## Numerical Examples

For practitioners to use the option pricing approach, it must be relatively transparent and easy to understand. We use a lattice method that requires only basic algebra and which can be solved on an Excel or Lotus spreadsheet. The results are identical to those that use much more complicated branches of mathematics such as stochastic calculus. The objective of the lattice is to model the present value of the project in a simple, but realistic way.

### Event Trees

The lattice that models the values of the underlying risky asset is called an event tree. It contains no decision nodes and simply models the evolution of uncertainty in the present value of the underlying risky asset. Suppose we are studying a project that has a present value (PV) of \$100, volatility of 15 percent per year, and an expected rate of return of 12 percent per year. The risk-free rate is 8 percent per year, and the cash outflow necessary to undertake the project, if we invest in it immediately, is \$105. To model a single source of uncertainty (changes in the value of the project) we can chose one of two types of event treeâ€”geometric or arithmetic. A geometric tree has multiplicative up-and-down movements that model a log-normal distribution of outcomesâ€”one that can go to values of plus infinity on the up side and to values of zero on the down side. We chose the geometric event tree because we believe the value of the project can never fall below zero. (Although we will not illustrate it here, an arithmetic tree has additive up and down movements, approaches a normal distribution, and can have values that go to either plus or minus infinity.)

Exhibit 20.5 An Event Tree (No Flexibility)

Exhibit 20.5 An Event Tree (No Flexibility)

Exhibit 20.5 illustrates the values that our project might take for each of five years, given a geometric event tree. The up-and-down movements are determined by the following formulas:4

Note that as we trace the down movements, they approach zero: Limd" =0

The objective probability of an up movement is 86.12 percent and the objective probability of a down movement is 13.88 percent.5 Using the tradi-

4 J. Cox, M. Rubinstein, and S. Ross, ''Option Pricing: A Simplified Approach," Journal of Financial Economics (September 1979), pp. 229-263.

e12-d f12-.8607

The formula for estimating the objective probability is u-d

0 0