So this term is the same expression that we had for figuring out what to do with a

gold mine where the operating cost was 200 and the operating rate or the maximum

rate at which you could extract gold was 10,000.

So this particular maximum, all it's saying is that if I continue to use the

current mine, meaning not upgrade, then I still have to decide whether I would

operate the mine or not, and that corresponds to the max of K16 minus 200

and 0, and if I decide to operate, I'm going to run it at the maximum possible

rate. And then I need to figure out what'll

happen in the future. The second term over here, this maximum,

refers to the fact that now I have the option of moving from this lattice point

to the lattice that corresponds to an upgraded mine.

So instead of staying on this lattice, I'll move to the upper lattice.

But in order to move the upper lattice, I have to pay a cost of $4 million.

So, K35 is the value that I would get, in this upgraded lattice, which is 20.73,

but I have to pay $4 million for it, and therefore, I have to decide whether it's

worth it. If you look at what happened over here,

the value here is 20.73, the value here is 16.94.

If I had decided to upgrade, I would have got 20.73 minus $4 million which is

16.73. In this particular state I can get $16.94

million simply by continuing. So its worth it for me to continue, and

not pay the extra $4 million. Same thing we go backwards.

Calculate out exactly the same calculations here we here.

What is the maximum profit that I could get by continuing with my current mine

operations? Or I can upgrade by paying $1 million.

The value here is 29.88. Which is 33.88 minus 4, so it appears

that at least in this particular state, we will jump up, and in a little bit I'm

going to show you how exactly the exercise boundary's going to be

calculated. We calculate this backwards, starting

from ten all the way down to time one, and at time one we get a time zero which

is the initial state we end up getting that the value of the gold mine with the

equipment option in place is $24.63 million.

And if you compare this to the lease without the equipment option, it's $24.07

million. So it turns out that it is having that

option, that equipment option is valuable.

And the value of that is approximately $0.6 million.

The thing to think about is that we can consider two different situations.

One situation where we have the option in place, and another situation where we are

forced to upgrade at time zero. If we are forced to upgrade at time zero,

we will get $27.02 million, which is the value that I get with the upgraded mike,

minus $4 million, so we would, I would only get $23.02 million.

Whereas, with an option in hand, I can get $24.63 million.

And the difference between these two comes from the fact with an option I have

the flexibility of when I excersize the option.

So it will turn out that the various states, when the price of gold is high,

and therefore I expect to make a lot of profit, it makes sense for me to pay the

extra operating cost of $40 per ounce in order to get the extra rate of

extraction, which is 2500 more. So in order to decide where we are going

to extract, we just compute the two value functions.

So, if you look at the genetic one over here, what we are doing is we are looking

at the value function of exercising my option versus the value function of

continuing. If the difference between them is

positive, I'm going to exercise. If the difference is negative I'm going

to not exercise. Negative or zero I won't exercise.

So if you calculate this out, you end up getting a exercise frontier, which is the

states at which you are going to exercise.