Generate fractal time series with non-normal returns distribution
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R Bump version to 1.4 Aug 17, 2016
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Asset price simulation using fractals and other approaches.

Simulating stock market prices and returns can be accomplished using a number of techniques. Most commonly, geometric brownian motion (aka a random walk) is used to simulate stock prices. Using this technique results in a normal distribution of price returns. As an alternative technique, it is possible to generate price series using fractals. The advantage is that price returns tend to have volatility that clusters, similar to actual returns.

The basic principle driving fractal generation of time series is that data is generated iteratively based on increasing levels of resolution. The initial series is defined by a so-called initiator pattern and then generators are used to replace each segment of the initial pattern. Regular, repeatable patterns can be produced by using the same seed and generators. By using a set of generators, non-repeatable time series can be produced. This technique is the basis of the fractal time series process in this package.


As random generators, most of the functions in this package follow the R convention of using r as a prefix to denote a random number generator. The available generators include:

  • rprices
  • rintraday

Stochastic processes

All price generators use a stochastic process to generate the sequence of random numbers.

Fractal time series

Mandelbrot not only pioneered fractals but also used them to investigate volatility and risk in financial markets.

Geometric brownian motion

GBM is another name for a random walk.

mygbm <- function(x) gbm(x, 40, .03/1440)


An OU process is typically used to simulate interest rates.

Daily price simulation

Any process can be used to generate prices. The rprices function attaches dates to the random sequence so it can be analyzed as an asset price series.

mygbm <- function(x) gbm(x, 40, .03/1440)
ps <- rprices(mygbm, obs=100)

Intraday price simulation

Generating intraday prices involves specifying a stochastic process along with some parameters indicating how many points to generating and how to map the points in the series to time. The trading_hours function knows about exchange holidays and trading hours to give a realistic price series.

th <- function(x) trading_hours(x,'cme')
seed <- rintraday(mygbm, 60, th)

> head(seed)
2015-02-23 40.00000
2015-02-24 39.95202
2015-02-25 40.03372
2015-02-26 40.13978
2015-02-27 40.17472
2015-03-02 40.04889

An end date can be specified instead of the number of observations. These definitions are typed via lambda.r, so it's not necessary to name either of these parameters, although for clarity you may choose to do so.

seed <- rintraday(mygbm, as.Date('2015-03-05'), th)
> range(index(seed))
[1] "2015-02-23" "2015-03-05"

To specify an explicit date range, use this signature:

seed <- rintraday(mygbm, start='2015-01-01', end='2015-03-01', th)
> range(index(seed))
[1] "2015-01-01" "2015-02-27"

Correlated random numbers

FractalRock now supports generating correlated random numbers. This usage involves simulating a seed series and then generating a set of correlated series.

First we create the trading hours generator, followed by a call to rintraday to generate the seed series.

th <- function(x) trading_hours(x,'cme')
seed <- rintraday(mygbm, obs=60, th)

Now we specify the correlation matrix.

cmat <- matrix(c(1,0,0, .8,1,0, .6,.4,1), ncol=3)

> cmat
     [,1] [,2] [,3]
[1,]    1  0.8  0.6
[2,]    0  1.0  0.4
[3,]    0  0.0  1.0

Finally, we generate the correlated price series.

z <- rintraday(seed, cmat)

> cor(z)
          [,1]      [,2]      [,3]
[1,] 1.0000000 0.8128455 0.6432900
[2,] 0.8128455 1.0000000 0.3788305
[3,] 0.6432900 0.3788305 1.0000000

Complete OHLC bars can be generated for correlated series as well. The OHLC option specifies the standard deviation for the HLC series. Instead of an xts object, a list of xts objects are returned, one for each time series.

th <- function(x) trading_hours(x,'cme')
seed <- rintraday(mygbm, obs=60, th)
cmat <- matrix(c(1,0,0, .8,1,0, .6,.4,1), ncol=3)
z <- rintraday(seed, cmat, ohlc=1, volume=100)

> head(z[[2]])
              close     open      low     high volume
2015-02-23 40.03911 40.18967 39.11649 40.34437    111
2015-02-24 39.94562 40.03911 39.18681 40.17525     94
2015-02-25 40.05002 39.94562 38.61717 40.39037    103
2015-02-26 40.10920 40.05002 39.41671 41.46493    109
2015-02-27 40.19329 40.10920 39.83513 40.58183     58
2015-03-02 40.04908 40.19329 39.72885 41.37340    128

Iterative function systems

Further Work

At a later date, implementation of the [modified] rescaled range statistic will be included to provide more analytical insight into the time series data produced by this package.


This package started off as a way to explore generating time series using fractals. It has since evolved to encompass other forms of asset price simulation.