## Binomial Lattices

In the binomial world, several basic similarities are worth mentioning. No matter the types of real options problems you are trying to solve, if the binomial lattice approach is used, the solution can be obtained in one of two ways. The first is the use of risk-neutral probabilities, and the second is the use of market-replicating portfolios. Throughout this book, the former approach is used. An example of the market-replicating portfolio approach is shown in Appendix 6B for the sake of completeness. The use of a replicating portfolio is more difficult to understand and apply, but the results obtained from replicating portfolios are identical to those obtained through risk-neutral probabilities. So it does not matter which method is used; nevertheless, application and expositional ease should be emphasized.

Market-replicating portfolios' predominant assumptions are that there are no arbitrage opportunities and that there exist a number of traded assets in the market that can be obtained to replicate the existing asset's payout profile. A simple illustration is in order here. Suppose you own a portfolio of publicly traded stocks that pay a set percentage dividend per period. You can, in theory, assuming no trading restrictions, taxes, or transaction costs, purchase a second portfolio of several non-dividend-paying stocks and replicate the payout of the first portfolio of dividend-paying stocks. You can, for instance, sell a particular number of shares per period to replicate the first portfolio's dividend payout amount at every time period. Hence, if both payouts are identical although their stock compositions are different, the value of both portfolios should then be identical. Otherwise, there will be arbitrage opportunities, and market forces will tend to make them equilibrate in value. This makes perfect sense in a financial securities world where stocks are freely traded and highly liquid. However, in a real options world where physical assets and firm-specific projects are being valued, financial purists would argue that this assumption is hard to accept, not to mention the mathematics behind replicating portfolios are also more difficult to apply.

Compare that to using something called a risk-neutral probability approach. Simply stated, instead of using a risky set of cash flows and discounting them at a risk-adjusted discount rate akin to the discounted cash flow models, one can instead easily risk-adjust the probabilities of specific cash flows occurring at specific times. Thus, using these risk-adjusted probabilities on the cash flows allows the analyst to discount these cash flows (whose risks have now been accounted for) at the risk-free rate. This is the essence of binomial lattices as applied in valuing options. The results obtained are identical.

Let's now see how easy it is to apply risk-neutral valuation. In any options model, there is a minimum requirement of at least two lattices. The first lattice is always the lattice of the underlying asset, while the second lattice is the option valuation lattice. No matter what real options model is of interest, the basic structure almost always exists, taking the form:

Inputs: S, X, a, T, rf, b u = eaVFt and d = e_aVFt = —

The basic inputs are the present value of the underlying asset (S), present value of implementation cost of the option (X), volatility of the natural logarithm of the underlying free cash flow returns in percent (a), time to expiration in years (T), risk-free rate or the rate of return on a riskless asset (rf), and continuous dividend outflows in percent (b). In addition, the binomial lattice approach requires two additional sets of calculations, the up and down factors (u and d) as well as a risk-neutral probability measure (p). We see from the equations above that the up factor is simply the exponential function of the cash flow volatility multiplied by the square root of time-steps or stepping time (Si). Time-steps or stepping time is simply the time scale between steps. That is, if an option has a one-year maturity and the binomial lattice that is constructed has 10 steps, each time-step has a stepping time of 0.1 years. The volatility measure is an annualized value; multiplying it by the square root of time-steps breaks it down into the time-step's equivalent volatility. The down factor is simply the reciprocal of the up factor. In addition, the higher the volatility measure, the higher the up and down factors. This reciprocal magnitude ensures that the lattices are recombining because the up and down steps have the same magnitude but different signs; at places along the future path these binomial bifurcations must meet.

The second required calculation is that of the risk-neutral probability, defined simply as the ratio of the exponential function of the difference between risk-free rate and dividend, multiplied by the stepping time less the down factor, to the difference between the up and down factors. This risk-neutral probability value is a mathematical intermediate and by itself has no particular meaning. One major error real options users commit is to extrapolate these probabilities as some kind of subjective or objective probabilities that a certain event will occur. Nothing is further from the truth. There is no economic or financial meaning attached to these risk-neutralized probabilities save that it is an intermediate step in a series of calculations. Armed with these values, you are now on your way to creating a binomial lattice of the underlying asset value, shown in Figure 6.3.

Starting with the present value of the underlying asset at time zero (S0), multiply it with the up (u) and down (d) factors as shown below, to create a binomial lattice. Remember that there is one bifurcation at each node, creating an up and a down branch. The intermediate branches are all recombining. This evolution of the underlying asset shows that if the volatility is zero, in a deterministic world where there are no uncertainties, the lattice would be a straight line, and a discounted cash flow model will be adequate because the value of the option or flexibility is also zero. In other words, if volatility (a) is zero, then the up (u = ea'^t) and down (d = jump sizes are equal to one. It is because there are uncertain ties and risks, as captured by the volatility measure, that the lattice is not a

straight horizontal line but comprises up and down movements. It is this up and down uncertainty that generates the value in an option. The higher the volatility measure, the higher the up and down factors as previously defined, the higher the potential value of an option as higher uncertainties exist and the potential upside for the option increases.

Chapter 7 goes into more detail on how certain real options problems can be solved. Each type of problem is introduced with a short business case. Then a closed-form equation is used to value the strategic option. A binomial lattice is then used to confirm the results. In the binomial approach, each problem starts with the lattice evolution of the underlying value, similar to what we have seen thus far. The cases conclude with a summary of the results and relevant interpretations. In each case, a limited number of timesteps are used to facilitate the exposition of the stepwise methodology. The reader can very easily extend the analysis to incorporate more time-steps as necessary.