A Scale-Up Option Valuation Using the Binomial Lattice Approach A firm has undertaken a project in which the firm has the option to invest in additional manufacturing and distribution resources (or scale up). The project’s current value is V0 = $10M. Anytime over the next three years, the firm

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A Scale-Up Option Valuation Using the Binomial Lattice Approach

A firm has undertaken a project in which the firm has the option to invest in additional manufacturing and distribution resources (or scale up). The project’s current value is [latex]V_{0}[/latex] = $10M. Anytime over the next three years, the firm can invest an additional [latex]I[/latex] = $3M and receive an expected 30% increase in net cash flows and, therefore, a 30% increase in project value. The risk-free interest rate is 6% compounded annually and the volatility of the project’s value is 30%. Use a binomial lattice with a one-year time increment to value the scale-up option as an American option. (You can scale up anytime you see fit to do so.)

Accepted Answer

We will solve the problem in two steps, the first of which is to determine how the value of the underlying asset changes over time and the second of which is to create the option valuation lattice.

Step 1. Binomial Approach: Lattice Evolution of the Underlying Asset. First, we determine the following parameters to build a binomial lattice that shows how the value of the project changes over time:

[latex]u = e^{0.30(1)} = 1.35[/latex]

[latex]d = \frac{1}{u} = 0.74[/latex]

[latex]q = \frac{1.06 - 0.74}{1.35 - 0.74} = 0.5241[/latex]

The equity binomial lattice looks like the one in Figure 13.25; the values are created in a forward multiplication of up and down factors, from left to right.

Step 2. Binomial Approach: Option Valuation Lattice. Once you obtain the lattice evolution of the underlying project, we can create the option valuation lattice in two steps: the valuation of the terminal nodes and the valuation of the intermediate nodes. The calculation proceeds in a backward manner, starting from the terminal nodes: The nodes at the end of the lattice are valued first, going from right to left.

All the required calculations and steps are shown in Figure 13.26. To illustrate, we will take a sample circled node (➊) in the figure:

Node ➊ reveals a value of $24.60, which is the value the firm can expect to achieve without any scaling up. Now, with the scale-up option, the project value can increase 30% at the expense of $3 million of investment, for a new value of 1.3(24.60) - 3 = $28.98 million. Since this figure is larger than $24.60, the firm will opt for scaling up the operation.
Similarly, the firm will reach the same decision for node ➋, which yields a scale-up result of $14.55.
Moving on to node ➌, we see that the firm again has one of two options: to scale up the operation at that point rather than in the future, or to keep the option open for the future in the hope that when the market is up, the firm will have the ability to execute the option. Mathematically, we can state the options as

Scale-up now (n = 2): 1.3($18.23) - $3 = $20.70

Keep the option open: C = [latex]\frac{\left(0.5241\right)\left(\$28.98\right)+ \left(0.4759\right) \left(\$14.55\right) }{\left(1 + 0.06\right) } = \$21.31[/latex]

Thus, the decision is to keep the option open. Finally at year 0, we compute

[latex]C=\frac{\left(0.5241\right)\left(\$15.51\right)+ \left(0.4759\right) \left(\$7.94\right) }{\left(1 + 0.06\right) }[/latex]

= $11.23

The end-result option value on the binomial tree shown in Figure 13.27 represents the value of the project with the scale-up option. The value of the scale-up option itself is

Option value = $11.23 - $10 = $1.23M

Therefore, the option to expand the project in the future is worth $1,230,000.

Question 13.12: A Scale-Up Option Valuation Using the Binomial Lattice Appro...

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