A binomial tree is a financial tool that seeks to present a graphical representation of different intrinsic values that an underlying asset or option can take in different periods. The first general approach was developed in 1979 by Cox, Ross, and Rubinstein and has since become a popular method of approximation for options. A binomial tree is a flexible model which can include a range of conditions, making it a popular tool for tracking options over time and a given interval. At each node of the ‘tree,’ the value of the option (the underlying bond or stock) will depend on the probability that the asset will grow or decrease. The model’s simplicity is highly beneficial in tracking values, but because it can only demonstrate one of two values (positive or negative growth), it is only an estimation. The mathematical formula behind the binomial tree is relatively simple, and I find it highly fascinating that the model can estimate option values with relative accuracy. Furthermore, the binomial model can capture ever-present volatility in the stock and option markets to derive correct approximations.
The binomial tree is often used in trading and investment; it allows investors to determine if and when an option should be exercised based on the probability of its growth or decrease in the model’s value prediction. It can price options on stocks with dividends, stock indices, currencies, and even futures (Guo & Liu, 2019). The so-called binomial options pricing model (BOMP) is the practical method of valuing options, utilizing the binomial tree graphical representation. Another similar but more complex method of valuing options, known as the Black-Scholes Model, is generally more helpful for complicated scenarios. However, despite being less complex and slower, practitioners in finance and investment prefer the binomial tree model because it is increasingly more accurate for long-term securities and dividends. The binomial tree model can be applied to both American and European type options, calculating for value. The binomial model is often the most preferential pricing method because of the possibility of implementing an efficient backward algorithm.
Binomial trees can differ in complexity, being one-step and two-step. When a multistep binomial treatment governs option price movements, each step is viewed separately to obtain the option’s current value (Korn & Muller, 2010). Overall, the conceptual simplicity of binomial trees offers significant efficiency in using the numerical method to approximate option prices. However, in a multi-dimensional context, they begin to have some drawbacks as not being as practically helpful as other models like Black-Scholes. Furthermore, one limitation noted is that conventional binomial trees often result in irregular convergence behavior, needing to control for the discretization error. The binomial tree approach evaluates options in a risk-neutral valuation; however, estimations show that no-arbitrage arguments and risk-neutral valuation led to the same price outcome.
Binomial trees are highly useful in investment and business contexts and trading. As an example, it can help calculate the value of investment opportunities in a startup. The investment opportunity provides the right to invest, not an obligation. Using tools such as the binomial tree, an investor can see how the value of an underlying asset can evolve. This can offer significant benefits in terms of long-term risk assessment and evaluation of the assets of any given firm or stock.
References
Guo, S., & Liu, Q. (2019). A simple accurate binomial tree for pricing options on stocks with known dollar dividends. The Journal of Derivatives, 26(4), 54–70.
Korn, R., & Müller, S. (2010). Binomial trees in option pricing—history, practical applications and recent developments. In Devroye, L., Karasözen, B., Kohler, M., Korn, R (Eds.), Recent developments in applied probability and statistics (pp. 59-77). Physica-Verlag HD.