Venture Capital & the Finance of Innovation - by A. Metrick & A. Yasuda
Binomial Trees
The appeal of valuing both financial and real options using binomial trees lies in their ability to evaluate more complex and nuanced situations. The Black-Scholes formula has many assumptions, which in some cases may not align with reality. For example, valuing an American Call using the Black-Scholes formula is incorrect if the stock pays dividends.
To simplify the construction of event and decision trees, the method of the “risk-neutral world assumption” is used, breaking the tree into two parts: 1. The base tree, which reflects the step-by-step changes in the price of the underlying asset (stock), and 2. The options tree, which, using a backward induction method, leads to the project’s NPV.
In the first stage, the main parameters are calculated using the risk-neutral world assumption:
Minor deviation:
It should be noted that the risk-neutral world assumption is not some unrealistic theory. This idea emerged from the replication method when the Black-Scholes formulas were derived, showing that the risk factor was absent in the final formula, meaning it should not be significant. Afterward, a mathematical solution was developed, yielding the same results as the replication method. More details:
In the next stage, trees are constructed:
In this specific example, the fair price of a European Call option is $54.91, which will be very close to the price calculated by the Black-Scholes formula and would become even closer if we increased the number of steps within the one-year period from 3.
However, we would not be able to use the Black-Scholes formula if, for example, we assumed the possibility of receiving a dividend in the second stage. In this case, the need to make a decision between events arises — to exercise the option and receive the dividend or to allow the stockholder to hold onto it. Which is more profitable?
This model can be complicated within Excel’s framework. Below is a photo of a 5-year option model that can be exercised either exactly after two years or after five years:
Here you can find excel models. On the 3rd link is a model of a real 5-year option that involves shifting the project’s start dates over time. On one hand, we lose monthly dividends; on the other, it allows us to start the project later after observing developments. So, we have 60 steps (months), with 60 exercising opportunities (one a month decision nodes).
Excel Models:
- 2-step option, without and with dividend consideration
- 60-step option, with the possibility of exercising twice
- 60-step option, with the possibility of exercising 60 times
Source:
Venture Capital & the Finance of Innovation – by A. Metrick & A. Yasuda