## Insider trading paradox

The underlying probability space

For convenience of the reader, the example is repeated at the bottom of this page.

Answer to Question 3.

We will show that the example is consistent with the laws of probability. We will present a probability space on which one can define an event A, random variables X and Y and numbers x1, y1 and y2 with the following properties.

1. P(A) = 0.5
2. X is independent of A
3. P(Y=y1 | X=x1) + P(Y=y2 | X=x1) = 1
4. P(A | Y=y1) = 0.7
5. P(A | Y=y2) = 0.8
Consider i.i.d. random variables Z(j), for j = 1,2,...,10, such that P(Z(j) = 1) = P(Z(j) = 0) = 0.5. Let
• A = {Z(10) = 1},
• X be the sum of Z(j) from j = 1 to j = 9,
• Y be the sum of Z(j) from j = 1 to j = 10,
• x1 = 7
• y1 = 7
• y2 = 8
It is elementary to check that the properties (1)-(5) are satisfied.

One may wonder if the "paradox" is a result of choosing a very unrealistic scenario. In other words, is the probability that X takes the value x1 or larger exceedingly small? It is equal to about 9%, which seems to be quite reasonable.

Answer to Questions 1, 2 and 4.

The information obtained from Investor II by Investor I does not change his assessment of the probability of A. Note that the event {X=x1 and P(A|Y) > 0.6} is equal to {X=x1} so

P(A | X=x1 and P(A|Y) > 0.6) = P(A | X=x1) = P(A) = 0.5.

Investor I should not participate in the investment.

Conclusion.

It turns out that it is logically consistent and fully rational for

• Investor II to consider the investment profitable and
• Investor I to disregard the second investor's assessment of the probability of A despite his own lack of knowledge of any event or random variable that is not independent from A.
This appears paradoxical in view of our everyday experience. Consider any investment opportunity depending on the outcome of some random action A. We all have access to great volumes of "insider" information (i.e., information which is not available to the public) and which is independent from A. For example, you know what you had for breakfast today, which books you read in the last two weeks, etc. We instinctively disregard all the information that falls into the "insider and independent" category as totally useless. In such circumstances, a friendly advice based on some information which is not independent of A is normally regarded as worthy of following. The "paradox" presented on this page shows that various pieces of information can interplay in a way that may contradict the intuition.

Insider trading paradox (example repeated for the convenience of the reader)

Two investors, Investor I and Investor II are friends and insider traders. Each one of them has access to information not available to the general public. Being friends, they share some information and advice but they are not totally open with each other. Investor I knows an "insider" who can provide him with the value of a quantity X, which to the general public appears to be a random variable. Likewise, Investor II can learn the value of a random variable Y by calling a secret source. Each investor knows what type of information the other one has (for example, Investor I knows that Investor II knows tha value of the random variable Y) but they do not know the information itself (for example, Investor II never tells Investor I the value of Y before it becomes public).

A new investment opportunity appeared on the horizon. If the Congress passes a certain law by the end of the year (let us call this event A) then an investment in a certain company would greatly increase in value. Otherwise, all the money would be lost. The probability of A is 0.5. The potential investment would be break-even if the probability A were 0.6 or greater. One could make a potentially very profitable investment if one could know that the probability of A is greater than 0.6. The value of this probability can be affected by the insider information, of course.

Investor I checked the value of X - it happens to be x1. He did it mostly out of habit, because he knows that X is independent of A. Hence, P(A|X=x1) = 0.5. The insider information is of no help to Investor I.

Soon after that, his friend, Investor II, called him and told him that, in view of his insider information, the investment is definitely profitable, i.e., the conditional probability of A given the true value of Y is greater than 0.6. However, Investor II did not tell Investor I the value of Y.

Since Investor II did not tell Investor I the value of Y, the first investor decided to check if the optimism of the second one is consistent with his own information. It turned out that given {X = x1}, the random variable Y can take only two values, y1 or y2. The conditional probabilities happen to be P(A|Y=y1) = 0.7 and P(A|Y=y2) = 0.8. Hence, Investor II must be right - no matter what the value of Y is, the investment seems to be profitable.