Each of the following questions should be answered by building a 15-period binomial model whose parameters should be calibrated to a Black-Scholes geometric Brownian motion model

第一题。
Asset  price   S   100  
Exercise price   K  110
Interest rate   r   0.02   
Volatility   sigma   0.3   
Dividend yield   q   0.01   
Time to maturity   T   0.25  
Number of time periods   N   15
For American Put Option， when is the earliest period at which it mightbe optimal to early exercise？答案要求保留两位小数

第二题
Compute the fair value of an American call option with strike=110 and maturity=10
periods where the option is written on a futures contract that expires after 15 periods(其他数据用第一题的数据，比如S，r, sigma, q)

第三题
在第二题的基础上 What is the earliest time period in which you might want to exercise the American futures option

第四题
Compute the fair value of a chooser option which expires after n=10 periods. At expiration, the owner of the chooser gets to choose (at no cost) a European call option or a European put option. The call and put each have strike K=100 and they mature 5 periods later, i.e. at n=15
(其他数据用第一题的数据，比如S，r, sigma, q)

最后补充上面四个题目 不要用BlackScholes模型， 必须用 binomial model 。 只要求得到答案（保留两位小数），不需要解答过程。

谢谢大家了