Consider a 3-month European put option on a non-dividend-paying stock, where the stock price is $60, the strike price is $60, and the risk-free rate is 3% per annum. Stock price will either move up by 10% or down by 5%, every month. Price the put with binomial trees.

Suppose that put options on a stock with strike prices $45 and $55 cost $4 and $9, respectively. Use these options to create a bear spread. At what stock price at maturity will you break even? In other words, at what stock price, will you make $0 profit?

A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 9%. The risk-free interest rate is 5%. What is the risk-neutral probability that the stock price will increase each period? (Report in % such as 12.34%.)