35. Several independent random walks plotted together showing the spread of the walkers widening as time progresses

36. The distribution of positions of 130 random walkers after 20 steps

The graph in Figure 34 of the position of a random walker already has a rather spiky, irregular appearance. Now suppose that the walker takes rather shorter steps but much more frequently, say steps of length every second or steps of length every second. The graph retains the same sort of overall appearance at a large scale, but smaller scale irregularity is also introduced (just as in the successive stages of the von Koch curve construction). By taking very small steps very rapidly the graph of the random walk takes on a fractal form, called the Brownian or Wiener process. For the process to remain interesting the steps taken at very small time intervals t need to be of length . (With shorter or longer steps the typical displacement from the origin, in other words the standard deviation, at any given time will become negligibly small or ridiculously large.) The Brownian process, Figure 37, has a fractal graph and, although random, always has box-counting dimension . It is statistically self-similar, in the sense that enlarging a small portion of the graph yields a graph of a similar overall appearance to the original, subject to statistical variations.

37. Graph of a Brownian process

Many real phenomena which depend on very frequent random events appear very like a random walk or Brownian process. One striking example is of financial data such as share prices or exchange rates. As early as 1900, Louis Bachelier (1870–1946) proposed in his PhD thesis Thйorie de la spйculation that share prices might behave in a similar way to a Brownian process. The prices of shares result from many individual investors estimating the future value of assets using information available to them at the time. Such information arrives randomly, and involves numerous factors, such as news or rumours of the behaviour of governments, companies, banks, etc. With virtually instantaneous trading possible, share prices are determined by a combination of a large number of small, seemingly random, upwards or downward steps in very short time intervals, with the consequence that a graph of share prices looks in many ways very similar to a Brownian graph. Indeed, graphs of share prices over a week, a year, or 20 years is completely defined by the ppeall have a very similar overall form, displaying strong statistical self-similarity. Moreover, box-counting on a graph of share prices invariably gives a dimension very close to Ѕ, the dimension of the Brownian graph. There are mathematical and economic reasons for this. The stock market behaves in a way so as not to permit arbitrage, that is the possibility of an investor making a risk-free profit. Fractal models of share prices with graphs of dimensions other than may allow an arbitrage opportunity.

The Brownian process underlies option pricing in finance. A call option is a contract that gives the buyer the right, but not the obligation, to buy an agreed quantity of a stock or shares from the seller at a specified price, termed the strike price, on (or, in some cases, before) a specified expiry date. The buyer pays a certain amount to purchase the option, hoping that the stock will increase in value to above the strike price, so he or she can buy the stock for below its actual value on the expiry date and make an overall profit. If the buyer decides not to exercise the option and not buy the stock, then his or her loss is limited to the purchase price of the option. On the other hand, the seller does not expect such a rise in value, or perhaps is prepared to accept a reduced profit should the shares rise steeply in return for immediate cash. An obvious problem that arises is how to determine a fair price for the option so both buyer and seller feel that they have a good chance of making a profit, otherwise they are unlikely to agree an option contract.

A widely used basis for determining option prices is the Black-Scholes model (here a ‘model’ means a ‘mathematical description’) proposed in 1973 by Fisher Black (1938–95) and Myron Scholes (1941–) for which Scholes and Robert Merton (1944–), who developed the model, were awarded the Nobel Prize for Economics in 1997. The Black-Scholes differential equation describes how, under certain assumptions, the value of an option varies with time and with the price of the stock. A fundamental assumption is that the logarithm of the stock price follows a random Brownian process along with a fixed underlying drift (i.e. constant rate of increase or decrease). It is also assumed that there is no arbitrage opportunity, that is the model does not allow a risk-free profit, and that it is possible to borrow or lend cash at a known risk-free interest rate. The equation may be solved to obtain a formula for the ‘natural’ price for an option in terms of the time to expiry, the initial price of the stock, the strike price, and the risk-free rate. The natural price is determined by the requirement that the rate of return on the investment equals the risk-free rate, otherwise there would be an opportunity for arbitrage using the imbalance of the rates. A key idea in the derivation of the Black-Scholes equation is that terms relating to the underlying Brownian process cancel, so that there is no random component in the final equation and formula.

Despite its appeal, the Brownian description fails to reflect some crucial aspects of share behaviour—in particular it would essentially never permit the rapid changes that have occurred in reality, such as the stock market crashes of 1929, 1973, 1987, 2001, and 2007. A Brownian process requires that price changes over, say, each weekly period have a normal, ‘bell-shaped’ distribution with the typical weekly variation being given by its standard deviation. However, these market crashes would represent a fall of over 4 times the standard deviation, which with a normal distribution would occur less than every 300 years. Study of the actual weekly changes in prices shows that the after the French mathematician (pe variation is far from normally distributed and the Brownian, or random walk, model has nothing like the variation in step lengths needed to account for this. Consequently, many other mathematical processes that behave much more like real price data have been proposed, for example by allowing considerable variation in the step lengths of random walk, and even the possibility of instantaneous ‘jumps’ to occur from time to time. Another approach, that allows for the volatility of the markets to vary widely at different times, is to introduce a ‘time warp’, with the time scale on a Brownian, or other, graph distorted by stretching and compression in a highly irregular manner. Mandelbrot summed up the necessary features of a financial model in a nice way, by referring to the Joseph effect (from the biblical prediction of seven prosperous years followed by seven lean ones) and the Noah effect (the sudden dramatic effect of a flood). The Joseph effect reflects that trends tend to persist, but the Noah effect means that when things change they can change suddenly and dramatically. Plausible financial models need to allow for both effects.

A great deal of effort continues to be devoted to finding good mathematical descriptions of stock market behaviour, since a model that could indicate future trends might allow the discoverer to make large profits. Fractal models are natural candidates, being erratic in the short term but perhaps in the longer term having some underlying regularity that could be detected.

Random walks and Brownian motion in the plane or space

For a random walk in the plane, the walker starts at the origin and takes steps of a given length to the north, south, east, or west with equal probability, that is with a 25 per cent chance of moving in each direction. The path traversed will include many reversed steps, loops, and crossings, see Figure 38. (A practical setting for this might be the square grid of streets in New York with each ‘step’ comprising a walk between two adjacent junctions with a random choice of direction on reaching each junction.) The plane random walk has many surprising properties. For example, it can be shown that with probability one (i.e. with certainty), the path will eventually visit every point on the plane grid. However, the average time taken to reach any given point on the grid, or even to return to the start, is infinite!

Again, we can reduce the length of the steps and increase their frequency. Taking steps at very small time intervals t of length in one of the four directions leads to a highly irregular continuous path in the plane that has many loops and self-crossings, Figure 39. This limiting path is called planar Brownian motion and is a statistically self-similar fractal of box dimension 2 but nonetheless occupying area 0, so is of negligible ‘size’ as a two-dimensional object. Exactly the same procedure can be followed in three dimensional space, with each step going north, south, east, west, up, or down with equal likelihood to get a random walk in space. Taking very rapid steps of very short length gives spatial Brownian motion. This very irregular, tangled, path still has dimension only 2; in fact taking a photograph from any direction gives a picture that is precisely a realization of plane Brownian motion.