18 http efinance org cn zlzhengxmu edu cn

第 18章 期权的风险管理与复制 郑振龙 厦门大学金融系 厦门大学金融 程研究中心 http: // efinance. org. cn zlzheng@xmu. edu. cn Copyright © 2013 Zhenlong Zheng

Example A bank has sold for $300, 000 a European call option on 100, 000 shares of a non-dividend paying stock S 0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% The Black-Scholes-Merton value of the option is $240, 000 How does the bank hedge its risk to lock in a $60, 000 profit? Copyright © 2013 Zhenlong Zheng 3

Naked & Covered Positions Naked position Take no action Covered position Buy 100, 000 shares today What are the risks associated with these strategies? Copyright © 2013 Zhenlong Zheng 4

Stop-Loss Strategy This involves: Buying 100, 000 shares as soon as price reaches $50 Selling 100, 000 shares as soon as price falls below $50 Copyright © 2013 Zhenlong Zheng 5

Stop-Loss Strategy continued Ignoring discounting, the cost of writing and hedging the option appears to be max(S 0−K, 0). What are we overlooking? Copyright © 2013 Zhenlong Zheng 6

期权价值的影响因素 Copyright © 2013 Zhenlong Zheng 7

18. 2 Delta 与期权的套期保值 Copyright © 2013 Zhenlong Zheng 8

期权的Delta（Δ） Copyright © 2013 Zhenlong Zheng 9

期权的Delta（Δ） 看涨期权价格 斜率 = D B A Copyright © 2013 Zhenlong Zheng 股价 10

期权Δ的性质 Copyright © 2013 Zhenlong Zheng 12

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期权Δ值与�� 之间的关系(cont. ) Copyright © 2013 Zhenlong Zheng 19

证券组合的Δ值 Copyright © 2013 Zhenlong Zheng 21

案例1 ：期权的Delta 中性保值 (cont. ) 周次 股价 Delta 购买的股 数 成本 (千 累计成本 (千 元) 元) 利息（千 元） 0 49. 00 0. 522 52, 200 2, 557. 8 2. 5 1 48. 12 0. 458 (6, 400) (308. 0) 2, 252. 3 2. 2 2 47. 37 0. 400 (5, 800) (274. 7) 1, 979. 8 1. 9 . . . 19 55. 87 1. 000 1, 000 55. 9 5, 258. 2 5. 1 20 57. 25 1. 000 0 0 5263. 3 Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 26

案例1 ：期权的Delta 中性保值 (cont. ) 周次 股价 Delta 购买的股 数 0 49. 00 0. 522 52, 200 1 49. 75 0. 568 2 52. 00 . . . . 成本 (千 累计成本 (千 元) 元) 利息（千 元） 2, 557. 8 2. 5 4, 600 228. 9 2, 789. 2 2. 7 0. 705 13, 700 712. 4 3, 504. 3 3. 4 . . 19 46. 63 0. 007 (17, 600) (820. 7) 290. 0 0. 3 20 48. 12 0. 000 (700) (33. 7) 256. 6 Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 27

18. 3 Theta与期权的套期保值 Copyright © 2013 Zhenlong Zheng 31

Θ 值的定义 Copyright © 2013 Zhenlong Zheng 32

Copyright © 2013 Zhenlong Zheng 34

无收益资产看涨期权Theta值与S的关系 Copyright © 2013 Zhenlong Zheng 35

无收益资产看涨期权Theta值与有效期 之间的关系 Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 36

Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 37

Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 38

18. 4 Gamma与期权的套期保值 Copyright © 2013 Zhenlong Zheng 40

期权Γ 定义 Copyright © 2013 Zhenlong Zheng 41

期权Γ 定义(cont. ) 期权 价格 C'' C' C 股价 S S' Copyright@2013 Zheng Copyright © 2013 Zhenlong Zheng 42

Γ 值的计算 Copyright © 2013 Zhenlong Zheng 43

Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 44

无收益资产看涨期权和欧式看跌期权Gamma 值与T-t的关系 Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 45

Copyright@2013 Zheng Copyright. Zhenlong © 2013 Zhenlong Zheng 46

证券组合的Γ 值 Copyright © 2013 Zhenlong Zheng 47

案例2 ：Gamma 中性(cont. ) Copyright © 2013 Zhenlong Zheng 50

Δ、Θ 和Γ 之间的关系(cont. ) 对于一个Δ 中性组合，, DP » Q Dt + ½GDS 2 DP DP DS DS 正Gamma 负Gamma Copyright © 2013 Zhenlong Zheng 52

Δ、Θ 和Γ 的符号 Copyright © 2013 Zhenlong Zheng 53

18. 5 Vega 、Rho与期权的套期保值 Copyright © 2013 Zhenlong Zheng 54

n的定义 Copyright © 2013 Zhenlong Zheng 55

Vega Copyright © 2013 Zhenlong Zheng 56

Copyright © 2013 Zhenlong Zheng 57

平价期权 c/S与波动率和期限的关系 Copyright © 2013 Zhenlong Zheng 60

n的计算 Copyright © 2013 Zhenlong Zheng 61

n的计算 Copyright © 2013 Zhenlong Zheng 62

n中性 Copyright © 2013 Zhenlong Zheng 63

rho 与套期保值 Copyright © 2013 Zhenlong Zheng 64

Copyright © 2013 Zhenlong Zheng 65

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18. 6 期权复制 Copyright © 2013 Zhenlong Zheng 67

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交易费用与期权复制 Copyright © 2013 Zhenlong Zheng 70

本讲参考资料 郑振龙和陈蓉，《金融 程》（第三版），高等教育 出版社，2012：第 14章 John C. Hull, Options, Futures and other derivatives, 8 th ed. , Pearson, 2012: Ch 18 Dan Passarelli, Trading Option Greeks: How Time, Volatility and other Pricing Factors Drive Profit, 2008 http: //efinance. org. cn/本科课程金融 程 http: //efinance. org. cn/硕士生课程金融 程 http: //reference. wolfram. com/applications/finance/Option s. html Copyright © 2013 Zhenlong Zheng 71

请提问 Any Questions？ Copyright@2013 Zheng Copyright © 2013 Zhenlong Zheng 72

The End. Copyright © 2013 Zhenlong Zheng 73