为避免参数法和半参数法的事前模型设定偏误和参数估计误差，本文引入无需做事前分布假设的非参数核估计法对VaR和CVaR进行估计，并基于VaR和CVaR的核估计量构建风险对冲模型，实现风险估计和风险对冲同时进行。在此基础上，本文进一步在核估计框架内引入期望效用理论比较最小方差、最小VaR和最小CVaR对冲策略的对冲效率，以期解决传统文献将风险下降比率作为风险对冲效率指标，却因风险度量指标不同而导致比较结果不一致的问题。最后，将核估计框架下的风险对冲模型和期望效用理论运用到沪深300股指期货现货的风险对冲问题，实证结果表明：最小CVaR对冲策略的对冲效率优于最小方差和最小VaR对冲策略，且四种效用函数给出的比较结果一致。
Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR) are two main popular risk measurement tools presently. However, the present study on risk hedging problem with VaR (CVaR) is mostly carried out under specific distribution assumptions, which is prone to resulting in model risk and limiting its scope of application in practice. In addition, the traditional literature only defines risk reducing ratio as the risk hedging efficiency index, but different risk measure indices often induce inconsistent or even contradictory results. Therefore we need to seek a hedging efficiency index which is independent of risk measure indices. Method:To overcome the shortcomings above and improve existing results, the nonparametric kernel estimation method is introduced to estimate VaR and CVaR without a distribution assumption, and then the risk hedging model is constructed based on the VaR and CVaR kernel estimators, which can avoid the ex-ante model risk and parametric estimation error. In addition, expected utility theory is further applied to compare the hedging efficiency of risk-minimizing hedging strategies so as to avoid the inconsistent or even contradictory comparison results that are often induced by different risk decline ratios in traditional literatures.Data:The historical data of CSI 300 stock index and its futures is collected to test the theorem above. The data window ranges from April 16, 2010 to February 11, 2015, a total of 1172 daily data.Results:The empirical results based on CSI 300 index futures and spot market data show that four kinds of utility functions used by financial economics confirm consistently that minimum CVaR hedging strategy is more efficient than the minimum variance and the minimum VaR hedging strategies.Future research:The purpose of this paper is to provide a research framework which studies hedging problem under distribution uncertainty using kernel estimation method and expected utility theory. The research framework can be easily applied to hedging efficiency problem of other derivatives or other risk measure indices. So this paper provides a new research perspective for other scholar's related research.
[1] Johnson L L. The theory of hedging and speculation in commodity futures[J]. The Review of Economic Studies, 1960, 27(3):139-151.
[2] Ederington L H. The hedging performance of the new futures markets[J]. The Journal of Finance, 1979, 34(1):157-170.
[3] Myers R J, Thompson S R. Generalized optimal hedge ratio estimation[J]. American Journal of Agricultural Economics, 1989, 71(4):858-868.
[4] 迟国泰, 余方平, 刘轶芳. 基于VaR的期货最优套期保值模型及应用研究[J]. 系统工程学报, 2008, 23(4):417-423.
[5] 迟国泰, 赵光军, 杨中原. 基于CVaR的期货最优套期保值比率模型及应用[J]. 系统管理学报, 2009, 18(1):27-33.
[6] Hung J C, Chiu C L, Lee M C. Hedging with zero-value at risk hedge ratio[J]. Applied Financial Economics, 2006, 16(3):259-269.
[7] Harris R D F, Shen Jian. Hedging and value at risk[J]. Journal of Futures Markets, 2006, 26(4):369-390.
[8] Cao Zhigang, Harris R D F, Shen Jian. Hedging and value at risk:A semi-parametric approach[J]. Journal of Futures Markets, 2010, 30(8):780-794.
[9] 刘晓倩, 周勇. 金融风险管理中ES度量的非参数方法的比较及其应用[J]. 系统工程理论与实践, 2011, 31(4):631-642.
[10] 刘晓倩, 周勇. 金融风险管理中VaR度量的光滑估计效率[J]. 应用数学学报, 2011, 34(4):752-768.
[11] 刘玉涛, 刘鹏, 周勇. 竞争风险下剩余寿命分位数的光滑非参数估计[J]. 应用数学学报, 2015, 38(1):109-124.
[12] 刘晓倩, 周勇. 加权复合分位数回归方法在动态VaR风险度量中的应用[J]. 中国管理科学, 2015, 23(6):1-8.
[13] 谢尚宇, 姚宏伟, 周勇. 基于ARCH-Expectile方法的VaR和ES尾部风险测量[J]. 中国管理科学, 2014, 22(9):1-9.
[14] Taylor J W. Using exponentially weighted quantile regression to estimate value at risk and expected shortfall[J]. Journal of Financial Econometrics, 2008, 6(3):382-406.
[15] Taylor J W. Estimating value at risk and expected shortfall using expectiles[J]. Journal of Financial Econometrics, 2008, 6(2):231-252.
[16] Cai Zongwu, Wang Xian. Nonparametric estimation of conditional VaR and expected shortfall[J]. Journal of Econometrics, 2008, 147(1):120-130.
[17] Yu Keming, Ally A K, Yang Shanchao, et al. Kernel quantile-based estimation of expected shortfall[J]. The Journal of Risk, 2010, 12(4):15.
[18] Wu Weibiao, Yu Keming, Mitra G. Kernel conditional quantile estimation for stationary processes with application to conditional value-at-risk[J]. Journal of Financial Econometrics, 2008, 6(2):253-270.
[19] Alemany R, Bolancé C, Guillén M. A nonparametric approach to calculating value-at-risk[J]. Insurance:Mathematics and Economics, 2013, 52(2):255-262.
[20] Rassoul A. Kernel-type estimator of the conditional tail expectation for a heavy-tailed distribution[J]. Insurance:Mathematics and Economics, 2013, 53(3):698-703.
[21] Yao Haixiang, Li Zhongfei, Lai Yongzeng. Mean-CVaR portfolio selection:A nonparametric estimation framework[J]. Computers & Operations Research, 2013, 40(4):1014-1022.
[22] Yao Haixiang, Li Yong, Benson K. A smooth non-parametric estimation framework for safety-first portfolio optimization[J]. Quantitative Finance, 2015, 15(11):1865-1884.
[23] 姚海祥, 李仲飞. 基于非参数估计框架的期望效用最大化最优投资组合[J]. 中国管理科学, 2014, 22(1):1-9.
[24] 黄金波, 李仲飞, 姚海祥. 基于CVaR核估计量的风险管理[J]. 管理科学学报, 2014, 17(3):49-59.
[25] 黄金波, 李仲飞, 周先波. VaR与CVaR的敏感性凸性及其核估计[J]. 中国管理科学, 2014, 22(8):1-9.
[26] 黄金波, 李仲飞, 周鸿涛. 期望效用视角下的风险对冲效率[J]. 中国管理科学, 2016, 24(3):9-17.
[27] 黄金波, 李仲飞, 姚海祥. 基于CVaR两步核估计量的投资组合管理[J]. 管理科学学报, 2016, 19(5):114-126.
[28] Li Qi, Racine J S. Nonparametric econometrics:theory and practice[M].Princeton:Princeton University Press, 2007.
[29] Jorion P. Value at risk:The new benchmark for managing financial risk[M].New York:McGraw-Hill, 2007.
[30] Rockafellar R T, Uryasev S. Optimization of conditional value-at-risk[J]. The Journal of Risk, 2000, 2:21-42.
[31] Rockafellar R T, Uryasev S. Conditional value-at-risk for general loss distributions[J]. Journal of Banking & Finance, 2002, 26(7):1443-1471.
