Before valuing the option implicit in WOE it is useful to see what we can say

Before valuing the option implicit in woe it is

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Before valuing the option implicit in WOE, it is useful to see what we can say by just applying common sense. To begin with, the mine should be opened only when the price of gold is sufficiently above the extraction cost of $350 per ounce. Because it costs $2 million to open the mine, the mine should not be opened whenever the price of gold is only slightly above $350. At a gold price of, say, $350.10, the mine wouldn’t be opened because the ten-cent profit per ounce translates into $5,000 per year (=50,000 ounces × $.10/ounce). This would not begin to cover the $2 million opening costs. More significantly, though, the mine probably would not be opened if the price rose to $360 per ounce, even though a $10 profit per ounce—$500,000 per year—would pay the $2 million opening costs at any reasonable discount rate. The reason is that here, as in all option problems, volatility (in this case the volatility of gold) plays a significant role. Because the gold price is volatile, the price has to rise sufficiently above $350 per ounce to make it worth opening the mine. If the price at which the mine is opened is too close to the extraction price of $350 per ounce, say at $360 per ounce, we would open the mine every time the price jogged above $360. Unfortunately, we would then find ourselves operating at a loss or facing a closing decision whenever gold jogged back down $10 per ounce (or only 3 percent) to $350. The estimated volatility of the return on gold is about 15 percent per year. This means that a single annual standard deviation movement in the gold price is 15 percent of $320, or $48 per year. Surely with this amount of random movement in the gold price, a threshold of, for example, $352 is much too low at which to open the mine. Similar logic applies to the closing decision. If the mine is open, we will clearly keep it open as long as the gold price is above the extraction cost of $350 per ounce because we are profiting on every ounce of gold mined. But we also won’t close the mine down simply because the gold price drops below $350 per ounce. We will tolerate a running loss because gold may later rise back above $350. If, alternatively, we closed the mine, we would pay the $1 million abandonment cost, only to pay another $2 million to reopen the mine if the price rose again. To summarize, if the mine is currently closed, then it will be opened—at a cost of $2 million— whenever the price of gold rises above the extraction cost of $350 per ounce. If the mine is currently operating, then it will be closed down—at a cost of $1 million—whenever the price of gold falls below the extraction cost of $350 per ounce. WOE’s problem is to find these two threshold prices at which it opens a closed mine and closes an open mine. We call these prices p and p , respectively, where: p > $350/ounce > p In other words, WOE will open the mine if the gold price option is sufficiently in the money and will close it when the option is sufficiently out of the money.
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We know that the more volatile the gold price, the further away p and p will be from $350 per ounce. We also know that the greater the cost of opening the mine, the higher p
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