# Binomial Model vs. Black Scholes

Joachim Kuczynski, 06 December 2022

There are two basic ways to evaluate real options. The first way is the evaluation with exact analytical approaches like the famous Black Scholes Merton model. They have their origin in financial option pricing and deliver exact results. But they have certain underlying conditions that do not match reality of real options many times. The second basic approach to evaluate real options are discrete approximative models like the Binomial Model from Cox, Ross and Rubinstein. They are approximations of the exact analytical solutions when they have the same underlying conditions. In my real option analysis I prefer the approximative approach because of many reasons:

• Volatility: The volatility, or risk respectively, defines the variance of the binomial tree branches (ups and downs). Volatility can change over time because of many reasons. But in the Black Scholes Merton model volatility is fixed for the considered time interval. Taking the binomial approach it is up to you to change volatility whenever you want. For sure, the binomial tree can become more complicated, e.g. if you have to change from a recombining to a non-recombining binomial tree. But with computer support also a more complicated event tree is no problem.
• Exercise price: The exercise price at a node is the time value of the corresponding future cash flows which do not depend on the market development and are not diversiviable (mainly investments, fixed expenses and fixed earnings). Each of these components can change in time. That means that the exercise price of the real option can be differerent in each period. In the Black Scholes Merton model the exercise price is the same over time. In case of changing exercise prices you have to use approximative (binomial) approaches.
• Discount rates: The discount rate includes many parameters like risk free rate, non-diversifiable market risks of the cash flow, the investor’s capital structure, the investor’s opportunity portfolio and tax shields. All these parameters can vary over time, and hence change the discount rate. If the discount rate changes, you might get a non-recombining binomial tree. Take care that the discount rate in each period has to match the expected values of the binomial tree branches in each period.
• Decision tree: The decision tree does not have to be a complete binomial tree as required at the approximation of the Black Scholes Merton approach. Some branches might not exist or there can be more than one possibility in a node of the binomial tree (tri- or multinomial trees). Sometimes these exceptions represent reality more accurately and can be calculated in a discrete model. But these cases are not an approximation of Black Scholes Merton model any more. The Black Scholes Merton model cannot handle such tasks.
• Time steps: Times between nodes of an event tree can be different at each link. You can adapt the times to your specific problem if required. The nodes of the event tree can for example be determined by the decision time of the investment project. At the Black Scholes approach the temporal development is fixed by the input parameters. There is no possibility to adapt it anyway. In the binomial approach you can adapt time steps as required by the corresponding problem.

In general, the (binomial) approximative approaches are much more flexible and can be adjusted to the specific problem. With specific input parameters the binomial model is an exact approximation of the Black Scholes Merton model. But many times the input parameters describing reality are different. Black Scholes Merton comes from financial option markets, where situations are less complex as at real options many times. The binomial approach is much more suitable for real option analysis. Because of the inaccuracy of many input variables at real options, the approximative character of the binomial model does not distort the result mostly.

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