Option Pricing Theory

An option pricing theory is any model or theory-based approach for calculating the fair value of an option. Today, the most commonly used models are the Black-Scholes model and the binomial model. Both theories have wide margins for error due to deriving their values from other assets, usually the price of a company's common stock.

Aside from using a company's stock price to determine an option's fair price, time also plays a significant role. Calculations involve time periods of several years or more. Marketable options require different valuation methods than non-marketable options. Real traded options prices are determined in the open market, and as with all assets, the value can differ from a theoretical value. However, having the theoretical value allows traders to assess the likelihood of profiting from trading those options.

The evolution of the modern-day options market is attributed to the 1973 pricing model published by Fischer Black and Myron Scholes. The Black-Scholes formula is used to derive a theoretical price for financial instruments with a known expiration date. However, this is not the only model. The Cox, Ross, and Rubinstein binomial options pricing model is also widely used.

The Black-Scholes model requires five input variables, the strike price of an option, the current stock price, the time to expiration, the risk-free rate, and the volatility. Direct observation of volatility is impossible, so it must be estimated or implied. Also, implied volatility is not the same as historical or realized volatility. 

Additionally, the Black-Scholes model assumes stock prices follow a log-normal distribution because asset prices cannot be negative. Other assumptions made by the model include the assumption there are no transaction costs or taxes, the risk-free interest rate is constant for all maturities, short selling of securities with use of proceeds is permitted, and there are no arbitrage opportunities without risk.

Clearly, some of these assumptions do not hold true all of the time. For example, the model also assumes volatility remains constant over the option's lifespan. Consistency is not the case because volatility fluctuates with the level of supply and demand.

Also, Black-Scholes assumes that the options are European Style, executable only at maturity. The model does not take into account the execution of American Style options before the expiration date. However, for practical purposes, this is one of the most highly regarded pricing models. On the other hand, the binomial model can handle both styles of options because it can check for the option's value at every point in time during its life.