本篇論文試圖透過報酬、波動度、機率密度三個面向，來檢驗其對於S&P 500指數的預測能力。藉由波動度指數(VIX)、隱含波動度指數(Implied VIX)以及隱含波動度指數的殘差項與報酬（實質波動度）來建立回歸式、衡量對報酬（波動度）預測的表現。針對風險中壢機率密度預測能力的衡量上，同時採用隨機波動度以及包含跳動的隨機波動度模型，作為選擇權的定價模型，試圖發現S&P 500指數珠隱含的跳動型態。除此之外，從2006年3月到2010年12月的樣本期間被分為三個子樣本，以檢測在不同的市場情況下，三個面向的預測能力是否會受到影響。實證結果指出：報酬的預測能力在市場極端波動的時候會消失，而波動度的預測能力則不受市場情況影響。在對報酬與波動的預測能力上，隱含波動度指數也確實能夠提供相較於波動度指數更多的資訊。將跳動納入選擇權定價模型後，S&P 500指數未來的動態過程更為穩定，於此同時，兩模型間針對未來預測所提供的資訊內涵在金融海嘯爆發後呈現更大的差異。

This paper intends to look at the forecasting ability for S&P 500 index (SPX) based on three channels, returns, volatility, and density. The performance of return (volatility) forecasting ability is measured by forming
the regressions with return (realized volatility), the VIX, the implied VIX, and the implied VIX residuals. In order to detect the jump in the SPX, the stochastic volatility (SV) and stochastic volatility with jump (SVJ) models are selected to be the option pricing model for generating the risk-neutral density of the SPX. Furthermore, the sample period between March 2006 and December 2010 is divided into three sub-sample periods to compare the different influence of market situations. The results indicate that the return forecasting ability will disappear in extremely volatile market situation; meanwhile, the volatility forecasting ability is independent among the three market situations, with the implied VIX remain providing incremental information. After considering jump components in pricing ptions,the dynamics process of the SPX is more stable. The information contents derived from two models are much more different after the occurrence of financial crisis.

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