Binomial Tree: Modeling Price Movements in Finance

A binomial tree represents the intrinsic values an option may take at different time periods. The option’s value at any node is contingent upon the probability of the underlying asset’s price either increasing or decreasing.

A binomial tree is a tool for pricing American and embedded options. While its simplicity makes it easy to model, the limitation lies in the binary nature of the possible values for the underlying asset within one period. Assessing the likelihood of option exercise is vital, with options having a higher probability of exercise if they hold positive value.

The binomial options pricing model (BOPM) relies on the binomial tree as its initial step. Key assumptions include two possible prices, no dividends from the underlying asset, a constant interest rate, and no taxes or transaction costs.

The Black-Scholes model offers an alternative method for valuing options, and while the binomial tree is slower in computation, it is considered more accurate, especially for longer-dated options and securities with dividend payments. The Black-Scholes model excels in handling complex and uncertain options, converging with the binomial model for European options without dividends as time steps increase.

Let’s consider a stock with a $100 price, a $100 option strike price, a one-year expiration date, and a 5% interest rate. At year-end, there’s a 50% chance the stock rises to $125 and a 50% chance it drops to $90. The option value is calculated as [(probability of rise * up value) + (probability of drop * down value)] / (1 + r), resulting in an option value of $11.90.

The binomial tree is employed in option pricing to model possible intrinsic values at different time periods, aiding in the assessment of option exercise likelihood.

Key assumptions include two possible prices, no dividends, a constant interest rate, and no taxes or transaction costs.