Cacfuté le 08.11.00
Futures Trading and the Gambler's Ruin Problem
This section explains how casinos make most of their money, as well as why the traders at Goldman Sachs make more money speculating than you do. It's not necessarily because they are smarter than you. It's because they have more money. (However, we will show how the well-heeled can easily lose this advantage.)
Many people assume that the futures price of a stock index, bond, foreign currency, or commodity like gold represents a fair bet. That is, they assume that the probability of an upward movement in the futures price is equal to the probability of a downward movement, and hence the mathematical expectation of a gain or loss is zero. They use the analogy of flipping a fair coin. If you bet $1 on the outcome of the flip, the probability of your winning $1 is one-half, while the probability of losing $1 is also one-half. Your expected gain or loss is zero. For the same reason, they conclude, futures gains and futures losses will tend to offset each other in the long run.
There is a hidden fallacy in such reasoning. Taking open positions in futures contracts is not analogous to a single flip of a coin. Rather, the correct analogy is that of a repeated series of coin flips with a stochastic termination point. Why? Because of limited capital. Suppose you are flipping a coin with a friend and betting $1 on the outcome of each flip. At some point either you or your friend will have a run of bad luck and will lose several dollars in succession. If one player has run out of money, the game will come to an end. The same is true in the futures market. If you have a string of losses on a futures position, you will have to post more margin. If at some point you cannot post the required margin, you will have to close out the contract. You are forced out of the game, and thus you cannot win back what you lost. In a similar way, in 1974, Franklin National and Bankhaus I. D. Herstatt had a string of losses on their interbank foreign exchange trading positions. They did not break even in the long run because there was no long run. They went broke in the intermediate run. This phenomenon is referred to in probability theory as the gambler's ruin problem [1].
What is a "fair" bet when viewed as a single flip of the coin, is, when viewed as a series of flips with a stochastic ending point, really a different game entirely whose odds are quite different. The probabilities of the game then depend on the relative amounts of capital held by the different players.
Suppose we consider a betting process in which you will win $1 with probability p and lose $1 with probability q (where q = 1 - p). You start off with an amount of $W. If your money drops to zero, the game stops. Your betting partner-the person on the other side of your bet who wins when you lose and loses when you win-has an amount of money $R. What is the probability you will eventually lose all of your wealth W, given p and R? From probability theory [1] the answer is:
| (q/p)W + R - (q/p)W | |
| Ruin probability = | |
| (q/p)W + R - 1 |
Ruin probability = 1 - [W/(W + R)], for p = q = .5.
You have $10 and your friend has $100. You flip a fair coin. If heads comes up, he pays you $1. If tails comes up, you pay him $1. The game ends when either player runs out of money. What is the probability your friend will end up with all of your money? From the second equation above, we have p = q = .5, W = $10, and R = $100. Thus the probability of your losing everything is:
1 - (10/(10 + 100)) =.909.
You will lose all of your money with 91 percent probability in this supposedly "fair" game.
The gambler's ruin odds are the important ones. True, the odds are stacked against the player in each casino game: heavily against the player for kino, moderately against the player for slots, marginally against the player for blackjack and craps. (Rules such as "you can only double down on 10s and 11s" in blackjack are intended to turn the odds against the player, as are the use of multiple card decks, etc.) But the chief source of casino winnings is that people have to stop playing once they've had a sufficiently large losing streak, which is inevitable. (Lots of "free" drinks served to the players help out in this process. From the casino's point of view, the investment in free drinks plays off splendidly.)
Note here that "wealth" (W or R in the equation) is defined as the number of betting units: $1 in the example. The more betting units you have, the less probability there is you will be hit with the gambler's ruin problem. So you if you sit at the blackjack table at Harrah's with a $1000 minimum bet, you will need to have 100 times the total betting capital of someone who sits at the $10 minimum tables, in order to have the same odds vis-à-vis the dealer.
A person who has $1000 in capital and bets $10 at a time has a total of W = 1000/10 = 100 betting units. That's a fairly good ratio.
While a person who has $10,000 in capital and bets $1000 at a time has W = 10000/1000 = 10 betting units. That's lousy odds, no matter the game. It's loser odds