## Unexpected Probability Parallels

A few weeks ago I wrote an article on Trader Edge titled "Political Diversions", which introduced two sites with different but effective methodologies for determining probabilistic election forecasts. One of those sites was Nate Silver's FiveThirtyEight.com, which is arguably the leading and most respected election modeling site on the Internet.  Silver is also the bestselling author of The Signal and the Noise: Why So Many Predictions Fail - but Some Don't.

About a month before the mid-term elections, I read one of Silver's articles in which he made a statement that forever changed my understanding of election probabilities and allowed me to see surprising parallels to option probabilities.

## Option Probabilities

Before getting to Silver's statement, let's review some basic option probability calculations. Many traders sell out-of-the-money credit spreads, which is a high-probability directional option strategy designed to earn income. A put credit spread involves selling an out-of-the-money put option with a strike price below the current price of the underlying security and buying a second put option that is further out of the money. The purchase of the second put option ensures the total risk of the strategy is limited, which also reduces required capital.

When selling credit spreads, it is important to know the probability the short option will expire worthless. This will happen when the price of the underlying security closes above the strike price of the short put option on the expiration date of the option.

Here is a specific example to demonstrate the probability calculation. Let's assume the following:

• Underlying stock price: \$55.00
• Strike price of Short Put Option: \$50.00
• Strike price of Long Put Option: \$45.00
• Time remaining until expiration: one month
• One-month, 1.00 Standard Deviation (SD) price change: \$5.00

Given this example, we want to know the probability that the current stock price (\$55.00) will close at or above the strike price of the short put option (\$50.00) at expiration. In other words, what is the probability that the stock price will drop by \$5.00 or less in the following month? To do this calculation, we need to first calculate the Z-score, which represents the difference between the current price and the strike price, divided by the standard deviation of the price of the stock over the coming month. (Stock price - Strike price)/(SD) = (\$55 - \$50)/(\$5) = +1.0.

The current stock price is exactly 1.00 SD above the strike price. To determine the probability the price remains above the strike price, we only need to look up the cumulative normal density function for a Z-score of +1.0, which will be available in any statistics textbook and can also be calculated with the NormDist function in Excel.

The resulting probability of the short put option expiring worthless [stock price (\$55) remaining at or above short strike price (\$50)] in one month is 84.13%.

So far so good, but the more interesting question is what happens to this probability over time, assuming the price of the stock and the expected level of volatility do not change? Let's do the same calculation one-day before option expiration.

• Underlying stock price: \$55.00
• Strike price of Short Put Option: \$50.00
• Strike price of Long Put Option: \$45.00
• Time remaining until expiration: one day
• One-Trade day, 1.00 Standard Deviation (SD) price change: \$1.066 = (\$5 / (22^0.5))

Note that the stock price remained the same, but the one SD move for one trade day (\$1.066) is far less than the one SD move for one month (22 trade days). Price volatility is proportional to the square root of time, so the expected price change over one trade-day equals the expected price change over one month (22 trade days) divided by the square root of the number of trade days in the month (assumed to be 22).

Given this example, we want to know the probability that the current stock price (\$55.00) will remain at or above the strike price (\$50.00) of the short put option at expiration (in one day). In other words, what is the probability that the stock price will drop by \$5.00 or less in the following day? To do this calculation, we need to again calculate the Z-score (stock price - Strike price)/(SD) = (\$50 - \$55)/(\$1.066) = +4.69.

The current stock price is now 4.69 SD above the strike price. The resulting probability of the short put option expiring worthless [stock price (\$55) remaining at or above short strike price (\$50)] in one day is 99.9994%.

This is what makes selling directional credit spreads appealing. Notice that the stock price and volatility remained the same, but the probability of the short put option expiring worthless increased to a virtual certainty.

## Election Probabilities

Let's look at a simplistic example of two candidates, one of which is leading the other candidate by a weighted-average of 55% to 45%, across all polls, one month before the election. In this example, the 55% represents the model forecast. What is the probability the leading candidate wins the election? To answer the question, we would need to know the standard deviation of the model forecast. If the standard deviation were 5%, the probability the leading candidate would win would be 84.13%. This is exactly the same calculation that we performed above in the option probability example.

The initial weighted-average poll result of 55% corresponds to the current stock price. The 50% winning threshold corresponds to the \$50 strike price and the 5% standard deviation corresponds to the \$5 standard deviation. The resulting probability the leading candidate holds on for victory in one month would have to be 84.13%.

However, there are some important differences. First, the \$55 stock price is known with certainty, but the 55% to 45% weighted-average polling result is only an estimate of the actual percentage for the entire likely-voting population. You have probably heard of the "margin of error" based on the sample size, but this statistical calculation understates the magnitude of this problem. Many other factors could affect the quality of the poll results that are not captured by the margin of error. First, poll results can be biased and polls are not independent. Silver's recent article titled "Here's Proof Some Pollsters are Putting A Thumb on the Scale" makes a compelling case of pollsters adjusting their results to match other poll results.

On October 15, 2014 (less than a month before the election) FiveThirtyEight.com gave the Republicans a 60% chance of gaining a Senate majority in the mid-term elections but Silver stated that the probability would be 75% if the poll numbers were free of bias with 100% certainty. Why? Because reducing the potential for bias would shrink the standard deviation, which would increase the Z-score, and increase the probability. Unfortunately, bias and the lack of independence cannot be eliminated and both factors increase the standard deviation and neither diminishes over time.

In a separate article, Silver also stated that the foresting model becomes more certain as time passes and the election date nears. This is exactly what happened in the options example above. As time passes and the election date nears, there are fewer opportunities for discrete "events" that could affect the opinions of voters. This includes debates, ad campaigns, missteps by candidates, etc. Similarly, as time passes, there are fewer opportunities for events that could significantly affect the price of the stock. As a result, the standard deviation of the stock price and the standard deviation of the election forecast both decline over time.  However, as mentioned above, the standard deviation of the stock price approaches zero as the expiration date approaches but the standard deviation of the election forecast does not, principally because of the uncertainty surrounding the polling results.

## Conclusion

I never realized that the standard deviation of an election model forecast decreases as the election date nears, just as the standard deviation of the stock price  declines as the expiration date approaches. I also never fully appreciated the magnitude of polling biases and systematic errors that far exceed the margin of error. There are important parallels between option and election probabilities, but there are also important differences.

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