Calculating the Probability of Option Payoff

In practice, the ideal goal should be to develop a realistic probability rather than relying on extremes.

Many random variables are at work, and when you calculate maximum profit or loss, it must be based on those best or worst outcomes even when other possibilities exist, ands are easy to overlook. This means that for many traders, calculations of likely or realistic outcomes are inaccurate in most cases. These perfect models of either profit or loss are not real-world or likely results:

The hardest aspect about random variables to understand is that they are idealized models of reality. That is, we sacrifice a precise model of reality for the facility of dealing with precise mathematical objects. [Chriss, Neil A. (1997). Black-Scholes and Beyond. New York: McGraw-Hill, p. 70]

The many possible outcomes may be kept in mind as well as breakeven and the maximum profit or loss. These three are rarely likely outcomes, just the extremes. Because options are not usually held until expiration (whether long or short), these expiration date-based extremes are not reliable. It is useful to know where breakeven lies, not to strive for it but to know where your position needs to go in order to become profitable (or to minimize or cut losses).

The yield expressed as a percentage varies in more than one way. For most people, it represents the net return based on the original amount traded; but it matters just as much to consider the time remaining to expiration. It matters equally to calculate outcomes based on actual number of days a position is left open before profit or loss are realized. To  make comparisons realistic, traders must annualize the return. The steps are not difficult. First, divide the dollar amount of profit by the original amount traded. Divide that percentage by the days the position was left open, and then multiple by 365 to get the annualized return.

This calculation, easily overlooked, is of great importance because traders have positions open for varying numbers of days. Annualizing is the only way to make valid comparisons of outcomes between two or more sets of trades.

To demonstrate how unrealistic it is to calculate outcomes based only on maximum profit or loss, consider how it all really works. Without delving too deeply into how traders assume, it is easy to start out believing that an option has a 50-50 chance of profit or loss. In other words, if this random assumption were true, traders might as well go to a casino and bet on red or black with every turn of the wheel. But in the real world, outcome is not 50-50.

The 50-50 assumption depends on all elements of the option containing equal value in the calculation. This means that moneyness, time and historical volatility each contain the same influence on outcomes. But traders know that all these factors vary considerably, and, in some cases, one element holds much greater influence than the others. For example, in writing short options, focusing on contracts expiring on one week or less increases the influence of time because time decay will be more rapid than longer-term options.

In defining risk realistically, it is necessary to understand how these attributes vary and may hold greater or lesser weight. This is the most accurate way to estimate true risk, but it remains an estimate. Some matters cannot be calculable, and probability of outcomes is one of those matters. It is comforting to calculate maximum profit or loss and breakeven because these are absolute and unchanging. But the assumption remains that the option will be held to last trading day, and this does not happen most of the time.

Therefore, many traders have turned to pricing models like Black Scholes. They desire some accurate, absolute, reliable system for better understanding probability. But pricing models, notably Black Scholes, contain so many flaws that they cannot accurately calculate outcomes. In fact, no model of  current value or risk can accurately predict future price movement or outcome. This is the reality, even though options traders constantly try to find the perfect predictor.

Models are only good for making calculations of future outcomes based on current levels of volatility, moneyness, and time, all of which cannot accurate make predictions on their own. The application of a pricing model must fail because the “model” itself is not the same as a model used elsewhere:

In physics or engineering, a theory predicts future values … 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]

The comparison between several options trades can never be completely accurate, because every trade, like every underlying, contains varying levels of risk in many forms. Among these variables, the biggest one of all is the human element, the tendency of traders to seek confirmation, to make inaccurate assumptions, or to place too much weight on maximum profit or loss and assume those are accurate measurements. But an honest evaluation of the varying risk elements within the option’s variables (such as selection of an expiration date whether long or short in the option) will help to weight the likelihood of one outcome over another.  This is where traders can make accurate outcome predictions, based on their own analysis and weighting of elements in the option, and not on the assumptions that models reveal very much of value.

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.