+(1-d) k/(n-1));`

`p2[k_]:=k/n (e+(1-d) (n-k)/(n-1));`

`pos=Table[{p1[i],1-p1[i]-p2[i],p2[i]},{i,0,n}];`

`m1=Table[pos[[i]][[1]]/pos[[i+1]][[3]],{i,1,n-1}];`

`m2=FoldList[Times,1,m1];`

`m=m2/(1+Total[m2]);`

`buy=Table[i=i+RandomVariate[MultinomialDistribution[1,pos[[i+1]]]][[1]]`

`-RandomVariate[MultinomialDistribution[1,pos[[i+1]]]][[3]],{x}];`

`sell=Table[j=j+RandomVariate[MultinomialDistribution[1,pos[[j+1]]]][[1]]`

`-RandomVariate[MultinomialDistribution[1,pos[[j+1]]]][[3]],{x}];`

`price=buy-sell;`

`volume=Max/@Transpose[{buy,sell}];`

`diff=Differences[price,20];`

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**Figure 1.**Trading process of Sony(6758) at 9:22 on 28 June 2022. The time is updated from left to right and from top to bottom. First, a trade is executed at JPY 11,500 in the upper left figure. Next, in the upper right diagram, a sell order for 100 shares is generated at JPY 11,510, and the trade is executed, and at the same time, a sell order for 100 shares is generated at JPY 11,520. The transaction proceeds in the same manner.

**Figure 2.**An image of a real trading board in a state of equilibrium immediately before a share of security is traded. In addition to displaying supply and demand with the large point limit prices on the board, potential supply and demand is displayed by the small points, and the traders are all in a state of subjective equilibrium. If nobody changes the subjective equilibrium for any reason, the market equilibrium is sustained without any trades being made.

**Figure 3.**An image of simple security market model. First, in Market Equilibrium 1 in which trades are not made, the subjective equilibrium of a trader who indicates the second sales quote changes for some reason, and she sells to the trader with the highest buy quote. The buying trader then changes into a seller, and simultaneously, the selling trader changes into a buyer; then, they line up with their respective voluntary quotes. As the price is voluntary, they are aligned by the order of the quotes, and the subsequent Market Equilibrium 2 is realized and so on.

**Figure 4.**(

**a1**) TOPIX return of every 20 ticks(8,897 records), morning, 1 December 2021; (

**a2**) simulation result with $\u03f5=0.003,\delta =0.05$; (

**a3**) fitting of both; (

**b1**) USD/JPY return of every 20 ticks (21,314 records), 7 July 2014; (

**b2**) simulation result with $\u03f5=0.005,\delta =0.05$; (

**b3**) fitting of both.

**Figure 5.**Results of Markov chain simulation. (

**a**) TOPIX, (

**b**) USD/JPY, (1) Numbers of buying shares, (2) Numbers of selling shares, (3) Prices, (4) Trading Volumes, and (5) Distribution of returns.

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Sano, K. A Binary Decision Model and Fat Tails in Financial Market. *Appl. Sci.* **2022**, *12*, 7019.
https://doi.org/10.3390/app12147019

**AMA Style**

Sano K. A Binary Decision Model and Fat Tails in Financial Market. *Applied Sciences*. 2022; 12(14):7019.
https://doi.org/10.3390/app12147019

**Chicago/Turabian Style**

Sano, Kazuo. 2022. "A Binary Decision Model and Fat Tails in Financial Market" *Applied Sciences* 12, no. 14: 7019.
https://doi.org/10.3390/app12147019