The binomial option pricing model is a method to price options by assuming that the underlying asset can move up or down by a certain percentage at each time step until the expiration date. The model is based on the assumption that the options can only be exercised at expiration. It is a discrete-time model that involves building a binary tree representing all the possible price movements of the underlying asset until the expiration date.
To value an option using the binomial model, the first step is to construct the binomial tree. The tree has a set number of levels representing the different expiration dates for the option and a set number of possible price movements at each level. The probability of the stock going up or down is calculated using the volatility of the stock and the time interval between each node in the tree. Once the binomial tree is constructed, the option value can be calculated by working backwards through the tree from the final expiration date, calculating the expected value at each node at each level.
The Black-Scholes model, on the other hand, is a continuous-time model that assumes the underlying asset has a random walk with constant volatility, and that the option can be exercised at any time before the expiration date. The model uses the concept of a risk-neutral probability measure to value the option. The key formulas used in the Black-Scholes model are:
$$\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} - rC = 0$$
where C is the price of the call option, S is the price of the underlying asset, Ο2 is the volatility of the underlying asset, r is the risk-free interest rate, and t is time.
The Black-Scholes model assumes that the underlying asset follows a log-normal distribution of prices and that the option can be exercised at any time before expiration. The model requires the input of five variables: the underlying asset price, the strike price, the time to expiration, the risk-free interest rate, and the volatility of the underlying asset.
One of the main differences between the binomial option pricing model and the Black-Scholes model is that the binomial model assumes that the underlying asset can only move up or down, while the Black-Scholes model assumes a continuous range of stock price movements. Another major difference is that the binomial model works well for American options, which can be exercised at any time before expiration, while the Black-Scholes model is mostly used for European options, which can only be exercised at expiration. Finally, the Black-Scholes model is a closed-form solution, meaning that it can be solved using a simple formula, while the binomial model requires a computationally intensive tree-building process.