# Why Does the Mathematical Probability So Often Turn Out NOT To Be the Real Life Probability?

I** am preparing a presentation for traders on trading options. They work in a large international energy corporate. These are traders working in treasury but not option specialists. This, you would expect, would be an intermediate-level course on options and options maths. I have always had a problem with giving courses like this as, so often, I say: that is the theoretical answer but invariably it will not be the real life answer. **

For example, there are a number of ways of calculating the fair value of an option’s premium. Very very often the answer is too low and often by a sizeable percentage. It is the fair value but that is not the trading price. There is no arbitrage to get the option premium to its *correct* value. There are too many buyers willing to pay too much for it. Dr Andrew Lo, the director of MIT’s Laboratory for Financial Engineering, hit on it in A Non-Random Walk Down Wall Street in 2002. **What is missing from the calculation is people.** People have a powerful computer-like brain. But this brain in not a purely rational computer. People have biases, intuition, fear, greed and many other human overlays to their decision making People put more value when buying a call option in the idea of paying a fixed premium and therefore fixed limited loss in exchange for a theoretically infinite profit potential than they should if they were truly rational and a computer. They find the concept of receiving a fixed amount of money to give the theoretically infinite potential profit so uncomfortable they price it much higher than its *fair value*. The commonly used maths and calculators still used today simply give consistently wrong answers. The price of things is simply not their value.

I found a very helpful new book to help me prepare the content of my options trading course. It addresses these mathematical short comings in a very interesting new way. My new course will not just be **Black and Scholes** and the **Greeks**. We will look to model as close to real life as possible. The answer is in The Mathematics of Options by Michael C. Thomsett. This 2017 book addresses the mathematical short comings of traditional option modelling in a very interesting and insightful way.

**So too with probability.**

Why is it that real life outcomes drift so much from theoretical probabilities? Take this example taken from the book:

*Calculating the probability of making a profit on a trade seems like a straightforward challenge, mathematically speaking. But in fact, the calculation itself might be so wrong that it misleads us into thinking our odds are better than they are.*

*The roll of a single die appears to contain a 67% chance of any one result occurring in a series of four rolls. This statistical assumption is easily perceived by first recognizing the one-in-six chance in a single roll of any one number resulting. For four rolls, the probability formula for this is:*

*4 * 1/6 = 67%*

*You know there is a one-in-six chance because there are six possible results—1, 2, 3, 4, 5, and 6. However, when two dice are employed, the calculation of probable outcomes is more complex. For example, an initial calculation using the same formula as above yields a similar result if the number of rolls is expanded. The factor used for two dice requires multiplying the one-die factor by itself:*

* 1/6 * 1/6 = 1/36*

*An application of** the formula assuming 24 rolls is:*

* 24 * 1/36 = 67%**As evident as this appears, it is not accurate.*

* To determine the true probability of any one outcome, it is necessary to calculate the probability of that outcome not occurring. Because there are many more chances of failure, the additive method above is not a reliable or accurate probability result. Thus, the odds of rolling any number with a single die are not accurate at 67%, but rather produce a result of 52%. This is apparent when you calculate the odds of any one number not being rolled in 4 attempts, and then subtracting that total from 1:*

* 1 – ( 5 ÷ 6 )4 = 52%*

*This is dramatically different than the initial calculation, revealing that with the odds of a number coming up are 52%, and not 67%.*

* Applying this to two dice with 24 rolls, a different and lower outcome is arrived at when the calculation determines the odds, lower in fact than for the single die calculation. The odds of a specific two-dice outcome not occurring in a series of 24 rolls and then subtracting from 1:*

* 1 – ( 35 ÷ 36 )24 = 49%*

* The fact that in 24 rolls, the probability of any one outcome resides at 49% rather than at 67%, vastly changes the probability, thus the risk level. The initial calculat**ion of a single value coming up 67% of the time out of 4 rolls, or a double value in 24 rolls, is inaccurate when the more advanced multiplicative formula is applied.*

With options trading, the same thinking applies to more accurately reveal the success of a specific strategy. Options traders tend to think of their trades not so much for hedging value, but in how a speculative result can be devised to “beat the market’ and develop a consistent method for generating profits. A realistic assessment of risks requires a process similar to that of calculating probability for dice rolls. If an options trader believes that profits are likely at the rate of 67% in ‘x’ trades, when the true probability is closer to a 50/50 result, the implications are clear. Such a trader may enter into a series of traders assumed to contain the same risk attributes, expecting to profit at a level far greater than actual probability. Although the example of dice is narrow and finite, the same observation applies for the greater level of variables involved in options trading.

**As Albert Einstein said ”God does not play dice with the universe.”**

I strongly recommend the books and am very pleased to have been introduced to Mr. Thomsett’s radical rethink in this new and exciting book. My student will get answer closer to the real world as a result.

*This article was originally published by Trevor Neil www.betagroup.co.uk/blog BETA Group(c)2017*