[1] BLACKF, SCHOLES M. The pricing of options and corporate liabilities[J]. The Journal of Political Economy, 1973, 81: 637-659. [2] MERTON R C. Theory of rational option pricing[J]. Bell Journal of Economics and Management Science, 1973, 4(1): 141-183. [3] RUBINSTEIN M, REINER E. Breaking down the barrier[J]. Risk, 1991, 4: 28-35. [4] HEYNEN P, KAT H. Partial barrier options[J]. Journal of Financial Engineering, 1994, 3: 253-274. [5] HEYNEN P, KAT H. Discrete partial barrier options with a moving barrier[J]. Journal of Financial Engineering, 1994, 5: 199-209. [6] CARR P. Two extension to barrier option valuation[J]. Applied Mathematical Finance, 1995, 2: 173-209. [7] XING L. Application of finite difference methods in stock option pricing[J]. Science Technology and Engineer, 2007, 7(19): 5192-5195. [8] CAROLE B, PHELIM B. Monte Carlo methods for pricing discrete Parisian options[J].The European Journal of Finance, 2011, 17(3): 169-196. [9] Derman E, Ergener D, Kani I. Forever hedged[J]. Risk, 1994, 7: 139-145. [10] CARR P, CHOU A. Hedging complex barrier options[R]. Cambridge, MA: Morgan Stanley and MIT Computer Science, 1997. [11] CARR P, ELLIS K, GUPTA V. Static hedging of exotic options[J]. Journal of Finance, 1998, 53(3): 1165-1190. [12] CARR P, PICRON J. Static hedging of timing risk[J]. Journal of Derivatives, 1999, 6 (3): 57-70. [13] TOMPKINS R. Static versus dynamic hedging of exotic options: An evaluation of hedge performance via simulation[J]. Journal of Risk Finance, 2002, 3: 6-34. [14] CVITANI J, PHAM H, TOUZI N. Super-replication in stochastic volatility models under portfolio constraints[J].Applied Probability Trust, 1999, 36(2): 523-545. [15] CVITANI J, PHAM H, TOUZI N. Hedging in discrete time under transaction costs and continuous-time limit [J]. Applied Probability Trust, 1999, 36(1): 163-178. [16] JUN D, KU H. Static hedging of chained-type barrier options[J]. North American Journal of Economics and Finance, 2015, 33: 317-327. [17] CARR P P, JARROW R A. The stop-loss start-gain paradox and option valuation: A new decomposition into intrinsic and time value[J]. The Review of Financial Studies, 1990, 3(3): 469-482. [18] RUBINSTEIN M, REINER E. Breaking down the barrier[J]. Risk,1991,4: 28-35. [19] 储国强,卫剑波,王琦.沪深300指数障碍期权的动态对冲研究[J].武汉金融,2014(12): 25-29.
附录 该期权产品参数为1年期,执行价格102,障碍价格118,无风险利率7%,波动率25%.值得注意的是,不同产品参数背景下,该最优触发价格和新的虚拟障碍价格外移边界不同,因此针对不同期权参数,触发价格和虚拟障碍价格不同,需要重新进行遍历计算.
表A1 单纯外移边界法对冲结果 Tab.A1 Hedge result of pure outward moving barrier boundary
外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数 0.5 118.5 1.194 4 1.868 1 13.004 2 19.506 2 7.538 8 10.704 1 1 119 1.206 3 1.813 8 11.716 2 16.296 2 7.384 10.314 7 1.5 119.5 1.219 1.776 5 11.074 3 14.175 7.302 4 10.023 2 2 120 1.232 3 1.748 8 10.640 2 13.489 7.235 4 9.773 2.5 120.5 1.246 1.728 7 10.304 12.940 8 7.176 7 9.539 7 3 121 1.259 9 1.715 7 10.017 9 12.423 3 7.124 8 9.352 5 123 1.443 1.524 8 16.871 7 8.899 9 5.436 1 6.713 7 6.43 124.43 1.5151 1.756 8 16.346 9 8.690 9 5.502 3 6.697
表A2 蝶式外移边界法对冲结果 Tab.A2 Hedge result of spread outward moving barrier boundary
外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数 0.5 118.5 1.008 3 1.961 3 12.393 3 20.353 9 7.585 6 10.938 1 119 1.154 1 1.850 6 11.717 7 17.226 4 7.391 9 10.425 1.5 119.5 1.184 2 1.761 8 11.244 1 15.326 9 7.307 8 10.111 7 2 120 1.205 4 1.760 8 10.875 2 14.156 7.242 1 9.854 3 2.5 120.5 1.223 1 1.735 10.541 2 13.566 1 7.185 3 9.631 3 3 121 1.239 1 1.716 8 10.220 2 13.056 3 7.133 7 9.438 9 5 123 1.297 7 1.698 4 9.072 1 11.557 9 6.979 3 8.908 4 6.43 124.43 1.339 4 1.725 7 8.484 1 10.845 6 6.903 9 8.650 6 8 126 1.386 8 1.784 7 8.097 1 10.325 3 6.851 4 8.470 3
表A3 价差外移边界法 Tab.A3 Hedge result of butterfly outward moving barrier boundary
外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数 0.5 118.5 1.191 1 1.870 1 13.006 2 19.532 7 7.54 10.711 1 119 1.203 1.815 6 11.725 5 16.354 4 7.384 4 10.321 5 1.5 119.5 1.215 8 1.778 11.094 1 14.239 8 7.302 9 10.031 4 2 120 1.229 1.749 9 10.673 3 13.572 3 7.236 1 9.782 8
续表A3
外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数
2.5 120.5 1.242 4 1.729 3 10.330 8 13.038 5 7.177 7 9.552 1 3 121 1.256 1.715 4 10.033 7 12.54 2 7.126 2 9.364 4 5 123 1.311 8 1.714 4 9.241 6 10.889 7 6.970 7 8.829 7 6.43 124.43 1.353 8 1.755 5 8.950 2 10.546 6 6.895 3 8.601 9 8 126 1.401 9 1.827 9 8.757 2 10.443 2 6.849 8.447 1
表A4 遍历法触发式外移边界法 Tab.A4 hedge result of traversal trigger outward moving barrier boundary
触发价格 外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数
-1 0.5 118.5 1.147 1 1.855 4 13.786 5 19.506 2 7.478 7 10.762 9 1 119 1.140 5 1.82 12.703 4 16.296 2 7.379 3 10.512 1.5 119.5 1.133 3 1.799 8 12.273 7 14.184 7 7.348 7 10.339 5 2 120 1.125 4 1.787 9 12.041 5 14.184 7 7.326 9 10.197 8 2.5 120.5 1.116 6 1.782 3 11.875 5 14.184 7 7.307 9 10.081 7 -2 0.5 118.5 1.107 4 1.794 8 13.93 2 19.506 2 7.235 6 10.374 1 1 119 1.101 2 1.757 2 12.511 3 16.296 2 7.125 10.061 8 1.5 119.5 1.094 4 1.734 4 11.785 7 14.175 7.085 6 9.824 9 2 120 1.086 9 1.720 3 11.315 6 13.48 9 7.055 8 9.616 5 2.5 120.5 1.078 4 1.713 1 10.964 8 12.940 8 7.029 4 9.435 2 3 121 1.069 1.712 2 10.696 2 12.422 3 7.007 1 9.294 8 4 122 1.047 4 1.726 3 10.369 8 11.432 4 6.971 8 9.079 7 -3 0.5 118.5 1.078 6 1.765 1 14.545 7 19.506 2 7.118 3 10.309 1 119 1.072 2 1.725 2 12.961 2 16.269 2 6.994 3 9.967 7 1.5 119.5 1.062 2 1.700 7 12.11 2 14.17 5 6.944 4 9.700 8 2 120 1.059 1.685 11.539 3 13.48 9 6.907 9 9.459 3 2.5 120.5 1.050 9 1.765 11.099 4 12.940 8 6.874 5 9.249 4 3 121 1.041 9 1.674 5 10.75 7 12.423 3 6.843 7 9.084 4 4 122 1.021 1 1.687 1 10.334 3 11.432 4 6.795 8.821 8 -4 0.5 118.5 1.047 2 1.738 3 15.196 6 19.506 2 7.007 5 10.275 2 1 119 1.041 1 1.694 7 13.473 7 16.269 2 6.86 9.901 8 1.5 119.5 1.034 4 1.667 2 12.520 2 14.175 6.791 3 9.617 3 2 120 1.027 2 1.649 5 11.859 4 13.489 6.7395 9.363 8 2.5 120.5 1.079 2 1.639 5 11.34 3 12.940 8 6.694 4 9.145 4 3 121 1.011 4 1.636 5 10.936 2 12.423 3 6.655 8.958 1 4 122 0.990 4 1.648 5 10.426 2 11.432 4 6.592 9 8.665 9
续表A4
触发价格 外移边界 新障碍价格 动态对冲平均成本 标准差 峰度 极大值 上5%分位数 上1%分位数
-5 0.5 118.5 1.021 3 1.718 2 15.767 3 19.506 2 6.931 8 10.239 6 1 119 1.015 6 1.673 8 13.977 1 16.296 2 6.778 9 9.870 1 1.5 119.5 1.009 4 1.645 8 12.982 6 14.175 6.706 4 9.594 4 2 120 1.002 8 1.627 6 12.287 3 13.489 6.652 9.346 6 2.5 120.5 0.995 4 1.617 3 11.737 3 12.940 8 6.605 7 9.134 5 3 121 0.987 2 1.614 3 11.297 7 12.423 3 6.565 7 8.951 3 4 122 0.968 6 1.626 5 10.729 4 11.432 4 6.531 3 8.663 1 4.5 122.5 0.958 3 1.640 5 10.565 4 10.958 4 6.478 2 8.554 7 5 123 0.947 3 1.658 9 10.453 5 10.606 7 6.459 9 8.469 2 8 126 0.869 8 1.828 10.222 9 10.594 1 6.411 5 8.211 8 10 128 0.809 1 1.994 9 10.019 8 10.330 1 6.432 8 8.168 -6 1 119 0.983 5 1.654 7 14.619 2 16.296 2 6.7187 9.852 8 2 120 0.971 8 1.607 8 12.854 6 13.48 9 6.590 7 9.329 3 2.5 120.5 0.965 1.597 3 12.274 8 12.940 8 6.543 9 9.118 8 3 121 0.957 4 1.594 2 11.806 4 12.423 3 6.504 1 8.937 3 3.5 121.5 0.949 2 1.597 6 11.447 1 11.916 7 6.470 1 8.785 9 4 122 0.940 2 1.606 7 11.183 3 11.432 4 6.442 1 8.652 9 4.5 122.5 0.930 6 1.621 10.992 1 10.985 4 6.419 8 8.547 2 5 123 0.920 4 1.639 8 10.852 7 10.606 7 6.402 7 8.463 1 5.5 123.5 0.906 6 1.662 7 10.749 7 10.652 6.389 1 8.392 9 -6.43 4 122 0.923 5 1.595 1 11.409 5 11.432 4 6.398 8.628 7
() () |