Probability, Expectation and Options
As you probably know by now, I trade options from a mathematical perspective. My basic philosophy is that I will place trades that have a positive expectation of making a profit and a high probability of success. This will lead to low variability in the short run and consistent profits in the long run. That's the same logic they used to build the casinos in Las Vegas and it seems to have worked out pretty well for them!
To understand expectation, first we have to understand a little bit about probability. Probability is the likelihood or chance that some event will occur. Intuitively, we're probably all aware that if we flip a fair coin the probability of it landing on heads is 50%. Another example would be when you roll a die, the probability of getting a 4 (or any of the 6 possible outcomes) is 1 out of 6 or 16-2/3%. Let's dig a little deeper and look at 3 related probability questions:
1) A man states that he has 2 children. What is the probability that he has at least one girl?
2) A man states that he has 2 children at least one of which is a boy. What is the probability that he has a girl?
3) A man states that he has 2 children and the first born is a boy. What is the probability that he has a girl?
When I ask these questions in my options class, or when doing individual one on one mentoring, I get lots of interesting answers. Let me tell you right off the bat that the answers are not all equal to 50% and, in fact, they are all different. And for those of you who are thinking of emailing me to let me know that a man can't have children, only women can, thank you, but I've already heard it.
How do we calculate a probability? Well, to find the probability of an event happening, we have to form a fraction. The denominator is equal to the total number of possible outcomes, and the numerator is equal to the number of ways the event can occur. Let's tackle the 3 problems described above.
For question #1, what are the possible ways the man can have 2 kids? Let's let B stand for boy and G for girl. Then we can list all the possible ways he can have 2 children. The complete list of first born and then second born is: BB, BG, GB, GG. So the denominator of our fraction is 4. Now, in how many of these outcomes, does he have a girl? Three; BG, GB, and GG. So the probability is ¾ or 75%.
In question #2 we are effectively eliminating one of the possible outcomes by saying that the man has a boy. Therefore, the remaining possibilities are: BB, BG, GB, since GG is no longer possible. Of these 3 possibilities, 2 have a girl, so the probability is 2/3 or 66-2/3%.
Question #3 is the one that students complain is a trick question. Well, yes and no. It's meant to illustrate how order is important. By specifying that the first born is a boy, we can eliminate GB which represents a girl and boy with the girl being the first born. In this case the denominator is 2 (only BB and BG) and the numerator is 1, so the probability is ½, or 50%.
So if the probability of winning a trade is greater than 50% does that mean we should do the trade? The answer is; not necessarily. What if the probability of winning a trade is 60% and you would make $100, but if you lost you would lose $200? If you made this trade 10 times you would expect to win $100 6 times, and lose $200 4 times for a net loss of $200 or $20 per trade. We refer to this trade as having an expectation of negative $20.
This expectation is also called the mathematical expectation, or expected value. It represents the average amount you can expect to earn if the trade is repeated many times. Mathematically, it's calculated as:
Expected Value = (prob. of winning X amount won) + (prob. of losing X amount lost)
In the example above:
Expected Value = .6 X 100 + .4 X (-200) = -20
I'll use 2 more examples to illustrate the concept of expected value. Suppose someone offered to play a game of dice with you on the following terms. If you roll 1 die and it lands on 4 you get $10. If it lands on anything other than 4, you lose $1. The expected value is calculated as:
1/6 X 10 + 5/6 X (-1) = .83 a positive expectation
Note that if you played this game you would expect to win 83 cents per roll, even though you would expect to lose (5 out of 6 times) on any particular roll. Think about that.
Next, consider the casino game of roulette. The game consists of a wheel with pockets numbered from 1 to 36 half of which are red and the other half black. In addition, there are 2 green pockets labeled 0 and 00 making a total of 38 pockets. There are 2 main types of bets. First, you can bet on a number and if it comes up you get paid 35 times what you bet. Since the probability of your number coming up is 1 out of 38, we can calculate the expected value as:
1/38 X 35 + 37/38 X (-1) = -.0526 a negative expectation for the player or conversely, a positive expectation for the casino, no surprise here!
The second type of bet is to bet on either red or black. Let's say we bet red (it would be the same for black). This bet pays off 1 to 1, so on a $1 bet, if red comes up you win a dollar and if black or green comes up you lose $1. Since there are 18 red numbers, the probability of red coming up is 18/38 and therefore the expected value is:
18/38 X 1 + 20/38 X (-1) = -.0526
Notice that it's the same negative expectation that we calculated when making the single number bet. So no matter which type of bet you make playing this form of roulette, you're going to lose in the long run. There are no honest systems that can consistently win at this game (don't believe the gambling systems that tell you otherwise.) Of course in the short run, anything is possible.
So what's actually going on? The real problem is that the casinos are charging $1 for a bet that's only worth .95 (to be more exact 1-.0526, but you get the picture.) Alternatively, we can say that if the casino is charging $1 to bet then they should pay out 37 to 1 in the single number bet and about $1.05 in the red/black bet.
While the overall expectation can't be changed without changing the payoffs, the pattern of wins and losses will be different between the player and the casino. In the single number game the player's pattern of winning will generally be many small losses accompanied by an occasional large win. That may be analogous to the pattern of an options buyer (without hedging.) On the other hand, the casino, which may be thought of as being analogous to an options writer, has a pattern of many small gains, followed by an occasional large loss.
Summing this all up as it relates to our options trading, it seems that we definitely want a positive expectation and it would make sense from a cash flow point of view if the probability of winning was not too low, the higher the better. Suppose in a particular trade, the expectation was positive, but the probability of winning was only 1/1000, it probably would not be a very good trade.
Okay, so now you know when to make the trade, right? If I offer you a trade with a 50% probability of winning and a positive expectation, is that enough information to take the trade? Here's the deal, you flip a coin. If it comes up heads I'll give you $1.1million and if it comes up tails you only have to give me $1million. So the probability is good, the expectation is positive and assuming you're sane, you wouldn't make this trade, because the risk is too great. What's the point? Money/risk management is also an important factor in trading, stay tuned for more on that in future articles.
As always, if you have any questions about my articles, have suggestions for future topics, or want more information about our options mentoring program, feel free to email me at: SFreifeld@tradingacademy.com or call me at: (888) OTA-2580 ext. 2010.
11. Know Thy Options!
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Stan Freifeld comes to us from the Floor of the American Stock Exchange where he traded options for his own account from 1994-2001. He was a Market Maker for the options on several popular equities including Dupont, Schering Plough, Walgreen's, CBS, U.S. Surgical and Biovail.When he is not trading or thinking about trading, Stan relieves his stress by playing competitive squash, competing in local road rallies with his Ferrari Cabriolet and tutoring local high school students for the SAT's. The bottom line is that Stan, a long time MENSA member, is an engaging teacher with an extraordinary background in options trading and risk management. He is helpful and patient by nature and equally at ease with all levels of traders, from complete novices to advanced pros and academics. He'll be happy to teach you to trade!