臺灣博碩士論文加值系統

本篇主要在研究，隱含波動率是否配適隨機波動率過程下的波動率期間結構模型。Heynen(1994)有類似的研究。但Heynen的研究方法使用時間序列模型，而 Black-Scholes公式倒推的隱含波動率具有風險中立世界的假設，與時間序列模型下的真實世界不一。我們以風險中立下的波動率隨機過程，且考慮到波動率風險，去建構波動率期間結構的模型，驗證S&P 500指數選擇權的隱含波動率是否能為模型解釋。
結論如下，首先，模型無法描述隱含波動率行為，這可能是因為我們的平均預期波動率假設不成立，或是此假設成立，但隱含波動率仍受其他變數影響。其次，我們由歷史資料估計出較大的波動率均數回歸參數，這表示波動率的期間結構曲線較平坦，當隱含波動率距離長期波動率水準越遠，則越有收斂的可能。第三，模型間彼此的波動率均數回歸參數不存在相等關係。

This study is aimed to answer this question:” Are implied volatilities consistent with the stochastic properties of underlying asset return?” The importance of this study is that we make the comparison between actual implied volatilities and underlying return volatility with certain specifications that few studies discuss. Though Heynen, Kemma and Vorst(1994) have the similar study, we develop term structure model in the risk-neutral world, rather than Heynen’s model in the real world. Moreover, unlike other studies, volatility risk premium is considered in our term structure model which is developed from the average of expected volatility assumption.
Three major findings are as follows. First, all models may not be well descriptive about implied volatility behavior, which suggests the average expected volatility assumption does not hold. Second, the estimated mean reversion parameter of volatility in the term structure model is larger. A larger mean reversion parameter suggests a flatter term structure curve and more possibility that spot volatility reverts to the long-term mean variance level. Third, there should be theoretical equivalent relation on the mean reverting parameter between any two models among three models in this study. However, the empirical result does not support the equivalent relation among models.

Chapter　1　
Introduction　3
1.1　Research　Background　and　Motivation　3
1.2　Objectives　of　This　Study　3
1.3　Structure　of　This　Study　4
1.4　Importance　of　This　Study　and　Major　Finding　5

Chapter　2　Literature　review　7
2.1　Implied　Volatility　and　Realized　Volatility　7
2.2　Some　Problems　of　Measurement　Errors　on　Implied　Volatility　7
2.3　The　Informational　Content　of　Implied　Volatility　8
2.3.1　Can　Implied　Volatility　Predict　Underlying　Asset　Movement?　8
2.3.2　Can　Implied　Volatility　Predict　Future　Volatility?　8
2.4　Deterministic　and　Stochastic　Volatility　10
2.5　Volatility　Process　11
2.5.1　AR(1)　Model　11
2.5.2　GARCH　Models　12

Chapter　3　Data　14
3.1　Interpolation　for　At-the-money　Exercise　Price　14
3.2　Data　Processing　in　Daily　Timeframe　14
3.3　Exogenous　Variable　:　Mean　of　Unconditional　Variance　15
3.4　Data　Missing　16
3.5　Sampling　Period　16

Chapter　4　Methodology　17
4.0　Elementary　Tests　and　Important　Introduction　17
4.0.1　Unit　Root　Test　17　
4.0.2　White　Noise　Test　18
4.0.3　An　Important　Assumption　in　This　Study　18
4.1　Term　Structure　Model　19
4.1.1　Stochastic　Volatility　Process　1　19
4.1.2　Stochastic　Volatility　Process　2　23
4.2　Time　Series　Model　:GARCH　24
4.3　The　Equivalent　Relation　between　Term-structure　Models　and　GARCH　
Model　26

Chapter　5　Results　28
5.1　GARCH　Model　28
5.1.1　Determination　ARMA(p,q)-GARCH(1,1)　28
5.1.2　The　Comparison　among　Candidate　Models　30
5.1.3　Test　of　the　Consistence　between　Implied　Volatility　and　GARCH　
Specification　31
5.2　Term　Structure　Model　33
5.2.1　Test　of　the　Consistence　between　Implied　Volatility　and　Stochastic　
Properties　34
5.2.2　The　Mean　Reversion　Parameter　36
5.2.3　Volatility　Risk　Premium　39
5.3　Test　on　the　Equivalent　Relation　between　Term-structure　Model　and　
GARCH　Model　40

Chapter　6　Conclusion　42

Reference　44

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