Tuesday, April 5, 2011

Investing or Gambling? Part 8: Mathematical Similarities

Mathematics plays an important role in the decision making of both investors and gamblers. While some of the analytical quantification methods are substantially different, we believe that there are two common techniques that inherently similar. These are:
  • Decision Trees, and
  • Probability Analysis.

Decision Trees
Whether written or conceived, both gamblers and investors continuously make (and often repeat) decisions based on possible outcomes. After each decision is reached, a new set of possibilities brings forth a new set of possible outcomes. Each decision node may have two or more potential choices that can then be made. The diagram below illustrates a sample two level set of possible decision paths.

Source: Decision Tree: How to do it
From a financial viewpoint, a stock market investor typically has three choices he can initially make: (1) buy a stock; (2) sell a stock; or (3) do nothing. Once these decisions are made, the choices will vary depending on the path taken. In the case of one who buys a stock, he now has at least 4 new options available: (a) continue to hold; (b) sell the position; (c) buy more; or (d) hedge. Had the investor sold a stock short initially, he too has various choices to make regarding his position. And, if the investor initially did nothing, then he would circle back to the initial set of choices to be made.

From a gambling viewpoint, a player makes decisions based on the type of game or activity he is entertaining. If the person is playing poker, he has the choice to: (1) fold; (2) pass; (3) raise; or (4) call after each different card is dealt. Each time, his decision will be based on his previous choice and the final possible outcomes. Lottery players have the choices of: (1) playing; (2) not playing; or (3) playing and buying a multiplier. If the person decides to play, he must then decide to: (a) ask for quick picks or (b) pick his own number. If he choose the latter, then he decides: birthday numbers; even odd; hot cold; etc. Lastly, if he wins a large prize, he must then begin making decisions based on the financial decision tree above.

Probability Distributions
Most of the decisions that the investor or gambler makes are based on the underlying probabilities of success. These probabilities are based on some type of mathematical model. In the financial world, a Normal Probability Density Function is typically used. Ranges of numbers are quantified, or counted, in terms of Standard Deviations as shown in the figure below.

Source: SPC Tools - Control charts
Let us return to the investor above who initially purchased a stock at decision point 1. His decision to sell, hold, or buy more will be determined by the the price change in his stock. If the change remains within 1 standard deviation (based on volatility), he will most likely hold. If it drops more than 2 or 3 standard deviations, he will probably sell. If new positive economic information about the stock is released, he will probably buy more.

Some financial instruments are priced and valued strictly on the amount of price change and probability of occurrence. In particular, Call and Put Stock Options are priced this way. As seen below, a call option price increases when the stock goes up, but this is based on a probability weighted amount. Thus, the stock price change movement must be positively larger than the initial premium paid in order for the investor to earn a profit.
Source: Option Pricing Models
Similarly, poker player make similar decisions based on probabilities. Consider a 5 card stud player that remains until all 5 cards are dealt. If he holds a pair, then he knows that approximately 42% of all hands will have pair. He also knows that his hand is better than 50% of hands, and that approximately 8% of all hands will beat his. Using this information, he will then decide to call, raise, or fold.

In this brief article, we illustrated how similar investors and gamblers are in decision making. Underlying each conscious decision is the understanding of the underlying mathematical probabilities of successfully earning a profit. While the investments or games may vary from simple (buy a stock or flip a coin), to complex (derivative options & swaps or poker and backgammon), nearly the same mathematical properties can be used to quantify success.

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