Black Scholes Option Pricing

The Black and Scholes Option Pricing Model is a specific approach designed for calculating the value of a stock option. This article briefly details that model to give some background information about it. A financial researcher named Fisher Black started on his own working to create a model for stock warrant valuation. The work he did included calculating a derivative to try to pinpoint how a warrant's discount rate varies relative to time and stock price. The result of Black's work was an equation that bore a significant resemblance to an existing model for predicting heat transfer.

Myron Scholes and Fisher Black

Shortly after Black's discovery regarding the similarity of his formula to the heat transfer equation, Myron Scholes joined Black in his scholarly research on the topic. Their work now produces an amazingly accurate model for option pricing. Their work piggybacked on that of an earlier scholar, and the Black Scholes Option Pricing Formula added proof that the risk free interest rate is the correct discount factor to work off of and that no assumptions are necessary regarding investors' risk preferences or aversion to risk [1].

Understanding the model requires thinking about it in its respective parts. The first part of the Black Scholes Model calculates the expected benefit associated with buying a stock outright. This can be found by multiplying the stock price by the change in call premium with respect to the change in underlying stock price. The second part of the formula comes up with the present value of paying the exercise price and thus choosing to exercise the stock on the options expiration day. The fair market value of the option is then calculated by taking the result of the second part of the formula and subtracting it from the first.

Assumptions Underlying Black Scholes Model

The Black Scholes formula for determining option pricing is based on a certain set of assumptions, as are many formulas of this type. The first assumption is that the stock in question does not pay out any dividends during the life of the option. Most companies do pay dividends to their stock holders (at least, those who hold preferred stock). This can come across as a flaw or a real limitation to the formula because dividends are so common. It is well known that higher dividends yield lower call premiums. One way financial researchers have gotten around this criticism is to simply subtract the discounted value of future dividends off the stock price [1].

Another assumption that has gotten a fair bit of attention through the years is the fact that European exercise terms are used in the model. These terms dictate that the stock option can only be used on the expiration date. By contrast, American exercise terms allow options to be exercise at any time during the life of the option. American terms are more valuable to stock holders with options because they are so flexible. A stock holder does not just have to hope for the right set of circumstances on that particular expiration day, but can simply follow the ebb and flow of the market cycles and exercise the option at any financially advantageous time. With this being said, most calls are actually exercised toward the end of their life.

Black and Scholes also assumed in their formula that markets are efficient, meaning individuals can't consistently predict the direction of the market or of a particular stock. They also assumed no commissions were charged, constant interest rates were present, and that returns were normally distributed [1].

[1] Retrieved 2010-05-30.

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