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14-March-2006
Probability, Markets and Decision Making, by Victor Niederhoffer

The theory of probability is at bottom nothing but common sense reduced to calculus: It enables us to appreciate with exactness that which accurate minds feel with a sort of instinct for which oftentimes they are unable to account. It teaches us to avoid the illusions which often mislead us. There is no science more worthy of our contemplation nor a more useful one for our system of education. -- P.S. Laplace, ~ 1800.

There is universal acceptance these days that knowledge of probability theory is useful in almost any field where decisions are made. The reason is that most of the things we are confronted with in life are not causal situations but common sense ones where our previous knowledge of circumstances and the intrinsic uncertainty of the situation calls for decisions based on probability. Thus, almost any scholarly field these days ranging from biology to economics to engineering to law  where the relevance of evidence, physics, computer science, networks, linguistics, is replete with probabilistic reasoning. See "Current and Emerging Research Opportunities in Probability" for nice summary of this.

There is also a general appreciation that probability has as wide a range of application in terms of its practical range as geometry. However, unlike geometry, probability theory depends upon a different way of thinking, one that's not intuitive to most of us, and one where we commonly make more mistakes in reasoning than in any other field.

To improve my own understanding of probability theory,  I have been reading some excellent books on it recently Understanding Probability Theory by Henk Tijms (a book of interesting problems with rigorous but easy to conquer mathematical reasoning), Elementary Probability Theory with Applications, by Larry Rabinowitz (a book with practical problems with worked out problems that just requires good reasoning and is completely assessable to anyone who likes numbers), Why Flip a Coin? by H.W. Lewis (an elementary non-mathematical tract with a discussion of the wide ranging areas where decision making under uncertainty comes into play from dating to war), Probability Theory by E.T. Jaynes (a thoroughly profound and deep work that breaks new ground in a philosophy of reasoning and decision making with highly technical statistical discussions), and Probability and Random Processes by G. Grimmett and D. Stirzaker (a complete advanced modern text along the lines of Feller but more modern and up to date and accessible). All accomplish their goals, and are highly recommended for those interested in such goals.

Needless to say, probability theory is crucial to correct decision making in finance and markets. Thus, it is not at all unlikely that I would find that many of the classical probability problems have direct applications to markets that to my knowledge have not been fully explored. I will consider two of these below the secretary of Sultan's Dowry problem (a problem in sequential decision making, and the Monty Hall problem (a problem relating to revision of beliefs based on new information).

Let's start with the Sultan's Dowry problem. The situation here is that a commoner is given the chance to marry one of say eight sultan's daughters. Each has a different dowry. And each will be presented to the commoner one at a time, with a thumbs up or down decision that cannot be reversed. The situation also comes into play from the other side as the dating game problem with the lovely lady being presented sequentially with a dozen suitors. How is she to choose the one that she wants to marry with or will have the best s#x with based on her previous experience. Or in more common terms, how does the boss choose the best secretary from a dozen that he's going to interview sequentially, or how dose he indeed travel from Austin to Dallas and decide how many gas stations to pass before choosing the one that has the lowest price.

Such problems are at the core that every market person is faced with. How does one decide when to sell a stock, especially if a need for money arises before the end of the period, or how does the day trader decide when to get out of his positions knowing that overnight positions are not accepted, or how does the pattern trader decide when to enter his trades given that he waits too long, he may miss the trade. How at a more general level does one pick the right time to sell premium of any kind considering that you'd like to sell at the high, but if you wait too long you might lose the whole thing.

The solution to such problems is very elegant and many of us in the office gained an appreciation for the closed form solution while the artful simulator, Mr. Downing, immediately went to the simulation solution and was able to work it out in about 30 seconds. As an aside here, all modern books on probability theory suggest that the key to gaining an understanding of it is to work with random number generators on the computer to simulate the answers to the common problems. I would agree except to say that's it's even better to work it out by hand with a random number table for the simplest one or two element formulations of the problems.

The solution to the problem is that with 8 wives, the commoner should interview 3 of the potential wives, find out her dowry. Okay, that means he's missed 3 . Then he should choose the first one that has a higher dowry than any of the first 3. I will present several solutions to this problem and they are well summarized by Wolfram.

The correct decision making to make is to go through interviewing three daughters just to gain information and then to start searching for the first one with a higher dowry among the remaining five.

The solution in brief starts with the knowledge that any daughter looked at has a chance of 1/8 to be the highest dowry of the eight daughters. However, to correctly choose her, she must not only be the highest but the highest of the first three that you interviewed to form a base that had no chance of being chosen must be higher than the subsequent ones you interviewed and passed over because they did not contain a higher score.

Thus, after going thru 3 dowries, the chances that the fourth one will be highest and correctly chosen is 1 in 8. The chance that the fifth one will be highest and correctly chosen is 1/8 times the chance that the highest of the four previous scores came among the first 3, i.e. 1/8 x 3/4. The chance that the sixth one will be correctly chosen is 1/8 times the chance that the daughter with the highest dowry was among the first 3 of the 5 daughters considered, i.e. 3/5 x 1/8. The chance that the seventh daughter will be correctly chosen is 1/8 times the chance the daughter with the highest dowry thru daughter 6 was among the first 3 of the six daughters previously considered, .i.e. 1/8x 3/6. The chance of winning by choosing the last daughter is the 1/8 chance that she's highest x the chance that the highest dowry of the first 7 was among the first 3, i.e. 1/8 x 3/7. In order to be correct, the commoner must choose the one and only that is highest of the 5 chances he has to be correct. Thus, each of the 5 chance calculations if exclusive and the correct answer for the probability of being correct is to add up the 5 probabilities. Such a sum has a beautiful closed form solution which for large numbers of daughters comes to 1/e (of course e), or 0.37.

The application to when to put on a trade is rather direct. Divide the day into 8 intervals. Let 3 of them pass to give you a foundation for choosing the best of the 3 that you let go. Then choose the next one that gives you a better entry (lower if you want to buy, or higher if you want to sell). The chances that you will be right according to the closed form solution verified by Mr. Downing is about 0.40. Indeed, the chance of finding the daughter with the greatest dowry out of 8 daughters is 0.32 if you stop after the first daughter and then look for the next one higher, 0.39, if you stop after the second daughter, and look for the next one higher, 0.41 for 3, 0.39 for 4, 0.32 for 5, 0.24 for 6, and 0.12 for 7. Thus, it doesn't make much difference if you wait to interview two, three, or four secretaries or brides or lovers out of 8, before you begin to clear for the action of choosing the next one that's higher. But there's quite a fall off if paralyzed by the fantastic potential of it all you wait for 5, 6, or 7 of the beauties to pass you by. A more realistic version of this problem might take into account the cost in time of interviewing all the candidates, and the chance that conditions might change.

Such reasoning is easily transferable to a wide range of trading decisions during the fray. (to be continued with the Monte Hall problem applications).

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