A stock index is currently at 2,000. Its volatility is 20%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 1%. Calculate values for u, d, and p when a six-month time step is used. What is the value of a 12-month American put option with a strike price of 2,050 given by a two-step binomial tree? Show how you derive the binomial trees for the index price and the option price.