# Study on the Stability of an Artificial Stock Option Market Based on Bidirectional Conduction

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Model

#### 2.1. Stock Trade Module and Its Limitations

#### 2.1.1. The Introduction of the Stock Trade Module

_{f}= 0.1. Similarly, stock holders can receive stock dividend which is generated by a stationary stochastic process and the equation of dividend process is denoted as Equation (1):

#### 2.1.2. The Limitations of the Stock Trade Module and the Countermeasure

Initial cash | 20 | 200 | 2,000 | 20,000 |
---|---|---|---|---|

Average | 98.47 | 98.41 | 98.27 | 98.27 |

Variance | 3.34 | 3.48 | 3.45 | 3.45 |

Before | After | |
---|---|---|

Average | 98.27 | 98.42 |

Variance | 3.45 | 3.54 |

#### 2.2. The Compound Poisson Process

Old dividend process | New dividend process | Old stock price | New stock price | |
---|---|---|---|---|

Average | 10.0 | 10.0 | 97.85 | 96.59 |

Variance | 0.074 | 0.1463 | 5.05 | 15.23 |

#### 2.3. The Option Trade Module

#### 2.3.1. The Setting of Option

#### 2.3.2. The Types of Option Traders

_{t}denotes the current option demand, s

_{t}denotes the current stock position, x

_{i,t-}

_{1}is the quantity of different types option in the previous time (t−1) and if call option was bought or put option was sold by agent the item is negative, if call option was sold or put option was bought the item is positive, all items are summed to obtain the option demand in the current time.

#### 2.3.3. The Option Market Maker

## 3. Experiments

#### 3.1. Experimental Parameters

Parameter | Parameter value | Parameter | Parameter value |
---|---|---|---|

$\overline{d}$ | 10.0 | $a$ | 0.95 |

$\rho $ | 0.95 | $b$ | 4.5053 |

$D({d}_{t})$ | 0.1463 | $a$ range | 0.7–1.2 |

$f$ | 6.3333 | $b$ range | −10.2945–19.7053 |

$e$ | 16.7732 |

Parameter | Parameter value | Parameter | Parameter value |
---|---|---|---|

lifetime | 60 | Leverage Ratio | 0.2 |

Option start time | 10000 | $\lambda $ | 0.01 |

Stocks per contract | 1.0 |

#### 3.2. The Stability of the Model

#### 3.3. The Effect of Option Market on Stock Price and Volume

**Figure 7.**The price time series of stock. (left: before the option is introduced; right: after the option is introduced).

Stock price without option | Stock price with option | Trade volume without option | Trade volume with option | |
---|---|---|---|---|

Average | 89.81 | 80.12 | 14.81 | 11.96 |

Variance | 10.40 | 20.31 | 230.12 | 115.98 |

#### 3.4. The Effect of Option Market on Stock Return and Volatility

_{t}denotes the current stock dividend. Figure 8 is stock price return time series which are chosen in a long stable price sequence. The sample size in Table 7 is 50,000.

**Figure 8.**The time series of stock logarithmic returns. (left: before the option is introduced; right: after the option is introduced).

Stock return without option | Stock return with option | |
---|---|---|

Average | 0.105579 | 0.120366 |

Variance | 0.000346 | 0.000937 |

Kurtosis | 22.1443 | 14.49292 |

Variable | Coefficient | Std. Error | t-Statistic | Prob. |
---|---|---|---|---|

C | 0.105579 | 5.28E−05 | 2000.599 | 0.0000 |

AR(1) | −0.138857 | 0.012371 | −11.22476 | 0.0000 |

MA(1) | −0.232591 | 0.012149 | −19.14488 | 0.0000 |

Variable | Coefficient | Std. Error | t-Statistic | Prob. |
---|---|---|---|---|

C | 0.119826 | 0.000857 | 139.7441 | 0.0000 |

AR(1) | 1.718534 | 0.011286 | 152.2778 | 0.0000 |

AR(2) | −0.51085 | 0.023445 | −21.7891 | 0.0000 |

AR(3) | −0.61926 | 0.01682 | −36.8165 | 0.0000 |

AR(4) | 0.409515 | 0.004987 | 82.11901 | 0.0000 |

MA(1) | −1.53815 | 0.012256 | −125.502 | 0.0000 |

MA(2) | 0.392182 | 0.023032 | 17.0278 | 0.0000 |

MA(3) | 0.16174 | 0.011836 | 13.66452 | 0.0000 |

**Figure 9.**The autocorrelation of the stock return residual series. (left: before the option is introduced; right: after the option is introduced).

**Figure 10.**The autocorrelation of the stock return residual squared series. (left: before the option is introduced; right: after the option is introduced).

Parameters | $c$ | ${\alpha}_{1}$ | ${\beta}_{1}$ |
---|---|---|---|

Before | 0.000168 | 0.429756 | −0.00509 |

After | 1.62E−05 | 0.058384 | 0.914057 |

_{1}and β

_{1}coefficients represent the strength of volatility persistence. After the option is introduced the α

_{1}value decreases, but the β

_{1}increases more, so in general, return volatility persistence becomes more obvious following the introduction of the option. Furthermore, more information has been transmitted from option market to stock market which is the reason of increases of volatility in stock market.

#### 3.5. The Effect of Various Information Length

Weighing coefficients | t-1 | t-2 | t-3 | t-4 | t-5 | t-6 |
---|---|---|---|---|---|---|

1 | 1.0 | / | / | / | / | / |

2 | 0.6 | 0.4 | / | / | / | / |

3 | 0.5 | 0.25 | 0.25 | / | / | / |

4 | 0.4 | 0.2 | 0.2 | 0.2 | / | / |

5 | 0.3 | 0.175 | 0.175 | 0.175 | 0.175 | / |

6 | 0.2 | 0.16 | 0.16 | 0.16 | 0.16 | 0.16 |

γ | 0.0 | 0.01 | 0.02 | 0.03 | 0.04 | 0.05 | 0.06 | 0.07 | 0.08 | 0.09 |
---|---|---|---|---|---|---|---|---|---|---|

Average | 89.59 | 89.23 | 89.11 | 65.55 | 56.28 | 45.67 | 37.17 | 30.34 | 23.89 | 21.23 |

Variance | 9.979 | 11.89 | 12.05 | 54.85 | 64.93 | 87.67 | 98.07 | 92.57 | 109.09 | 97.02 |

Information length | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Average | 62.48 | 72.65 | 79.15 | 84.32 | 87.46 | 88.65 |

Variance | 30.81 | 23.17 | 24.78 | 24.92 | 21.22 | 16.67 |

#### 3.6. The Effect of Various Proportions of Option Traders

Proportion | Average | Variance | Proportion | Average | Variance |
---|---|---|---|---|---|

0.25-0.25-0.25-0.25 | 87.197 | 63.410 | 0-0-0.6-0.4 | 91.510 | 13.661 |

0.15-0.15-0.35-0.35 | 88.038 | 48.726 | 0-0-0.5-0.5 | 89.389 | 16.170 |

0.05-0.05-0.45-0.45 | 89.737 | 16.555 | 0-0-0.4-0.6 | 89.394 | 14.506 |

0-0-0.9-0.1 | 98.431 | 12.975 | 0-0-0.3-0.7 | 89.125 | 14.016 |

0-0-0.8-0.2 | 97.487 | 12.718 | 0-0-0.2-0.8 | 89.371 | 14.180 |

0-0-0.7-0.3 | 96.462 | 13.062 | 0-0-0.1-0.9 | 89.359 | 16.088 |

## 4. Conclusions and Future Work

## Acknowledgements

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**MDPI and ACS Style**

Yang, H.-J.; Sun, G.-P.
Study on the Stability of an Artificial Stock Option Market Based on Bidirectional Conduction. *Entropy* **2013**, *15*, 700-720.
https://doi.org/10.3390/e15020700

**AMA Style**

Yang H-J, Sun G-P.
Study on the Stability of an Artificial Stock Option Market Based on Bidirectional Conduction. *Entropy*. 2013; 15(2):700-720.
https://doi.org/10.3390/e15020700

**Chicago/Turabian Style**

Yang, Hai-Jun, and Gui-Ping Sun.
2013. "Study on the Stability of an Artificial Stock Option Market Based on Bidirectional Conduction" *Entropy* 15, no. 2: 700-720.
https://doi.org/10.3390/e15020700