BS模型下的雪球定价

标的资产价格遵循Black-Scholes随机微分方程

\[ dS = \mu Sdt+\sigma SdW \]

其中 \(\mu\), \(\sigma\) 是常数， \(dW\) 是一个维纳过程。

标准雪球,挂钩中证500

雪球合约

起始日

2022-08-03

到期日

2023-08-03

期初标的价格

6472.52

敲入观察日

每日

敲入障碍价格

5178.016(80%)

敲入赔付

-\(\text{Max}(S_0 - S_T, 0)\)

敲出观察日

每月

敲出障碍价格

6472.52 (100%)

利息率

25.63% (按自然日/365天年化)

类型

向上敲出向下敲入

敲出观察日 :

2022-09-05, 2022-10-10, 2022-11-03, 2022-12-05, 2023-01-04, 2023-02-03

2023-03-03, 2023-04-06, 2023-05-04, 2023-06-05, 2023-07-03, 2023-08-03

定价测试

测试案例A1 - 未敲入

定价参数

valuation date

2022-08-03

End of day valuation

risk free rate

2.00%

b

-4.00%

\(b = r - q, F = e^{-qT}S\)

\(\sigma\)

22%

定价结果

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

-196.0527043

-196.00972802 \(\pm\) 1.05E+00

4.30E-02

6213.619

-148.0428289

-148.24677039 \(\pm\) 1.67E+00

2.04E-01

6278.344

-104.4854665

-104.57889976 \(\pm\) 1.39E+00

9.34E-02

6343.070

-65.3566709

-65.29542971 \(\pm\) 6.90E-01

6.12E-02

6407.795

-30.5691389

-30.08973053 \(\pm\) 7.90E-01

4.79E-01

6472.520

0.0323706

-0.17733488 \(\pm\) 1.00E+00

2.10E-01

6537.245

26.6515177

27.01615853 \(\pm\) 1.02E+00

3.65E-01

6601.970

49.5457148

49.84049483 \(\pm\) 8.23E-01

2.95E-01

6666.696

69.0093770

69.07223885 \(\pm\) 7.81E-01

6.29E-02

6731.421

85.3611400

85.3839709 \(\pm\) 5.43E-01

2.28E-02

6796.146

98.9340196

99.11063364 \(\pm\) 5.88E-01

1.77E-01

image info

测试案例B

定价参数

valuation date

2023-01-05

End of day valuation

risk free rate

2.00%

b

-4.00%

\(b = r - q, F = e^{-qT}S\)

\(\sigma\)

22%

定价结果 B1: 未敲入

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

427.325062

427.9228192 \(\pm\) 1.38E+00

5.98E-01

6213.619

492.7445585

493.1424038 \(\pm\) 1.47E+00

3.98E-01

6278.344

550.9069733

551.2859545 \(\pm\) 1.07E+00

3.79E-01

6343.070

601.9983288

602.2868404 \(\pm\) 7.99E-01

2.89E-01

6407.795

646.2981965

646.5486041 \(\pm\) 6.78E-01

2.50E-01

6472.520

684.1838004

684.4620590 \(\pm\) 8.51E-01

2.78E-01

6537.245

716.1099193

716.2424969 \(\pm\) 7.60E-01

1.33E-01

6601.970

742.6105229

742.6392973 \(\pm\) 8.23E-01

2.88E-02

6666.696

764.2603656

764.1403718 \(\pm\) 8.55E-01

1.20E-01

6731.421

781.6548032

781.7166434 \(\pm\) 5.46E-01

6.18E-02

6796.146

795.3989672

795.5211704 \(\pm\) 6.30E-01

1.22E-01

image info

定价结果 B2: 已敲入

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

38.7057021

38.3458455 \(\pm\) 7.10E-01

3.60E-01

6213.619

128.6549666

128.3170770 \(\pm\) 9.48E-01

3.38E-01

6278.344

215.6031344

215.5695826 \(\pm\) 6.06E-01

3.36E-02

6343.070

298.5808561

298.6465002 \(\pm\) 7.57E-01

6.56E-02

6407.795

376.6242609

376.4077947 \(\pm\) 5.65E-01

2.16E-01

6472.520

448.8585180

448.6964908 \(\pm\) 5.25E-01

1.62E-01

6537.245

514.5598903

514.0085162 \(\pm\) 6.46E-01

5.51E-01

6601.970

573.2317933

572.9944114 \(\pm\) 8.47E-01

2.37E-01

6666.696

624.6220783

624.5185256 \(\pm\) 8.50E-01

1.04E-01

6731.421

668.7400067

669.0810299 \(\pm\) 4.65E-01

3.41E-01

6796.146

705.8501155

706.0451325 \(\pm\) 4.84E-01

1.95E-01

image info

反向雪球

雪球合约

起始日

2022-08-03

到期日

2023-08-03

期初标的价格

6472.52

敲入观察日

daily

敲入障碍价格

7767.024 （120%）

敲入赔付

-\(\text{Max}(S_T - S_0, 0)\)

敲出观察日

monthly

敲出障碍价格

6472.52 （100%）

票息率

15.72%（按自然日/365天年化）

类型

向下敲出向上敲入

敲出观察日 :

2022-09-05, 2022-10-10, 2022-11-03, 2022-12-05, 2023-01-04, 2023-02-03

2023-03-03, 2023-04-06, 2023-05-04, 2023-06-05, 2023-07-03, 2023-08-03

定价测试

测试案例 C

定价参数

valuation date

2022-08-03

End of day valuation

risk free rate

2.00%

b

-4.00%

\(b = r - q, F = e^{-qT}S\)

\(\sigma\)

22%

定价结果

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

65.3448933

65.3452647 \(\pm\) 2.98E-01

3.71E-04

6213.619

56.6940233

56.7330567 \(\pm\) 4.18E-01

3.90E-02

6278.344

46.0518268

46.0270761 \(\pm\) 4.22E-01

2.48E-02

6343.070

33.1932399

33.1540734 \(\pm\) 4.53E-01

3.92E-02

6407.795

17.9173506

17.6653259 \(\pm\) 4.85E-01

2.52E-01

6472.520

0.0507756

-0.0731201 \(\pm\) 6.13E-01

1.24E-01

6537.245

-20.5433362

-20.5894126 \(\pm\) 9.12E-01

4.61E-02

6601.970

-43.9599523

-44.1733741 \(\pm\) 7.55E-01

2.13E-01

6666.696

-70.2579891

-70.3019983 \(\pm\) 4.40E-01

4.40E-02

6731.421

-99.4516585

-99.8852855 \(\pm\) 8.58E-01

4.34E-01

6796.146

-131.5212902

-131.7481609 \(\pm\) 8.76E-01

2.27E-01

image info

测试案例 D

定价参数

valuation date

2023-01-05

End of day valuation

risk free rate

2.00%

b

-4.00%

\(b = r - q, F = e^{-qT}S\)

\(\sigma\)

22%

定价结果 D1: 未敲入

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

480.6164666

480.5418820 \(\pm\) 3.08E-01

7.46E-02

6213.619

468.2669310

468.1959089 \(\pm\) 3.39E-01

7.10E-02

6278.344

452.6208759

452.9667710 \(\pm\) 6.03E-01

3.46E-01

6343.070

433.2865243

433.6995642 \(\pm\) 5.45E-01

4.13E-01

6407.795

409.9354642

410.2911129 \(\pm\) 4.63E-01

3.56E-01

6472.520

382.3143624

382.4486253 \(\pm\) 4.55E-01

1.34E-01

6537.245

350.2504395

350.5597632 \(\pm\) 4.22E-01

3.09E-01

6601.970

313.6566351

313.8583732 \(\pm\) 3.93E-01

2.02E-01

6666.696

272.5210791

272.4193765 \(\pm\) 4.14E-01

1.02E-01

6731.421

226.8982120

226.8036417 \(\pm\) 5.31E-01

9.46E-02

6796.146

176.8869928

176.6121000 \(\pm\) 9.45E-01

2.75E-01

image info

定价结果 D2: 已敲入

标的价格

pv (积分法)

pv (10M蒙卡)

误差

6148.894

457.9406897

457.9956641 \(\pm\) 2.88E-01

5.50E-02

6213.619

438.2313091

438.1506878 \(\pm\) 2.11E-01

8.06E-02

6278.344

413.9737716

413.9394246 \(\pm\) 3.48E-01

3.43E-02

6343.070

384.9055300

384.8481964 \(\pm\) 3.57E-01

5.73E-02

6407.795

350.9203744

350.8858678 \(\pm\) 4.53E-01

3.45E-02

6472.520

312.0694606

312.0674783 \(\pm\) 7.53E-01

1.98E-03

6537.245

268.5470142

268.5783158 \(\pm\) 8.07E-01

3.13E-02

6601.970

220.6706281

220.9279611 \(\pm\) 8.80E-01

2.57E-01

6666.696

168.8425323

169.2444951 \(\pm\) 6.48E-01

4.02E-01

6731.421

113.5154808

113.7365403 \(\pm\) 8.22E-01

2.21E-01

6796.146

55.1471061

55.0962627 \(\pm\) 8.87E-01

5.08E-02

image info

NOTE

1. 误差  = |pv (积分法) - pv (10M蒙卡)| ;
2. 根据中心极限定理，蒙特卡洛计算的pv是正态分布的。pv是通过10次一万次蒙特卡洛计算出来的， 取10次计算的平均值作为最终pv(相当于10M蒙特卡洛)，标准差作为误差项;