In this module, we'll go through the Simplico Gold Mine example that we introduced in the video modules. The first half of this module is about the operating option, meaning that the only thing that I'm trying to value is a lease over ten periods and every year I have the option of either shutting down the mine or operating it at the maximum possible rate. In the next half of this module we are going to value an equipment upgrade option that is also valid over the ten years that we are trying to figure out when we will exercise it. And what is the net revenue gain that I would get from that option? So here are the details of the model. The current price of gold is $400 per ounce. It can go up by a factor of 1.2 or go down with a factor of 0.9. The interest rate is 10%. And therefore, from there I can calculate out what the risk-neutral probability of the up state is, and the risk-neutral probability of the down state is. So that's going to be Q and one minus Q, respectively. The extraction rate, and cost are as follows. The cost is $200 per ounce. The extraction rate is 10,000 ounces per year. So in order to decide when to operate this plant and when to shut it down, I need to have a lattice of gold. Whenever the gold price is greater than $200, I'm going to operate the plant. And I'm going to operate it at 10,000 ounces per year. Whenever the price of gold is below $200 I'm going to shut it down and not incur any costs. Here is the gold lattice, it starts with $400, it goes up to 480 or goes down to 360 and so on. These are in thousands of dollars for ups. Down here you'll see States where the price of gold is lower than the $200 required to extract gold. And therefore in these States we are going to exercise the option of shutting down the mine and incurring no cost. In all the other States where the price of gold is greater than $200, we are going to exercise the option of operating the plant, and getting the profits that we get from it. So here are, here is a value of the gold mine lease, in millions of dollars. At the beginning of the tenth year we return the gold mine back to the owners and therefore the value at the beginning of the tenth year from the future is going to be zero. Now we start looking at what is going to happen to a state before. So if we look at a particular state over here, all we are trying to do, is we are using the formula that we had developed in the video modules. If I decide to operate the gold mine, I'm going to get K19 minus cost, which is the price at the current point, minus cost, times the maximum possible rate. But if this term, K19 minus cost, the current price minus the cost of operating the mine happens to be less than zero, I will elect not to run the mine. I will exercise the option of shutting it down, and get a zero from this term. So this max of K19 minus cost and zero times the rate tells you what is the one year revenue from operating the mine in the optimal fashion. And that might include shutting it down. This next term, Q time offset K33 minus 11 and 1 minus Q times offset K3301 basically computes what is the expected value from the future. It's Q times the value in the upstate plus 1 minus Q times the value in the downstate, at time T plus 1, and all of it is divided by 1 plus the interest rate to bring it down to current dollar terms. So same thing is being done overe here at previous times. Exactly the same thing and at the end of the day you end up getting that the value of this option turns out to be 24.07 million dollars. That is total value that you could generate from leasing this market. That's the fair price for the mine under the circumstances that we are considering. Now let's go on to consider the case with an equipment option in place. So the story is the same. The gold price dynamics remain the same. You've got $400 per ounce today. They go up by a factor of 1.2, go down by a factor .9, and the risk neutral probability is 0.6. The interest rate is 10%. This lattice is exactly the same as it was in the previous page. So now what we want to do, is figure out what is the value of an option that allows us to upgrade the equipment any time between time zero, which is now, and time ten, when I return the mine back to its owner. As we detailed in the video module, in order to compute this, I want to think of it as an American option on two different mines. So I've got the mine which is operating as it does currently, with $200 per ounce as the operating cost, 10,000 as the maximum extraction rate. At any given time I can exercise my upgrade option and move to another mine, which has $240 per ounce as the operating cost, but 12,500 as the maximum possible rate. In order to do this, we showed in the video modules, I have to first calculate out the value of a gold mine with the equipment option already in place. And that's what I'm dong over here. This is the value of a gold mine lease with the cost equal to 240 and the rate equal to 12,500. This is the value where the equipment upgrade has already happened. Again, the calculation is the same as before. You start with all zeroes and then you back calculate. You back calculate as K16, which is the cost of gold, or price of gold, minus the cost of operation, which in this case is 240, times the maximum rate, which is 12,500. And Q time the upstate plus 1 minus Q the times table divided by the interest rate, which is 1 plus the interest rate. Calculate it backwards and you end up getting various values of the various states in this particular latice. Now we want to value what happens to the gold mine with an equiment option in place. You start off with 0s here. In the beginning of the 10th year you have to give the gold mine back, whether you upgraded the equipment or not, and therefore the value you end up getting is 0. Now lets see what happens to this particular state. It's a complicated formula, so lets go through it one by one. There are two different MAXs, there's one term over here. 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.
