期权及其他衍生证券定价的二叉树模型的出发点是假设标的资产的价格波动服从二项分布(Binomial Distribution)
为什么用二叉树model标的资产价格的波动?
利用二叉树模型为风险资产估值包括以下三个步骤:
一个简单的例子
思路:
构造(复制)无风险组合:
到期时无风险组合的价值:
价值 | ||
---|---|---|
股票的头寸 | ||
期权的头寸 | ||
组合 |
确定(组合无风险):
无风险组合的价值
确定(组合获得无风险收益):
构造(复制)无风险组合:
确定(组合无风险):
确定(组合获得无风险收益):
令,可表示为: 。
问题:是概率吗?
假设我们生活在风险中性世界:
用风险中性笔概率为股票和期权估值
价值 | 期望 | ||
---|---|---|---|
风险中性概率 | 不适用 | ||
股票 | |||
期权 |
在风险中性世界:
因此,期权费为:
风险中性定价理论有两个最基本的假设:
无风险的套利机会出现时
从风险中性世界进入到风险厌恶或者风险喜好的世界时:
Delta ()表示期权价格变化量与标的资产价格变化量之比
每个节点对应的不同
,
Girsanov定理
对其他资产,用同样的方法构造二叉树,不同之处仅仅在于的计算:一般地,令
一些已知的结论:
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课后阅读:教材第十一章第三节、十二章第一节、第六章、第七章相关内容
练习
Explain the no-arhitrage and risk-neutral valuation approaches to valuing a European option using a one-step binomial tree.
What is meant by the delta of a stock option?
A stock price is currently $40. It is known that at the end of one month it will he either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39?
A stock price is currently $100. Over each of the next two 6-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-year European call option with a strike price of $100?
For the situation considered in the above problem, what is the value of a 1-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put-call parity.
A stock price is currently $50. It is known that at the end of 6 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a 6-month European call option on the stock with an exercise price of $48. Verify that no-arhitrage arguments and risk-neutral valuation arguments give the same answers.
A stock price is currently $40. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.