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Understanding Option Models

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Many option traders do not understand the theory behind the Black-Scholes model. The actual idea is quite simple. The underlying assumption is that the stock price moves as a random walk. There is no assumption of directional movement. The only assumption is that the recent historical volatility will persist. Note that the volatility is stock specific. Different stocks will have different volatilities.

However the volatility can be calculated for each stock based on the recent daily price movement. The calculation is based on the formula for finding the standard deviation of the daily price moves. For option use the formula is modified by taking the natural log of the relative return.

There is a graphic which shows the probability of the stock winding up at each price up or down. The scale at the bottom is in units of standard deviations. In other words it is a volatility scale. The current price of the stock is assumed to be at the zero standard deviation level. Note that the highest probability is right in the middle, where the stock is today. The probability falls off as the stock price moves farther from the middle in either direction.



For example the graphic has a vertical line drawn on it at -1 standard deviation. The value of a put option with a strike price set at -1 standard deviation would be based on the area under the curve to the left of the vertical line. The exact formula need not concern us here. But suffice it to say that the more area under the curve the more the put would be worth.

The area under the curve can be used to calculate the probability that a given option trade will wind up making money. As a rough rule of thumb a move of more than 1 standard deviation (in either direction) is only a 16% probability. A move of 2 standard deviations is only about a 5% probability.

Naturally this has important implications for safety margin when we sell options or sell a credit spread. The farther out of the money the safer an option sale is. However it is important to remember that it depends greatly on how volatile the underlying stock is. If a stock is more volatile then we need a greater margin of safety. If it is less volatile then a reduced margin of safety may be all that is needed.

Personally I use proprietary software which is not publicly available. It tells me the probability that a given option will expire worthless. To me it is very important to keep the probability of loss as low as possible. I am a big believer in having many winners and as few losers as possible. That way I can stay diversified and let the many small winners overcome any possible losers. That strategy enhances the probability that any given month will wind up with a profit.

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