利用股價與選擇權的數據來估計GARCH選擇權定價模型

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(1)୯ҥᆵ᡼εᏢᆅ౛Ꮲଣ଄୍ߎᑼᏢ‫س‬ റγፕЎ Department of Finance College of Management. National Taiwan University Doctoral Dissertation. ճҔިሽᆶᒧ᏷៾‫ޑ‬ኧᏵٰ՗ी GARCH ᒧ᏷៾‫ۓ‬ሽኳࠠ Using Stock and Options Data to Estimate the GARCH Options Pricing Model. ᎄֻЎ Hung-Wen Cheng. ࡰᏤ௲௤Ǻഡ‫܍‬ቺ റγ Цᝬ፵ റγ Advisor: Cheng-Der Fuh, Ph.D. Yaw-Huei Wang, Ph.D.. ύ๮҇୯ 100 ԃ 6 Д June, 2011.

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(3) ᇞᖴ!. ҁጇፕЎૈ໩ճֹԋǴᕇள೚ӭΓ‫ڐޑ‬շၟЍ࡭ǴӧԜ᝘΢‫ך‬ന၈ኑ‫ޑ‬གᖴǶ! २ӃǴགᖴ‫ࡰޑך‬ᏤԴৣ!ഡ‫܍‬ቺ௲௤Ǵӧ‫ך‬᠐റγ੤೭ϖԃ๏ϒ‫ࡰޑ‬ᏤϷගឫ! Ǵ஥ሦ‫ך‬຾Ε଄୍πำሦୱǴவӵՖ‫פ‬ୢᚒǵ࣮ୢᚒϷགྷୢᚒ‫ډ‬ӵՖှ،ୢᚒǴᕴ ࢂૈҔభԶܰᔉ‫ޑ‬БԄᡣ‫ך‬ΑှǴ೭ჹ‫ך‬ӧࣴ‫ز‬΢ࢂ΋εշ੻ǶନᏢೌ΢‫ޑ‬௲ᏤǴ ഡԴৣΨӧғࢲ΢๏ϒ‫ࡐך‬ӭᝊ຦‫ޑ‬Γғ࿶ᡍǴ๱ჴ‫ؼ੻ڙ‬ӭǶ‫ځ‬ԛǴाགᖴ‫ך‬ӧ ύࣴଣ಍ी‫ޑ܌‬Դᗥ!ሌቼখ௲௤Ǵ஥ሦ‫ך‬຾Εीໆ࿶ᔮᏢሦୱǴሌԴৣჹ଺ࣴ‫ز‬ᝄ ᙣϷଓ‫ֹ؃‬ऍ‫ޑ‬ᄊࡋǴ٬‫ך‬ӧ଺ࣴ‫ز‬ၸำύૈ‫׳‬ಎჴЪ‫׳‬ᙣ཈‫ֹޑ‬ԋǶନࣴ‫ز‬ϐᎩ! ǴሌԴৣΨ๏Α‫ࡐך‬ӭғࢲ΢‫ޑ‬ᔅշǴ٬‫ڙך‬ඁሥӭǶӆ‫ޣ‬Ǵाགᖴ΢ੇҬεଯભ ߎᑼᏢଣ!஭પߞ௲௤ȐDibsmft!DibohȑǴ஥ሦ‫ך‬຾Ε଄୍ჴ᛾ሦୱǴӧᏢ཰΢๏ ϒ‫ࡐך‬ӭ଄୍‫ޕ‬᛽Ϸ୯ሞᢀǴӧࣴ‫ز‬΢ा‫ֹ؃‬ऍ‫܄ঁޑ‬Ǵ٬‫ך‬ᏢಞϷԋߏࡐӭǴԶ ғࢲ΢Ψ๏ϒ‫ࡐך‬ӭ‫ޑ‬ԵᡍᆶࡷᏯǴᡣ‫ך‬Ꮲಞ‫଺ډ‬ҺՖ٣೿ሡӄΚаॅаၲ‫ֹډ‬ऍ ნࣚࣁҞ኱Զᗌ຾Ǵ੿ࢂҭৣҭ϶‫ޑ‬ӳұՔǶ! ӧፕЎα၂ਔǴགᖴα၂‫ہ‬঩஭හ݅௲௤ǵЦᝬ፵௲௤ǵယλ⣬௲௤ϷᄃѤ॔ ௲௤๏ϒᝊ຦‫ࡌޑ‬᝼٬ளፕЎ‫׳‬уֹ๓ǶӧѠ᡼εᏢ଄ߎ‫܌‬൩᠐ය໔Ǵགᖴ‫س‬΢໳ ၲ཰௲௤ǵᆅύ໐௲௤ǵचࢃ໚௲௤Ϸഋ཰ჱ௲௤‫ࡰޑ‬ᏤϷᔅշǶ੝ձाགᖴ໳ ၲ཰௲௤ࠄΠଯ໢‫ךܭ‬ஆᘶ΢ठຒǴࣁஆᘶቚబΑ೚ӭ॥ߍǶΨाགᖴ‫ޑך‬ӝ๱‫ޣ‬ ߲లඁ௲௤Ϸ݅ࡌ‫ד‬௲௤Ǵӧ૸ፕࣴ‫ޑز‬ၸำύǴᐟวࡐӭО޸٬ள‫ॺך‬ፕЎૈ‫׳‬ ֹ᏾‫ޑ‬և౜ǶӆٰǴाགᖴ‫ৣޑך‬л݅γ຦௲௤ϷЦϘ‫ک‬௲௤Ǵϩ٦ࡐӭдॺ‫ޑ‬ཀ ii.

(4) ‫ـ‬ᆶ࿶ᡍǴᡣ‫ך‬Ͽ‫و‬΋٤ঌ݉ၡǶаϷቺঢ়‫ߪک‬मǴ΋ଆ૸ፕӝբፕЎǴӚБόӕ ‫ޑ‬ཀ‫ߦـ‬٬‫ॺך‬Ԗόᒱ‫ޑ‬ว౜ǶќѦǴགᖴ᜽։ǵ࢙‫کݒ‬Ҏ௴Ǵӧ‫ך‬ၶ‫ډ‬ೀ౛ၗ਑ ΢‫֚ޑ‬ᜤਔǴᔅշ‫ך‬ှ،‫ޑך‬ୢᚒǶགᖴጰဠ٣ߏ‫ד‬ᗶ‫ک‬ඁ๭Ǵӧ‫ך‬ԖྠඊϷ֚ᘋ ਔǴ‫פ‬գॺ೿ૈ๏ϒ‫࣬ך‬྽‫ޑ‬ᔅշǶགᖴඵ߿ǴӧፕЎ΢‫ޑ‬ᔅԆǶᗋԖǴचಛᏢ‫!ۆ‬ ǵߪ๮Ꮲߏǵ҅ࣤᏢߏǵ‫ػ‬৖Ꮲߏǵਦ෕ᏢߏǵᗶஜᏢߏǵࡏՉᏢߏǵᇖદᏢ‫ۆ‬ǵ ‫۝ے‬Ꮲߏǵ៼৓Ꮲߏǵ‫ދ‬ඁᏢ‫ۆ‬ǵVodmfξ݅ǵඳቺǵηᆗǴ଄റᝣౚ໗ԋ঩ॺ჏࠶! ǵ஖፣ǵֻ◖ǵ໚ྣǴӕৣߐ‫ދࡿޑ‬Ꮲ‫ۆ‬ǵԟ‫ޱ‬ǵ౰ᙦǵཁ໚ǵԮ൫ǵγᆜǴᚐ჏! ǵࡘᆝǵᆢণǵ‫ࡏד‬ǵ࢙ᆺǵϘᆢǵаၲаϷύࣴଣ಍ी‫ޑ܌‬ӕϘॺ฻Ǵᖴᖴգॺ ᡣ‫ޑך‬റγ੤ғఱкᅈӣᏫǶ! གᖴᒃང‫ޑ‬ԴஇȋૠඁǴჹ‫ޑך‬х৒ᆶбрǴᔅ‫៝ྣך‬ӳ‫ٽޑॺך‬ηӂ໋Ǵᡣ ‫ૈך‬คࡕ៝ϐኁ‫ޑ‬໩ճֹԋᏢ཰ǶΨाགᖴ‫ٽ‬ηӂ໋‫ޑ‬рғǴ٬‫׳ך‬ԖೢҺЪ‫׳‬Ԗ ୏Κ‫ޑ‬۳߻‫و‬ǶനࡕǴགᖴ‫ך‬Рǵ҆ǵۢРϷۢ҆‫ޑ‬Ѝ࡭ᆶႴᓰǴஒԜፕЎ᝘๏գ ॺǴаൔเჹ‫ޑך‬Ꭶ‫ػ‬ϐৱǶ! ! ᎄֻЎ!!!!!!!!!!ᙣᇞ‫!ܭ‬ Ѡ᡼εᏢ଄୍ߎᑼࣴ‫!܌ز‬ ҇୯΋ԭԃϤДΒΜΎВ. ! iii.

(5) ύЎᄔा!. ೭ঁࣴ‫ز‬௢ᏤΑ྽ѝҔި౻ኧᏵȐTUȑǵѝҔᒧ᏷៾ኧᏵȐPUȑϷҔި౻‫ک‬ᒧ᏷៾ ኧᏵԖ֖ȐT,P,Fȑ‫ؒ܈‬Ԗ֖ᇤৡ໨ȐT,Pȑਔ‫ ޑ‬HBSDI)2-2*!ᒧ᏷៾ሽ਱ኳࠠ՗ी ‫ޑ‬ᅌ߈੝‫܄‬Ƕӧεኬҁ౛ፕΠ‫ޑ‬ᅌ߈ᡂ౦ኧᇥܴΑѝҔᒧ᏷៾ኧᏵȐPUȑ཮Ꮴठወ ӧӦୃᇤ‫ک‬คਏ౗‫ޑ܄‬՗ीǴϸϐǴҔި౻‫ک‬ᒧ᏷៾ኧᏵԖ֖ᇤৡ໨ȐT,P,Fȑ཮ౢ ғεठ΢К‫ځ‬дҺՖ΋ᅿБ‫׳ݤ‬Ԗਏ‫ޑ܄‬คୃ՗ीǶ೭٤่݀೏Ԗज़ኬҁኳᔕ‫ࣴޑ‬ ‫ز‬᛾ჴΑǶӢԜǴϟ‫!ܭ‬T,P,F ‫ ک‬TU ‫ޑ‬՗ीᇤৡࢂჴ፦‫܄‬ӦᏤठᡉ๱Ӧόӕ॥ᓀᆅ ౛่݀Ƕ೭٤ᇤৡεεቹៜΑ‫܌‬௦ҔБ‫ޑݤ‬॥ᓀᆅ౛ࡰ኱)ӵᒧ᏷៾‫!ޑ‬efmubt ‫ک‬ hbnnbt ॶ*ଯၲ!91&Ƕҗ‫ܭ‬೭ HBSDI ᒧ᏷៾ኳࠠࢂ࣬ჹӦज़‫ڋ‬Ϸόૈਂਆჴ᛾౜Ⴝ ȐୖԵ Fohmf ‫ ک‬Nvtubgb!)2::3*ȑǴ‫ॺך‬Ї຾΋ঁᇤৡ໨‫ډ‬೭ᒧ᏷៾‫ۓ‬ሽኳࠠǴॷ ສ‫܌‬ሡ‫ޑ‬քᅉ‫ډ‬೭ঁ՗ीၸำǴҗԜౢғΑനεԖਏ౗‫ޑ܄‬คୃ՗ीǶΨ൩ࢂᇥǴ ኧᏵӭࢂ‫׳‬ӳ‫ޑ‬ǴՠࢂѝԖ྽ኧᏵࢂ҅ዴ‫ޑ‬೏ᔈҔਔǶ! ! ᜢᗖຒ;!HBSDI!ᒧ᏷៾ኳࠠǴᅌ߈ՉࣁǴ՗ी‫ޑ‬Ԗਏ౗‫܄‬ϷୃᇤǴ॥ᓀᆅ౛!. iv.

(6) Abstract. This study derives asymptotic characteristics of GARCH(1,1) options price model estimators when using stock data only (ST), using option data only (OT), and using stock and options data with (S+O+E) or without an error term (S+O). The asymptotic variance in large sample theory shows that the OT method results in potentially biased and inefficient estimators, whereas S+O+E generates unbiased estimators which are substantially more efficient than either ST (S+O) or OT. These results are confirmed by finite sample simulation studies. Hence, the difference in estimation between S+O+E and ST is substantial and results in significantly different risk management consequences. These errors substantially impact risk management metrics as options deltas and gammas vary by as much as 80%, depending on the method used. Since the GARCH option models are relative restrictive and cannot capture the empirical phenomena (cf. Engle and Mustafa (1992)), we introduce an error term to the options pricing model, lending needed slack to the estimation process and resulting in unbiased estimates that are maximally efficient. That is, more data is better, but only if the data set is appropriately applied.. Keywords: GARCH option model, asymptotic behavior, estimator efficiency and bias, risk management. v.

(7) Ҟ ᒵ α၂‫ہ‬঩཮ቩ‫ۓ‬ਜ. i. ᇞᖴ. ii. ύЎᄔा. iv. मЎᄔा. v. 1. Introduction. 1. 2. An Introduction of Pricing Error. 8. 3. Parameter Estimation under GARCH Option Pricing models 3.1 Model setup and OT specification. 10 10. 3.1.1 GARCH(1,1) stock and options pricing models. 10. 3.1.2 QMLEs for ST and asymptotic results. 13. 3.1.3 QMLEs for OT and asymptotic results. 14. 3.1.4 Numerical computation for asymptotic bias and mean square errors. 18. 3.2 The S+O+E specification. 20. 3.2.1 QMLEs and asymptotic results. 20. 3.2.2 Numerical computation for asymptotic mean square errors. 23. 3.2.3 Numerical findings and direct comparisons in finite sample studies. 25. 3.3 Risk management implications. 27. 3.4 Asymptotic behavior for ST, OT, and S+O+E. 29. 3.4.1 Asymptotic behavior for ST. 30. 3.4.2 Asymptotic behavior for OT. 32. 3.4.3 Asymptotic behavior for S+O+E. 43. vi.

(8) 4. Robustness Checking. 48. 4.1 Model setup and ST specification. 48. 4.1.1 GARCH(1,1) stock and options pricing models. 48. 4.1.2 QMLEs and asymptotic results. 50. 4.2 The S+O+E specification. 51. 4.2.1 QMLEs and asymptotic results. 52. 4.2.2 Numerical computation for asymptotic mean square errors. 54. 4.2.3 Numerical findings and direct comparisons in finite sample studies. 56. 4.3 Risk management implications. 57. 5. Conclusion. 59. Appendix A. 61. Appendix B. 73. B.1 Asymptotic behavior for ST. 73. B.2 Asymptotic behavior for S+O+E. 79. References. 86. vii.

(9) List of Figures. Figure 1. |BiasOT| Graphs. 89. Figure 2. dMSEOT Graphs. 91. Figure 3. MS+O+E Graphs with Different ρ. 93. Figure 4. MS+O+E Graphs with ρ = 0. 94. Figure 5. MS+O+E Graphs with Different ρ. 96. Figure 6. MS+O+E Graphs with ρ = 0. 97. viii.

(10) List of Tables. Table 1. Simulated Parameter Estimates and Errors. 98. Table 2. Risk Management Metrics under Different Data Specifications. 99. Table 3. Simulated Parameter Estimates and Errors. 100. Table 4. Risk Management Metrics under Different Data Specifications. 101. ix.

(11) Chapter 1 Introduction A number of papers have sought to develop methods of appropriately and empirically accurately measuring stock price volatility, a factor that is critical in numerous fields of study including options pricing and risk management. It is a value that is necessary for calculation of hedge ratios and key risk metrics such as the options delta and gamma and, as a result, lies at the very core of traditional risk measurement, options pricing, and hedging strategies. Early work in this field focuses on application of stock price data (henceforth the ST data specification) while more recent efforts have sought to apply both stock and options data to the estimation problem under the assumption that the more data is applied, the more accurate are the resulting estimates. Empiricists have sought to do so in both the GARCH and stochastic volatility (SV) settings, generally applying these data without allowing for error in the options pricing model (henceforth the S+O method). Ultimately, GARCH models have been shown to provide better empirical fit and characteristics, making it an important class of models to consider. However, when it comes to estimating these models, application of the dual dataset becomes difficult owing to the restrictive nature of the specification, leading some to believe that the flexibility of SV models makes it a more desirable environment for estimation. Since there are 1.

(12) consequences of the likelihood principle and Engle and Mustafa (1992), S+O method is found to be similar to ST method under GARCH models. That is, more data is not always better, and traditional methods leave estimation of this important class of models severely handicapped. To resolve this issue, we develop additional two data specifications, a third data specification that applies options data only (OT) and a fourth data specification that applies stock and options data but includes an error term (S+O+E) such that options data need not match precisely the options pricing model. 1 We derive a quasi-maximum likelihood estimator (QMLE) for volatility under the GARCH(1,1) specification, both through analytical derivation of asymptotic behavior and numerical simulation, that the OT method generates inefficient, and more importantly, estimate bias that is economically and statistically significant. After relaxing this important modeling constraint, S+O+E generates asymptotically unbiased estimates that are the most efficient of the four data specifications. Most importantly, we then apply S&P500 stock and options daily data from January 2007 to the end of 2007 to generate commonly used risk and hedging metrics, i.e. the options delta and gamma. We find that those calculated using the S+O+E method are substantially different from those arrived at using stock data alone, indicating that the impact of including options data is economically significant and should be taken into account when determining hedging strategies. Eraker (2004) offers several advantages using both stock and options data under SV model: A primary advantage is that risk premiums relating to volatility and jumps can 1. More details about price errors see Engle and Mustafa (1992), Jacquier and Jarrow (2000), Eraker (2004) and Johannes, Polson, and Stroud (2009) and others.. 2.

(13) be estimated. Secondly, the one-to-one correspondence of options to the conditional returns distribution allows parameters governing the shape of this distribution to potentially be very accurately estimated from option prices. For example, Eraker, Johannes, and Polson (2003) suggest that estimation from stock data alone requires fairly long samples to properly identify all parameters. Hopefully, the use of option prices can lead to very accurate estimates, even in short samples. Moreover, the use of option prices allows, and in fact requires, the estimation of the latent stochastic volatility process. Since volatility determines the time variation in relative option prices, there is also a strong potential for increased accuracy in the estimated volatility process. Finally, joint estimation also raises an interesting and important question: Are estimates of model parameters and volatility consistent across both markets? How to get the same results under GARCH models? This is the essential question to be addressed in this paper. Other papers apply both stock and options data to estimate volatility, such as Chernov and Ghysels (2000), Jacquier and Jarrow (2000), Pan (2002), Jones (2003), Aїt-Sahalia and Kimmel (2007), and Johannes, Polson, and Stroud (2009), each addressing the estimation issue under different assumptions. However, all of these papers do so under an SV or Black-Scholes (BS) rather than GARCH specification. Chernov and Ghysels (2000), Pan (2002), Jones (2003), and Aїt-Sahalia and Kimmel (2007) apply an S+O model whereas Jacquier and Jarrow (2000), Eraker (2004) and Johannes, Polson, and Stroud (2009) apply an S+O+E model. In addition, Chernov and Ghysels (2000) and Pan (2002) use generalized method of moments (GMM) estimators, Jacquier and Jarrow (2000), Jones (2003), Eraker (2004), and Johannes, Polson, and Stroud (2009) use Bayesian inference estimators, whereas Aїt-Sahalia and Kimmel (2007) uses maximum likelihood estimators 3.

(14) (MLE). Moreover, each of these papers focuses primarily on the computational aspects of their model, leaving open the important issues of the statistical properties and empirical implications of the models. Our paper explores these issues explicitly. Importantly, Lehar, Scheicher, and Schittenkopf (2002) analyzes GARCH v/s SV models and finds that GARCH models dominate in terms of fit to observed prices. Given this attractive property, it is fruitful to develop a GARCH model that applies both data sets in an efficient, unbiased way: the task we explore here. Unfortunately, GARCH models that apply both stock and options data are scarce. Engle and Mustafa (1992) is among the earliest of these efforts. We follow this early work in that we also consider the role of error in the options pricing model but differ in that their work focuses on the application nonlinear least square (NLS) estimators for minimum of loss function toward the estimation of implied volatility, that is, this does so under an S+O specification. They find that the persistence of volatility shocks implied by options is found to be similar to that estimated from historical data on the index itself, that is, S+O method is similar to ST method. Christoffersen and Jacobs (2004) also uses NLS estimators and the S+O specification with both stock and options data under GARCH model. The focus of that study is on the accuracy of options pricing models and their ability to describe observed options prices. It does not address the estimation quality of the model nor does it seek to differentiate its data specifications from others. In contrast, our study is the first to derive the asymptotic characteristics of estimators then test these results using empirical data under the different data specifications. Our theoretical construct builds upon the GARCH(1,1) setups of Heston and Nandi (2000), which propose a class of GARCH models that allow for a closed-form solution for 4.

(15) the price of a European call option. We apply this model to the application of different data inclusion specifications and address asymptotic behavior using QMLE methods (cf. Lee and Hansen (1994) and Lumsdaine (1996)). For the S+O specification, we revise Aїt-Sahalia and Kimmel’s (2007) log-likelihood function to follow a GARCH(1,1), which has only one random source, and derive the asymptotic behavior of the QMLE. From this is a consequence of the likelihood principle, we find that the asymptotic behavior of the QMLE for S+O method is equal to that for ST method under GARCH models, that is, S+O method is similar to ST method. Thus, we don’t display S+O method in this paper. Specifically, for the OT specification, we apply the log-likelihood function of Duffie, Pedersen, and Singleton (2003) to derive the asymptotic behavior of the QMLE. Our theoretical findings show that the OT method generates biased estimates and further partly results in higher estimation variance and mean squared error than applying stock data alone, the ST specification. Applying data and Monte Carlo simulations, we confirm these findings for all four variables that we seek to estimate. These findings alone are perhaps not surprising. The GARCH specification is a restrictive one under which the application of the dual dataset can tend to obtain helpless the model. In contrast, the SV class of models introduces an error term into the volatility measure, thereby providing considerable slack in the model and allowing for the application of a more comprehensive dataset. The intuition behind our S+O+E method is the same. By allowing for additional slack, this time in the options pricing formula itself, we hope to provide the slack necessary for the dual dataset to generate unbiased, maximally efficient estimates for the GARCH class of models. Specifically, the S+O+E specification assumes that Ct CtHN et . The error term et is 5.

(16) assumed to be distributed N (0, 2 ) and is correlated with the error term of stock return.2 Put plainly, we allow for options price data to err from the theoretical options price. Under this specification, we find that, for all four variables that we seek to estimate, the asymptotic mean squared error is lower than that of ST specification. That is, inclusion of an error term guarantees that the specification will dominate stock data only in large sample theory, where as applying stock and options data without an error term does not. Importantly, S+O+E also generates asymptotically unbiased estimates. These results are confirmed by our simulations in finite sample studies. Indeed, the OT method generates estimate bias, standard deviation, and mean squared error that are several times higher than those of S+O+E. ST, while generally not substantially biased, also generates estimates with standard deviations and mean squared errors that are several times higher than that of S+O+E. We conclude that the use of option prices can lead to very accurate estimates not only in short samples but also in long samples. To test the implications of these differences in estimate quality, particularly in the risk management setting, we apply 12 months of stock and options data and show empirically that ST and S+O+E generate substantially different critical risk metrics. We calculate options delta and gamma using both the Black-Scholes and GARCH options pricing models. We find that estimates vary considerably depending on the data specification used. Delta estimates differ by as much as 80% which gamma estimates may differ by more than 60%. Although these differences are not systematically related to the. 2. In the information point of view, when good or bad news occur in financial market, these maybe affect the stock price and option price simultaneously. Then, these lead to the emergence of the correlation between stock error and price error.. 6.

(17) moneyness of the option, they are nonetheless considerable. We conclude that the unbiased, more efficient estimates derived from our S+O+E method have a concrete and economically important impact, a notion managers would do well to keep in mind as they implement risk management practices. Finally, we apply the Duan’s (1995) options pricing formula for a robustness check. The results again show in this study, both through analytical derivation of asymptotic behavior and numerical simulation, that S+O+E method generates more efficient and unbiased estimates. These errors substantially impact risk management metrics as options deltas and gammas vary by as much as 90%. These results are consistent with our general findings. The remainder of this paper is organized as follows: Chapter 2 introduces price error. Chapter 3 presents the main results. Chapter 4 proposes a robustness check. Chapter 5 concludes. All proofs of the results are relegated to Appendix.. 7.

(18) Chapter 2 An Introduction of Pricing Error Consider T observations of a contingent claim's market price, Ct , for t 1,, T .We think of Ct as a limited liability derivative, like a call or put option. Formally, we can assume that there exists an unobservable equilibrium or arbitrage free price ct for each observation. Then observed price Ct should be equal to the theoretical price ct . There is a basic model f ( X t , ) for the equilibrium price ct . The model depends on vectors of observables X t and parameters . We assume that the parameters are constant over the sample span. The model is an approximation, even though it was theoretically derived as being exact. There is an unobservable pricing error, et . A quote Ct may also sometimes depart from equilibrium. The error then has a second component t , which can be thought of as a market error. t and et are not identified without further assumptions. In this paper, we merge these two errors into one common pricing error et . Formally,. Ct f ( X t , ) et .. (2.1). This implies a multiplicative error structure on the level, which guarantees the positivity of the call price for any error distribution. 8.

(19) The introduction of a non-zero error et is justified. First, simplifying assumptions on the structure of trading or the underlying stochastic process made to derive tractable models. They result in errors, possibly biased and non i.i.d. For example, Renault and Touzi (1996) and Heston (1993), show this within the context of stochastic volatility option pricing models. Renault (1997) shows that even a small non-synchroneity error in the recording of underlying and option prices can measurement can cause skewed Black-Scholes implied volatility smiles. Bakshi, Cao, and Chen (1997) show that adding jumps to a basic stochastic volatility process further improves pricing performance. Bossaerts and Hillion (1997) show that the assumption of continuous trading also leads to smiles while Platen and Schweizer (1994)'s hedging model causes time varying skewed smiles in the Black-Scholes model. In all of the above cases, the model errors are related to the inputs of the model. Second, in typical models, the rational agents are unaware of market or model error and know the parameters of the model. Such models could be biased in the ‘larger system’ consisting of expression (2.1).. 9.

(20) Chapter 3 Parameter. Estimation. under. GARCH Option Price models 3.1. Model setup and OT specification First, we describe the general stock and option pricing models applied in this paper. Then, we derive QMLE and asymptotic results for the ST and OT specifications, noting the bias and estimator inefficiencies of the OT method. Numerical results confirm these characteristics.. 3.1.1. GARCH(1,1) stock and option pricing models We adopt the generalized setup used by Heston and Nandi (2000), which propose a class of GARCH models that allow for a closed-form solution for the price of a European call option, where the data-generating process for the stock price S is:. yt ln St ln St 1 r ht ht1/2 zt , under P measure,. (3.1). ht

(21) zt 1 ht1/2 1 ht 1 ,. (3.2). 2. 10.

(22) where r is the risk free rate and is the price of risk. The variance equation (3.2) is in fact a nonlinear asymmetric (NAGARCH) configuration (cf. Engle and Ng (1993)). The process remains stationary with finite mean and variance if 2 1. We may consider process (3.2) as running indefinitely or we may assume initial values y0 and h0 , with the latter drawn from the stationary distribution applied by Bollerslev (1986), Nelson (1990), Bougerol and Picard (1992), and others. Let t be the sigma-field generated by. yt , yt 1, and let 0 (0 0 0 , 0 ) represent the true parameter vector. Assume that 0 4 is in the interior of , a compact, convex parameter space. Specifically, for any vector ( , ) , 0 L U , 0<L U , 0<L U ,. 1 L U , and U ( U )2 U 1 .3 Assume also that zt t is i.i.d., drawn from 2 a symmetric, uni-modal density, bounded in a neighborhood of 0, with mean 0, and variance 1. In addition, assume that ht is independent of zt , zt 1, . The corresponding model under local risk neutralization reads. 1 yt ln St ln St 1 r ht ht1/2 ztQ , under Q measure 2. (3.3). ht

(23) ztQ 1 Q ht1/2 1 ht 1 ,. (3.4). 2. where Q . 1 1 and ztQ zt ht1/2 . Then, the GARCH option pricing 2 2 . formula is described as:. Since U ( U 1/ 2)2 U 1 implies Q2 1 , the process (3.4) under Q measure also remains stationary with finite mean and variance. These conditions easy to be arrived from our estimative parameters. 3. 11.

(24) CtHN (T , K , St , ht 1; ) e r (T t ) EtQ $&max

(25) ST K ,0 %' 1 1 # $ K i! f * (i! 1) % e r (T t ) f * (1) * Re ( )d! * 0 2 " i ! f (1) & ' 1 1 # $ K i! f * (i! ) % e r (T t ) K * Re ( )d! , 0 i! & ' 2 " +. +. f * (+) EtQ $&e+xT %' St+ eAt Bt ht1 ( ). where. ln(1 2 Bt+1 ) , At At 1 +r Bt 1 2 +. +. +. xt ln(St ). ,. ,. (3.5). AT+ BT+ 0. 2 (+ Q )2 1 Q + Bt +( Q ) Bt 1 2 2 2(1 2 Bt+1 ). +. ,. ,. Re( x) is the real part of x , T is the maturity date, and K is exercise price. From the derivative of real part doesn’t exist, we consider a complex number + m !i , m 0,1 to rewrite (3.5) such that the partial differentiation of CtHN ( ). At+ , A1m,t (!, ) A2m,t (!, )i. and. A1m,t (!, ) A1m,t 1(!, ) mr B1m,t 1(!, ) . Bt+ , B1m,t (!, ) B2m,t (!, )i. ln

(26) [1 2 B1m,t 1(!, )]2 4 2 B22m,t 1(!, ) 4. exists. Then, ,. where. ,. 2 B2m,t 1 (! , ) 1 A2m,t (!, ) A2m,t 1 (!, ) !r B2m,t 1 (! , ) tan 1 , 2 1 2 B1m,t 1 (! , ). (1 2 B1m,t 1(! , ))[(m Q )2 ! 2 ] 4 B2m,t 1(! , )(m Q )! 1 1 B1m,t (! , ) m( Q ) Q2 B1m,t 1(! , ) 2 2 2[(1 2 B1m,t 1(! , ))2 4 2 B22m,t 1(! , )]. and. B (!, )[(m Q )2 ! 2 ] (m Q )! (1 2 B1m,t 1(!, )) 1 B2m,t (!, ) ! ( Q ) B2m,t 1(!, ) 2m,t 1 2 (1 2 B1m,t 1 (! , ))2 4 2 B22m,t 1(!, ). we rewrite (3.5) as. on S&P 500 index data. Hence, the parameter space is enough large.. 12. ,. . Hence,.

(27) $ 1 1 # e X11,t (! , ) sin

(28) X 21,t (! , ) % C ( ) St ( * d! ) ! (& 2 " 0 )' $ 1 1 # e X10,t (! , ) sin

(29) X 20,t (! , ) % r (T t ) e K( * d! ) , ! (& 2 " 0 )' HN t. (3.6). X1m,t (!, ) r(T t )m A1m,t (!, ) B1m,t (!, )ht 1 ( ). where. and. X 2m,t (!, ) ! ln(St / K ) A2m,t (!, ) B2m,t (!, )ht 1 ( ) , m 0,1.. 3.1.2. QMLEs for ST and asymptotic results We now turn our attention to estimating the parameters in the model. The base case ST uses only stock data. Specifically, ht is the conditional variance of yt with respect to. t 1 . The estimation model utilizes (3.1) and (3.2), applying estimated parameter values ( , )

(30) 1,2 ,3 ,4 . The error terms zt are computed as z0 . z1 . y0 r h0 , h01/2. y1 r h1 , , where yt , t 0,, T are observed data. The process ht is not h11/2. observed but is constructed recursively using estimated parameter values, z0 , and an appropriate startup value, h0 , to be discussed in detail later. QMLE is obtained by maximizing, conditional on h0 , as follows: T 2 yt r ht ( )

(31) 1 L ( y0 ,, yT , h0 ; ) L ( ) . - ln

(32) ht ( ) 2T t 1 ht ( ) ST T. ST T. p. (3.7). That is, ˆTST arg max LTST ( ) . This estimator is consistent as ˆTST .0 and is . asymptotically Normal as 13.

(33) A. 1/2 1/2 ˆST HST (T 0 )~ N (0, I4 ), 0 FST 0T. (3.8). 1 0 / L (0 ) /L (0 ) /L (0 ) I H E T , , and where FST 0 E ST 0 4 0 ' ' / / / / 0 2 ST T. ST T. ST T. 0 1 0 0. 0 0 1 0. 0 0 . 0 1. A full proof appears in Section 3.5 as Theorem 3.1. In the interest of computational simplicity, assume that zt is Normal so that. FST 0 HST 0 , though our general intuition remains the same under the more relaxed aforementioned specification for zt . The asymptotic covariance matrix VST and asymptotic mean square errors MSEST are:. MSEST (0 ) VST (0 ) FST 10 HST 0 FST 10 . 1 , FST 0. (3.9). $. 1. 2 /ht (0 ) /ht (0 ) % ). 2 / ' & 2ht (0 ) ht (0 ) /. where FST 0 HST 0 E (. 3.1.3. QMLEs for OT and asymptotic results For OT, we simply have Ct CtHN , t 1 T , since the pricing formula is assumed to match the observed data exactly. Duffie, Pedersen, and Singleton (2003) provide a treatment for the log-likelihood function when only options data is applied in this fashion. Let St be an unobservable stock price. Expressing the stock and options price vector as a function of the state variable vector, we have: Ct f (St 0 ) for a differentiable function f that is easily computed. At a given parameter vector , we may now express the 14.

(34) state variable as a function of observed asset prices as follows: St ( ) f 1 (Ct 0 ) assuming invertibility (which is not an issue in our application). Letting C (C1,, CT ) denote the sequence of observed vector of reference option prices, standard change-of-variable arguments lead to the likelihood T. P(C0 ) 2 P

(35) St ( ) 1 St 1 ( )0 t 1. 1 , det Df

(36) St ( )0 . (3.10). where. det Df

(37) St ( )0 /Ct /S(tS(t() )0 ) HN. 1 1 #e * 2 " 0 e r (T t ). X11,t (! , ). sin

(38) X 31,t (! , ) 1 # d! * e X11,t (! , ) cos

(39) X 31,t (! , ) d!. !. ". 0. K 1 # X10,t (! , ) cos

(40) X 30,t (! , ) d! e St ( ) " *0. and. X 3m,t (!, ) ! ln

(41) St ( ) / K A2m,t (!, ) B2m,t (!, )ht 1 ( ) , m 0,1. Then, the log-likelihood function for discrete data of the asset price vector C sampled at dates 0 t T has the form. LOT T ( ) . Yt 2 ( ) % 1 T $ 1 (ln

(42) Jt ( ) ln

(43) ht ( ) ) . T t 1 & ht ( ) ' 2. (3.11). where Jt ( ) det Df (St ( )0 ) and Yt ( ) ln St ( ) ln St 1 ( ) r ht ( ) . And, the QMLE for ˆTOT arg max LOT T ( ) . Note, then, that this estimator is asymptotically biased . 1. p /2 LOT ( ) 1 T 1 /Jt (0 ) . Investigating the bias since: ˆTOT 1 .0 , where 1 , 0 T 0 T t 1 Jt (0 ) / // '. 15.

(44) 1. /2 LOT 1 T 1 /Jt (0 ) T (0 ) in particular, we have that BiasOT (0 ) . Full proofs can T t 1 Jt (0 ) / // ' be found in Section 3.5 as Theorem 3.2. Theorem 3.2 also shows that the estimator is asymptotically Normally distributed A. as. 1/2 1/2 ˆOT HOT (T 1 )~ N (0, I4 ), 0 FOT 0T. where. /2 LOT ( ) FOT 0 E T 0 // '. and. /LOT ( ) HOT 0 Var T 1/2 T 0 . Again, assume that zt is Normal. The asymptotic covariance / . matrix VOT and asymptotic mean square errors MSEOT for the OT case are:. 1 1 FOT 0 HOT 0 FOT 0. (3.12). 2 MSEOT (0 ) VOT (0 ) BiasOT (0 ) ,. (3.13). VOT (0 ) and. where. $ 1 /ht (0 ) /ht (0 ) % $ 1 /Yt (0 ) /Yt (0 ) % E( HOT 0 (0 ) E ( 2 ) / ' ' / ' )' & 2ht (0 ) / & ht (0 ) /. $ 1 /2 Jt (0 ) % $ 1 /Jt (0 ) /Jt (0 ) % E( 2 HOT 0 (0 ) FOT 0 (0 ) E ( ) / ' )' & Jt (0 ) // ' ' & Jt (0 ) /. .. Then,. and. the. difference of asymptotic mean square errors between OT and ST dMSEOT is:. dMSEOT (0 ) MSEOT (0 ) MSEST (0 ) .. (3.14). Using these results, which follow from Lemmas 3.1, 3.2, 3.5, and 3.6 in Section 3.5, we can compare the magnitude of mean square errors in large sample theory, a lower asymptotic mean square errors indicating better estimation. Namely, if dMSEOT (0 ) 0 , then. MSEOT MSEST and using options data only specification is more efficient than using 16.

(45) stock only. Since the covariance matrix V is a 43 4 matrix, we estimate each of the four parameters separately, holding the other three constants. When 0 , 0 , and 0 are known and is unknown, we have the asymptotic bias 1. /2 LOT 1 T 1 /Jt (0 ) T (0 ) . Bias, then, is non-zero in is BiasOT (0 | 0 , 0 , 0 ) 2 T t 1 Jt (0 ) / / magnitude. Investigating estimate mean square errors, we find that:. dMSEOT (0 | 0 , 0 , 0 ) MSEOT (0 ) MSEST (0 ) .. (3.15). Note that dMSEOT may be positive or negative, where a positive results means that results are less efficient than using stock data alone. Since the dMSEOT depends on true parameters, we don’t compare these values. Thus, we will calculate these values by numerical simulation in Section 3.1.4. As illustrated later, dMSEOT is in fact sometimes positive. Similarly, when 0 , 0 , and 0 are known and is unknown, we have 1. /2 LOT ( ) 1 T 1 /Jt (0 ) which is again non-zero in BiasOT (0 | 0 , 0 , 0 ) T 2 0 T t 1 Jt (0 ) / / magnitude and. dMSEOT (0 | 0 , 0 , 0 ) MSEOT (0 ) MSEST (0 ) .. (3.16). As demonstrated later, dMSEOT is sometime positive definite and the estimator is sometimes less efficient than that which is found using the ST method. Similarly, when 0 , 0 , and 0 are known and is unknown, we have 1. /2 LOT ( ) 1 T 1 /Jt (0 ) which is again non-zero in BiasOT (0 | 0 ,0 , 0 ) T 2 0 T t 1 Jt (0 ) / / magnitude and 17.

(46) dMSEOT (0 | 0 ,0 , 0 ) MSEOT (0 ) MSEST (0 ) .. (3.17). As demonstrated later, we find that dMSEOT is sometimes positive definite. Finally, when 0 , 0 , and 0 are known and is unknown, we find similar to the 1. /2 LOT 1 T 1 /Jt ( 0 ) T ( 0 ) , a non-zero previous case that BiasOT ( 0 | 0 ,0 , 0 ) 2 T t 1 Jt (0 ) / / entity and. dMSEOT ( 0 | 0 ,0 ,0 ) MSEOT ( 0 ) MSEST ( 0 ) .. (3.18). Again, as for the case where is unknown, we show that dMSEOT is sometimes positive.. 3.1.4. Numerical computation for asymptotic bias and mean square errors We now generate numerical results to test and illustrate these asymptotic findings. We presume that parameter true values are (λ, ω0, α0, β0, γ0) = (0.1746, 6.792×10-9, 6.546×10-8, 0.9914, 351.945), and the risk-free rate is fixed at 5%. These parameters are estimated using S&P 500 daily index data from January 1996 to the end of 2007. We use these parameters to run our tests. First, we investigate and calculate analytically the aforementioned estimate bias. Graphs in Figure 1 show the absolute value of bias divided by true value in the area surrounding true parameter values. Since the orders of magnitude for the four parameters are quite different, we graph the bias of ω on the left, that of α, that of β, and that of γ on the right. True parameters are circled in each graph. Then, in each panel, one variable is varied while the other three are treated as known. Specifically, in Panel A, ω is varied, in B α, in C β, and in D γ. 18.

(47) [Insert Figure 1 here] Note that, in all graphs, bias is decidedly non-zero and non-trivial for all four parameters. Though not shown here, the absolute value of bias is positive for all four parameters for the entire span of possible parameter values.4 In Panel A, the bias for ω is sometimes large and sometimes small with ω locally in the region around the true parameter values but that for ω in the true parameter values is non-zero in magnitude. There are the same results for those for α, β, and γ. The bias for ω is always higher than that of others. The order of these values is the bias for ω, that for α, that for γ, and that of β. Corresponding graphs Panels B, C, and D are similar to each other in shape, though their x-axes differ. All in all, using OT, bias is non-zero for each variable estimated, regardless of the true parameter values implemented. In contrast, neither ST nor S+O+E generate asymptotic bias in any variable. Shifting our attention to the efficiency of the estimator, graphs of dMSEOT are shown in Figure 2. Remember that, the more positive this value, the more efficient the estimator. [Insert Figure 2 here] Once again, in Panel A, ω is varied, in B α , in C β, and in D γ. Looking at Panel A, dMSEOT for ω is almost negative and sometimes positive for all four parameters in the area surrounding true parameters but that for ω in the true parameter values is always negative. There are the same results for those for α, β, and γ. dMSEOT for α is always lower than that of others. The order of these values is dMSEOT for γ, that for β, that for ω, and that of α.. 4. Each the parameter for ω (α, β, γ) ranges from 0 to 1 such that ( 0.5)2 1 .. 19.

(48) Graphs in Panels B, C, and D are again similar in shape. In three panels, dMSEOT is sometimes positive for all four parameters. In summary, dMSEOT is sometimes positive for all parameters in the area surrounding true parameters. As such OT does not generally produce efficient estimators. As aforementioned, it furthermore generates significant bias. We conclude that OT is not an optimal estimation method given the restrictive nature of GARCH models. As a result, we seek to develop a method that will allow for asymptotic unbias and efficient estimation of this important class of models.. 3.2. The S+O+E specification We now turn our attention to a new specification that takes both stock and options data into account, but which allows for an error term in the options pricing formula. Then, we derive QMLE and asymptotic results for the S+O+E specification. Numerical results confirm these characteristics.. 3.2.1. QMLEs and asymptotic results For this method, we allow that Ct CtHN ( ) et where et ut and t 1,, T . i.i.d .. Assume that ut ~ N (0,1) and 4 0 . For the purpose of calculating the QMLE, let us assume zt and ut , with correlation( zt ut ) 5 where 1 5 1 , have a bi-Normally. 20.

(49) distribution, that is,. z 0 1 5 5 ~ N , . Let Gt [St Ct ] be a vector of u 5 1 0. t t. observable stock and option prices, respectively. Then, the joint density is as follows: T. P(G; ) P(S , C; ) P(C | S; ) P(S; ) 2 P(Ct | St ; )P(St | St 1; ) t 1. T. 2 t 1. 1 2" 1 5 2. 2 HN $ y r h ( ) 2

(50) Ct CtHN ( ) % . (3.19) exp ( 2(1 15 2 )

(51) t ht ( t) 25 yt rh (ht)( ) Ct Ct ( ) ) 2 ht ( ) t & '. The log-likelihood function for discrete data on the asset price vector Gt sampled at dates. 0 t T has the form:. LTS OE ( ) ln

(52) P(G0 ) T.

(53). 2 HN $ 1 y r h ( ) 2

(54) Ct CtHN ( ) % . - (2ln 2" 1 5 2 ln(ht ( )) (1 15 2 )

(55) t ht ( t) 25 yt rh (ht)( ) Ct Ct ( ) 2 ) t 2T t 1 & '. (3.20) And, the QMLE for ˆTS O E arg max LTS O E ( ) . . p. Unlike the OT case, this estimator is consistent as ˆTS OE .0 and is A. asymptotically Normally distributed as HS 1/2OE0 FS OE0T1/2 (ˆTS OE 0 )~ N (0, I4 ) , where. /2 LS O E (0 ) /LTS OE (0 ) /LTS OE (0 ) FS O E 0 E T H E and T . A full proof S O E 0 / / ' // ' appears as Theorem 3.3 in Section 3.5. Again, assume that zt is Normal so that. 5. This simple idea likes Eraker (2004) to joint stock error and price error but differ in that he assumes that the relation between stock error and price error is zero and there is the relation between price error at time t and price error at time t-1.. 21.

(56) FS OE 0 HS OE 0 . The asymptotic covariance matrix VS O E and asymptotic mean square error MSES O E for the S+O+E case are:. MSES O E (0 ) VS O E (0 ) FS 1O E 0 HS O E 0 FS 1O E 0 . 1 FS O E 0. . , 1 FST 0 M S O E (0 ). (3.21). where M S OE (0 ) FS OE 0 FST 0 and MS O E (0 ) . $ 1 /CtHN (0 ) /ht (0 ) % $ /C HN (0 ) /CtHN (0 ) % 5 2 $ 1. 2 /ht (0 ) /ht (0 ) % 2 5 1 2 E ( 2 E ( 1/2 E( t . ) ) 2 2 2 / ' )' (1 5 ) & ht (0 ) / / ' (1 5 ) & / / ' )' 1 5 (& 4ht (0 ) ht (0 ) /. These results follow from Lemmas 3.1, 3.2, 3.13, and 3.14 in Section 3.5. As before, we can compare the magnitude of asymptotic mean square errors, again a lower asymptotic mean square errors indicating better estimation. Here, if M S O E 4 0 , then. MSES OE MSEST . Namely, we now investigate M S O E where the more positive, the more efficient the estimator. When 0 , 0 , and 0 are known and is unknown: M S O E (0 | 0 , 0 , 0 ) . 2 $ 1 52. 2 /ht (0 ) % E ( ) 1 5 2 &( 4ht2 (0 ) ht (0 ) / '). (3.22). $ /C HN ( ) 2 % $ 1 /CtHN (0 ) /ht (0 ) % 2 5 1 t 0 ( E E. ). ( ) (1 5 2 ) & ht1/2 (0 ) / / ' 2 (1 5 2 ) ( / )' &. Note that MS+O+E may be positive or negative, where a positive result means that results are more efficient than using stock data alone. In ρ = 0 case, we easy to see that MS+O+E is positive definite from (3.22), indicating that S+O+E generates more efficient estimates than ST. This method makes it the most desirable data specification of the two. In ρ ≠ 0 case, we don’t compare these values since the MS+O+E depend on true parameters. Thus, we. 22.

(57) will calculate these values by numerical simulation in Section 3.2.2. As illustrated later, MS+O+E is in fact generally positive. When 0 , 0 , and 0 are known and is not: 2 $ 1 52. 2 /ht (0 ) % E ( 2 ) 2 1 5 &( 4ht (0 ) ht (0 ) / '). M S O E (0 | 0 , 0 , 0 ) . (3.23). $ /C HN ( ) 2 % $ 1 /CtHN (0 ) /ht (0 ) % 2 5 1 0 ( t. E E ). ) / ' 2 (1 5 2 ) ( / (1 5 2 ) &( ht1/2 (0 ) / )' &. In ρ = 0 case, MS+O+E is positive definite, and in ρ ≠ 0 case, as demonstrated later, MS+O+E is positive definite. Similarly, when 0 , 0 , and 0 are known and is not: M S O E (0 | 0 ,0 , 0 ) . 2 $ 1 52. 2 /ht (0 ) % E ( ) 1 5 2 &( 4ht2 (0 ) ht (0 ) / '). (3.24). $ /C HN ( ) 2 % $ 1 /CtHN (0 ) /ht (0 ) % 2 5 1 0 t (. E E ). ) / ' 2 (1 5 2 ) ( / (1 5 2 ) (& ht1/2 (0 ) / & '). Again, as for the case where is unknown, we show that MS+O+E is always positive. Finally, when 0 , 0 , and 0 are known and is not: 2 $ 1 52. 2 /ht ( 0 ) % M S O E ( 0 | 0 , 0 ,0 ) E ( ) 1 5 2 &( 4ht2 (0 ) ht (0 ) / '). (3.25). $ /C HN ( ) 2 % $ 1 /C ( 0 ) /ht ( 0 ) % 2 5 1 0 ( t. E E ). ) / ' 2 (1 5 2 ) ( / (1 5 2 ) (& ht1/2 (0 ) / )' & HN t. Again, as for the case where is unknown, we show that MS+O+E is always positive.. 3.2.2. Numerical computation for asymptotic mean square errors We now generate numerical results to test and illustrate these asymptotic findings. As before, we use these parameters (λ, ω0, α0, β0, γ0) = (0.1746, 6.792×10-9, 6.546×10-8, 0.9914, 351.945) and the risk-free rate is fixed at 5% to run our tests.. 23.

(58) First, we investigate and calculate analytically the efficiency of the estimator, graphs of MS+O+E, in varied ρ. Remember that, the more positive this value, the more efficient the estimator. Graphs in Figure 3 show that the value of MS+O+E in the true parameter values and ρ from -0.9 to 0.9. Then, in each panel, one variable is unknown while the other three are treated as known. Specifically, in Panel A, ω is varied, in B α, in C β, and in D γ. [Insert Figure 3 here] Looking at all graphs, MS+O+E is decidedly non-zero and non-trivial for all four parameters. And, MS+O+E is positive for all four parameters. Specifically, MS+O+E is always minimum in ρ = 0 and increases as the absolute value of ρ increases. In all cases, S+O+E generates more efficient estimates than ST. Graphs of MS+O+E are shown in Figure 4, in the area surrounding true parameter values. We only consider ρ = 0 case since MS+O+E is minimum in this case. Since the orders of magnitude for the four parameters are quite different, we graph the MS+O+E of ω, α, β, and γ on the left to the right. True parameters are circled in each graph. Then, in each panel, one variable is varied while the other three are treated as known. Specifically, in Panel A, ω is varied, in B α, in C β, and in D γ. [Insert Figure 4 here] Note that, in all graphs, MS+O+E is decidedly non-zero and non-trivial for all four parameters. Thought not shown here, MS+O+E is positive for all four parameters for the entire span of possible parameter values.6 In Panel A, MS+O+E is always positive for all four. 6. Each parameter for ω (α, β, γ) ranges from 0 to 1 such that ( 0.5)2 1 .. 24.

(59) parameters in the area surrounding true parameters. While MS+O+E for ω decreases with ω locally in the region around the true parameter values, those for α, β, and γ are increasing. In Panel B, MS+O+E for ω, α, β, and γ increase with α locally in the region around the true parameter values. Corresponding graphs Panels C and D are similar to each other in shape, though their x-axes differ. However, MS+O+E is always positive in each case. In summary, MS+O+E is always positive for all parameters in the area surrounding true parameters. We conclude that S+O+E generates more efficient estimates than ST in all cases. This method also generates asymptotically unbiased estimates, making it the most desirable data specification of the two. We also conclude that the use of option prices can lead to very accurate estimates, even in long samples.. 3.2.3. Numerical findings and direct comparisons in finite sample studies We generate parameter estimates using a Monte Carlo method, comparing the bias and variance characteristics of the three data specifications in finite sample studies. Specifically, we simulate 30 days of stock and/or option prices and then run 1,000 iterations over each period.7 We again presume that parameter true values are (λ, ω0, α0, β0, γ0) = (0.1746, 6.792×10-9, 6.546×10-8, 0.9914, 351.945). For the S+O+E case, we additionally assume that η = 1 and that ρ = 0. As a robustness check, we re-run these tests for a variety of calibrations of ρ and η and find no qualitative differences. We estimate our four parameters for out-of-the-money (S0/K = 0.9), at-the-money (S0/K = 1.0), and in-the-money (S0/K = 1.1) cases. Results for 30 days are presented in. 7. We have also re-run all tests using 90 and 360 simulated days with qualitatively identical results.. 25.

(60) Table 1 where we report the absolute value of estimate bias (estimate less true value), standard deviation of the estimate (SD), and mean squared errors (MSE). [Insert Table 1 here] When ω is unknown, we find that S+O+E arrives at estimates within 4.355x10-9 of the true value. In contrast, estimates using ST and OT present biases on the order of roughly 1.1 to 4.4 times higher, respectiviely. Note that OT has the stronger bias regardless of the moneyness of the options. When options are in-the-money, standard deviations are lowest for S+O+E, with ST and OT again about 1.3 times and 1.1 times higher, respectively. With regard to MSE, results are even more staggering with S+O+E exhibiting the lowest values and ST and OT generating errors that are about 9 times and 10 times higher. Note that OT seems to perform particularly poorly when options are in-the-money. When estimating α, we similarly find that estimation bias is lower for S+O+E than for ST and OT, with the latter of these again having by the largest bias. Standard deviations are also lowest for S+O+E, with ST and OT again about 5 times and 3 times higher for most variables, respectively. MSE exhibits the same behavior as before, with S+O+E exhibiting by the lowest values, with magnitudes of difference similar to before. Note that OT presents particularly poor results with options out-of-the-money. When estimating β, we similarly find that estimation bias is far lower for S+O+E than for ST and OT, with the latter of these again having by far the largest bias. Standard deviation and MSE exhibit the same behavior as before, with S+O+E exhibiting by far the lowest values, with magnitudes of difference similar to before. Once again, OT presents particularly poor results with options out-of-the-money.. 26.

(61) Estimation results for γ are consistent with those of the other three parameters. S+O+E exhibits the smallest bias, the lowest standard deviation, and the lowest MSE of the three methods. Again, OT exhibits by far the worst performance along all four metrics and again particularly poor when options are out-of-the-money. The prevalence of biased estimates is striking for OT and is a particular strength for S+O+E. In summary, we conclude that the use of option prices can lead to very accurate estimates, even in short samples. This result is consistent with that of Eraker (2004).. 3.3. Risk management implications In this section we document that errors and bias in estimation may have substantial repercussions as relates to risk management benchmarks and practices. To illustrate, we obtain daily stock and options data from the Center for Research in Security Prices (CRSP) and the Option Metrics for the period from January 2007 to the end of 2007. For stock prices (St), we use the S&P 500 index, and for options data (Ct), we use the price of a short-maturity at-the-money call options where the price is measured as the midpoint of the last reported bid-ask spread. We assume that h0 ( ) / (1 2 ) , and for ease of interpretation, let the risk-free rate equal 0%. Applying the 12 months of stock and options data, we find key parameters to be (λ, ω0, α0, β0, γ0) = (0.1821, 6.847×10-9, 6.669×10-8, 0.9911, 342) for ST and (λ, ω0, α0, β0, γ0) = (0.1821, 6.029×10-9, 7.166×10-8, 0.9879, 402) for S+O+E. We omit the OT specification as it has been demonstrated that this method sometimes produces inefficient and, more importantly, biased estimates. As demonstrated in the following discussion, while these parameters may not appear to differ greatly, the resulting risk management implications are quite significant. 27.

(62) We then use these parameter estimates to calculate options deltas and gammas, measuring options stock price sensitivity and convexity, respectively. We calculate deltas and gammas for both the Black-Scholes and GARCH options pricing models, so that we have a total of four risk management metrics. For the former, we have that 6BS 7(d1). 0 0 ln(S0 / K ) (r 8 2 / 2)T +(d1 ) and 9 where d1 and 8 2 (cf. Duan 1 0 02 0 S08 T 8 T BS. (1995)). For the latter, we find that 6GARCH e rT E0Q[. and 9GARCH . N S ST 1 1{ST :K} ] ; e rT - T ,i 1{ST ,i :K} S0 N i 1 S0. /6GARCH (ST ) 6ˆ GARCH (S0 k ) 6ˆ GARCH (S0 k ) (cf. Engle and Rosenberg < /ST 2k. (1995)). We calculate these metrics for a variety of levels of moneyness, ranging from 0.9 (out-of-the-money) to 1.1 (in-the-money), and times to maturity, ranging from 30 days to 180 days. Results are provided in Table 2. Columns labeled I present values for ST while those labeled II present values for S+O+E. First, consistent with the findings of Engle and Rosenberg (1995) and Duan (1995), we find that GARCH and Black-Scholes deltas and gammas may differ but not systematically so and not to a large degree. Our focus is on the difference between these measures across the different estimation methods ST and S+O+E, not between the models GARCH vs Black-Scholes. Hence, note that in columns labeled III we present the quotient of each value in I divided by the corresponding value from II, less 1. For example, the upper right-most value in the area labeled III is -0.8412 = 0.0101/0.0634 – 1. That is the GARCH delta using ST is about 84% lower than the GARCH delta calculated using S+O+E. We see that, although there does not appear to be a 28.

(63) systematic relation, deltas and gammas may be significantly different depending on the method of estimation used. [Insert Table 2 here] First consider delta. Black-Scholes values from range about 10% higher for ST than for S+O+E when options are in the money to as much as 84% lower when they are out of the money. The difference tends to be negative when options are out of the money and positive when they are in the money. The trend is the same for GARCH deltas, although the magnitude ranges from 8.29% higher when options are in the money to 84.12% lower when they are out of the money. As a result, replicating and hedging portfolios will be significantly different based on the estimation method used, regardless of whether the agent applies a Black-Scholes or GARCH options pricing model. For gammas, there does not appear be a pattern in the difference related to the moneyness of the options. However, the magnitude of the differences ranges from -66.31% to 49.84%, indicating that gammas are more strongly impacted by the method of estimation used than deltas. As a result, the updating dynamics dictated to maintain hedges will be substantially different depending on the data specification employed. The magnitude of these differences suggests that managers and investors would do well to keep this in mind when calculating risk management metrics for options.. 3.4. Asymptotic behavior for ST, OT, and S+O+E Proofs apply Brown (1971) results regarding Central Limit Theorem analogs for martingale differences. All lemma proofs in this section are deferred to the Appendix A.. 29.

(64) 3.4.1. Asymptotic behavior for ST We seek to prove the following: Theorem 3.1. The estimator of ST is ˆTST argmax LTST ( ) . It is consistent such that . p. D. ˆTST .0 and asymptotically Normal such that HST 1/20 FST 0T 1/2 (ˆTST 0 ) . N (0, I 4 ), where /2 LST ( ) /LST ( ) /LTST (0 ) FST 0 E T 0 and HST 0 E T T 0 . / / ' // ' . To begin, the log-likelihood function for ST is given by (3.7) and it follows that: T. /LTST ( ) 1 / 2T t 1. T. $ 1

(65) yt =t ( ) 2 2 y = ( ) % /h ( ) ( 1 1/2 t 1/2 t ) t , ( ) ( ) h h h (& t )' / t t ( ) ht ( ). /2 LTST ( ) 1 // 2T t 1. 2 $ 1 2

(66) yt =t ( ) 2 y = ( ) 2 2 % /ht ( ) /ht ( ) ( 2 1 ) 3/2 t 1/2 t h h h ht ( ) ) / / ( ) ( ) t t ( ) ht ( ) &( t '. T $ 1

(67) yt =t ( ) 2 1 2 y = ( ) % /2h ( ) ( 1 1/2 t 1/2 t ) t , ht ( ) ht ( ) ht ( ) ) // 2T t 1 ( ht ( ) & '. T $ 1 6

(68) yt =t ( ) 2 /3 LTST ( ) 1 8 y = ( ) 4 2 % /h ( ) /ht ( ) /ht ( ) 5/2 t 1/2 t 2 ) t - ( 3 2 // / / / 2T t 1 ( ht ( ) ht ( ) ht ( ) ht ( ) ht ( ) ) / & ' T 2 $ 1 2

(69) yt =t ( ) 3 2 y = ( ) 2 2 % /2ht ( ) /ht ( ) ) 3/2 t 1/2 t - ( 2 1 ht ( ) ht ( ) ht ( ) ) // / 2T t 1 ( ht ( ) ht ( ) & ' T 2 3 $ %

(70) y =t ( ) 2 yt =t ( ) ) / ht ( ) , 1 1 1 t -( ht1/2 ( ) ht1/2 ( ) ) // / 2T t 1 ( ht ( ) ht ( ) & '. (3.26). (3.27). (3.28). where =t ( ) r ht ( ) . Consider the asymptotic Normality of the first derivative and the limit of the observed information matrix in (3.26) and (3.27), using Lemmas 3.1 and 3.2, respectively: Lemma 3.1. The form given by (3.26) evaluated at 0 is asymptotically Gaussian, 1/2 1/2 H ST 0T. /LTST (0 ) D . N

(71) 0, I 4 , /. $ 1 /LST ( ) /LST ( ) . 2 /ht (0 ) /ht (0 ) % where HST 0 E T T 0 T 0 E ( 2 ). / / ' / ' )' (& 2ht (0 ) ht (0 ) / 30.

(72) Lemma 3.2. The observed information matrix given by (3.27) evaluated at 0. / LT (0 ) /2 LTST (0 ) p . I4 , where FST 0 E . // ' // ' 2 ST. 1 ST 0. converges in probability to F. Then, evaluating the third derivative of the likelihood function in (3.28), we seek to show that it is uniformly bounded in a neighborhood around the true parameter value 0 . The neighborhood N

(73) 0 around the true value 0 defined as. N

(74) 0 | 0 L 0 U , 0 L 0 U , 0 L 0 U , L 0 U ,. U ( U 1/ 2)2 U 1.. (3.29). Lemma 3.3. There exists N

(75) 0 , for all 1 i, j, k 4 , for which sup. N (0 ). a. s . /3 LTST ( ) g (L ,U , L ,U , L , U , L , U , T ) . M # as T .# where M is /i / j /k. constant. In order to prove Lemma 3.3, without loss of generality, consider the case i j k . The next lemma establishes that the individual terms of the third derivative.

(76) / L. 3 ST T. / / 3

(77) in (3.28) are uniformly bounded in the neighborhood N

(78) 0 .. Lemma 3.4. With N

(79) 0 defined in (3.29), then for any t,. L sup ht ( ) Ht. (3.30). N (0 ). and. sup hit ( ) Hit , for i 1,2,3 ,. (3.31). N (0 ). where hit ( ) , Ht , and Hit are given by, 31.

(80) hit ( ) ,. t /i ht ( ) , Ht , max U 2U2 , h0 U 2U ( U )2 -

(81) U ( U )2 U i 1 ( yt i r)2 , i / 1 U ( U ) U L i 1. t 1. H1t , - Ht i

(82) U ( U )2 U , H2t , 2-1 U ( yt i3 r) H1t i H1t i

(83) U ( U )2 U , i 1. i 1. t 1 i 1. 2. . i 1. L. 2 2 t 1 i 1 and H3t , 3- 2 U ( yt i4 r) H13t i 2 U ( yt i3 r) H2t i H1t i H2t i

(84) U ( U )2 U . L L i 1 . Proof of Theorem 3.1. From Lemma 3.3, we have that $0% ( ) ST 2 ST ST ˆST (0) /LT (T ) < /LT (0 ) / LT (0 ) (ˆTST 0 ) (0) / / // ( ) &0'. and 1. /2 LST ( ) /LTST (0 ) . ˆTST 0 < T 0 / // . Combining with Lemmas 3.1 and 3.2, we complete the proof of Theorem 3.1.. 3.4.2. Asymptotic behavior for OT We seek to prove the following: Theorem 3.2. The estimator of OT is ˆTOT argmax LOT T ( ) . It is asymptotically biased such 1. p /2 LOT ( ) 1 T 1 /Jt (0 ) , and asymptotically that ˆTOT 1 .0 , where 1 , 0 T 0 T t 1 Jt (0 ) / // ' A /2 LOT 1/2 1/2 ˆOT T (0 ) F T . N I ( )~ (0, ), where Normal such that HOT . F E and T 0 OT 0 1 4 OT 0 // '. /LOT ( ) HOT 0 Var T 1/2 T 0 . / . 32.

(85) To begin, the log-likelihood function for OT is given by (3.11) and it follows that: Y 2 ( ) /ht ( ) Yt ( ) /Yt ( ) % /LOT 1 T $ 1 /Jt ( ) 1 1 T ( ) -( t ), T t 1 &( Jt ( ) / 2 ht ( ) ht2 ( ) / ht ( ) / ') /. $ % 1 /Jt ( ) /Jt ( ) 1 /2 Jt ( ) 1 1 2Yt 2 ( ) /ht ( ) /ht ( ) ( 2 ) / ' / / ' Jt ( ) // ' 2 ht2 ( ) ht3 ( ) / L ( ) 1 ( Jt ( ) / ), -( ) 2 2 2 // ' T t 1 1 1 / / / / / Y Y Y ( ) h ( ) 2 Y ( ) h ( ) Y ( ) Y ( ) Y ( ) ( ) ( ) / 1 t t t t t ( ) t2 t 2t t (& 2 ht ( ) ht ( ) // ' ht ( ) / / ' / ' ht ( ) // ' )' ht ( ) / 2 OT T. T. $ 2 /Jt ( ) /Jt ( ) /Jt ( ) % 3 /2 Jt ( ) /Jt ( ) 1 /3 Jt ( ) 2 ( J 3 ( ) / ) / ' / J ( ) / / ' / J ( ) / / ' / t t t ( ) 2 2 2 ( 1 3Y ( ) /ht ( ) /ht ( ) /ht ( ) 3 1 2Yt ( ) / ht ( ) /ht ( ) ) ( 3 4t ) / / ' / 2 ht2 ( ) ht3 ( ) // ' / ) ( ht ( ) ht ( ) ( ) Y 2 ( ) /3ht ( ) 6Yt ( ) /ht ( ) /ht ( ) /Yt ( ) /3 LOT 1 1 1 T ), T ( ) t2 3 - (( ) // ' / T t1 2 ht ( ) ht ( ) // ' / ht ( ) / / ' / ( ) 2 ( 3Yt ( ) /2ht ( ) /Yt ( ) 3 /ht ( ) /Yt ( ) /Yt ( ) 3Yt ( ) / Yt ( ) /ht ( ) ) 2 2 ( 2 ) / ' / ht ( ) / ht ( ) // ' / ) ( ht ( ) // ' / ( 3 /2Y ( ) /Y ( ) Y ( ) /3Y ( ) ) t t t ( ) t ht ( ) // ' / (& ht ( ) // ' / )'. (3.32). (3.33). (3.34). where /X (! , ) cos

(86) X 31,t (!, ) /X (! , ) % /Jt ( ) 1 # X11,t (! , ) $ sin

(87) X 31,t (!, ) ( )d! * e cos

(88) X 31,t (! , ) 11,t sin

(89) X 31,t (!, ) 31,t 0 " ! ! / / / &( ') /X (! , ) /X (! , ) % K 1 # X10,t (! , ) $ cos

(90) X 30,t (!, ) /St ( ) e r (T t ) cos

(91) X 30,t (! , ) 10,t sin

(92) X 30,t (!, ) 30,t e ( ) d!, / / / St ( ) " *0 St ( ) (& )'. / Jt ( ) 1 e // ' " *0 2. # X (! , ) 11,t. $ sin

(93) X 31,t (! , ) /X (! , ) /X11,t (!, ) /X 31,t (!, ) /X 31,t (!, ) /2 X11,t (!, ) % ( cos

(94) X 31,t (! , ) 11,t. ) / / ' / / ' // ' ) ! ( ( )d! ( cos

(95) X 31,t (! , ) ) /X (! , ) /X11,t (! , ) /2 X 31,t (!, ) ( ) sin

(96) X 31,t (! , ) 2 31,t / ! / ' // ' (& )'. $ cos

(97) X 30,t (! , ) /X10,t (!, ) /S ( ) /S ( ) /St ( ) /2 St ( ) 2sin

(98) X 30,t (! , ) /St ( ) /X 30,t (!, ) % t ( ) 2 t 2 St ( ) St ( ) / / ' ) ' / / / / ' // ' ( ( ) /X (! , ) /X10,t (!, ) /2 X10,t (!, ) /X 30,t (!, ) /X 30,t (!, ) K 1 # X10,t (! , ) ( ) e cos

(99) X 30,t (! , ) 10,t . e r (T t ) ) d! , * ( 0 St ( ) " / / ' // ' / / ' ) ( ) ( 2 ) ( sin

(100) X (! , ) 2 /X10,t (! , ) /X 30,t (!, ) / X 30,t (!, ) 30,t ) ( / / ' // ' & '. 33.

(101) $ /X (! , ) /X 31,t (! , ) /X11,t (! , ) % /X11,t (! , ) /X11,t (! , ) /X11,t (! , ) 3 31,t ( sin

(102) X (! , ) ) / / ' / / / ' / 31,t ( ) cos

(103) X 31,t (! , ) /2 X11,t (! , ) /X11,t (! , ) /2 X 31,t (! , ) /X 31,t (! , ) /3 X11,t (! , ) ) ( ! 3 3 ( ) 3 / Jt ( ) 1 # X11,t (! , ) ( / // ' / // ' / // ' )d! e ( // ' / " *0 /3 X 31,t (! , ) ) /2 X 31,t (! , ) /X11,t (! , ) /2 X11,t (! , ) /X 31,t (! , ) ( 3 3 ) // ' / // ' / ( cos

(104) X 31,t (! , ) sin

(105) X (! , ) // ' / ) t 31, ( /X 31,t (! , ) /X11,t (! , ) /X11,t (! , ) /X 31,t (! , ) /X 31,t (! , ) /X 31,t (! , ) ) ! ( . 3 ) (& )' / / / ' / / / ' 2 2 $ /3St ( ) % ( , ) X ! X ( ! , ) / / / S ( ) /St ( ) /St ( ) /S ( ) /St ( ) 10,t 10,t 3 2 t 4 t ( cos

(106) X (! , ) ) 30,t // ' / // ' / / / ' / ( // ' / ) ( /2 St ( ) /X10,t (! , ) St ( ) /X (! , ) /X10,t (! , ) /St ( ) ) /S ( ) /X 30,t (! , ) /X 30,t (! , ) 3 t 3 10,t ( 3 ) // ' / / / ' / / / ' / ) ( ( cos

(107) X (! , ) /X (! , ) /S ( ) /S ( ) ) /S ( ) /St ( ) /St ( ) / 2 S ( ) /St ( ) 30,t 10,t t t ( ) 4 t 2 t 4 ( ) / St2 ( ) / ' / / / ' / // ' / ( ) /X10,t (! , ) /St ( ) /X 30,t (! , ) /2 X10,t (! , ) /St ( ) ( ) /2 St ( ) /X 30,t (! , ) 3 2 ( sin

(108) X 30,t (! , ) 6 ) / / ' / // ' / // ' / ( ) 2 ( ) S ( ) / X 30,t (! , ) /St ( ) /S ( ) /St ( ) /X 30,t (! , ) t 2 t ( ) K 1 # X10,t (! , ) ( // ' / / / ' / ) d! , e e r (T t ) ( 4sin

(109) X 30,t (! , ) /S ( ) /X (! , ) /S ( ) ) St ( ) " *0 30,t t t ( ) St2 ( ) / / ' / ( ) ( ) 3 2 ( ! , ) ( ! , ) ( ! , ) ( ! , ) ( ! , ) ( X X X X X X ! , ) / / / / / / ( ) 30,t 30,t 30,t 30,t 10,t 30,t 3 ( ) / / / ' / // ' / // ' ( sin

(110) X 30,t (! , ) ) 2 / X 30,t (! , ) /X10,t (! , ) /X10,t (! , ) /X 30,t (! , ) /X10,t (! , ) ( ) 3 3 ( ) // ' / / / ' / ( ) ( ) /X10,t (! , ) /X 30,t (! , ) /X 30,t (! , ) / 2 X10,t (! , ) /X10,t (! , ) 3 3. ( ) / / ' / // ' / ( cos

(111) X (! , ) ) 30,t 3 2 ( ) / X 30,t (! , ) /X 30,t (! , ) /X10,t (! , ) /X10,t (! , ) /X10,t (! , ) / X10,t (! , ) ( ). 3 // ' / / / ' / // ' & '. /Yt ( ) /h ( ) 1 /St ( ) 1 /St 1 ( ) . t , / St ( ) / St 1 ( ) / /. /2Yt ( ) 1 /St ( ) /St ( ) 1 /2 St ( ) 1 /St 1 ( ) /St 1 ( ) 2 2 // ' St ( ) / / ' St ( ) // ' St 1 ( ) / / '. /2h ( ) 1 /2 St 1 ( ). t , St 1 ( ) // ' // '. /3Yt ( ) 2 /St ( ) /St ( ) /St ( ) 3 /2 St ( ) /St ( ) 1 /3St ( ) 3 2 // ' / St ( ) / / ' / St ( ) // ' / St ( ) // ' /. 2 /St 1 ( ) /St 1 ( ) /St 1 ( ) 3 /2 St 1 ( ) /St 1 ( ) St3 1 ( ) / / ' / St2 1 ( ) // ' /. /3ht ( ) 1 /3St 1 ( ). , St 1 ( ) // ' / // ' /. /X1m,t (!, ) /A1m,t (!, ) /B1m,t (!, ) /h ( ) ht 1 ( ) B1m,t (!, ) t 1 , / / / / 34.

(112) /2 X1m,t (! , ) /2 A1m,t (! , ) /2 B1m,t (! , ) /B (! , ) /ht 1 ( ) ht 1 ( ) 2 1m,t // ' // ' // ' / / 2 / h ( ) B1m,t (! , ) t 1 , // '. /3 X1m,t (! , ) /3 A1m,t (! , ) /3 B1m,t (! , ) /2 B (!, ) /ht 1 ( ) ht 1 ( ) 3 1m,t // ' / // ' / // ' / // ' / 2 3 / h ( ) /B1m,t (! , ) / ht 1 ( ) 3 t 1 B1m,t (!, ) , // ' / // ' /. /X3m,t (!, ) /B (!, ) /h ( ) ! /St ( ) /A2m,t (!, ) ht 1 ( ) 2m,t B2m,t (!, ) t 1 , / St ( ) / / / / /2 X 3m,t (! , ) ! /St ( ) /St ( ) ! /2 St ( ) /2 A2m,t (!, ) 2 // ' St ( ) / / ' St ( ) // ' // ' 2. /B2m,t (! , ) /ht 1 ( ) /2 B2m,t (! , ) /2h ( ) ht 1 ( ) B2m,t (! , ) t 1 , / / ' // ' // '. and. /3 X 3m,t (! , ) 2! /St ( ) /St ( ) /St ( ) 3! /2 St ( ) /St ( ) ! /3St ( ) 3 2 // ' St ( ) / / ' / St ( ) // ' / St ( ) // ' / /3 A2m,t (! , ) /2 B2m,t (! , ) /ht 1 ( ) /2ht 1 ( ) /B2m,t (! , ) 3 3 / // ' / // ' / // ' 3 3 / B2m,t (! , ) / h ( ) ht 1 ( ) B2m,t (! , ) t 1 . // ' / // ' / Consider the asymptotic Normality of the first derivative and the limit of the observed information matrix in (3.32) and (3.33), using Lemmas 3.5 and 3.6, respectively: Lemma 3.5. The form given by (3.32) evaluated at 0 is asymptotically Gaussian, OT 1 T 1 /Jt (0 ) D 1/2 1/2 /LT (0 ) HOT T . N

(113) 0, I 4 , 0 T t 1 Jt (0 ) / /. $ 1 /ht ( ) /ht ( ) % $ 1 /Yt ( ) /Yt ( ) % /LOT ( ) where HOT 0 Var T 1/2 T 0 E ( 2 ) E( ). / / ' ' & 2ht ( ) / / ' ' & ht ( ) /. 35.

(114) Lemma 3.6. The observed information matrix given by (3.33) evaluated at 0 converges. in. probability. to. FOT 10. p /2 LOT T (0 ) . I4 // '. ,. where. /2 LOT T (0 ) FOT 0 E // ' $ 1 /2 Jt (0 ) % $ 1 /Jt (0 ) /Jt (0 ) % $ 1 /ht (0 ) /ht (0 ) % $ 1 /Yt (0 ) /Yt (0 ) % . E( ) E ( J 2 ( ) / ) E ( 2h2 ( ) / ) E ( h ( ) / ( ) ' ' ' / / / / / ' )' J & t 0 ' & t 0 ' & t 0 ' & t 0. Then, evaluating the third derivative of the likelihood function in (3.34), we seek to show that it is uniformly bounded in a neighborhood around the true parameter value 0 . Lemma 3.7. There exists N

(115) 0 , for all 1 i, j, k 4 , for which sup. N (0 ). a. s . /3 LOT T ( ) g (L ,U , L ,U , L , U , L , U , T ) . M # as T .# where M is /i / j /k. constant. In order to prove Lemma 3.7, again, we only consider the case i j k . We want to 3 show that the individual terms of the third derivative

(116) /3 LOT T / /

(117) in (3.34) are. uniformly bounded in the neighborhood N

(118) 0 . It has to apply these results of Lemmas 3.8-3.13. First, we prove that these lemmas. Lemma 3.8. With N

(119) 0 defined in (3.29), then for any t, m 0,1,. B1m,t (!, ) . !2 2. b1m,t (!, ). (3.35). and. !. B2m,t (!, ) b2m,t (!, ) , 2. (3.36). where 36.

(120) b1m,t (!, ) b1m,t 1(!, ) . (1 ! 2b1m,t 1(!, )) $&1

(121) m(2 Q 1) Q2 b1m,t 1(!, )%' 2

(122) Q2 m(2 Q 1) b22m,t 1(! , ) 2 (m Q )b2m,t 1(! , ) (1 ! 2b1m,t 1(!, ))2 2! 2b22m,t 1(!, ). b2m,t (!, ) 2 Q 1 b2m,t 1 (! , ) . 2(m Q )(1 ! 2b1m,t 1 (! , )) [(m Q )2 ! 2 ]b2m,t 1(!, ) , (1 ! 2b1m,t 1 (! , ))2 2! 2b22m,t 1 (! , ). ,. and. b1m,T (!, ) b2m,T (!, ) 0 . Lemma 3.9. With N

(123) 0 defined in (3.29), then for any t, m 0,1, i 1, 2 , and k 1,2,3 ,. 0 sup b1m,t (!, ) b1mU ,. (3.37). b2mL sup b2m,t (!, ) b2mU ,. (3.38). N (0 ). N (0 ). and. sup bimk ,t (!, ) bimkU ,. (3.39). N (0 ). where bimk ,t (! , ) . /k bim,t (! , ) and the constants, b1mU , b2mL , b2mU , bimkU , are functions of / k. L , U ,L ,U , L , U , L , U , . Lemma 3.10. With N

(124) 0 defined in (3.29), then for any t, m 0,1, and k 1,2,3 ,. !2 2. b1mU sup B1m,t (!, ) 0 ,. sup B1mk ,t (!, ) . N (0 ). (3.40). N (0 ). !2 2. !. b1mkU ,. (3.41). !. (3.42). b sup B (!, ) b2mU , 2 2mL N (0 ) 2m,t 2. and. !. sup B2mk ,t (!, ) b2mkU . 2 N (0 ). (3.43) 37.

(125) Lemma 3.11. With N

(126) 0 defined in (3.29), then for any t, i 1, 2 , and m 0,1,. sup A1m,t (!, ) mr(T t ) . L. N (0 ). 2. !2 ,. (3.44). $ b % sup A2m,t (!, ) (! r U 2mU 2c" ) (T t ) , 2 N (0 ) & '. (3.45). and. sup Aim1,t (!, ) . N (0 ). sup Aim2,t (!, ) . N (0 ). sup Aim3,t (!, ) . N (0 ). ! 3 i 2. ! 3 i 2. ! 3 i 2. Aim1,t ,. (3.46).

(127) A. ! 2 Aim22,t ,. (3.47).

(128) A. ! 2 Aim32,t ! 4 Aim33,t ,. (3.48). im21,t. im31,t. where c is a positive integer,. A1m1,t

(129) U b1m1U U b1m1U U2 b2mU b2m1U (T t ) ,. b2 A1m21,t U b1m2U 1m2 1U U b1m2U U2 b2mU b2m2U U2 b22m1U (T t ) , b2mL A1m22,t 2

(130) U b1m1U U2 b2m1U

(131) U b1m1U U2 b2mU b2m1U (T t ) ,. 3b b A1m31,t U b1m3U 1m1U2 1m2U U b1m3U 3U2 b2m1U b2m2U U2 b2mU b2m3U (T t ) , b2mL . A1m32,t. $ b12m1U % 2 2 2 (4( b2 U b1m2U U b2mU b2m2U U b2m1U )(U b1m1U U b2mU b2m1U )) ) (T t ) , ( 2mL 2 ( ) b (2(U b1m1U U2 b2m1U )( 1m2 1U U b1m2U U2 b2mU b2m2U U2 b2m1U ) ) b2mL (& )'. 38.

(132) A1m3,t 8(U b1m1U U2 b2m1U )(U b1m1U U2 b2mU b2m1U )2 (T t ) ,. b A2m1,t U b2m1U U b2m1U U2 1m1U (T t ) , b2mL b A2m21,t U b2m2U U b2m2U 1m2U (T t ) , b2mL b A2m22,t 2U U b2m1U 1m1U

(133) b1m1U U b2mU b2m1U (T t ) , b2mL b b b b b A2m31,t U b2m3U U b2m3U 2m2U 1m1U 2 2m1U 1m2U ,t i 1m3U (T t ) , b2mL b2mL . A2m32,t. $ % b1m2U (4U U b2m2U )

(134) b1m1U U b2mU b2m1U b2mL ( ) (T t ) , and ( 2 )) b1m1U 2 (2U U b2m1U b1m1U b b b b U 2 m1U U 2 mU 2 m2U (& )' b2mL 1m2U Lb22mL . 2 b A2m33,t 8U2 U b2m1U 1m1U

(135) b1m1U U b2mU b2m1U (T t ) . b2mL . Assume that the state variable as a function of observed asset price,. St ( ) f 1(Ct ; ) , satisfies SLt St ( ) SUt and Sit ( ) SiUt , where Sit ( ) . /i St ( ) , / i. i 1,2,3 , and SLt , SUt , and SiUt don’t depend on these parameters. Lemma 3.12. With N

(136) 0 defined in (3.29), then for any t and m 0,1, sup X1m,t (!, ) . L. N (0 ). sup X1m1,t (!, ) . N (0 ). 2. !2 ,. !2 2. (3.49). X1m1,t ,. (3.50). 39.

(137) sup X1m2,t (!, ) . N (0 ). sup X1m3,t (!, ) . N (0 ). !2.

(138) X 2. 1m2,t. !2.

(139) X 2. 1m3,t. ! 2 A1m22,t ,. (3.51). ! 2 A1m32,t ! 4 A1m33,t ,. (3.52). sup X 3m,t (!, ) ! X 3m,t 2c" (T t ) ,. (3.53). !. (3.54). N (0 ). sup X3m1,t (!, ) . N (0 ). sup X3m2,t (!, ) . N (0 ). 2. X3m1,t ,. !.

(140) X 2. 3m2,t. ! 2 A2m22,t ,. (3.55). ! 2 A2m32,t ! 4 A2m33,t ,. (3.56). and. sup X3m3,t (!, ) . N (0 ). !.

(141) X 2. 3m3,t. where c is a positive integer,. X1m1,t A1m1,t b1m1U Ht 1 b1mU H1t 1 , X1m2,t A1m21,t b1m2U Ht 1 2b1m1U H1t 1 b1mU H2t 1 , X1m3,t A1m31,t b1m3U Ht 1 3b1m2U H1t 1 3b1m1U H2t 1 b1mU H3t 1 , b 1 X 3m,t r U 2mU (T t ) ln(SUt / K ) b2mU Ht 1 , 2 2 X 3m1,t . 2S1Ut A2m1,t b2m1U Ht 1 b2mU H1t 1 , SLt. X 3m2,t . 2S12Ut 2S2Ut A2m21,t b2m2U Ht 1 2b2m1U H1t 1 b2mU H2t 1 , and SLt2 SUt. X 3m3,t. 4S13Ut 6S2Ut S1Ut 2S3Ut 3 A2m31,t b2m3U Ht 1 3b2m2U H1t 1 3b2m1U H2t 1 b2mU H3t 1 . SLt SLt2 SLt 40.

(142) Lemma 3.13. With N