## Evaluating Real Investment Opportunities

Options theory can be used to evaluate investment opportunities that are not pure financial instruments. We shall illustrate this by again considering our gold mine lease problems Now, however, the pr ice of gold is assumed to fluctuate randomly, and this fluctuation must be accounted for in our evaluation of the lease prospect.

Example 12.7 (Simplico gold mine) Recall the Simplico gold mine from Chapter 2 Gold can be extracted from this mine at a rate of up to 10,000 ounces per year at a cost of \$200 per ounce Currently the market price of gold is \$400 per ounce, but we recognize that the price of gold fluctuates randomly. The term structure of interest rates is assumed to be flat at 10% As a convention, we assume that the price obtained for gold mined in a given year is the price that held at the beginning of the year; but all cash flows occur at the end of the year. We wish to determine the value of a 10-year lease of this mine

We represent future gold prices by a binomial lattice Each year the price either increases by a factor of 1.2 (with probability 75) or decreases by a factor of 9 (with probability .25). The resulting lattice is shown in Figure 12.10

How do we solve the problem of finding the lease value by the methods developed for options pricing? The trick is to notice that the gold mine lease can be regarded as a financial instrument It has a value that fluctuates in time as the price of gold fluctuates. Indeed, the value of the mine lease at any given time can only be

 0 1 2 3 4 5 6 7 8 9 10 400 0 480 0 576.0 691 2 829 4 995.3 11944 1433 3 17199 2063 9 2476.7 360 0 4.32.0 518 4 622.1 7465 895.8 1075 0 1289 9 1547 9 1857 5 324 0 .388 8 466.6 559 9 671 8 806 2 967 5 1161.0 139.3.1 291 6 349 9 419.9 503.9 6047 725 6 8707 1044 9 262 4 .314.9 .377 9 453 5 544 2 65,3 0 783 6 Gold price (dollars) 2.36 2 283.4 340.1 408 1 4898 587.7 2126 255 1 191 3 306.1 2296 172 2 367.3 275 5 206 6 155.0 440 8 .330 6 247,9 186.0 1395

FIGURE 12 10 Gold price lattice. Each year the price either increases by a factor of 1 2 or decreases by a factor of 9 The resulting possible values each year are shown in spreadsheet form

FIGURE 12 10 Gold price lattice. Each year the price either increases by a factor of 1 2 or decreases by a factor of 9 The resulting possible values each year are shown in spreadsheet form a function of the price of gold and the interest rate (which we assume is fixed) In other words, the lease on the gold mine is a derivative instrument whose underlying security is gold Therefore the value of the lease can be entered node by node on the gold price lattice

The lease values on the lattice are determined easily for the final nodes, at the end of the 10 years: the values are zero there because we must return the mine to the owners. At a node representing 1 year to go, the value of the lease is equal to the profit that can be made from the mine that year, discounted back to the beginning of the year. For example, the value at the top node for year 9 is 10,000(2,063 9- 200)/l I = 16 94 million For an earlier node, the value of the lease is the sum of the profit that can be made that year and the risk-neutral expected value of the lease in the next period, both discounted back one period. The risk-neutral probabilities are q = (1.1— 9)/(1.2 — 9) = and i —q — The lease values can therefore be calculated by backward recursion using these values (At nodes where the price of gold is less than \$200, we do not mine ) The resulting values are indicated in Figure 12.11. We conclude that the value of the lease is \$.24,074,548 (showing all the digits).

 0 1 2 3 4 5 6 7 8 9 10 24.1 27 8 31 2 34 2 36 5 37.7 37.1 34.1 27 8 16.9 0.0 17.9 20 7 23 3 25 2 26.4 26 2 24 3 20 0 123 0 0 12 9 15 0 167 179 18 1 170 14.1 8.7 0 0 88 10 4 115 120 115 9 7 61 00 5 6 67 74 74 64 4 1 0 0 Lease value (millions) 3.2 4 0 4 3 3 9 2.6 00 1 4 2.0 2 1 1.5 0 0 04 0 7 0 7 0.0 0 0 0 1 00 0.0 0 0 00

FIGURE 12 11 Simplico gold mine. The value of the lease is found by working backward If the price of gold is greater than \$200 per ounce, it is profitable to mine; otherwise no mining is undertaken

Many readers wiil be able to see from this example that they have a deeper-understanding of investment than they did when they began to study this book. Earlier, in Chapter 2, we discussed the Simplico gold mine under the assumption that the price of gold would remain constant at \$400 over the course of the lease, We also assumed a constant 10% interest rate These assumptions, which are fairly commonly employed in problems of this type, were probably not regarded as being seriously incongruous by most readers, Now, however, we see that they are not just a simplification, but an actual inconsistency. If the price of gold were known to be constant, gold would act as a risk-free asset with zero rate of return. This is incompatible with the assumption that the risk-free rate is 10% Indeed, in our lattice of gold prices we must select tt, d, and R such that it > R > d

Now that we have "mastered" the Simplico gold mine, it is time to move on to even greater challenges. (If you think you have really mastered the Simplico mine, try Exercise 8.)

Example 12.8 (Complexico gold mine*) 3 T he Complexico gold mine was discussed in Chapter 5. in this mine the cost of extraction depends on the amount of gold remaining Hence if you lease this mine, you must decide how much to mine each period, taking into account that mining in one period affects future mining costs. We also assume now that the price of gold fluctuates according to the binomial lattice of the previous example

The cost of extraction in any year is \$500z2/a\ where a- is the amount of gold remaining at the beginning of the year and z is the amount of gold extracted in ounces. Initially there are xo = 50,000 ounces of gold in the mine, We again assume that the term structure of interest rates is Hat at 10%. Also, the profit from mining is determined on the basis of the price of gold at the beginning of the period, and in this example all cash flows occur at the beginning of the period,

To solve this problem we must do some preliminary analysis At the final time the value of the lease is clearly zero. If we are at a node representing the end of year 9, we must determine the optimal amount of gold to mine during the tenth year. Accordingly, we must compute the profit

where g is the price of gold at that particular node From the calculations of Example 5 5 we know that the maximization gives g'Xi)

2,000

This shows that the value of the lease is proportional to x9, the amount of gold remaining. We therefore write VgUo) = KcjXq, where 