The Importance Of Differentiating Between Historic And Implied Probability

Includes: SPY

Last week I came across an article on Seeking Alpha with the intriguing title "How To Choose Your Own Probability Of Success."

Because it was published just one day after an article I had written that suggested a very different probability (~34% vs. 20.77%) for the same stock on the same option expiration date, I felt I should both comment on the article and clarify my own views on the concept of "probability."

For outcomes that are normally distributed, probability calculations are dependent on the mean and the standard deviation (i.e., "volatility" in the financial markets). It stands to reason, therefore, that since volatility can be estimated either from past stock price data (historical volatility) or from the market's view of future volatility based on current options pricing (implied volatility), then it is very important to know whether one is assessing the probability of what may occur in the future based on past history or based on a consensus view by market participants.

In other words, as with volatility, probability estimates are either historic or implied. Yet surprisingly few, if any, software applications highlight this distinction when displaying a number for "probability".

To get a more practical sense as to why all of this matters, consider the hypothetical option trade I mentioned in my article last week. Summarizing this trade, if the SPY ETF closes below $138 a share on March 16 (options expiration for March), the hypothetical investor will realize a gain for the month of about 1%.

One week has now passed since the trade was "executed." From today's standpoint, what is the probability that 3 weeks from now (March 16) SPY will close below $138 a share?

A very good (and free) probability calculator that is typical of most such calculators provided by brokerage firms and software vendors can be found here courtesy of and TradeKing.

To use the calculator, simply enter SPY in the symbol box and select the "Go" button. The current information needed to calculate the probability is automatically loaded. All you need to do is change the expiration date (by selecting from the dropdown box) and enter the target price of $138 in each of the two boxes on the right. Then select the "Calculate" button.

When I did this just now (your results will differ depending on when you do this), I got a probability of 67.48% (don't you just love 4-digit accuracy?) that SPY will close below $138 a share on March 16. Cool.

But there's also a dropdown box on the left that is pre-populated with something referred to as "ATM Volatility." If you select the dropdown box, you are presented with more than a dozen choices; if I select "HV 10D" the probability that SPY will close below $138 jumps up to 73.68%. Even better!

But if I select "HV 150D" the probability drops to 59.22%. Not so cool.

And to top things off there is even a "Custom" choice that allows you to enter your own volatility number if you don't like any of the choices ... It would seem that in a very real sense you can indeed "Choose Your Own Probability of Success" for any price over any time period using this tool.

As if this weren't confusing enough, there's another problem with tools such as this that base probability only on implied volatility; there is no place on the / TradeKing calculator for you to enter the expected mean return over the period in which you wish to calculate probability.

To illustrate the importance of this omission, consider this example problem: Given that the standard deviation (i.e., "volatility") of the height of an adult American male is about 3", what is the probability that the next adult American male you pass on the street is taller than 6' 0"? Clearly, there is insufficient information to answer this; what's missing is the average (or mean) height of an adult American male.

Because probability calculators are based on option pricing models that for the most part incorporate the Nobel Prize winning concept that the fair value of options does not depend on expectations of price movements of the underlying, it is easy to see why many software applications don't include expected change in the price of the underlying as input for their probability calculations.

However, if we are going to play in the arena of normal distributions (which is also a key assumption in these pricing models) then we must consider the mean as well as standard deviation in our probability calculations.

Now some may argue that volatility strike skew (whereby the put and call options for the same underlying strike price and expiration date have different implied volatilities) takes into account the market's expectation of future return. But since there is also no way to input two volatilities (one value for the call and one for the put) into these calculators, the argument is moot.

So how did I arrive at my estimates of probability in last week's article? The Excel workbook I used considers the historic mean and standard deviations of the price changes of the underlying over the time period of interest (in this case, 19 trading days between Feb and Mar expiration).

I'm not comfortable relying on the combined consensus of a market that consists of traders, investors, and (increasingly) computer algorithms to distill and factor the net conditional probabilities of the Greek debt situation, the future of the eurozone, the Iran oil embargo, American monetary and fiscal policy changes, etc into a single probability value.

I prefer instead to rely on "having been there before" with all of these uncertainties, and that the combined history of these and other socioeconomic and political market factors over the last 20+ years have resulted in the current mean and standard deviation. I further believe that these values will continue to be adequate for my purposes for at least the next 20 days.

Right or wrong, it's comforting to me to know that if there is a price move in SPY greater than 1 standard deviation in the coming month, it will have been an event that has occurred only about 32% of the time over the last 20 years.

I have no such confidence in the track record of a market consensus of probability, because I have no information to demonstrate that the market's constantly changing "methodology" for assigning probability is any more accurate than assuming longer-term historic price movements around the mean will continue just a bit longer into the future.

If you would like a copy of the Excel workbook I developed for determining historic probabilities as a means to compare with implied probabilities, feel free to contact me using the email address or phone number listed in my profile.

Disclosure: I am long SPY. I hold both long and short option positions in SPY.

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