Louis Bachelier Visits the New York Stock Exchange Louis Bachelier, resurrected for the moment, recently visited the New York Stock Exchange at the end of May 1999. He was somewhat puzzled by all the hideous concrete barriers around the building at the corner of Broad and Wall Streets. For a moment he thought he was in Washington, D.C., on Pennsylvania Avenue. Bachelier was accompanied by an angelic guide named Pete. "The concrete blocks are there because of Osama bin Ladin," Pete explained. "He’s a terrorist." Pete didn’t bother to mention the blocks had been there for years. He knew Bachelier wouldn’t know the difference. "Terrorist?" "You know, a ruffian, a scoundrel." "Oh," Bachelier mused. "Bin Ladin. The son of Ladin." "Yes, and before that, there was Abu Nidal." "Abu Nidal. The father of Nidal. Hey! Ladin is just Nidal spelled backwards. So we’ve gone from the father of Nidal to the son of backwards-Nidal?" "Yes," Pete said cryptically. "The spooks are never too
creative when they are manufacturing the boogeyman of the moment. If you
want to understand all this, read about ‘Goldstein’ and the daily
scheduled ‘Two Minutes Hate’ in George Orwell’s book "1984? Let’s see, that was fifteen years ago," Bachelier said. "A historical work?" "Actually, it’s futuristic. But he who controls the present controls the past, and he who controls the past controls the future." Bachelier was mystified by the entire conversation, but once they got inside and he saw the trading floor, he felt right at home. Buying, selling, changing prices. The chalk boards were now electric, he saw, and that made the air much fresher. "Look," Bachelier said, "the Dow Jones average is still around!" "Yes," nodded Pete, "but there are a lot of others also. Like the S&P500 and the New York Stock Exchange Composite Index." "I want some numbers!" Bachelier exclaimed enthusiastically. Before they left, they managed to con someone into giving them the closing prices for the NYSE index for the past 11 days. "You can write a book," Pete said. "Call it Bachelier didn’t pay him any mind. He had taken out a pencil and paper and was attempting to calculate logarithms through a series expansion. Pete watched in silence for a while, before he took pity and pulled out a pocket calculator. "Let me show you a really neat invention," the angel said.
Bachelier’s Scale for Stock Prices Here is Bachelier’s data for eleven days in May. We have
the calendar date in the first column of the table; the NYSE Composite
Average, S(t), in the second column; the log of S(t) in the third column;
the change in log prices, x(t) = log S(t) – log S(t-1) in the fourth
column; and x(t) |

Date |
S(t) |
log S(t) |
x(t) |
x(t) |

May 14 |
638.45 |
6.459043 |
||

May 17 |
636.92 |
6.456644 |
-.002399 |
.000005755 |

May 18 |
634.19 |
6.452348 |
-.004296 |
.000018456 |

May 19 |
639.54 |
6.460749 |
.008401 |
.000070577 |

May 20 |
639.42 |
6.460561 |
-.000188 |
.000000035 |

May 21 |
636.87 |
6.456565 |
-.003996 |
.000015968 |

May 24 |
626.05 |
6.439430 |
-.017135 |
.000293608 |

May 25 |
617.34 |
6.425420 |
-.014010 |
.000196280 |

May 26 |
624.84 |
6.437495 |
.012075 |
.000145806 |

May 27 |
614.02 |
6.420027 |
-.017468 |
.000305131 |

May 28 |
622.26 |
6.433358 |
.013331 |
.000177716 |

sum of all x(t) |

What is the meaning of all this? The variables x(t), which are the one-trading-day changes in log prices, are the variables in which Bachelier is interested for his theory of Brownian motion as applied to the stock market: x(t) = log S(t) – log S(t-1). Bachelier thinks these should have a normal distribution.
Recall from Part 4 that a
normal distribution has a location parameter The location parameter In fact, it is not quite zero. Essentially there is a
drift in the movement of the stock index So what Bachelier is doing with the data is trying to
estimate Recall from Part 2 that if today’s price
is (log P – But now, we are writing our stock index price as
(log One way of estimating the scale c itself. (This is called a maximum likelihood
estimator for the standard deviation.)Adding up the terms in the right-hand column in the table gives us a value of .001229332. And there are 10 observations. So we have variance = Taking the square root of this, we have standard deviation = So Bachelier’s changing probability interval for log
(log To get the probability interval for the price ( Since the current price on May 28, from the table, is 622.26, this interval becomes: (622.26 exp(– .0110875 T "This expression for the probability interval tells us the probability distribution over the next T days," Bachelier explained to Pete. "Now I understand what you meant. He who controls the present controls the past, because he can obtain past data. While he who masters this past data controls the future, because he can calculate future probabilities!" "Umm. That wasn’t what I meant," the angel replied. "But never mind." Over the next 10 trading days, we have T (622.26 (.965545), 622.26 (1.035683)) = (600.82, 644.46). This probability interval gives a price range for plus or
minus one scale parameter (in logs) To get a (622.26 exp(– (2) .0110875 T which gives us a price interval over 10 trading days of (580.12, 667.46).
Volatility In the financial markets, the scale parameter Here we have measured the scale However, market custom would dictate two criteria by which volatility is quoted: - quote volatility at an
*annual*(not daily) rate;
- quote volatility in
*percentage*(not decimal) terms.
To change our daily volatility annual Then we convert this to percent (by multiplying by 100 and calling the result "percent"): annual The New York Stock Exchange Composite Index had a historical volatility of 17.74 percent over the sample period during May. Note that an Notice that 256 trading days give us a probability
interval that is only 16 times as large as the probability interval for 1
day. This translates into a Hausdorff dimension for time (in the
probability calculation) as D = log(16)/log(256) = ½ or 0.5, which is just
the Bachelier-Einstein square-root-of-T (T The way we calculated the scale
Fractal Sums of Random Variables Now for the fun part. We have been looking at random
variables Under the assumption these random variables were normal,
we estimated a scale parameter In order to estimate Were our calculations proper and valid? Do they make any
sense? The answer to these questions depends on the issue of In answering this question we want to focus on ways we
can come up with a location parameter Suppose we have
Does the distribution of the sum m
and scale c. If each of the X has a location
_{i}m and scale c, whether normal or Cauchy, can that
information be translated into a location and a scale for the sum
X?The answer to all these questions is We will use the notation "~" as shorthand for "has the same distribution as." For example, X means X
X D_{n}where X Think of what this definition means. If their
distribution is stable, then the sum of Does this remind you of fractals? Fractals are
geometrical objects that look the same at different scales. Here we have
random variables whose probability distributions look the same at
different scales (except for the add factor D Let’s define two more terms.[2]
symmetric stable distributions.
The scale parameter C
C So if we have For the normal or Gaussian distribution, a = 2. So for ntimes as
large.^{1/ 2} For the Cauchy distribution, a =
1. So for ntimes as large.^{1/ 1} = n Thus if, for example, Brownian particles had a Cauchy
distribution, they would scale not according to a T law!Notice that we can also calculate a Hausdorff dimension
for symmetric stable distributions. If we divide a symmetric stable random
variable n copies of X/n. So the Hausdorff dimension is^{1/ a
}D = log N/ log c = log This gives us a simple interpretation of a . The parameter a is simply the Hausdorff dimension of a symmetric stable distribution. For the normal, the Hausdorff dimension is equal to 2, equivalent to that of a plane. For the Cauchy, the Hausdorff dimension is equal to 1, equivalent to that of a line. In between is a full range of values, including symmetric stable distributions with Hausdorff dimensions equivalent to the Koch Curve (log 4/log 3) and the Sierpinski Carpet (log 8/log3).
Some Fun with Logistic Art Now that we’ve worked our way to the heart of the matter, let’s take a break from probability theory and turn our attention again to dynamical systems. In particular, let’s look at our old friend the logistic equation: x(n+1) = where x(n) is the input variable, x(n+1) is the output
variable, and In Part
1, we looked at a particular version of this equation where k = 4. In
general, The dynamic behavior of this equation depends on the
value Instead, we are going to look at this equation when we
substitute for z(n+1) = k z(n) [1 – z(n)]. Complex numbers z have the form z = x + where That means when we iterate x + i y = k (x + i y) [ 1 – (x + i y)]. (Note that I have dropped the notation x(n) and y(n) and
just used The output To complete the transformation of the logistic equation,
we let k = A + B i, giving as our final form: x + i y = (A + B i) (x + i y) [ 1 – (x + i y)]. Now we multiply this all out and collect terms. The result is two equations in x and y: x = A (x-x As in the real version of the logistic equation, the behavior of the equation depends on the multiplier k = A + B i (that is, on A and B), as well as the initial starting value of z = x + i y (that is, on x(0) and y(0) ).
Julia Sets Depending on x The square root of this number is called the x which implies the modulus of When the equation is iterated, some starting values
diverge to infinity and some don’t. Each value for Let’s do an example. Let We keep We iterate the equation 256 times. If, at the end of 256
iterations, the modulus of To see the demonstration, be sure Java is enabled on your web browser and click here. We can create a plot that looks entirely different by
making a different color assignment. For the next demonstration, we again
iterate the dynamical system 256 times for different starting values of
z after 256 iterations. For example, if the square of the
modulus of z is greater than .6, but less than .7, the point z(0)
is assigned a light red color. Hence the colors in the Julia set
indicate the value of the modulus of z at the end of 256
iterations.To see the second demonstration of the same equation, but with this alternative color mapping, be sure Java is enabled on your web browser and click here So, from the complex logistic equation, a dynamical
system, we have created a fractal. The border of the Julia set is
determined by Meanwhile, we have passed from mathematics to art. Or maybe the art was there all along. We just had to learn how to appreciate it.
Notes [1] This is the stock market equivalent of the Interest Parity Theorem that relates the forward price F(t+T) of a currency, T-days in the future, to the current spot price S(t). In the foreign exchange market, the relationship can be written as: F(t+T) = S(t) [1 + r (T/360)]/[1+r*(T/360)] where r is the domestic interest rate (say the dollar interest rate), and r* is the foreign interest rate (say the interest rate on the euro). S is then the spot dollar price of the euro, and F is the forward dollar price of the euro. We can also use this equation to give us the forward value F of a stock index in relation to its current value S, in which case r* must be the dividend yield on the stock index. (A more precise calculation would disaggregate the "dividend yield" into the actual days and amounts of dividend payments.) This relationship is explored at length in Chapter 5,
"Forwards, Swaps, and Interest Parity," in J. Orlin Grabbe,
[2] The definitions here follow those in Gennady
Samorodnitsky and Murad S. Taqqu, [3] This is Theorem VI.1.1 in William Feller, [4] If Y = n independent copies of Y,Y X.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 29, July 19,
1999 |