Random Fractals and the Stock Market

Surrogates of the Stock Market - Trading Time

The cartoon with turning points (4/9, 2/3) and (5/9, 1/3) is a unifractal cartoon of Brownian motion.

It does not give a convincing forgery of real stock market data: it lacks the global dependence and long tails of observed data.

By changing the (dt_i, dY_i), we can obtain convincing forgeries of real data.

Now we shall see there is an elementary way to change the way we measure time so these two effects, global dependence and long tails, are disentangled.

First, some notation.

Plots of Y vs t are called price vs clock time graphs.

To mimick real data, these graphs must be multifractal.

We shall find a way to rescale time to a new variable, T, so the plot of Y vs T is a unifractal.

This time is called trading time.

We shall show that every multifractal cartoon can be reexpressed as a unifractal cartoon in (multifractal) trading time.

The Price vs Trading Time graph exhibits global dependence, but not long tails; the Clock Time vs Trading Time graph exhibits long tails.

General Principle Take the generator price increments dY₁, dY₂, and dY₃. There is a unique number D satisfying |dY₁|^D + |dY₂|^D + |dY₃|^D = 1. The trading time generators are dT₁ = |dY₁|^D, dT₂ = |dY₂|^D, and dT₃ = |dY₃|^D. While the trading time vs clock time (T vs t) graph is multifractal, the price vs trading time graph (Y vs T) is unifractal.

Example 1 We take the multifractal generator with turning points (1/4, 1/2) and (3/4, 1/4), calculate its trading time generator, and look at the correlations in the price vs trading time graph.

Example 2 We take the multifractal generator with turning points (.2, .7) and (.6, .4), calculate its trading time generator, and look at the correlations in the price vs trading time graph.

Graph in three dimensions The conversion of price vs clock time into price vs trading time can be thought of as stretching and shrinking the time scale so the large jumps are absorbed into the time rescaling, leaving a graph that emphasizes long-term dependence. A graph in three dimensions, designed with the help of our students, makes this effect transparent.

Return to Surrogates of the Stock Market.

	General Principle Take the generator price increments dY₁, dY₂, and dY₃. There is a unique number D satisfying \|dY₁\|^D + \|dY₂\|^D + \|dY₃\|^D = 1. The trading time generators are dT₁ = \|dY₁\|^D, dT₂ = \|dY₂\|^D, and dT₃ = \|dY₃\|^D. While the trading time vs clock time (T vs t) graph is multifractal, the price vs trading time graph (Y vs T) is unifractal.
	Example 1 We take the multifractal generator with turning points (1/4, 1/2) and (3/4, 1/4), calculate its trading time generator, and look at the correlations in the price vs trading time graph.
	Example 2 We take the multifractal generator with turning points (.2, .7) and (.6, .4), calculate its trading time generator, and look at the correlations in the price vs trading time graph.
	Graph in three dimensions The conversion of price vs clock time into price vs trading time can be thought of as stretching and shrinking the time scale so the large jumps are absorbed into the time rescaling, leaving a graph that emphasizes long-term dependence. A graph in three dimensions, designed with the help of our students, makes this effect transparent.