Bill Gates expects the price of Apple stock (AAPL) to decrease in the next 3 months from the current price of $130. Bill decides to speculate on his forecast by using American put options on Apple with a strike of $125. He expects the volatility of Apples stock returns to be 25% p.a. for the following 2 months and increases to 30% p.a. afterwards. Also, he expects a dividend of $1.8 per share to be paid in 2 months from today and a dividend of $2.5 to be paid in 5 months from today. The continuous compounding risk free rate is currently 1.3% p.a.

1. a) Use a three-step binomial tree to calculate the option price. Round u and d to 6 decimal digits at each step. For all other calculations, round to 4 decimal digits. Please show all your workings.

2. b) Without recalculating the option price, estimate the change in option price if the stock price increases by $1 instantly. Round to 4 decimal digits. Please show all your workings.

3. c) What are the upper-bound and lower-bound to the value of an otherwise equivalent American call option, based on put-call parity? Round to 4 decimal digits. Please show all your workings.