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In this paper, we take the advantage of high frequency data to develop option pricing model and select the Realized GARCH model to describe the volatility of assets, use NIG distribution to describe the distribution of underlying assets, and also build the Realized-GARCH-NIG model to price the option. Finally, we obtain the dynamic option pricing model based on the Realized-GARCH-NIG approach. To verify the effect of the dynamic option pricing model based on the Realized-GARCH- NIG approach, this paper provides the empirical analysis between the dynamic option pricing model based on the Realized-GARCH-NIG approach and the B-S option pricing model. The results show that the option value obtained from the dynamic option pricing model based on the Realized-GARCH-NIG approach is more accurate and effective than the B-S option pricing model.

Option is a derivative financial tool, which now has become an important risk management tool for investors. The estimation and forecast of asset volatility have a great influence on option value, which have become the important direction of current option pricing research. Black and Scholes [

Efficient estimation on asset volatility and distribution characteristics is an important factor affecting the option pricing model. As high frequency data is widely used in recent years, many econometricians start to take the advantages of high frequency data into account, and directly model the volatility of asset with high frequency data. The realized volatility model proposed by Andersen [

Different from previous option pricing research, we take the advantages of high frequency data into account, and use high frequency data to make research of option pricing model to describe the volatility of assets with Realized GARCH model integrating high frequency data. To capture the skewness, sharp peak and fat tail distribution characteristics, we use the normal inverse gaussian (NIG) distribution to describe the distribution of assets rather than the normal distribution and build Realized-GARCH-NIG model to price the option. Final, we obtain the dynamic option pricing model based on the Realized-GARCH-NIG approach. In this paper we use the NASDAQ 100 Index Option data as research example, give an empirical analysis between the dynamic option pricing model based on the Realized-GARCH-NIG approach and the B-S option pricing model.

The structures of this article are shown in the order as follows: Section 2 is Research design, Section 3 is Empirical Methods, Section 4 is results and discussion, and Section 5 is conclusion.

The B-S option pricing model proposed by Black and Scholes [

As the B-S option model,

This paper can obtain

If the return of risk-free asset

So, this paper can obtain

The function shows that the current price of derivative product is the discounted value of the price under risk neutral measure Q at T. According to equivalent martingale measure principle, the option value can be regarded as the function of the underlying asset, so the option that can be obtained under the underlying asset condition is as follows:

where, r_{f} is risk-free interest rate, E^{Q} is the expectation under risk neutral measure, T is the expiration date, _{T} is the price of the underlying asset i at the expiration date , and K is the exercise price of option.

Duan [

where _{t} of the return follows standard normal distribution, and x_{t} can be any one of realized volatility measure, and errors caused by market intraday microstructure noise and non-trading time (closing) are adjusted by coefficients

where,

when

If X follows NIG distribution, it is simplified as:

where in,

This paper utilizes Realized-GARCH-NIG model to describe the volatility process and distribution of assets. Duan provided a method of converting asset distribution into risk neutral distribution in his research, and this paper utilizes Realized-GARCH-NIG model to price the option by reference to Duan’s method. According to the risk neutral pricing principle, option price can be expressed as:

This paper selects the data from the database of the Chicago Board Options Exchange, and selects the NASDAQ 100 Index Option data as the research samples from June 01, 2012 to May 30, 2013. To verify the effectiveness of option pricing model proposed in this paper, this paper selects the B-S option pricing model to compare with the dynamic option pricing model based on the Realized-GARCH-NIG approach. Because of the complicated model that we used in this paper, we use the Monte-Carlo method (simulation times: 10000) to analyse the option price.

It can be seen from

To quantitatively compare the pricing effect of model, this paper compares the model with three different error measurement methods. These three errors are root mean square error (RMSE), average absolute error (AAE) and average relative error (APRE) respectively. Calculation functions are as follows:

Log | AIC | BIC | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | 0.0461 | 0.0170 | 0.2603 | 0.0594 | 0.0261 | 0.0611 | 0.0575 | 0.0154 | 375.72 | 480.44 | 334.53 |

Median | 0.0407 | 0.0136 | 0.2210 | 0.0456 | 0.0187 | 0.0589 | 0.0587 | 0.0111 | 345.61 | 410.23 | 324.32 |

Std | 0.0870 | 0.0299 | 0.0631 | 0.0059 | 0.0558 | 0.0049 | 0.0571 | 0.0382 | 24.76 | 74.43 | 84.93 |

Minimum | 0.0356 | 0.0010 | 0.0470 | 0.0285 | 0.0232 | 0.0507 | 0.0461 | 0.0097 | 305.33 | 368.66 | 263.74 |

Maximum | 0.0691 | 0.0296 | 0.7317 | 0.0933 | 0.0771 | 0.2019 | 0.1578 | 0.2677 | 510.94 | 652.44 | 552.98 |

Log | AIC | BIC | |||||
---|---|---|---|---|---|---|---|

Mean | 0.1527 | 0.5507 | 0.0645 | 0.0489 | 334.4 | 279.1 | 356.1 |

Median | 0.1838 | 0.5541 | 0.0512 | 0.0474 | 364.7 | 205.5 | 285.9 |

Std | 0.0645 | 0.0456 | 0.0285 | 0.4547 | 51.3 | 48.5 | 54.5 |

Minimum | 0.0655 | 0.0943 | 0.0176 | 0.0182 | 258.7 | 208.1 | 227.6 |

Maximum | 0.2558 | 0.9012 | 0.1052 | 0.0970 | 585.6 | 355.6 | 471.6 |

RMSE | AAE | ARPE | |
---|---|---|---|

the B-S option pricing model | 53.07 | 62.19 | 0.94 |

the dynamic option pricing model based on the Realized-GARCH-NIG approach | 40.62 | 42.85 | 0.29 |

where,

Above empirical results indicate that the B-S option pricing model can’t accurately describe the volatility and the characteristics of fat tail, sharp peak and skewness distribution. These defects result the deviation of option price. In this paper, the Realized GARCH model integrating high frequency data selected reflects unique advantages and the empirical results indicate that the Realized-GARCH-NIG model built in this paper is more accord with actual market situation and it can effectively describe the volatility of assets. Compared with the B-S option pricing mode, the dynamic option pricing model based on the Realized-GARCH-NIG approach is slightly deviated from actual option price and more accord with actual market situation, and can improve the accuracy and effectiveness of option pricing model.

In previous option pricing researches, low frequency data and normal distribution are often used to estimate the dynamic process of assets, those inaccurate estimation methods have a larger influence on option value estimation. Based on the advantages of high frequency data, NIG distribution, this paper utilizes the Realized- GARCH-NIG model to describe the volatility and the characteristics of fat tail, sharp peak and skewness distribution, and gets the dynamic option pricing model based on the Realized-GARCH-NIG approach. Choice the NASDAQ 100 Index Option data as research example, give an empirical analysis between the dynamic option pricing model based on the Realized-GARCH-NIG approach and the B-S option pricing model. Empirical results indicate that the dynamic option pricing model based on the Realized-GARCH-NIG approach is more accord with actual market situation, and can effectively improve the accuracy of option pricing model. With the development of financial markets, the demand of option products will certainly continue to emerge, and the option pricing model proposed in this paper provides a good reference in option pricing research. This paper will further research the application of high frequency data in option pricing model, and research high frequency nonlinear option pricing model and realization method in a dynamic market process.

Zhang Hong Lei thanks the support of the Natural Science Foundation of Sichuan Education Department (16ZB0538).

Honglei Zhang,Yixiang Tian,Gaoxun Zhang, (2016) Dynamic Option Pricing Model Based on the Realized-GARCH-NIG Approach. Open Journal of Social Sciences,04,66-71. doi: 10.4236/jss.2016.43011