摘要
作为描述金融市场动态变化的重要指标，波动率对参与者的投资决策、
资产管理、风险对冲等行为都有极为深刻的影响。学界与业界度量波动率的
方法主要有基于模型和无模型两类，其中芝加哥期权交易所 (CBOE)发布
的无模型波动率指数(VIX)，因其可以实时动态地精准衡量市场波动程度和
投资者情绪，已被当作美国股票市场乃至全球金融市场的“晴雨表”。与此同
时，CBOE发布的VIX衍生品可以用来对冲标普500指数的投资风险，其
在极大丰富金融投资市场的同时，也为波动率指数的研究带来机遇和挑战。
本文提出并分析一类具有混频数据和混合结构的仿射GARCH类模型，
该模型通过引入波动率成分、跳跃成分和已实现波动率，进而可以有效捕
捉金融时间序列长记忆性和尖峰厚尾等特征，并可进一步提升波动率预测
能力。基于此，本文首次得到多因子 GARCH类模型条件方差与 VIX的
关系，并结合模型仿射结构推导出 VIX衍生品定价公式。此外，我们开创
性地探索仿射结构对 VIX衍生品定价的影响，并进一步比较 EGARCH、
GJR-GARCH与NGARCH等非仿射模型在VIX衍生品上的定价表现。
本文内容结构安排如下。首先，我们在第一章中介绍本文研究背景、动
机、内容和贡献等，并在第二章中介绍衍生品定价模型和风险中性测度等相
关基础知识。在第三章中，我们使用广义仿射已实现波动率模型严格地推导
出 VIX期限结构表达式与 VIX期货定价公式。本章还通过实证研究表明，
引入已实现波动率能有效地提升上述模型的 VIX衍生品定价表现。在第四
章中，我们进一步将波动率成分与跳跃成分引入上述已实现波动率模型，并
严格推导出VIX期货定价解析解。本章实证研究发现，已实现波动率、波动
率成分与跳跃成分都可有效提升 VIX预测及其期货定价能力，且这三个成
分在上述提升中具有显著互补性。第五章进一步研究非仿射模型在 VIX衍
生品定价上的表现。我们发现，尽管仿射GARCH模型在高价值期权表现突
出，这类非仿射模型尤其是EGARCH模型具有更为显著且稳健的优越定价
表现。最后，我们在第六章中简要综述本文研究结果并探讨未来发展方向。
关键词： VIX衍生 ;GARCH;实现波动率;波动率成 ;跳成
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基于GARCH类模型的VIX衍生品定价
Abstract
Volatility, as an important indicator describing the dynamic changes of
the?nancialmarket,hasaprofoundimpactonparticipants’decisionssuchas
investmentstrategies,assetmanagement,andriskhedging. Model-based and
model-free volatility measures are the two approaches popularized by both
academia and industry, and the Chicago Board Options Exchange (CBOE)
volatilityindex(VIX),whichcanmeasuremarketvolatilityandinvestorsen-
timent accurately and dynamically in real-time, has e?ectively become the
barometer in the US stock market and even the global ?nancial market. At
thesametime,theVIXderivativesreleasedbyCBOEo?eradditionalinvest-
mentandhedgingopportunitiesfor?nancialmarketparticipants,whichbring
tremendouschallengesandopportunities forvolatilityresearch,togetherwith
ampleinvestmentoutlets.
This thesis proposes and analyses a wide class of GARCH-type models
with mixed frequency data and hybrid structure. These models incorporate
volatilitycomponents,jumps,andrealizedvolatilitytoe?ectivelycapturethe
long memory and thick tail of ?nancial time series and then signi?cantly im-
prove their volatility forecasting abilities. This thesis is the ?rst work that
analytically obtains the links between conditional variances of multi-factor
GARCH models and VIX, and it further establishes the closed-form pricing
formula of VIX derivatives through the connections. In addition, we empir-
ically explore the role of a?ne structure in the VIX derivatives pricing and
comparethepricingperformanceofEGARCH,GJR-GARCH,andNGARCH
models.
This thesis is organized as follows. First of all, we describe in Chapter
1 the background, motivation, contents, and contribution of our studies, and
presentinchapter2thederivativespricingmodelsandrisk-neutralmeasures.
II

Abstract
In chapter 3, we apply the generalized a?ne realized volatility model to ob-
taintheVIXtermstructureexpressionandVIXfuturespricingformula. Our
empiricalstudiesinthischapter?ndthatincludingrealizedvolatilitycansub-
stantiallyimprovetheforecastingofVIXandthepricingofitsfutureswithin
these models. Chapter 4 nests multiple volatility components and jump in
the generalized a?ne realized volatility model and studies their contribution
in describing VIX and its futures data. To this end, we theoretically derive
the closed-form solution for VIX futures pricing and empirically show that
therealizedvolatility,volatilitycomponents,andjumpcanprovideaneviden-
t improvement in VIX forecasting and its futures pricing; more importantly,
these three models features are complements rather than substitutes. Chap-
ter5furtherinvestigatestheVIXderivativesvaluationperformanceo?eredby
thenona?nedynamics. We?ndstrongempiricalevidencethatthevaluation
superiority of the nona?ne speci?cation, especially the EGARCH model, is
signi?cant and robust. However, we also ?nd evidence in favor of the a?ne
model inpricingVIXoptionsofhighvalues. Finally,inchapter6,wepresent
abriefconclusionofthisthesisandadiscussionoffutureresearch.
Keywords: VIXDerivatives,GARCH,RealizedVolatility,VolatilityCompo-
nent,Jump
III
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