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The problem of Option Math


Option traders may be divided into two categories. First are those relying on instinct or casual observation. This group tends to speculate on directional movement, future volatility, value, and on potential profitability of trades. The second group is involved deeply with math of trading and depends on what is perceived as certainty or near certainty.

This more “scientific” approach is believed to be conservative and reliable. But is it?


Option people tend to be mathematical by nature. Most traders use math to pricing, payoff expectation, calculated of net returns, risk identification, and overall probability. Math comes into play when considering offsetting positions, short exposure, hedging,  and conversion (of complex relationships between options and underlying securities, into actionable decisions and probabilities.


In viewing the many ways that traders view the math involved, it at first appears to be sensible and rational to analyze trades and their risks based on finite mathematical study. Experienced traders, in fact, do not rely on the relatively simple mathematical study of strategies (or their maximum profit and loss and breakeven calculations). These calculations are helpful but limited. They assume what happens to an option if it is held open until last trading day. However, experienced traders know that most options are closed before this day, so the basic calculations provide only a “worst case” or “best case” outcome that rarely occurs. These outer perimeters of outcome do not address the needs of the experienced trader, who will not find the math required online. Experienced traders have moved beyond the calculation of possible outcomes or the deeply flawed Black-Scholes pricing model that has captured the imagination of academics and theorists.


In fact, even these basic calculations so popularly found in options books often are deeply flawed for numerous reasons:

  1. They do not address real-world outcomes. For example, holding options until last trading day is a rarity, so maximum profit or loss is unlikely to ever be realized.
     
  2. Most popular calculations do not include allowances for trading costs, time value of money, collateral requirements for short positions, or dividends.
     
  3. The methods of calculation often are simply wrong as well. For example, when websites offer analysis of implied volatility (itself only an estimate), why don’t they reveal their methods? It is because the variables employed can easily be adjusted to report any desired outcome.
     
  4. The calculation of probability itself is also wrong in most cases. Most calculations are performed using the additive method (the odds of an outcome being realized), and not on the more accurate mutilative method (the odds of an outcome not being realized). It might seem that both methods would produce the same outcome, but they do not. A significant difference, in fact, is derived with each method, and the additive method overstates the estimated positive results of trades. [See Reehl (2005), The Mathematics of Options Trading. New York: McGraw-Hill, pp. 44-45; and Thomsett (2017), The Mathematics of Options, New York: Palgrave Macmillan, pp. 247-249]

The flaws in commonly applied math should be of great concern to all traders, but they are only rarely addressed or even discussed. The problems of multiple variables makes options math particularly elusive, and as a result it is difficult to pin down a precise method for any of the mathematical applications involved. However, traders may experience more positive outcomes by recognizing the flaws in math that prevent certainty in estimates. Unfortunately, too many traders have become convinced that basic calculations of volatility, probability, and profitability are reliable. This is not true, and therein lies the real flaw: Choosing to believe a set of assumptions that are not only inaccurate, but often entirely untrue.
 

A favorite practice of some portfolio managers and traders is to engage in a deep study of implied volatility. This calculation is based on estimates about future volatility (which is impossible to know) and may rely on calculations and spreadsheet formulas provided by online experts and industry observers. But how have they calculated volatility? Since none of these sites explain their methods, how can anyone rely on the results? The fact is that implied volatility is as flaws as the Black-Scholes pricing model. Traders replying on a study of historical volatility in the underlying security, at least have certainty on their side. Historical volatility is a firm value derived from recent price movement, and not from estimates.


Critics of historical volatility argue that it is based on past price behavior and cannot be used to estimate future price behavior. However, implied volatility used estimates based on assumptions about something no one can know, which is the future. What is more reliable, the certainty of recent history or the unknown factors of the future? Neither is perfect. However, option value is directly derived from underlying volatility (which explains why they are called derivatives).


The desire among traders to match strategies to risk tolerance, should serve as a starting point for deciding which strategies are most suitable, and what level of capital should be placed at risk. This is sensible because it applies to every individual, and it is up to the individual trader to match strategies with acceptable levels of risk. This is more realistic than accepting definitions of implied volatility, probability, or the Black-Scholes pricing model to pick strategies and contracts. If these methods are imperfect, that may be acceptable. But if they are based on outright inaccuracies, what value do they provide? Relying on inaccurate assumptions or flawed calculations is more than ill-advised; it sets thew trader up for failure.


The simple calculation of probability of an outcome usually is based on a flawed set of calculations, something realized by 17th century mathematicians Blaise Pascal and Pierre de Fermat in studying outcomes of dice rolls. However, even today, the deep flaws on probability and other options math is too easily overlooked. For the modern trader, acknowledging common flaws in options math is a smart starting point for improving the profitability of trading. It comes down to the knowledge of the flaws in most mathematical options conclusions and beginning to make a more informed comparison between actual risks and personal risk tolerance.
 

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 website at Thomsett Publishing as well as on Seeking Alpha, LinkedIn, Twitter and Facebook.

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