A company has granted 1,000,000 options to its employees. The stock price and strike price are both $20. The options last 10 years and vest after 3 years. The stock price volatility is 30%, the risk-free rate is 5%, and the company pays no dividends. Use a four-step tree to value the options. Assume that there is a probability of 4% that an employee leaves the company at the end of each of the time steps on your tree. Assume also that the probability of voluntary early exercise at a node, conditional on no prior exercise, when
(a) the option has vested and
(b) the option is in the money, is
here S is the stock price, K is the strike price, T is the time to maturity, and a = 2.