A stock is now selling for $100. Each month it will either increase or decrease in value by 10% (u = 1.1, d = 0:9). The riskless rate with annual compounding is 12.682503% per year. Note that 1.12682503 = 1.01^12. A European put option on this stock with an exercise price of $105 matures in three months (n = 3).

(a) (5 points) Draw the tree of values for the stock price.

(b) (10 points) For each stock price and time to maturity (i.e., at all nodes of the tree), determine the value of the European put.

(c) (10 points) Assume that the put option is American (exercise price $105 and maturity in three months). Determine the value of this put for each stock price and time to maturity. When (at which tree nodes) would it be optimal to exercise this put option?