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 Apple's 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.

 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 working.