Options Pricing

So, how is the estimated “value” of a stock option determined by an option trader? An automobile’s estimated value can be found by looking it up on the Internet or in the “blue book”. A house’s value can be estimated by comparing houses sold in the surrounding neighborhood. But how can an option trader estimate the value of a stock option?

Back in the early 1970s a couple of smart guys named Fischer Black and Myron Scholes developed a methodology for estimating option prices known as the Black-Scholes option pricing model. The mathematics behind the development of the algorithm is somewhat complicated, but the Black-Scholes options pricing model is not too difficult to use for calculating option prices. The model only requires a few variables for calculation: current stock price, time to option expiration, risk-free interest rate, option strike price and the standard deviation of the stock price.

How Good Is It?

For options with strike prices close to the price of the underlying stock price, the Black-Scholes model works quite well, but for options with strike prices far out-of-the-money or far in-the-money, it does not work so will in practice. The reason the Black-Scholes model does not work so well for in/out-the-money options is due primarily to one of the assumptions made for developing the model, namely, the model assumes log normally distributed returns. Lognormal distributions frequently occur in the real world, and the Central Limit Theorem aids in explaining why they occur so frequently. In layman’s terms the Central Limit Theorem basically states that when a bunch of random “things” are added up, the resulting probability distribution will approximate a lognormal distribution. But, returns for the stock market have shown to not be lognormal; hence when used in the real world there is a problem with the Black-Scholes option-pricing model.

Volatility Smile
Due to the lognormal assumption problem with the Black-Scholes model, a phenomena known as the “volatility smile” is observed. Many option traders consider implied volatility a very important parameter for stock option trading. Implied volatility is determined using the Black-Scholes model by determining the volatility that should be “plugged” into the model to return the same price for an option as the current price that is observed in the market. Theoretically, an options pricing model should observe implied volatilities that are similar across all option strike prices for a particular stock or index, but this is not the case with the Black-Scholes option model. For the Black-Scholes option model, the further away the strike price is from the current stock price, the greater the value of the implied volatility observed. A plot of the implied volatilities versus strike price generates a graph having the shape of a smile; hence the term “volatility smile” was coined.

Calculate?
Other models have been developed for calculating far in/out-the-money option prices, but the models are generally difficult to implement, may require manual “tweaking” to make them work and/or only work well for certain market conditions. So until a comprehensive and easy-to-use option-pricing model is developed, Black-Scholes will remain the option-pricing model of choice, even though it has a nasty smile.

[tags]Black Scholes Pricing Model, options pricing, implied volatility, stock option trading[/tags]