Question

There are two parts to this question first part:

Consider a production facility, where the present value of expected future cash inflows from production, V = 80, may fluctuate in line with the random fluctuation in demand (u = 1.4, d = 0.71 per period and the risk-free rate, r = 5%). Suppose management has the option in two years, to contract to half the scale and half the value of the project (c = 50%), and recover $40m (Rc = $40m). Thus, in year 2 management has the flexibility either to maintain the same scale of operations (i.e., receive project value, V, at no extra cost) or contract the scale of operations and receive the recovery amount, whichever is highest. What are the pay-offs of this option at the end nodes (thus in the different states after 2 periods)?

The payoffs, F, of the option in the end note states are respectively: F = 0 , F = 0, F = 20

The payoffs, F, of the option in the end note states are respectively: F = 0 , F = 0, F = 14

The payoffs, F, of the option in the end note states are respectively: F = 196 , F = 100, F = 51

The payoffs, F, of the option in the end note states are respectively: F = 157 , F = 80, F = 41

Second part:

Consider again the production facility (from question above). Again, suppose that management has the option in two years, to halve the scale and the value of the project and recover some value. Thus, in year 2 management has the flexibility either to maintain the same scale of operations or contract the scale of operations, whichever is highest.

For this question, assume the end node pay-offs are 0, 20, 50. Calculate the option value by discounting with the risk neutral probability of 0.5 and a risk free rate of 5%. What is the option value?

Answer 1

Part 1:

This is a classic binomial model problem. Let us start with a few notations.

• represents the present value of expected future cash inflows from production V in case it moves up in both the periods at the end of the second period. Hence,

• represents the present value of expected future cash inflows from production V in case it moves up in the first period and moves down in the second period. represents the present value of expected future cash inflows from production V in case it moves down in the first period and moves up in the second period. In both cases,

• represents the present value of expected future cash inflows from production V in case it moves up in both the periods. Hence,

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In all the three states at the end of two periods, given the value is more than 40, it doesn't make sense to exercise the option in any of the states.. Hence, the option doesn't hold any value here. F = 0 for all the three states.

Part 2:

In this case, if we assume and be the option values at the end of two periods in case the value goes up in both, values goes up in first and down in second, the value goes down in first and up in second, and it goes down in both the periods respectively, we can write:

If and represents the option value in up and down states respectively at the end of period 1,

If represents the option value today,

Hence, the value of the option today is $20.40.