本研究的主要目的為探討隱含波動率曲面是否具有可預測的效果，參照Goccalves and Guidolin (2006)所使用的VAR兩階段預測方式，對台指選擇權進行實證。

    首先對每日在市場交易的選擇權之隱含波動率配適平滑公式，以價性、
到期期間為解釋變數，隱含波動率為被解釋變數，利用簡單迴歸估計出平滑公式的係數，並將所求出來的係數代入VAR模型對迴歸係數做預測，再利用修正過的迴歸係數做為更新平滑公式的係數，並且對隱含波動率曲面做預測，探討相同的預測方式在台指選擇權是否依然具有預測的效果。

    實證結果發現，利用平滑公式配適隱含波動率所得到的係數，會隨著時間變動而變化，具有隨狀態時間改變的性質。利用二階段的預測方式，可以增加橫斷面模型對隱含波動率曲面的配適效果，隱含波動率曲面具有可預測性；然而隨著預測的期間增加，預測曲面的效果會迅速降低，甚至產生對係數過度配適的問題。

One key stylized fact in the empirical option pricing literature is the existence of an implied volatility surface (IVS). The usual approach consists of fitting a linear model linking the implied volatility to the time to maturity and the moneyness, for each cross section of options data. However, recent empirical evidence suggests that the parameters characterizing the IVS change over time.
In this paper we study whether the resulting predictability patterns in the IVS coefficients may be exploited in practice. We propose a two-stage approach to modeling and forecasting the TAIEX option IVS. In the first stage we model the surface along the cross-sectional moneyness and time-to-maturity dimensions, similarly to Dumas et al. (1998). In the second-stage we model the dynamics of the cross-sectional first-stage implied volatility surface coefficients by means of vector autoregression models.
We find that not only the TAIEX implied volatility surface can be success fully modeled, but also that its movements over time are predictable in a statistical sense. However, when the fitted implied volatileity surface one week later, the VAR-type model’s prediction errors grow larger than another. The time passing is an important cause of overfitting at the movements of IVS.

目錄
第一章 緒論
第一節 研究背景與動機......................................1
第二節 研究目的............................................3
第三節 研究架構與流程......................................4
第二章 文獻回顧
第一節 影響波動率曲面的因素................................6
第二節 波動率曲面配適與預測之相關文獻.....................11
第三章 研究方法
第一節 配適隱含波動率曲面之平滑公式.......................18
第二節 隱含波動率曲面參數之預測...........................28
第三節 利用橫斷面的平滑公式預測隱含波動率.................34
第四節 預測績效分析.......................................36
第四章 實證結果分析
第一節 資料來源與處理.....................................37
第二節 配適隱含波動率曲面.................................42
第三節 預測隱含波動率曲面.................................49
第四節 隱含波動率曲面之預測分析...........................51
第五章 結論與建議
結論......................................................62
對後續研究之建議..........................................63
參考文獻..................................................65



表次目錄

表3-1：存續期間(交易日)分類表.............................24
表3-2：價位分類表.........................................26
表4-1：隱含波動率隨到期期間、價內程度之基本統計表.........41
表4-2：隱含波動率曲面參數之基本統計量. ...................43
表4-3：隱含波動率曲面參數原始序列單根檢定結果.............50
表4-4：VAR模型最適落階期選取表............................50
表4-5：簡單平滑公式配適樣本內誤差衡量表...................54
表4-6：不同模型下的預測誤差衡量表(預測第1期)..............54
表4-7：不同模型下的預測誤差衡量表(預測第5期)..............54
表4-8：不同價性、到期期間之隱含波動率預測誤差衡量表(預測第1期).....57
表4-9：不同價性、到期期間之市場價格預測誤差衡量表(預測第1期).....58
表4-10：不同價性、到期期間之隱含波動率預測誤差衡量表(預測第5期).....59
表4-11：不同價性、到期期間之市場價格預測誤差衡量表(預測第5期).....60










圖次目錄

圖1-1：研究架構圖..........................................4
圖3-1：波動率曲面圖模擬圖.................................22
圖3-2：delta價性分類與K/S-1價性分類關係圖.................27
圖4-1a：台指選擇權成交量比較圖(買權) .....................38
圖4-1b：台指選擇權成交量比較圖(賣權) .....................38
圖4-2：平均每日合約數目分布圖(交易量高於50筆) ............39
圖4-3：樣本觀察值分布圖...................................40
圖4-4：利用簡單迴歸估計平滑公式係數之時間序列圖...........45
圖4-5：迴歸係數之自我相關圖...............................45
圖4-6：迴歸係數之交叉相關圖...............................46
圖4-7：曲面係數配適之斷面圖...............................48
圖4-8：利用簡單迴歸估計平滑公式係數之時間序列圖(20個交易日觀察值)..52
圖4-9：隱含波動率曲面圖...................................61

杜化宇 (Anthony H. Tu) 任紀為 (Chi-Wei Jen)，外匯選擇權的定價與馬可夫鏈蒙地卡羅法的應用，風險管理學報 第七卷 第三期 2005年 11月

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