Modelling Prices Using Brownian Motion.

As stated in my previous post I'm going to rewrite my data generation code for future use in online training of neural nets, and the approach I'm going to take is to combine the concept of Brownian motion and the ideas contained in my earlier Creation of Synthetic Data post.

The basic premise, taken from Brownian motion, is that the natural log of price changes, on average, at a rate proportional to the square root of time. Take, for example, a period of 5 leading up to the "current bar." If we take a 5 period simple moving average of the absolute differences of the log of prices over this period, we get a value for the average 1 bar price movement over this period. This value is then multiplied by the square root of 5 and added to and subtracted from the price 5 days ago to get an upper and lower bound for the current bar. If the current bar lies between the bounds, we say that price movement over the last 5 periods is consistent with Brownian motion and declare an absence of trend, i.e. a sideways market. If the current bar lies outside the bounds, we declare that price movement over the last 5 bars is not consistent with Brownian motion and that a trend is in force, either up or down depending on which bound the current bar is beyond. The following 3 charts show this concept in action, for consecutive periods of 5, 13 and 21, taken from the Fibonacci Sequence:

where yellow is the closing price, blue is the upper bound and red the lower bound. It is easy to imagine many uses for this in terms of indicator creation, but I intend to use the bounds to assign a score of price randomness/trendiness over various combined periods to assign price movement to bins for subsequent Monte Carlo creation of synthetic price series. Interested readers are invited to read the above linked Creation of Synthetic Data post for a review of this methodology.

The rough working Octave code that produced the above charts is given below.

clear all

data = load("-ascii","eurusd") ;
% length = input( 'Enter length of look back for calculations: ' ) ;
% sq_rt = sqrt( length ) ;
close = data(:,7) ;
abslogdiff = abs( [ 0 ; diff( log(close) ) ] ) ;

lngth = 3 ;
sq_rt = sqrt( lngth ) ;
sma3 = sma(abslogdiff,lngth) ;
ub = exp(shift(log(close),lngth).+(sma3.*sq_rt)) ;
lb = exp(shift(log(close),lngth).-(sma3.*sq_rt)) ;

all_ub = ub ;
all_lb = lb ;

lngth = 5 ;
sq_rt = sqrt( lngth ) ;
sma5 = sma(abslogdiff,5) ;
ub = exp(shift(log(close),lngth).+(sma5.*sq_rt)) ;
lb = exp(shift(log(close),lngth).-(sma5.*sq_rt)) ;

all_ub = all_ub .+ ub ;
all_lb = all_lb .+ lb ;

lngth = 8 ;
sq_rt = sqrt( lngth ) ;
sma8 = sma(abslogdiff,lngth) ;
ub = exp(shift(log(close),lngth).+(sma8.*sq_rt)) ;
lb = exp(shift(log(close),lngth).-(sma8.*sq_rt)) ;

all_ub = all_ub .+ ub ;
all_lb = all_lb .+ lb ;

lngth = 13 ;
sq_rt = sqrt( lngth ) ;
sma13 = sma(abslogdiff,lngth) ;
ub = exp(shift(log(close),lngth).+(sma13.*sq_rt)) ;
lb = exp(shift(log(close),lngth).-(sma13.*sq_rt)) ;

all_ub = all_ub .+ ub ;
all_lb = all_lb .+ lb ;

lngth = 21 ;
sq_rt = sqrt( lngth ) ;
sma21 = sma(abslogdiff,lngth) ;
ub = exp(shift(log(close),lngth).+(sma21.*sq_rt)) ;
lb = exp(shift(log(close),lngth).-(sma21.*sq_rt)) ;

all_ub = ( all_ub .+ ub ) ./ 5 ;
all_lb = ( all_lb .+ lb ) ./ 5 ;

 plot(close(1850:2100,1),'y',ub(1850:2100,1),'c',lb(1850:2100,1),'r')
%plot(close,'y',ub,'c',lb,'r')