我們在此篇論文從隨機波動跳躍模型來估計出無風險密度函數 (risk-neutral density)的參數，以及利用GJR模型來模擬實際密度函數(real-world density)。接著利用這兩個函數就可以進一步獲得S&P 500指數的價格密度函數(SPD)，我們同時也測試SPD的斜率是否符合風險趨避假設。我們接著分別探討選擇權、股票、期貨市場的因素來解釋SPD的斜率，這些因素包含了價外的Put/Call的未平倉比率、價外Put 及Call的交易量、VIX指數、消費者情緒指數、投資人情緒指數、期貨交易量。這些因素都是和投資人對於市場的不同信念，我們要檢驗這些信念歧異是否確實會影響到價格密度函數。

We first estimate the parameter of risk-neutral density from stochastic volatility jump (SVJ) model and those of GJR model to simulate real-world density. We can therefore obtain the state price density (SPD) of S&P500 from the two densities and test the risk aversion assumption that connected to the slope of SPD. We then investigate the factors from options, equity, and futures markets to explain the slope of SPD. The factors include the open interest ratio of the OTM put to call, the OTM call and put trading volume, CBOE’s VIX index, consumer sentiment index, investors sentiments index, and the volume of futures. These factors are all connected with different investor beliefs. Whether heterogeneous beliefs influence the state price density is tested.

Contents
I. Introduction.......................................... 1

II. Literature Review..................................... 3

III. Methodology........................................... 5
3.1 State Price Density (SPD)................................. 5
3.2 Stochastic Volatility (SV) model.......................... 6
3.3 Stochastic Volatility Jump (SVJ) model.................... 6
3.4 Risk Neutral Density (RND)................................ 7
3.5 Parametric Estimation..................................... 8
3.6 Glosten, Jagannathan & Runkle (GJR) model................. 8
3.7 Smoothing denisty to Real-World Density................... 9
3.8 Regression between S&P 500 SPD and heterogeneous beliefs.. 10

IV. Data.................................................. 11

V. Empirical Results..................................... 13
5.1 The Shape of S&P 500 SPDs................................. 13
5.1 Summary Statistics and Correlation Matric................. 17
5.3 Heterogeneous Beliefs to Investors........................ 27
VI. Conclusion............................................ 30

VII. References............................................... 31

VIII. Appendix................................................ 35


List of Figures
Figure 1 State Price Density.................................. 14
Figure 2 Measure of Risk-Neutral Skewness and Heterogeneous Belief .............................................................. 21


List of Tables
Table 1 Descriptive Statistics of Variables................... 18
Table 2 Options Data by Moneyness (K/S)....................... 22
Table 3 Options Data by Maturity.............................. 24
Table 4 Correlation Between Regression Variables.............. 26
Table 5 Heterogeneous Beliefs Results......................... 29

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