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
$$
1-\exp [-a(S / K-1) / T]
$$
where $S$ is the stock price, $K$ is the strike price, $T$ is the time to maturity, and $a=2$.