Fooled by Randomness (25 page)

More particularly, where jurors (and lawyers) tend to make mistakes, along with the rest of us, is in the notion of joint probability. They do not realize that evidence compounds. The probability of my being diagnosed with respiratory tract cancer and being run over by a pink Cadillac in the same year, assuming each one of them is 1/100,000, becomes 1/10,000,000,000—by multiplying the two (obviously independent) events. Arguing that O. J. Simpson had 1/500,000 chance of not being the killer from the blood standpoint (remember the lawyers used the sophistry that there were four people with such blood types walking around Los Angeles) and adding to it the fact that he was the husband of the person and that there was additional evidence, then (owing to the compounding effect) the odds against him rise to several trillion trillion.

“Sophisticated” people make worse mistakes. I can surprise people by saying that the probability of the joint event is lower than either. Recall the availability heuristic: with the Linda problem rational and educated people finding the likelihood of an event greater than that of a larger one that encompasses it. I am glad to be a trader taking advantage of people’s biases but I am scared of living in such a society.

An Absurd World

Kafka’s prophetic book,
The Trial,
about the plight of a man, Joseph K., who is arrested for a mysterious and unexplained reason, hit a spot as it was written before we heard of the methods of the “scientific” totalitarian regimes. It projected a scary future of mankind wrapped in absurd self-feeding bureaucracies, with spontaneously emerging rules subjected to the internal logic of the bureaucracy. It spawned an entire “literature of the absurd”; the world may be too incongruous for us. I am terrified of certain lawyers. After listening to statements during the O. J. trial (and their effect) I was scared, truly scared, of the possible outcome—my being arrested for some reason that made no sense probabilistically, and having to fight some glib lawyer in front of a randomness illiterate jury.

We said that mere judgment would probably suffice in a primitive society. It is easy for a society to live without mathematics—or traders to trade without quantitative methods—when the space of possible outcomes is one-dimensional. One-dimensional means that we are looking at one sole variable, not a collection of separate events. The price of one security is one-dimensional, whereas the collection of the prices of several securities is multi-dimensional and requires mathematical modeling—we cannot easily see the collection of possible outcomes of the portfolio with a naked eye, and cannot even represent it on a graph as our physical world has been limited to visual representation in three dimensions only. We will argue later why we run the risk of having bad models (admittedly, we have) or making the error of condoning ignorance—swinging between the Carybde of the lawyer who knows no math to the Scylla of the mathematician who misuses his math because he does not have the judgment to select the right model. In other words, we will have to swing between the mistake of listening to the glib nonsense of a lawyer who refuses science and that of applying the flawed theories of some economist who takes his science too seriously. The beauty of science is that it makes an allowance for both error types. Luckily, there is a middle road—but sadly, it is rarely traveled.

Examples of Biases in Understanding Probability

I found in the behavioral literature at least forty damning examples of such acute biases, systematic departures from rational behavior widespread across professions and fields. Below is the account of a well-known test, and an embarrassing one for the medical profession. The following famous quiz was given to medical doctors (which I borrowed from the excellent Deborah Bennett’s
Randomness
).

Most doctors answered 95%, simply taking into account the fact that the test has a 95% accuracy rate. The answer is the conditional probability that the patient is sick and the test shows it—close to 2%. Less than one in five professionals got it right.

I will simplify the answer (using the frequency approach). Assume no false negatives. Consider that out of 1,000 patients who are administered the test, one will be expected to be afflicted with the disease. Out of a population of the remaining 999 healthy patients, the test will identify about 50 with the disease (it is 95% accurate).The correct answer should be that the probability of being afflicted with the disease for someone selected at random who presented a positive test is the following ratio:

Number of afflicted persons
________________________
Number of true and false positives

here 1 in 51.

Think of the number of times you will be given a medication that carries damaging side effects for a given disease you were told you had, when you may only have a 2% probability of being afflicted with it!

We Are Option Blind

As an option trader, I have noticed that people tend to undervalue options as they are usually unable to correctly mentally evaluate instruments that deliver an
uncertain
payoff, even when they are fully conscious of the mathematics. Even regulators reinforce such ignorance by explaining to people that options are a
decaying
or
wasting
asset. Options that are out of the money are deemed to
decay,
by losing their premium between two dates.

I will clarify next with a simplified (but sufficient) explanation of what an option means. Say a stock trades at $100 and that someone gives me the right (but not the obligation) to buy it at $110 one month ahead of today. This is dubbed a
call
option. It makes sense for me to
exercise
it, by asking the seller of the option to deliver me the stock at $110, only if it trades at a higher price than $110 in one month’s time. If the stock goes to $120, my option will be worth $10, for I will be able to buy the stock at $110 from the option writer and sell it to the market at $120, pocketing the difference. But this does not have a very high probability. It is called
out-of-the-money,
for I have no gain from exercising it right away.

Consider that I buy the option for $1. What do I expect the value of the option to be one month from now? Most people think 0. That is not true. The option has a high probability, say 90%, of being worth 0 at expiration, but perhaps 10% probability to be worth an average of $10. Thus, selling the option to me for $1 does not provide the seller with free money. If the seller had instead bought the stock himself at $100 and waited the month, he could have sold it for $120. Making $1 now was hardly, therefore, free money. Likewise, buying it is not a wasting asset. Even professionals can be fooled. How? They confuse the expected value and the most likely scenario (here the expected value is $1 and the most likely scenario is for the option to be worth 0). They mentally overweigh the state that is the most likely, namely, that the market does not move at all. The option is simply the weighted average of the possible states the asset can take.

There is another type of satisfaction provided by the option seller. It is the steady return and the steady feeling of reward—what psychologists call
flow.
It is very pleasant to go to work in the morning with the expectation of being up some small money. It requires some strength of character to accept the expectation of bleeding a little, losing pennies on a steady basis even if the strategy is bound to be profitable over longer periods. I noticed that very few option traders can maintain what I call a “long volatility” position, namely a position that will most likely lose a small quantity of money at expiration, but is expected to make money in the long run because of occasional spurts. I discovered very few people who accepted losing $1 for most expirations and making $10 once in a while, even if the game were fair (i.e., they made the $10 more than 9.1% of the time).

I divide the community of option traders into two categories:
premium sellers
and
premium buyers.
Premium sellers (also called option sellers) sell options, and generally make steady money, like John in Chapters
1
and
5
. Premium buyers do the reverse. Option sellers, it is said, eat like chickens and go to the bathroom like elephants. Alas, most option traders I encountered in my career are
premium sellers—
when they blow up it is generally other people’s money.

How could professionals seemingly aware of the (simple) mathematics be put in such a position? As previously discussed, our actions are not quite guided by the parts of our brain that dictate rationality. We think with our emotions and there is no way around it. For the same reason, people who are otherwise rational engage in smoking or in fights that get them no immediate benefits; likewise people sell options even when they know that it is not a good thing to do. But things can get worse. There is a category of people, generally finance academics, who, instead of fitting their actions to their brains, fit their brains to their actions. These people go back and unwittingly cheat with the statistics to justify their actions. In my business, they fool themselves with statistical arguments to justify their option selling.

What is less unpleasant: to lose 100 times $1 or lose once $100? Clearly the second: Our sensitivity to losses decreases. So a trading policy that makes $1 a day for a long time then loses them all is actually pleasant from a hedonic standpoint, although it does not make sense economically. So there is an incentive to invent a story about the likelihood of the events and carry on such strategy.

In addition, there is the risk ignorance factor. Scientists have subjected people to tests—what I mentioned in the prologue as risk taking out of underestimating the risks rather than courage. The subjects were asked to predict a range for security prices in the future, an upper bound and a lower bound, in such a way that they would be comfortable with 98% of the security ending inside such range. Of course violations to such bound were very large, up to 30%.

Such violations arise from a far more severe problem: People overvalue their knowledge and underestimate the probability of their being wrong.

One example to illustrate further option blindness. What has more value? (a) a contract that pays you $1 million if the stock market goes down 10% on any given day in the next year; (b) a contract that pays you $1 million if the stock market goes down 10% on any given day in the next year due to a terrorist act. I expect most people to select (b).

PROBABILITIES AND THE MEDIA (MORE JOURNALISTS)

A journalist is trained in methods to express himself rather than to plumb the depth of things—the selection process favors the most communicative, not necessarily the most knowledgeable. My medical doctor friends claim that many medical journalists do not understand anything about medicine and biology, often making mistakes of a very basic nature. I cannot confirm such statements, being myself a mere amateur (though at times a voracious reader) in medical research, but I have noticed that they almost always misunderstand the probabilities used in medical research announcements. The most common one concerns the interpretation of evidence. They most commonly get mixed up between
absence of evidence
and
evidence of absence,
a similar problem to the one we saw in
Chapter 9
. How? Say I test some chemotherapy, for instance Fluorouracil, for upper respiratory tract cancer, and find that it is better than a placebo, but only marginally so; that (in addition to other modalities) it improves survival from 21 per 100 to 24 per 100. Given my sample size, I may not be confident that the additional 3% survival points come from the medicine; it could be merely attributable to randomness. I would write a paper outlining my results and saying that there is no evidence of improved survival (as yet) from such medicine, and that further research would be needed. A medical journalist would pick it up and claim that one Professor N. N. Taleb found evidence that Fluorouracil
does not help,
which is entirely opposite to my intentions. Some naive doctor in Smalltown, even more uncomfortable with probabilities than the most untrained journalist, would pick it up and build a mental block against the medication, even when some researcher finally finds fresh evidence that such medicine confers a clear survival advantage.

CNBC at Lunchtime

The advent of the financial television channel CNBC presented plenty of benefits to the financial community but it also allowed a collection of extrovert practitioners long on theories to voice them in a few minutes of television time. One often sees respectable people making ludicrous (but smart-sounding) statements about properties of the stock market. Among these are statements that blatantly violate the laws of probability. One summer during which I was assiduous at the health club, I often heard statements such as “the real market is only 10% off the highs while the average stock is close to 40% off its highs,” which is intended to be indicative of deep troubles or anomalies—some harbinger of bear markets.

There is no incompatibility between the fact that the average stock is down 40% from the highs while the average of all stocks (that is, the market) is down 10% from its own highs. One must consider that the stocks did not all reach their highs
at the same time.
Given that stocks are not 100% correlated, stock A might reach its maximum in January, stock B might reach its maximum in April, but the average of the two stocks A and B might reach its maximum at some time in February. Furthermore, in the event of negatively correlated stocks, if stock A is at its maximum when stock B is at its minimum, then they could both be down 40% from their maximum when the stock market is at its highs! By a law of probability called distribution of the maximum of random variables, the maximum of an average is necessarily less volatile than the average maximum.