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Chapter 6 二叉树模型介绍

Chapter 6 二叉树模型介绍. 一个简单的二叉树模型. 股票的现价为 $20 三个月之后股票的价格或为 $22 或为 $18. Stock Price = $22. Stock price = $20. Stock Price = $18. 一份看涨期权. 一份基于该股票的三个月到期的看涨期权,其执行价格为 $ 21. Stock Price = $22 Option Price = $1. Stock price = $20 Option Price=?. Stock Price = $18 Option Price = $0.

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Chapter 6 二叉树模型介绍

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  1. Chapter 6二叉树模型介绍

  2. 一个简单的二叉树模型 • 股票的现价为 $20 • 三个月之后股票的价格或为 $22 或为 $18 Stock Price = $22 Stock price = $20 Stock Price = $18

  3. 一份看涨期权 一份基于该股票的三个月到期的看涨期权,其执行价格为$ 21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0

  4. 22D – 1 18D 构造无风险资产组合 • 考虑一个资产组合: 持有 D份股票 成为一份看涨期权的空头 • 当 22D – 1 = 18D or D = 0.25,资产组合是无风险的

  5. 资产组合的估值( 无风险利率为 12% ) • 无风险组合为: 持有 0.25份股票 成为一份看涨期权的空头 • 三个月后组合的价值为 22´0.25 – 1 = 4.50 • 组合在时刻0的价值为 4.5e – 0.12´0.25 = 4.3670

  6. 期权的估值 • 资产组合为 持有 0.25份股票 成为一份看涨期权的空头 组合在时刻0的价值为4.3670 • 股票的价值是 5.000 (= 0.25×20 ) • 从而,期权的价格为 • 0.633 (= 5.000 – 4.367 )

  7. Su ƒu S ƒ Sd ƒd 推广到一般情形 • 一个依赖于股票的衍生证券,到期时间为 T

  8. 推广到一般情形(continued) • 考虑一个组合:持有D份股票,成为一份衍生证券的空头 • 当 D满足下面的条件时,组合为无风险: SuD – ƒu = SdD – ƒd or SuD – ƒu SdD – ƒd

  9. 推广到一般情形(continued) • 组合在时刻 T的价值为 SuD – ƒu • 组合在时刻0的价值为 (SuD – ƒu )e–rT • 组合在时刻0的价值又可以表达为 SD – f • 从而 ƒ = SD – (SuD – ƒu)e–rT

  10. 推广到一般情形(continued) • 于是,我们得到 ƒ = [ p ƒu + (1 – p )ƒd ]e–rT 其中

  11. Su ƒu S ƒ Sd ƒd Risk-Neutral Valuation • ƒ = [ p ƒu + (1 – p )ƒd ]e-rT • 变量 p和(1– p ) 可以解释为股票价格上升和下降的风险中性概率 • 衍生证券的价值就是它的到期时刻的期望收益的现值 p (1– p )

  12. 最初例子的修正 Su = 22 ƒu = 1 p • 由于 p是风险中性概率,所以 20e0.12 ´0.25 = 22p + 18(1 – p ); p = 0.6523 • 或者,我们可以利用公式 S ƒ Sd = 18 ƒd = 0 (1– p )

  13. Su = 22 ƒu = 1 0.6523 S ƒ Sd = 18 ƒd = 0 0.3477 期权的估值 期权的价值为 e–0.12×0.25 [0.6523´1 + 0.3477´0] = 0.633

  14. 24.2 22 19.8 20 18 16.2 两步二叉树模型 • 每步长为3个月

  15. 欧式看涨期权的估值 24.2 3.2 D • 在节点 B的价值 = e–0.12´0.25(0.6523´3.2 + 0.3477´0) = 2.0257 • 在节点 A的价值 = e–0.12´0.25(0.6523´2.0257 + 0.3477´0) = 1.2823 22 B 19.8 0.0 20 1.2823 2.0257 A E 18 C 0.0 16.2 0.0 F

  16. 72 0 D 60 B 48 4 50 4.1923 1.4147 A E 40 C 9.4636 32 20 F 一个看跌期权的例子:X=52

  17. 72 0 D 60 B 48 4 50 5.0894 1.4147 A E 40 C 12.0 32 20 F 美式期权该如何估值?

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