Black-Scholes vs Binomial Option Pricing Models Explained

Introduction to Option Pricing Models

There are different variables that typically influence the pricing of options. This paper examines the attributes of two widely accepted models used for pricing options: the Black-Scholes model and the Binomial model. Both models rest on the same theoretical assumptions and foundations, including risk-neutral valuation and the geometric Brownian motion theory of stock price behavior.

Option pricing theory has become one of the most powerful tools in commerce and finance. The famous Black-Scholes equation is an effective model used for option pricing. It was named after its pioneers—Black, Scholes, and Merton—who introduced it in 1973 and were awarded the Nobel Prize in Economics in 1997 for their discovery. Mathematically, it can be described as a final value problem for a second-order parabolic equation.

In this context, an option is a contract that gives the owner the right, but not the obligation, to purchase (call option) or sell (put option) an asset—most commonly a stock or parcel of shares in a company—at a pre-specified price termed the strike price, denoted "E," by a particular expiration date "T," at which point payoffs are received (Macbeth & Merville, 1979). The fundamental problem is determining a fair price to charge for granting these rights.

European options can only be exercised when the expiration date T has been reached, whereas American options may be exercised at any time up to and including the expiry date. For the American call option, the value V depends on the current market price of the underlying asset S and the time t remaining until the option expires, giving V = V(S, t) (Ehrhardt & Mickens, 2007).

The Black-Scholes Model

The Black-Scholes model is used to calculate a theoretical call price that excludes dividends paid during the life of the option. It relies on five key determinants of an option's price: the stock price, the strike price, volatility, time to expiration, and short-term interest rates.

The primary advantage of this model is its speed—it allows one to calculate a very large number of option prices in a short period of time. Its main limitation, however, is that it cannot accurately price options with an American-style exercise feature, as it calculates the option price only at a single point in time rather than across multiple potential exercise points.

The Binomial Model

The sister model to the Black-Scholes model is the Binomial, or two-state, model. It has attracted considerable attention and acclaim due to its ability to illustrate the essential ideas behind option pricing theory using relatively little mathematics, and its capacity to value many complex existing options. The precise origin of this model is somewhat unclear, and over the years an extensive body of research has been devoted to improving it.

In this model, the compounded risk-free rate per annum is denoted r. A risky asset is priced at S, which has the possibility of moving up to a state "+" with value uS, or moving down (Chance, 1998). The model breaks the time to expiration into a potentially large number of time intervals, or steps. At each step, an assumption is made that the stock price moves either up or down by an amount calculated using the asset's volatility and time to expiration.

This process produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents the possible paths the stock price could take over time. The major advantage of the Binomial model is that it can be used accurately for American options, since it is possible to check at each node of the tree whether early exercise is optimal. Its main limitation is its relatively low speed compared to the Black-Scholes model.