You have purchased a lease for the Little Bear Oil well. This well has initial reserves of 100 thousand barrels of oil. In any year you have three choices of how to operate the well: (a) you can not pump, in which case there is no operating cost and no change in oil reserves; (b) you can pump normally, in which case the operating cost is $\$ 50$ thousand and you will pump out $20 %$ of what the reserves were at the beginning of the year; or (c) you can use enhanced pumping using water pressure, in which case the operating cost is $\$ 120$ thousand and you will pump out $36 %$ of what the reserves were at the beginning of the year. The price of oil is $\$ 10$ per barrel and the interest rate is $10 %$. Assume that both your operating costs and the oil revenues come at the beginning of the year (through advance sales). Your lease is for a period of 3 years.

(a) Show how to set up a trinomial lattice to represent the possible states of the oil reserves.

(b) What is the maximum present value of your profits, and what is the corresponding optimal pumping strategy?