Option Payoff Probability

In the real world the single modeling of outcomes based on remaining open until expiration day, fail to consider several unknown and unanticipated random variables.

The easily calculated breakeven is easily calculated. Above or below that level, either profit or loss can mean many different things. When either are at maximum, you know the outcome exists within a range of possibilities. When maximum profit or loss are unlimited, it is more difficulty to identify probability beyond a rather simplified assumption: The greater the possible degree of profit or less, the lower the probability. In other words, in most instances, an “unlimited” profit or loss will not be realized in the extreme. A smaller (rather than a larger) profit or loss is more rational to assume as the likely outcome. This is all guesswork, of course, because whether options traders like it or not, the probability itself can never be calculated accurately. Being aware of the range of possibilities is the best anyone can expect.

Not only the percentage of outcome, but time itself, only add to the random variables of possible outcomes. This means that a small, medium or large level (of profit or loss) might be realized in a few hours, a day, several days, weeks or months. No matter how many applications of formulation are applied, there are many possibilities that, at lest, you can apply what you consider the most likely, more limited range.

This is the reality, but it is easy to overlook that by applying some flowed logic. For example, many traders believe that without considering other factors, profit or loss is a 50-50 proposition. Half will be profitable, and the other half will not. This is true for flipping a coin and expecting a 50-50 outcome of heads or tails, but options are not flat, two-sided items. They are not even tangible. The many variables include moneyness, time, volatility, and overall market conditions. Further complicating this, the random variables are not always equal in how they influence outcomes. Even if you can articulate all the matters influencing option profit or loss, how do you weight the elements at play?

Because there is no clear answer to this, every trader ends up having to make a few assumptions about the random variables. For example, you might assume the following:

1. The closer to the money, the more reliable the assumption that profit or loss stand a 50-50 chance of being realized. The farther away, the loss likely the 50-50 outcome.
2. The shorter the time to expiration, the greater the odds of loss due to time decay, and the less likely the odds of profit (for long positions). This assumption is flipped for short positions.
3. The greater the volatility in both underlying and option, the more difficult it will be to predict outcomes on either side.

There is a degree of logic to these assumptions, but every options trader has experienced examples in which even the most conservative and logical assumption is betrayed by unexpected surprises and price movements. This is a reality, but it still does not explain why pricing models often do not reveal anything of value. Uncertainty defines options trading, and no pricing model based on current and historical observations can really predict the future.

It may be desirable to wish a pricing model exists that would do so, but it does not. Despite popular beliefs that pricing models are predictive, they never are, and this is the stark reality of trading, whether in options or other instruments. Models just predict future volatility and price based on current volatility and price. It’s like guessing that tomorrow’s temperature will be 72 because today’s high was 70 and it has risen by two degrees every day for the past week. Everyone knows this is not reliable, but that is precisely how pricing models are applied to predict. If we know the flaws in weather prediction based on “trends,” why not apply the same limitation to trading?
 

The great problem of trying to predict future option pricing is that options are not like other topics of analysis:

In physics or engineering, a theory predicts future values. In finance, you’re lucky if your model can predict the future sign correctly. So what’s the point? Models in finance, unlike those in physics, don’t predict the future; mostly they relate the present value of one security to another. In science, when you say a theory is right, you mean that it’s mathematically consistent and true – that is, it explains and predicts its corner of the universe. In finance, right is used to mean merely consistent: many models are right but usually none of them are true. [Derman, Emanuel (2007). Sophisticated vulgarity. Risk, 20(7), 93]

A comparison between several different trades or potential trades is rarely identical. This is where probability analysis becomes useful. It is possible, as an alternative to seeking a perfect pricing model, to develop a means for quantifying the likelihood of profitability when it is greater in one trade than in another. There is no guarantee of a positive outcome, but there is a better probability when the two estimates are vastly different.

This is not only the best any trader can hope for. It is a reliable system for making sound judgments and for overcoming the greatest random variable of all: the human element. Every trader is subject to biases not based on rational analysis, but on opinion, false assumption or irrational preferences. No one can realistically identify a means for ensuring profitable trades all the time, but  modeling can be useful in applying that desirable but rate aspect of trading: objective analysis.

Michael C. Thomsett is a widely published author with over 80 business and investing books, including the best-selling Getting Started in Options, coming out in its 10th edition later this year. He also wrote the recently released The Mathematics of Options. Thomsett is a frequent speaker at trade shows and blogs on his websiteat Thomsett Guide as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.