Consider a model with only one time period.Assume that there exist a stock and a cash bond in the mo

题目

Consider a model with only one time period.Assume that there exist a stock and a cash bond in the model.The initial price of the stock is $40.The investor believes that with probability 1/5 the stock price will drop to $20 and with probability 4/5 the stock price will rise to $60 at the end of the time period.The cash bond has an initial price of $60 and it will with certainty deliver $80 at the end of the period.
What is a fair price for a European call with a maturity at the end of the time period and a strike price of $48?
What is a fair price for a European put with a maturity at the end of the time period and a strike price of $32?

答案

欧式看涨期权问题可以改写为：
无风险收益率为1/3（＝（80-60）/60）,某项资产（即欧式看涨期权）现价x,该资产期末价值为：1/5的概率是0（因为1/5的概率期末标的股票价格跌至20,此时不行权,期权期末价值为0）,4/5的概率收益是60-48＝12（4/5概率股价到60,减去行权价格48,即为期权期末价值）.求x的合理值.
只要让期权期末价值的现值等于期权现价x即可.
因此x＝（1/5*0+4/5*12）/（4/3）＝7.2.
同理,欧式看跌期权价格＝〔1/5*（32-20）+4/5*0〕/（4/3）＝1.8.