Assumptions of the Black-Scholes Model For Options Trading

The Black-Scholes Model is a formula for calculating the fair value of an option contract, where an option is a derivative whose value is based on some underlying asset.

In its early form the model was put forward as a way to calculate the theoretical value of a European call option on a stock not paying discrete proportional dividends. However it has since been shown that dividends can also be incorporated into the model.

In addition to calculating the theoretical or fair value for both call and put options, the Black-Scholes model also calculates option Greeks. Option Greeks are values such as delta, gamma, theta and vega, which tell option traders how the theoretical price of the option may change given certain changes in the model inputs. Greeks are an invaluable tool in portfolio hedging.

1) No Dividends

The original Black-Scholes model did not take into account dividends. Since most companies do pay discrete dividends to shareholders this exclusion is unhelpful. Dividends can be easily incorporated into the existing Black-Scholes model by adjusting the underlying price input. You can do this in two ways:

1. Deduct the current value of all expected discrete dividends from the current stock price before entering into the model or
2. Deduct the estimated dividend yield from the risk-free interest rate during the calculations.

You will notice that my method of accounting for dividends uses the latter method.

2) European Options

A European option means the option cannot be exercised before the expiration date of the option contract. American style options allow for the option to be exercised at any time before the expiration date. This flexibility makes American options more valuable as they allow traders to exercise a call option on a stock in order to be eligible for a dividend payment. American options are generally priced using another pricing model called the Binomial Option Model.

3) Efficient Markets

The Black-Scholes model assumes there is no directional bias present in the price of the security and that any information available to the market is already priced into the security.

4) Frictionless Markets

Friction refers to the presence of transaction costs such as brokerage and clearing fees. The Black-Scholes model was originally developed without consideration for brokerage and other transaction costs.

5) Constant Interest Rates

The Black-Scholes model assumes that interest rates are constant and known for the duration of the options life. In reality interest rates are subject to change at anytime.

6) Asset Returns are Lognormally Distributed

Incorporating volatility into option pricing relies on the distribution of the asset’s returns. Typically, the probability of an asset being higher or lower from one day to the next is unknown and therefore has a 50/50 probability. Distributions that follow an even price path are said to be normally distributed and will have a bell-curve shape symmetrical around the current price.

It is generally accepted, however, that stocks – and many other assets in fact – have an upward drift. This is partly due to the expectation that most equities will increase in value over the long term and also because a stock price has a price floor of zero. The upward bias in the returns of asset prices results in a distribution that is lognormal. A lognormally distributed curve is non-symmetrical and has a positive skew to the upside.