## Gamblers, Zero Sets, and Fractal MountainsHenry and Thomas are flipping a fair coin and betting $1 on the outcome. If the coin comes up heads, Henry wins a dollar from Thomas. If the coin comes up tails, Thomas wins a dollar from Henry. Henry’s net winnings in dollars, then, are the total number of heads minus the total number of tails. But we saw all this before, in Part 3. If we let x(n) denote Henry’s net winnings, then x(n) is determined by the dynamical system: x(n) = x(n) + 1, with probability p = ½ The graph of 10,000 coin tosses in Part 3 simply shows the fluctuations in Henry’s wealth (starting from 0) over the course of the 10,000 coin tosses. Let’s do this in real time, although we will restrict ourselves to 3200 coin tosses. Let’s plot Henry’s winnings for a new game that lasts for 3200 flips of the coin. You can quickly see the results of many games with a few clicks of your mouse. Make sure Java is enabled on your web browser, and click here. There are three things to note about this demonstration: - Even though the odds are even for each coin clip, winnings or losses can at times add up significantly. Even though a head or a tail is equally probable for each coin flip, there can be a series of "runs" that result in a large loss to either Henry or Thomas. This fact is important in understanding the "gambler’s ruin" problem discussed later.
- The set of points where x(n) comes back to x(n) = 0 (that is, the
points where wins and losses are equalized), is called the
**zero set**of the system. Using**n**as our measure of time, the time intervals between each point of the zero set are independent, but form clusters, much like Cantor dust. To see the zero set plotted for the coin tossing game, make sure Java is enabled on your web browser and click here. The zero set represents those times at which Henry has broken even. (Make sure to run the series of coin flips multiple times, to observe various patterns of the zero set.) - The fluctuations in Henry’s winnings form an outline that is suggestive of mountains and valleys. In fact, this is a type of "Brownian landscapes" that we see around us all the time. To create different "alien" landscapes, for, say, set decorations in a science fiction movie, we can change the probabilities. The effects in three dimensions, with a little color shading, can be stunning.
Since we will later be discussing motions that are not Brownian, and distributions that are not normal (not Gaussian), it is important to first point out an aspect of all this that is somewhat independent of the probability distribution. It’s called the Gambler’s Ruin Problem. You don’t need nonnormal distributions to encounter gambler's ruin. Normal ones will do just fine.
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 [1].gambler's ruin problemWhat 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 (
An Example 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 1 - (10/(10 + 100)) =.909. You will lose all of your money with 91 percent probability in this supposedly "fair" game. Now you know how casinos make money. Their bank account is bigger than yours. Eventually you will have a losing streak, and then you will have to stop playing (since the casinos will not loan you infinite capital). 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" ( 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.
Gauss vs. Cauchy We measure probability with our one-pound jar of jam. We can distribute the jam in any way we wish. If we put it all at the point x = 5, then we say "x = 5 with certainty" or "x = 5 with probability 1." Sometimes the way the jam is distributed is determined by
a simple function. The f(x) = [1/(2p ) Here f(x) creates the nice bell-shaped curve we have seen
before ( The jam (probability) is smeared between the horizontal
line and the curve, so the So we can calculate the probability density for each value of x using the function f(x). Here are some values:
At the center value of x = 0, the probability density is highest, and has a value of f(x) = .3989. Around 0, the probability density is spread out symmetrically in each direction. The entire one-pound jar of jam is smeared underneath the
curve between – ¥ and + ¥ . So the total probability, the The probability that Instead of writing this integral in the usual
mathematical fashion, which requires using a graphic in the
I(a,b) f(x) dx. I(a,b) f(x) dx, then, is the area under the f(x) curve
from I(- ¥ ,¥ ) f(x) dx = 1. A little more notation will be useful. We want a shorthand way of expressing the probability that x < b. But the probability that x < b is the same as the probability that -¥ < x < b. So this value is given by the area under the curve from -¥ to b. We will write this as F(b):
F(b) = I(-¥ ,b) f(x) dx = area under curve from minus infinity to b. Here is a picture of F(b) when b = 0: For any value (Note that since the F(x) takes values between 0 and 1, corresponding to our one-pound jar of jam. Hence F(-¥ ) = 0, while F(+¥ ) = 1. The probability between F(b) – F(a). The probability x > 1 – F(b). Now. Here is a different function for spreading
probability, called the g(x) = 1/[p (1 +
x Here is a picture of the resulting Cauchy curve: It it nice and symmetric like the normal distribution, but is relatively more concentrated around the center, and taller in the tails than the normal distribution. We can see this more clearly by looking at the values for g(x):
At every value of Note that at –3, for example, the probability density of
the Cauchy distribution is g(-3) = .0318, while for the normal
distribution, the value is f(-3) = .0044. There is more than 7 times as
much probability for this extreme value with the Cauchy distribution than
there is with the normal distribution! (The calculation is .0318/.0044 =
7.2.) Relative to the normal, the Cauchy distribution is
To see a more detailed plot of the normal density minus the Cauchy density, make sure Java is enabled on your web browser and click here. As we will see later, there are other distributions that
have more probability in the tails than the normal, and
Location and Scale So far, as we have looked at the normal and the Cauchy
densities, we have seen they are centered around zero. However, since the
density is defined for all values of f(x) = [1/(2p ) Here is a picture of the normal distribution after the
location has been moved from For the Cauchy density, the corresponding alteration to
include a location parameter g(x) = 1/[p (1 +
( In each case, the distribution is now centered at
m". The reason is simple. For the
Cauchy distribution, a mean doesn’t exist. But a location parameter, which
shows where the probability distribution is centered, does. For the normal distribution, the location parameter
Similarly, for the Cauchy distribution the standard
deviation (or the variance, which is the square of the standard deviation)
doesn’t exist. But there is a c
corresponds to the standard deviation. But a scale parameter c
is defined for the Cauchy and for other, leptokurtic distributions for
which the variance and standard deviation don’t exist ("are
infinite").Here is the normal density written with the addition of a
scale parameter f(x) = [1/( We divide ( Here is a picture of the normal distribution for
difference values of The blue curve represents For the Cauchy density, the addition of a scale parameter gives us: g(x) = 1/[ Just as we did with the normal distribution, we divide
( Operations with Most of the probability distributions we are interested in in finance lie somewhere between the normal and the Cauchy. These two distributions form the "boundaries", so to speak, of our main area of interest. Just as the Sierpinski carpet has a Hausdorff dimension that is a fraction which is greater than its topological dimension of 1, but less than its Euclidean dimension of 2, so do the probability distributions in which we are chiefly interested have a dimension that is greater than the Cauchy dimension of 1, but less than the normal dimension of 2. (What is meant here by the "Cauchy dimension of 1" and the "normal dimension of 2" will be clarified as we go along.)
Notes [1] See Chapter 14 in William Feller, J. Orlin Grabbe is the author of International Financial Markets, and is an internationally recognized derivatives expert. He has recently branched out into cryptology, banking security, and digital cash. His home page is located at http://www.aci.net/kalliste/homepage.html . -30-
from The Laissez Faire City
Times, Vol 3, No 27, July 5,
1999 |