基于时变波动率的期权定价与避险策略研究
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摘要
期权的定价和避险是期权研究中最重要两个的方面。Black-Scholes公式由于简单明了且易于计算,故深受金融分析者的欢迎,成为实践中指导期权定价及避险的重要工具。但人们发现,将波动率看作常数并不能与实际市场很好地吻合,一系列关于股票波动的实证分析表明,波动率实际上并不是常数。因此研究学者在此基础上主要发展了两类时变波动率模型:GARCH族期权定价模型,SV随机波动率定价模型。
国外的学术界和实务界,都对时变波动率的期权定价和避险进行了深入、系统的理论与实证研究。但我国由于金融衍生品发展步伐缓慢,无论在学术研究还是实务应用,相关的研究都还不够,尤其在实证方面,缺乏全面、系统的研究和分析。
本文对波动率变化下的期权定价和避险策略进行了系统深入的研究,期望对国内未来期权的进一步发展,提供理论参考和实证借鉴。
论文的第一章为绪论,说明本文的研究背景、研究目的及论文框架。第二章为文献探讨,从B-S定价模型及其缺陷开始,介绍了大量的放松各种限制条件的模型;重点介绍了放松波动率为常数假设的相关文献,回顾了国内外学者对期权定价方面的实证研究,包括所使用的模型及研究结果。第三章为模型阐述,详细介绍了GARCH族期权定价模型,及SV随机波动率期权定价模型。介绍了这些模型在定价上的特点,各自分别弥补了B-S模型的哪些方面的不足;同时探讨了不同参数对定价结果的影响。第四章为实证分析,对我国权证的价格分别进行了B-S模型与随机波动性模型的绩效评估,分析了理论模型价格与市场实际价格间差异的原因,提出了改善我国权证定价效率的建议。第五章分析了各类避险模型的避险效果,实证研究了波动率不变和随机波动率情况下,各种模型的实际避险效果。第六章为结论与建议,并对未来可能的研究方向提出了建议。
应用GARCH族期权定价模型和SV随机波动率期权定价模型,对我国权证市场的数据进行了全面的实证分析,得到的关于期权定价的主要结论有:
1.总体而言,市场实际价格高于理论模型价格,且幅度较大。偶尔也存在理论模型价格高于市场实际价格的情况,但比例非常低。
2.模型之间的相互比较:部分GARCH类模型表现好于SV(HESTON)模型和B-S模型。SV模型没有在任何一只权证的定价中表现最优,而B-S模型和GARCH,NGARCH,EGARCH,GJR-GARCH模型则分别有表现最优的结果,这说明GARCH模型是较好的定价模型。
3.对于认购权证,随着到期日的临近,理论模型结果与市场实际价格间的定价差异逐步缩小;而对认沽权证则正好相反。差异的原因主要是因为权证存续期间,股市基本是单边的牛市。因此多数认购权证到期时深度价内,而认沽权证则深度价外;且认沽权证到最后理论模型的价格都接近0,而实际的市场价格仍较高,因此误差较大。
由于市场实际价格高于理论模型价格幅度较大,论文分析了其中的可能原因:股改权证数量有限,供应量太小;创设机制对券商的要求较高,限制了权证的有效供应量,造成供不应求;以及不存在卖空机制,投资者无法通过套利操作使认购权证价格回归合理价位,影响B-S公式定价的合理性。对应的政策建议包括:交易所应尽早推出备兑权证和股指期货,丰富衍生产品;完善平衡供需机制,引入持续发行机制和自由发行机制,建立做市商制度;实行部分抵押或动态对冲制度,从而增加权证供给,完善权证定价套利机制,提高市场定价效率。
论文接下来对避险策略进行了理论分析和实证研究。在统一均值方差框架下,对存在交易成本时的四种避险策略进行了系统的实证比较。在比例交易成本和常数方差的情形下,实证结论为:Whalley-Wilmott效用最大化策略优于其他策略,同样的标准差时该策略的交易成本最小;其后依序分别是Delta固定避险带策略,以标的资产价格变化为基础的策略,Leland避险模型和间断的B-S策略。同时得到的结论还有:随着波动率σ上升,无风险利率r下降,以变动为基础的方法相对以时间为基础的方法优势明显。采用随机波动率,相对于常数波动率而言,避险表现更差。
尽管从理论上分析,以Delta变动为基础的策略拥有明显的优势;但以时间为基础的策略由于易于执行,从实际操作角度来看也很有优势,它们不需要经常监控市场。当标的资产价格在前一日的收盘价与第二日的开盘价之间有较大的跳空时,以变动为基础的策略比以时间为基础的策略面临的风险更大。
综合而言,本文最重要的结论是:GARCH族模型和SV随机波动率模型在定价方面有一定的优势;但在避险方面没有优势,增加了避险成本。
论文的主要创新点包括:
1.对NGARCH,EGARCH和GJR-GARCH期权定价模型进行了全面的分析和讨论,并将这些模型系统、完整的应用于我国权证市场的实际定价研究中。
2.结合我国市场部分权证为股本权证的特殊情况,对GARCH族定价模型和SV随机波动率模型做了相应修正,使它们的定价结果更能反映我国权证市场的真实现状。
3.针对理论模型价格与市场实际价格间的差异,提出了普遍因素和中国因素的分析框架,认为除了权证流动性、价内外程度等这些不同国家市场中均存在的普遍影响因素外,我国权证市场存在特别的中国因素。在中国因素中重点分析了全额担保制度这一被忽视的重要原因。
4.目前国内对避险策略进行研究的不多,而基于波动率时变情况下的避险策略研究则较为少见。论文丰富了该项研究,并分析了多种因素变化时不同模型的实际避险效果。
5.在国内外的研究中,完整而全面的分析了GARCH族期权定价模型(包括:GARCH,NGARCH,EGARCH和GJR-GARCH期权定价模型)与SV随机波动率期权定价模型(包括:Hull & White随机波动性模型,Heston随机波动性模型),分析了这些主流的时变期权模型在定价上的不同特点和实际运用中的定价差异,是目前国内外这一研究领域中较为全面系统的文章。
总体而言,本文是国内较全面系统的对权证定价和避险进行理论和实证研究的文章,期望为后来的研究打下基础,为国内实务界提供参考。
Option pricing and risk management are the most important fields of option research.After Black & Scholes published the famous option pricing model in 1973, option pricing theory has became an academic research focus. But the assumption of constant of volatility don’t match the market real situation. Many positive research on stock volatility shows that the volatility is variable. As a result of many unduly simplified assumptions of B-S model, many scholars started to modify the B-S model, such as the“stochastic volatility option model”,“stochastic interest rate option model”and“stochastic volatility and poisson jump diffusion option model”. Many scholars devoted themselves to investigate the issue about whether the free-restricted models outperform B-S model. Most empirical results indicated that the free-restricted models outperform B-S model.
Option pricing and risk management under time varing variance has been deeply developed on theory and empirical aspect of overseas. In the domestic, the relative research is scanty because of the underdevelopment of the financial derivative, especially on empirical research.
Considerable theoretical work has been devoted to option replication in the presence of transaction cost, and several competing methods have been advocated to improve the dynamic hedging risk-return tradeoff. Very little is known on the subject from an empirical standpoint.
This paper attempts to provide a systematic comparison of the four popular methods of option hedging in the presence of transaction cost within a unified mean-variance framework, and using an extensive data set of simulated asset prices. In the presence of proportional transaction costs and constant volatility, the optimal control or delta move-based approach clearly dominates other types of strategies, including time-based strategies and strategies based on moves in the underlying asset.
This paper examines the out-of-sample performance of two common extensions of the Black-Scholes framework, namely a GARCH and a stochastic volatility option pricing model. It attempts to employ the models to empirically examine the pricing of our stock market related warrants. When analyzing the observed prices, GARCH clearly dominates both stochastic volatility and the benchmark B-S model.
This paper study the option pricing and risk management under time varing variance systemically and deeply first time. The research expects to provide some useful information for further study.
The innovation of this paper including :
1. The most comprehensive analysis on NGARCH, EGARCH and GJR-GARCH model and apply into Chinese market for the first time.
2. Considering the local market situation, change the GARCH and SV model. The modified model can match real option market better.
3. Propose the framework of general factor and Chinese factor based on the price difference. The Chinese factor is the important one causing the difference.
4. There is seldom research on hedging strategy. The paper is the first time on this field and analysis multi-factor hedging strategy.
5. About the GARCH and SV model, this paper is the most comprehensive one in this field.
The first chapter is exordium, which introduce the background, objective and framework. The second chapter discuss the literature based on B-S model and the modules of reducing the restrict on B-S model. The third chapter introduces many model, including GARCH family model and stochastic volatility option pricing model. It also discuss the effect of alterable parameters. The fourth chapter is empirical analysis based on B-S model and stochastic volatility option pricing model. It discuss the reason of difference and bring forward to some suggestion of how to improve efficiency.The fifth chapter concentrate on the impact of all kinds of risk management, including the constant of volatility and stochastic volatility. The last chapter presents the conclusion and potential research aspect.
The empirical conclusions are:
1. Overall, the real market price is much higher than the model price. The higher situation is seldom.
2. Some GARCH model performance is better than HESTON model and B-S model. That means GARCH family is a proper choice of pricing model.
3. For buying option, the gap of model price and the real market price is smaller along with the duration is coming. The advantage of move-based methods over time-based methods increases with reduced drift of the underlying asset and increased volatility of the underlying asset. Move-based strategies are hurt by the introduction of stochastic volatility.
Because the difference is large, the paper analysis the several reasons, including the lack of option, the strict restrict of security company, no oversell system ect, . These factors impact the B-S formula applicability. The relative suggestion is to increase the supply of the derivative and consummate the institution.
Under the united mean-variance framework, this paper analysis four kinds of hedging strategy. The conclusion is: Whalley-Wilmott is the best one, Delta is the second, Leland is the third one and B-S model is the last one.
Overall, the conclusion is that the more complex option pricing models, such as GARCH and SV can improve on the B-S methodology only for the purpose of pricing, but not for dynamic hedging.
国外的学术界和实务界,都对时变波动率的期权定价和避险进行了深入、系统的理论与实证研究。但我国由于金融衍生品发展步伐缓慢,无论在学术研究还是实务应用,相关的研究都还不够,尤其在实证方面,缺乏全面、系统的研究和分析。
本文对波动率变化下的期权定价和避险策略进行了系统深入的研究,期望对国内未来期权的进一步发展,提供理论参考和实证借鉴。
论文的第一章为绪论,说明本文的研究背景、研究目的及论文框架。第二章为文献探讨,从B-S定价模型及其缺陷开始,介绍了大量的放松各种限制条件的模型;重点介绍了放松波动率为常数假设的相关文献,回顾了国内外学者对期权定价方面的实证研究,包括所使用的模型及研究结果。第三章为模型阐述,详细介绍了GARCH族期权定价模型,及SV随机波动率期权定价模型。介绍了这些模型在定价上的特点,各自分别弥补了B-S模型的哪些方面的不足;同时探讨了不同参数对定价结果的影响。第四章为实证分析,对我国权证的价格分别进行了B-S模型与随机波动性模型的绩效评估,分析了理论模型价格与市场实际价格间差异的原因,提出了改善我国权证定价效率的建议。第五章分析了各类避险模型的避险效果,实证研究了波动率不变和随机波动率情况下,各种模型的实际避险效果。第六章为结论与建议,并对未来可能的研究方向提出了建议。
应用GARCH族期权定价模型和SV随机波动率期权定价模型,对我国权证市场的数据进行了全面的实证分析,得到的关于期权定价的主要结论有:
1.总体而言,市场实际价格高于理论模型价格,且幅度较大。偶尔也存在理论模型价格高于市场实际价格的情况,但比例非常低。
2.模型之间的相互比较:部分GARCH类模型表现好于SV(HESTON)模型和B-S模型。SV模型没有在任何一只权证的定价中表现最优,而B-S模型和GARCH,NGARCH,EGARCH,GJR-GARCH模型则分别有表现最优的结果,这说明GARCH模型是较好的定价模型。
3.对于认购权证,随着到期日的临近,理论模型结果与市场实际价格间的定价差异逐步缩小;而对认沽权证则正好相反。差异的原因主要是因为权证存续期间,股市基本是单边的牛市。因此多数认购权证到期时深度价内,而认沽权证则深度价外;且认沽权证到最后理论模型的价格都接近0,而实际的市场价格仍较高,因此误差较大。
由于市场实际价格高于理论模型价格幅度较大,论文分析了其中的可能原因:股改权证数量有限,供应量太小;创设机制对券商的要求较高,限制了权证的有效供应量,造成供不应求;以及不存在卖空机制,投资者无法通过套利操作使认购权证价格回归合理价位,影响B-S公式定价的合理性。对应的政策建议包括:交易所应尽早推出备兑权证和股指期货,丰富衍生产品;完善平衡供需机制,引入持续发行机制和自由发行机制,建立做市商制度;实行部分抵押或动态对冲制度,从而增加权证供给,完善权证定价套利机制,提高市场定价效率。
论文接下来对避险策略进行了理论分析和实证研究。在统一均值方差框架下,对存在交易成本时的四种避险策略进行了系统的实证比较。在比例交易成本和常数方差的情形下,实证结论为:Whalley-Wilmott效用最大化策略优于其他策略,同样的标准差时该策略的交易成本最小;其后依序分别是Delta固定避险带策略,以标的资产价格变化为基础的策略,Leland避险模型和间断的B-S策略。同时得到的结论还有:随着波动率σ上升,无风险利率r下降,以变动为基础的方法相对以时间为基础的方法优势明显。采用随机波动率,相对于常数波动率而言,避险表现更差。
尽管从理论上分析,以Delta变动为基础的策略拥有明显的优势;但以时间为基础的策略由于易于执行,从实际操作角度来看也很有优势,它们不需要经常监控市场。当标的资产价格在前一日的收盘价与第二日的开盘价之间有较大的跳空时,以变动为基础的策略比以时间为基础的策略面临的风险更大。
综合而言,本文最重要的结论是:GARCH族模型和SV随机波动率模型在定价方面有一定的优势;但在避险方面没有优势,增加了避险成本。
论文的主要创新点包括:
1.对NGARCH,EGARCH和GJR-GARCH期权定价模型进行了全面的分析和讨论,并将这些模型系统、完整的应用于我国权证市场的实际定价研究中。
2.结合我国市场部分权证为股本权证的特殊情况,对GARCH族定价模型和SV随机波动率模型做了相应修正,使它们的定价结果更能反映我国权证市场的真实现状。
3.针对理论模型价格与市场实际价格间的差异,提出了普遍因素和中国因素的分析框架,认为除了权证流动性、价内外程度等这些不同国家市场中均存在的普遍影响因素外,我国权证市场存在特别的中国因素。在中国因素中重点分析了全额担保制度这一被忽视的重要原因。
4.目前国内对避险策略进行研究的不多,而基于波动率时变情况下的避险策略研究则较为少见。论文丰富了该项研究,并分析了多种因素变化时不同模型的实际避险效果。
5.在国内外的研究中,完整而全面的分析了GARCH族期权定价模型(包括:GARCH,NGARCH,EGARCH和GJR-GARCH期权定价模型)与SV随机波动率期权定价模型(包括:Hull & White随机波动性模型,Heston随机波动性模型),分析了这些主流的时变期权模型在定价上的不同特点和实际运用中的定价差异,是目前国内外这一研究领域中较为全面系统的文章。
总体而言,本文是国内较全面系统的对权证定价和避险进行理论和实证研究的文章,期望为后来的研究打下基础,为国内实务界提供参考。
Option pricing and risk management are the most important fields of option research.After Black & Scholes published the famous option pricing model in 1973, option pricing theory has became an academic research focus. But the assumption of constant of volatility don’t match the market real situation. Many positive research on stock volatility shows that the volatility is variable. As a result of many unduly simplified assumptions of B-S model, many scholars started to modify the B-S model, such as the“stochastic volatility option model”,“stochastic interest rate option model”and“stochastic volatility and poisson jump diffusion option model”. Many scholars devoted themselves to investigate the issue about whether the free-restricted models outperform B-S model. Most empirical results indicated that the free-restricted models outperform B-S model.
Option pricing and risk management under time varing variance has been deeply developed on theory and empirical aspect of overseas. In the domestic, the relative research is scanty because of the underdevelopment of the financial derivative, especially on empirical research.
Considerable theoretical work has been devoted to option replication in the presence of transaction cost, and several competing methods have been advocated to improve the dynamic hedging risk-return tradeoff. Very little is known on the subject from an empirical standpoint.
This paper attempts to provide a systematic comparison of the four popular methods of option hedging in the presence of transaction cost within a unified mean-variance framework, and using an extensive data set of simulated asset prices. In the presence of proportional transaction costs and constant volatility, the optimal control or delta move-based approach clearly dominates other types of strategies, including time-based strategies and strategies based on moves in the underlying asset.
This paper examines the out-of-sample performance of two common extensions of the Black-Scholes framework, namely a GARCH and a stochastic volatility option pricing model. It attempts to employ the models to empirically examine the pricing of our stock market related warrants. When analyzing the observed prices, GARCH clearly dominates both stochastic volatility and the benchmark B-S model.
This paper study the option pricing and risk management under time varing variance systemically and deeply first time. The research expects to provide some useful information for further study.
The innovation of this paper including :
1. The most comprehensive analysis on NGARCH, EGARCH and GJR-GARCH model and apply into Chinese market for the first time.
2. Considering the local market situation, change the GARCH and SV model. The modified model can match real option market better.
3. Propose the framework of general factor and Chinese factor based on the price difference. The Chinese factor is the important one causing the difference.
4. There is seldom research on hedging strategy. The paper is the first time on this field and analysis multi-factor hedging strategy.
5. About the GARCH and SV model, this paper is the most comprehensive one in this field.
The first chapter is exordium, which introduce the background, objective and framework. The second chapter discuss the literature based on B-S model and the modules of reducing the restrict on B-S model. The third chapter introduces many model, including GARCH family model and stochastic volatility option pricing model. It also discuss the effect of alterable parameters. The fourth chapter is empirical analysis based on B-S model and stochastic volatility option pricing model. It discuss the reason of difference and bring forward to some suggestion of how to improve efficiency.The fifth chapter concentrate on the impact of all kinds of risk management, including the constant of volatility and stochastic volatility. The last chapter presents the conclusion and potential research aspect.
The empirical conclusions are:
1. Overall, the real market price is much higher than the model price. The higher situation is seldom.
2. Some GARCH model performance is better than HESTON model and B-S model. That means GARCH family is a proper choice of pricing model.
3. For buying option, the gap of model price and the real market price is smaller along with the duration is coming. The advantage of move-based methods over time-based methods increases with reduced drift of the underlying asset and increased volatility of the underlying asset. Move-based strategies are hurt by the introduction of stochastic volatility.
Because the difference is large, the paper analysis the several reasons, including the lack of option, the strict restrict of security company, no oversell system ect, . These factors impact the B-S formula applicability. The relative suggestion is to increase the supply of the derivative and consummate the institution.
Under the united mean-variance framework, this paper analysis four kinds of hedging strategy. The conclusion is: Whalley-Wilmott is the best one, Delta is the second, Leland is the third one and B-S model is the last one.
Overall, the conclusion is that the more complex option pricing models, such as GARCH and SV can improve on the B-S methodology only for the purpose of pricing, but not for dynamic hedging.
引文
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[7] Andersen, T.G. and B.E. Srensen (1996),“GMM Estimation of a Stochastic Volatility Model:A Monte Carlo Study,”Journal of Business & Economic Statistics, 14, 328-352.
[8] Ang, A. and J. Chen (2002),“Asymmetric Correlation of Equity Portfolios,”Journal of Financial Economics, 63, 443-494.
[9] Baillie, R.T., T. Bollerslev and H.O. Mikkelsen (1996),“Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity,”Journal of Econometrics, 74, 3-30.
[10] Bakshi, G, C. Cao and Z. Chen (1997),“Empirical Performance of Alternative Option Pricing Models,”Journal of Finance, 52, 2003-2049.
[11] Bandi, F. and J.R. Russell (2004),“Microstructure Noise, Realized Variance, and Optimal Sampling,”Working paper, Graduate School of Business, University of Chicago.
[12] Dimitris Psychoyios, George Skiadopoulos & Panayotis Alexakis, 2003, A review of stochastic volatility process: properties and implications, The Journal of RiskFinance.
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[3] Dupire, B., 1994, Pricing with A smile, Risk 7, 18-20.
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[5] Christie, A., 1982, The stochastic behavior of common stock variances: value, leverage, and interest rate effects, Journal of Financial Economics 10, 407-432.
[6] Andersen, T.G. and J. Lund (1997),“Estimating Continuous-Time Stochastic Volatility Models of the Short term Interest Rate Diffusion,”Journal of Econometrics, 77, 343-377.
[7] Andersen, T.G. and B.E. Srensen (1996),“GMM Estimation of a Stochastic Volatility Model:A Monte Carlo Study,”Journal of Business & Economic Statistics, 14, 328-352.
[8] Ang, A. and J. Chen (2002),“Asymmetric Correlation of Equity Portfolios,”Journal of Financial Economics, 63, 443-494.
[9] Baillie, R.T., T. Bollerslev and H.O. Mikkelsen (1996),“Fractionally Integrated Generalized Autoregressive Conditional Heteroskedasticity,”Journal of Econometrics, 74, 3-30.
[10] Bakshi, G, C. Cao and Z. Chen (1997),“Empirical Performance of Alternative Option Pricing Models,”Journal of Finance, 52, 2003-2049.
[11] Bandi, F. and J.R. Russell (2004),“Microstructure Noise, Realized Variance, and Optimal Sampling,”Working paper, Graduate School of Business, University of Chicago.
[12] Dimitris Psychoyios, George Skiadopoulos & Panayotis Alexakis, 2003, A review of stochastic volatility process: properties and implications, The Journal of RiskFinance.
[13] Barndorff-Nielsen, O.E. and N. Shephard (2004a),“Power and Bipower Variation with Stochastic Volatility and Jumps,”Journal of Financial Econometrics, 2, 1-37.
[14] Barrett, C. (1999),“The effects of real exchange rate depreciation on stochastic producer prices in low-income agriculture,”Agricultural Economics 20, 215-230.
[15] Henrotte P. Transaction Costs and Duplication Strategies[Z]. Working Paper, 1993, Stanford University and HEC.
[16] Bates, D.S. (2003)“Empirical Option Pricing: A Retrospection,”Journal of Econometrics, 116,387-404.
[17] Battle, C. and J. Barquin (2004),“Fuel Prices Scenario Generation Based on a Multivariate Garch Model for Risk Analysis in a Wholesale Electricity Market,”International Journal of Electrical Power and Energy Systems, 26, 273-280.
[18] Jarrow, R., 1994, Derivative Securities Markets, Market Manipulation and Option Pricing Theory, Journal of Financial and Quantitative Analysis 29, 241-261.
[19] Bera, A.K. and S. Kim (2002),“Testing Constancy of Correlation and other Specifications of the BGARCH Model with an Application to International Equity Returns,”Journal of Empirical Finance, 7, 305-362.
[20] B Beran, J. (1994) Statistics for long-memory processes. New York: Chapman & Hall. Berkowitz, J. (2001),“Testing Density Forecasts with Applications to Risk Management”Journal of Business and Economic Statistics, 19, 465-474.
[21] Black, F. (1976),“Studies of Stock Market Volatility Changes,”Proceedings of the merican Statistical Association, Business and Economic Statistics Section, 177-181.
[22] B Black, F. and M. Scholes (1973),“The Pricing of Options and Corporate Liabilities,”Journal of Political Economy, 81, 637-654.
[23] Bollerslev, T., R.Y. Chou and K.F. Kroner (1992),“ARCH Modeling in Finance: A Selective Review of the Theory and Empirical Evidence,”Journal of Econometrics, 52, 5-59.
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[26] Bollerslev, T. and H.O. Mikkelsen (1999),“Long-Term Equity Anticipation Securities and Stock Market Volatility Dynamics,”Journal of Econometrics, 92, 75-99.
[27] Bollerslev, T. and J.H. Wright (2001),“Volatility Forecasting, High-Frequency Data, and Frequency Domain Inference,”Review of Economic and Statistics, 83, 596-602.
[28] Bontemps, C. and N. Meddahi (2005),“Testing Normality: A GMM Approach,”Journal of Econometrics, forthcoming.
[29] Braun, P.A., D.B. Nelson and A.M. Sunier (1995),“Good News, Bad News, Volatility, and Betas,”Journal of Finance, 50, 1575-1603.
[30] Brooks, C. (1997),“GARCH Modelling in Finance: A Review of the Software Options,”The Economic Journal, 107, 1271-1276.
[31] Calvet, L. and A. Fisher (2002),“Multifractality in Asset Returns: Theory and Evidence,”Review of Economics and Statistics, 84, 381-406.
[32] Christoffersen, P.F., and F.X. Diebold, (1996),“Further Results on Forecasting and Model Selection under Asymmetric Loss,”Journal of Applied Econometrics, 11, 561-572.
[33] Christoffersen, P.F. and F.X. Diebold (1997),“Optimal Prediction under Asymmetric Loss,”Econometric Theory, 13, 808-817.
[34] Danielsson, J. and J.F. Richard (1993),“Accelerated Gaussian Importance Sampler with Application to Dynamic Latent variable Models,”Journal of Applied Econometrics, 8,S153-S173.
[35] Davidson, J. (2004),“Moment and Memory Properties of Linear Conditional Heteroskedasticity Models, and a New Model,”Journal of Business and Economic Statistics, 22, 16-29.
[36] Diebold, F.X. and R.S. Mariano (1995),“Comparing Predictive Accuracy,”Journal of Business and Economic Statistics, 13, 253-265.
[37] Diebold, F.X. and M. Nerlove (1989),“The Dynamics of Exchange Rate Volatility: A Multivariate Latent Factor ARCH Model,”Journal of Applied Econometrics, 4, 1-21.
[38] Ding, Z. and C.W.J. Granger (1996),“Modeling Volatility Persistence of Speculative Returns: A New Approach,”Journal of Econometrics, 73, 185-215.
[39] Duan, J.-C. (1995),“The GARCH Option Pricing Model,”Mathematical Finance, 5, 13-32.
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