本篇論文主要探討考慮跳躍風險的資產價格過程，以兩種誤差項捕捉市場上的不規則變動，分別為布朗寧運動項及常見萊維跳躍項，其變異數均類比Heston and Nandi的GARCH(1,1)動態模型。以往傳統模型僅以單一因子捕捉市場波動，面對較不尋常的劇烈變動並無法掌握得很好。十幾年來，在國內外許多學者文獻實證結果上均證明出，使用常見之萊維跳動項作為另一個隨機因子的情況下，能顯著的捕捉經濟上大波動，使得模型有更好的解釋能力。在模型的參數估計方面，由於並不是所有的萊維跳動項的分配具有可供計算的分配形式，使得估計上有所困難，本文使用粒子濾波演算法，解決此類未知變數的問題。最後，實際帶入市場資料，比較與標準無跳躍項之GARCH模型的差異後可知，推廣的兩隨機因子資產價格過程更能精確地刻畫股價報酬的波動變化。

This paper mainly discusses the asset price process with jump risk. This model catches the irregular fluctuation in the market with two kinds of error terms, a brownian motion increment and a common Lévy jump increment respectively. Their variances similarly follow to that of Heston and Nandi’s GARCH(1,1) dynamic model.
The traditional models only use a single factor to capture market fluctuation. But these can not grasp the unusual dramatic situation well. Over the past few years, many empirical results of the literatures show that using a well-known Lévy jump increment as another random factor can significantly capture the economic fluctuations. So that the model has a better ability to explain. In terms of the model parameter estimation, since the filtering density may not be analytical, making it difficult to estimate. In this paper, we use the particle filter algorithm to solve the problem of such unknown variables.
Finally, with the real market data, the empirical result compares our model with standard GARCH model that does not have a jump component in the process. We can see that the promotion of the two stochastic factors asset price process can describe the fluctuation of stock price more accurately.

Abstract ii
Acknowledgements iii
Chapter 1 Introduction 1
Chapter 2 Model 4
2.1 Lévy Increment 4
2.2 Asset Price Process 5
2.3 Pure Lévy jump Increment 6
2.3.1 Merton Jump 6
2.3.2 Variance Gamma 7
2.4 Affine GARCH Dynamic 7
2.5 Change of Measure 9
Chapter 3 Estimating Methodology 15
3.1 Particle Filtering 15
3.2 Log-Likelihood Function of Daily Index Return 18
3.3 Parameter Estimation Constrains 18
Chapter 4 Empirical Analysis 19
4.1 Data 19
4.2 Estimation Result 22
Chapter 5 Conclusion 26
Appendix 27
References 34

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