more by chaos than by reason – more Law than Newton.
CHAPTER 2
Going Random
“We are floating in a medium of vast extent, always drifting uncertainly, blown to and fro; whenever we think we have a fixed point to which we can cling and make fast, it shifts and leaves us behind; if we follow it, it eludes our grasp, slips away, and flees eternally before us. Nothing stands still for us. This is our natural state and yet the state most contrary to our inclinations. We burn with desire to find a firm footing, an ultimate, lasting base on which to build a tower rising up to infinity, but our whole foundation cracks and the earth opens into the depth of the abyss.”
“Random; a dark field where dark cats are chased with laser guns; better than sex; like gambling; a little bit of math, some finance, lot of hypotheses, a lot of assumptions, more art than science; an attempt to predict or explain financial markets using mathematical theory; the art of collecting rent from the real economy; mathematical rationalisation for the injustices of capitalism; much like math, physics, and statistics helped meteorologists in building technology to predict weather, we quants do the same for markets; well, I could tell you but you don't have the necessary brain power to understand it *Stands up and leaves*.”
Quantitative finance is about using mathematics to understand the evolution of markets. One approach to prediction is to build deterministic Newtonian models of the system. Alternatively, one can make probabilistic models based on statistics. In practice, scientists usually use a combination of these approaches. For example, weather predictions are made using deterministic models, but because the predictions are prone to error, meteorologists use statistical techniques to make probabilistic forecasts (e.g., a 20 % chance of rain). Quants do the same for the markets, but then bet large amounts of money on the outcome. This chapter looks at how probability theory is applied to forecast the financial weather.
In 1724, after the collapse of his French monetary experiment, John Law supported himself in Venice by gambling. He would sit at a table at the Ridotto casino with 10,000 gold pistole coins arranged in stacks like casino chips, and offer any challenger the chance to make a wager of a single pistole. If they rolled six dice and got all sixes, then they could keep the lot. Law knew the odds of this happening were only 1 in 46,656 (6 multiplied by itself 6 times). So people always lost, but would go away happy at having gambled with the notorious John Law.
A key concept from probability theory is the idea of expected value, which equals the payout multiplied by the probability. For Law's gamble, this was 10,000 multiplied by 1/46,656, or 0.21 gold pistoles. Since the stake was 1 pistole, Law had an edge (a fair payout would have been 46,656 coins instead of 10,000). It was his money, after all, so he wanted to make a profit. We'll see later that he could still have made money even if he had offered the punters better odds, odds giving them the positive expectation. The solution to this apparent paradox is that he would have to do his gambling via a financial vehicle, a hedge fund, and he'd have to be betting with other people's money.
The connection between basic probability theory and something like the stock market becomes clear when we consider the result of a sequence of coin tosses, as in Figure 2.1. Here the paths start at the left and branch out to the right with time. If the coin comes up heads, you win one point, but if it is tails, you lose a point. The heavy line shows one particular trajectory, known as a random walk, against the background of all possible trajectories. At each time step, the path takes a random step up or down. Most paths remain near the center. Figure 2.2 shows how the final distribution looks after 14 time steps. The mean or average displacement is zero, and over 20 % of the paths end with no displacement. If this were a plot of price changes for a stock, and the horizontal axis represented time in days, we would say that the expected value of the stock after 14 days would be unchanged from its initial value.
Figure 2.1 Coin toss results
The black line shows one possible random walk, with a vertical step of plus 1 (up) or minus 1 (down) at each iteration. The light gray lines are an overlay of all possible paths through 14 iterations. The plot shows how the future becomes more uncertain as the possible paths multiply.
Figure 2.2 A histogram showing the final distribution after 14 iterations
The range is −14 to 14, but over 20 % of paths end with no change in position (center bar). The shape approximates the bell curve or normal distribution from classical statistics.
After n iterations, the maximum deviation from 0 is equal to n – so after 14 steps, the range is from −14 to 14. But most paths stay near the center, so the average displacement is much smaller.27 A longer random walk, of 100 steps, is shown by the solid line in Figure 2.3. The light-gray lines are the bounds for possible paths: the upper bound is the path with an increase of 1 at every step, while the lower bound is the path with a decrease of 1 at every step (the probability of these paths is extremely low, since they are the same as tossing a coin and getting heads 100 times in a row). In the background, the density of the grayscale at any point corresponds to the probability of a random walk going through that point. Note how this probability density spreads out with time, rather like an idealized, turbulence-free version of a plume of smoke emitted from a chimney. Random walks sound wild, but on average they are very well-behaved.
Figure 2.3 100-Step random walk
The solid line is a random walk of 100 steps, starting at 0, with a displacement of plus or minus 1 at each step. The light-gray lines show the upper and lower bounds, corresponding to the paths in which the displacement at every step is plus or minus 1, respectively. The density of the grayscale at any point corresponds to the probability of a random walk going through that point. This is highest for paths with small displacement. The probability of a path entering the white area is very low, or zero outside the light-gray lines.
Such computations become unwieldy when there are a very large number of games or iterations; however, in 1738 the mathematician Abraham de Moivre showed that after an infinitely large number of iterations, the results would converge on the so-called normal distribution, or bell curve. This is specified by two numbers: the mean or average and the standard deviation, which is a measure of the curve's width.28 About 68 % of the data fall within one standard deviation of the mean, and about 95 % are within two standard deviations. The homme moyen of statistics, this formula got its name because of its ubiquity in the physical and social sciences. The distinguishing feature of the normal distribution is that, according to the central limit theorem, which was partially proven by de Moivre, it can be used to model the sum of any random processes, provided that a number of conditions are met. In particular, the separate processes have to be independent of one another, and identically distributed. So, for example, if 18th-century astronomers made many measurements of the position of Saturn in the night sky, then each measurement would be subject to errors, but they could hope
