2. Consider a three-step binomial tree model for the price of a stock paying no divi- dend. Suppose the stock price at time 0 is $100. Assume the stock's annual volatility is 25%. The risk-free interest rate is 12% per annum compounded monthly (a) Price a vanilla American put option on the stock with strike price $100 ma- turing in one year and determine the probability to early exercise in the risk- neutral world [10 marks (b) Price a vanilla European call option on the stock with strike price $90 maturing in nine months. 110 marks 2. Consider a three-step binomial tree model for the price of a stock paying no divi- dend. Suppose the stock price at time 0 is $100. Assume the stock's annual volatility is 25%. The risk-free interest rate is 12% per annum compounded monthly (a) Price a vanilla American put option on the stock with strike price $100 ma- turing in one year and determine the probability to early exercise in the risk- neutral world [10 marks (b) Price a vanilla European call option on the stock with strike price $90 maturing in nine months. 110 marks