Suppose that Alphabet stock is trading at $1,000 on Wednesday, 10/25/2017. An at-the-money call option exists that expires 3 weeks from that date, on 11/15/2017. Assume that every week, Alphabet stock will either increase or decrease by a factor of 1.05. That is, in one week, the price will either be $1,000*1.05 = $1,050 or $1,000/1.05 = $952.38. You can assume that the risk-free rate is 5% per year. You can assume that this is either the continuously compounded rate or the APY.
a. At the end of three weeks, what possible prices can the stock take?
b. At the stock prices listed in (a), what is the value of the call option at expiration?
c. Draw this scenario as a multi-step binomial tree.
d. At any particular node, what is the risk-neutral probability that the stock price will increase? Write this probability, as well as the probability of a price decrease, on the tree from part (c).
e. Complete the tree by finding the price of the call option at every node.