# The Black Scholes Model – Helping Traders Value Options

Most options traders have heard of the Black Scholes Model but few really know much about it. Following are some key bullet points about the model, its use and history:

**Origin** – The Black Scholes model was developed in 1973 by Fisher Black and Myron Scholes. Their paper was published that year in the Journal of Political Economy under the title The Pricing of Options and Corporate Liabilities. Robert C. Merton published a follow up paper further expanding the understanding of the model. Merton and Scholes received the 1997 Nobel Prize for their work. Fisher Black was ineligible because Nobel prizes are not awarded posthumously.

**Main Contribution** – The main contribution of the Black Scholes model was the recognition that two parties with different expectations for the performance of a stock could still agree on a fair price for the option given that it was traded. The key factor in the valuation was the volatility of the stock. Expectation does not enter into it.

**Factors in the Model** – There are several factors in the model they are:

- Stock Price
- Strike Price
- Time to Expiration
- Interest Rates expressed as carrying cost
- Future Volatility of the underlying stock

Of the listed factors, only Volatility is not known with a high degree of certainty. Volatility is also the most important factor. It should also be mentioned that for stocks which pay dividends the dividends and ex-dividend date are also factors which effect the valuation of options. The original Black Scholes paper only considered call options which did not pay dividends. However other researchers have added various methods for valuing options with dividends.

**Usage** – The original usage of the model was to calculate a fair value for a call option. The practitioner would plug in a value for each of the main parameters of the model and it would calculate the fair value. The key to using the model has always been to come up with a good estimate of future volatility. The goal of a trader is to buy an option below its fair value and sell it if it is above.

**Key Model Assumptions** – One important assumption made in the Black Scholes model is that the stock price path follows a log normal random walk. To a large extent this price distribution is a fair model of stock price behavior. However many have pointed out the the distribution of the tails of the stock price changes are fatter than the tails of the log normal.

**Estimating Volatility** – Most practitioners estimate volatility by calculating the log normal standard deviation of the stock over some recent time period. Then they take the recent volatility and extrapolate it going forward. Some make adjustments based on whether they believe volatility will rise or fall in the future.

**Implied Volatility** – In the beginning of the model people used it to calculate fair value. However people quickly realized that one could run the model backwards. Basically the idea is to plug in the current option price and run the model backwards to calculate what the market’s current estimate of volatility must be. This is called Implied Volatility and can be understood as the volatility implied by the current options price. A high Implied Volatility can be understood to be a high price. A low one is a low priced option.

**The Skew** – There usually are many options traded on a given stock, each with its own implied volatility. Rarely are they all exactly equal. Most of the time options at different strikes show a fairly smooth curve based on the strike. This curve is called the skew and sometimes affectionately referred to as the smile. The skew is believed to represent the market’s estimate of the probability of fat tails. Thus strike prices that are far away from the current price of the stock usually exhibit the largest skew. Smart traders look for deviations from the skew in individual options which may indicate mispricing.

*Philip J. McDonnell is a full time option trader and author of Optimal Portfolio Modeling published by Wiley Trading.*

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