基于随机波动率模型的上证50ETF期权定价研究
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摘要
传统上,期权定价主要基于Black-Scholes (B-S)模型。但B-S模型不能描述时变波动率以及解释"波动率微笑"现象,导致期权定价存在较大的误差。随机波动率模型克服了B-S模型的这些缺陷,能够合理地刻画波动率动态性和波动率微笑。基于此,本文考虑随机波动率模型下的期权定价问题,并针对我国上证50ETF期权进行实证分析。为了解决定价模型的参数估计问题,采用上证50ETF及其期权价格数据,建立两步法对定价模型的参数进行估计。该估计方法保证了定价模型在客观与风险中性测度下的一致性。采用2016年1月到2017年10月的上证50ETF期权价格数据为研究样本,对随机波动率模型进行了实证检验。结果表明,无论是在样本内还是样本外,随机波动率模型相比传统的常数波动率B-S模型都能够获得明显更为精确和稳定的定价结果,B-S模型的定价误差总体偏大且呈现较高波动,凸显了随机波动率对于期权定价的重要性。另外,随机波动率模型对于短期实值期权的定价相比对于其它期权的定价要更精确。
Traditionally, the pricing of options is mainly based on the Black-Scholes(B-S) model. However, it has been well-documented in the finance literature that the B-S model isn't able to describe time-varying volatility and to explain volatility smile/smirk, which leads to substantial pricing errors.To overcome the drawbacks of the B-S model, the stochastic volatility model which can account for the volatility dynamics and volatility simle/smirk has been reasonably proposed. This paper concentrates on the pricing of SSE 50 ETF options under the stochastic volatility model. To estimate the model parameters in a way that maintains the internal consistency of the objective and risk-neutral measures,this paper proposes a two-step approach by using SSE 50 ETF and its option prices data. Using the actual market data on the SSE 50 ETF option prices from January 2016 to October 2017, an empirical investigation is carried out. The results show that the stochastic volatility model produces much more accurate and stable option pricing results than the traditional constant volatility of B-S model both inand out-of-sample, and the B-S model performs badly in the pricing of SSE 50 ETF options with high pricing errors, which suggests the importance of stochastic volatility in option pricing. Moreover, the stochastic volatility model provides more accurate pricing results for the short-term and in-the-money options than for the others.
Traditionally, the pricing of options is mainly based on the Black-Scholes(B-S) model. However, it has been well-documented in the finance literature that the B-S model isn't able to describe time-varying volatility and to explain volatility smile/smirk, which leads to substantial pricing errors.To overcome the drawbacks of the B-S model, the stochastic volatility model which can account for the volatility dynamics and volatility simle/smirk has been reasonably proposed. This paper concentrates on the pricing of SSE 50 ETF options under the stochastic volatility model. To estimate the model parameters in a way that maintains the internal consistency of the objective and risk-neutral measures,this paper proposes a two-step approach by using SSE 50 ETF and its option prices data. Using the actual market data on the SSE 50 ETF option prices from January 2016 to October 2017, an empirical investigation is carried out. The results show that the stochastic volatility model produces much more accurate and stable option pricing results than the traditional constant volatility of B-S model both inand out-of-sample, and the B-S model performs badly in the pricing of SSE 50 ETF options with high pricing errors, which suggests the importance of stochastic volatility in option pricing. Moreover, the stochastic volatility model provides more accurate pricing results for the short-term and in-the-money options than for the others.
引文
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[2] Hull J C, White A D. The pricing of options on asset with stochastic volatilities[J]. Journal of Finance, 1987, 42(2):281-300.
[3] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. Review of Financial Studies, 1993, 6(2):327-343.
[4] Jones C. The dynamics of stochastic volatility:Evidence from underlying and options markets[J].Journal of Econometrics, 2003, 116(1):181-224.
[5] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P 500:Evidence from realized volatility, daily returns, and option prices[J]. Review of Financial Studies, 2010, 23(8):3141-3189.
[6] Ferriani F, Pastorello S. Estimating and testing non-affine option pricing models with a large unbalanced panel of options[J]. Econometrics Journal, 2012, 15:171-203.
[7] Kaeck A, Alexander C. Volatility dynamics for the S&P 500:Further evidence from non-affine,multi-factor jump diffusions[J]. Journal of Banking&Finance, 2012, 36:3110-3121.
[8] Durham G B. Risk-neutral modeling with affine and nonaffine models[J]. Journal of Financial Econometrics, 2013, 11(4):650-681.
[9]吴鑫育,杨文昱,马超群,汪寿阳.基于非仿射随机波动率模型的期权定价研究[J].中国管理科学,2013,21(1):1-7.
[10] Cheng A, Gallant A R, Ji C, Lee B S. A Gaussian approximation scheme for computation of option prices in stochastic volatility models[J]. Journal of Econometrics, 2008, 146(1):44-58.
[11] Amengual D, Xiu D. Resolution of policy uncertainty and sudden declines in volatility[R]. FamaMiller Working Paper, 2017.
[12]郑挺国,宋涛.中国短期利率的随机波动与区制转移性[J].管理科学学报,2011,14(1):38-49.
[13]郑挺国,刘金全.随机波动和跳跃下的短期利率动态[J].系统工程理论实践,2012,32(11):2372-2380.
[14]郝梦,杜子平.基于GARCH-GH模型的上证50ETF期权定价研究[J].数学的实践与认识,2017, 47(5):289-296.
[15]王鹏,杨兴林.基于时变波动率与混合对数正态分布的50ETF期权定价[J].管理科学,2016, 29(4):149-160.
[16]杨兴林,王鹏.基于时变波动率的50ETF参数欧式期权定价[J].数理统计与管理,2018, 37(1):162-178.
[17]方艳,张元玺,乔明哲.上证50ETF期权定价有效性的研究:基于B-S-M模型和蒙特卡罗模拟[J].运筹与管理,2017, 26(8):157-166.
[18] Bakshi G, Cao C, Chen Z W. Empirical performance of alternative option pricing models[J]. Journal of Finance, 1997, 52(5):2003-2049.
[19] Bakshi G, Cao C, Chen Z W. Pricing and hedging long-term options[J]. Journal of Econometrics,2000, 94(1):277-318.
[20] Christoffersen P, Heston S L, Jacobs K. The shape and term structure of the index option smirk:Why multifactor stochastic volatility models work so well[J]. Management Science, 2009, 55(12):1914-1932.
[21] Chernov M, Ghysels E. A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation[J]. Journal of Financial Economics, 2000,56(3):407-458.
[22] Pan J. The jump-risk premia implicit in options:Evidence from an integrated time-series study[J].Journal of Financial Economics, 2002, 63:3-50.
[23] Eraker B. Do stock prices and volatility jump? Reconciling evidence from spot and option prices[J].Journal of Finance, 2004, 59(3):1367-1404.
[24] Johannes M S, Polson N G, Stroud J R. Optimal filtering of jump diffusion:Extracting latent states from asset prices[J]. The Review of Financial Studies, 2009, 22(7):2759-2799.
[25] Hurn A S, Lindsay K A, McClelland A J. Estimating the parameters of stochastic volatility models using option price data[J]. Journal of Business Economic Statistics, 2015, 33(4):579-594.
[26] Duan J C, Yeh C Y. Jump and volatility risk premiums implied by VIX[J]. Journal of Economic Dynamics&Control, 2010, 34:2232-2244.
[27] Kanniainen J, Lin B, Yang H. Estimating and using GARCH models with VIX data for option valuation[J]. Journal of Banking&Finance, 2014, 43:200-211.
[28] Yang H, Kanniainen J. Jump and volatility dynamics for the S&P 500:Evidence for infinite-activity jumps with non-affine volatility dynamics from stock and option markets[J]. Reveiw of Finance,2017, 21(2):811-844.
[29]吴恒煜,朱福敏,温金明.带杠杆效应的无穷纯跳跃Levy过程期权定价[J].管理科学学报,2014, 17(8):74-94.
[30] Chernov M, Gallant A R, Ghysels E, Tauchen G T. Alternative models for stock price dynamics[J].Journal of Econometrics, 2003, 116:225-257.
[31] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models[J]. Journal of Business and Economic Statistics, 1994, 12:371-389.
[32] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models with fat-tails and correlated errors[J]. Journal of Econometrics, 2004, 122:185-212.
[33]吴鑫育,周海林,汪寿阳,马超群.基于EIS的杠杆随机波动率模型的极大似然估计[J].管理科学学报,2013, 16(1):74-86.
[34]郑振龙,郑国忠.高阶矩风险溢酬:信息含量及影响因素[J].数理统计与管理,2017, 36(3):550-570.
[35] Broadie M, Chernov M, Michael J. Model specification and risk premia:Evidence from futures options[J]. Journal of Finance, 2007, 62(3):1453-1490.
[36] Eraker B, Johannes M, Polson N. The impact of jumps in volatility and returns[J]. Journal of Finance, 2003, 53(3):1269-1300.
[37] Duan J C, Fulop A. Estimating the structural credit risk model when equity prices are contaminated by trading noises[J]. Journal of Econometrics, 2009, 150:288-296.
[38] Pitt M K, Malik S, Doucet A. Simulated likelihood inference for stochastic volatility models using continuous particle filtering[J]. Annals of the Institute of Statistical Mathematics, 2014, 66(3):527-552.
[39] Gordon N J, Salmond D J, Smith A F M. Novel approach to nonlinear/non-Gaussian Bayesian state estimation[J]. IEE Proceedings F(Radar and Signal Processing), 1993,140(2):107-113.
[40] Malik S, Pitt M K. Particle filters for continuous likelihood evaluation and maximization[J]. Journal of Econometrics, 2011,165:190-209.
[2] Hull J C, White A D. The pricing of options on asset with stochastic volatilities[J]. Journal of Finance, 1987, 42(2):281-300.
[3] Heston S L. A closed-form solution for options with stochastic volatility with applications to bond and currency options[J]. Review of Financial Studies, 1993, 6(2):327-343.
[4] Jones C. The dynamics of stochastic volatility:Evidence from underlying and options markets[J].Journal of Econometrics, 2003, 116(1):181-224.
[5] Christoffersen P, Jacobs K, Mimouni K. Volatility dynamics for the S&P 500:Evidence from realized volatility, daily returns, and option prices[J]. Review of Financial Studies, 2010, 23(8):3141-3189.
[6] Ferriani F, Pastorello S. Estimating and testing non-affine option pricing models with a large unbalanced panel of options[J]. Econometrics Journal, 2012, 15:171-203.
[7] Kaeck A, Alexander C. Volatility dynamics for the S&P 500:Further evidence from non-affine,multi-factor jump diffusions[J]. Journal of Banking&Finance, 2012, 36:3110-3121.
[8] Durham G B. Risk-neutral modeling with affine and nonaffine models[J]. Journal of Financial Econometrics, 2013, 11(4):650-681.
[9]吴鑫育,杨文昱,马超群,汪寿阳.基于非仿射随机波动率模型的期权定价研究[J].中国管理科学,2013,21(1):1-7.
[10] Cheng A, Gallant A R, Ji C, Lee B S. A Gaussian approximation scheme for computation of option prices in stochastic volatility models[J]. Journal of Econometrics, 2008, 146(1):44-58.
[11] Amengual D, Xiu D. Resolution of policy uncertainty and sudden declines in volatility[R]. FamaMiller Working Paper, 2017.
[12]郑挺国,宋涛.中国短期利率的随机波动与区制转移性[J].管理科学学报,2011,14(1):38-49.
[13]郑挺国,刘金全.随机波动和跳跃下的短期利率动态[J].系统工程理论实践,2012,32(11):2372-2380.
[14]郝梦,杜子平.基于GARCH-GH模型的上证50ETF期权定价研究[J].数学的实践与认识,2017, 47(5):289-296.
[15]王鹏,杨兴林.基于时变波动率与混合对数正态分布的50ETF期权定价[J].管理科学,2016, 29(4):149-160.
[16]杨兴林,王鹏.基于时变波动率的50ETF参数欧式期权定价[J].数理统计与管理,2018, 37(1):162-178.
[17]方艳,张元玺,乔明哲.上证50ETF期权定价有效性的研究:基于B-S-M模型和蒙特卡罗模拟[J].运筹与管理,2017, 26(8):157-166.
[18] Bakshi G, Cao C, Chen Z W. Empirical performance of alternative option pricing models[J]. Journal of Finance, 1997, 52(5):2003-2049.
[19] Bakshi G, Cao C, Chen Z W. Pricing and hedging long-term options[J]. Journal of Econometrics,2000, 94(1):277-318.
[20] Christoffersen P, Heston S L, Jacobs K. The shape and term structure of the index option smirk:Why multifactor stochastic volatility models work so well[J]. Management Science, 2009, 55(12):1914-1932.
[21] Chernov M, Ghysels E. A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation[J]. Journal of Financial Economics, 2000,56(3):407-458.
[22] Pan J. The jump-risk premia implicit in options:Evidence from an integrated time-series study[J].Journal of Financial Economics, 2002, 63:3-50.
[23] Eraker B. Do stock prices and volatility jump? Reconciling evidence from spot and option prices[J].Journal of Finance, 2004, 59(3):1367-1404.
[24] Johannes M S, Polson N G, Stroud J R. Optimal filtering of jump diffusion:Extracting latent states from asset prices[J]. The Review of Financial Studies, 2009, 22(7):2759-2799.
[25] Hurn A S, Lindsay K A, McClelland A J. Estimating the parameters of stochastic volatility models using option price data[J]. Journal of Business Economic Statistics, 2015, 33(4):579-594.
[26] Duan J C, Yeh C Y. Jump and volatility risk premiums implied by VIX[J]. Journal of Economic Dynamics&Control, 2010, 34:2232-2244.
[27] Kanniainen J, Lin B, Yang H. Estimating and using GARCH models with VIX data for option valuation[J]. Journal of Banking&Finance, 2014, 43:200-211.
[28] Yang H, Kanniainen J. Jump and volatility dynamics for the S&P 500:Evidence for infinite-activity jumps with non-affine volatility dynamics from stock and option markets[J]. Reveiw of Finance,2017, 21(2):811-844.
[29]吴恒煜,朱福敏,温金明.带杠杆效应的无穷纯跳跃Levy过程期权定价[J].管理科学学报,2014, 17(8):74-94.
[30] Chernov M, Gallant A R, Ghysels E, Tauchen G T. Alternative models for stock price dynamics[J].Journal of Econometrics, 2003, 116:225-257.
[31] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models[J]. Journal of Business and Economic Statistics, 1994, 12:371-389.
[32] Jacquier E, Polson N G, Rossi P E. Bayesian analysis of stochastic volatility models with fat-tails and correlated errors[J]. Journal of Econometrics, 2004, 122:185-212.
[33]吴鑫育,周海林,汪寿阳,马超群.基于EIS的杠杆随机波动率模型的极大似然估计[J].管理科学学报,2013, 16(1):74-86.
[34]郑振龙,郑国忠.高阶矩风险溢酬:信息含量及影响因素[J].数理统计与管理,2017, 36(3):550-570.
[35] Broadie M, Chernov M, Michael J. Model specification and risk premia:Evidence from futures options[J]. Journal of Finance, 2007, 62(3):1453-1490.
[36] Eraker B, Johannes M, Polson N. The impact of jumps in volatility and returns[J]. Journal of Finance, 2003, 53(3):1269-1300.
[37] Duan J C, Fulop A. Estimating the structural credit risk model when equity prices are contaminated by trading noises[J]. Journal of Econometrics, 2009, 150:288-296.
[38] Pitt M K, Malik S, Doucet A. Simulated likelihood inference for stochastic volatility models using continuous particle filtering[J]. Annals of the Institute of Statistical Mathematics, 2014, 66(3):527-552.
[39] Gordon N J, Salmond D J, Smith A F M. Novel approach to nonlinear/non-Gaussian Bayesian state estimation[J]. IEE Proceedings F(Radar and Signal Processing), 1993,140(2):107-113.
[40] Malik S, Pitt M K. Particle filters for continuous likelihood evaluation and maximization[J]. Journal of Econometrics, 2011,165:190-209.
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