Brownian Bands

For the past few weeks I have been working on the idea presented in my earlier modelling prices using brownian motion post and below is a video of what I have come up with, which I'm calling Brownian Bands. The video shows these bands on the EURUSD forex pair with look back lengths of 1 to 10 bars inclusive, then 15 to 50 in increments of 5, then 50, 100 and 200 and finally 50, 100 and 200 on the whole time series. Each look back length band is the average of all bands from 1 up to and including the look back length band - the details of the implementation can be seen in the code boxes below. The blue band is the upper band, the red is the lower and the green is the mid point of the two.

           
Non-embedded view.

This following chart shows an adaptive look back length implementation, the look back length being determined by a period vector input to the function, which in this case is the dominant cycle period.

These adaptive Brownian bands can be compared with the equivalent adaptive Bollinger bands shown in this next chart.

For my intended purpose of separating out returns into distribution bins the Brownian bands seem to be superior. There are 3 distinct bins: above the upper band, below the lower band and between the bands, whereas with the Bollinger bands the vast majority of returns would be lumped into just one bin - between the bands. Of course the Brownian bands could be used to create a system indicator just as Bollinger bands can be, and if any readers test this idea I would be intrigued to hear of your results. However, should you choose to do this there is a major caveat you should be aware of. The calculation of the Brownian bands is based on the natural log of actual price, which means you should be very wary of test results on any back adjusted price series such as futures and adjusted stock prices. In fact I would go as far to say that the code given below should not be used as is on such price series. It is for this reason that I have been using forex price series in my own tests so far, but I will surely get around to adjusting the given code in due course.

Now for the code. This first code box shows a hard-coded, unrolled loop version for look back lengths up to and including 25 for the benefit of a reader who requested a clearer exposition of the method than was given in my earlier post, which was vectorised Octave code. All the code below is C++ as used in Octave .oct files.

DEFUN_DLD ( brownian_bands_25, args, nargout, 
"-*- texinfo -*-\n\
@deftypefn {Function File} {} brownian_bands_25 (@var{price,period})\n\
This function takes a price vector input and a value for the tick\n\
size of the instrument and outputs upper and lower bands based on\n\
the concept of Brownian motion. The bands are the average of look\n\
back periods of 1 to 25 bars inclusive of the relevant calculations.\n\
@end deftypefn" )

{
octave_value_list retval_list ;
int nargin = args.length () ;

// check the input arguments
if ( nargin != 2 )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }

if ( args(0).length () < 21 )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
   
if ( args(1).length () != 1 )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }   

if ( error_state )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
// end of input checking   

ColumnVector price = args(0).column_vector_value () ;
double tick_size = args(1).double_value() ;
ColumnVector abs_log_price_diff = args(0).column_vector_value () ;
ColumnVector upper_band = args(0).column_vector_value () ;
ColumnVector lower_band = args(0).column_vector_value () ;
ColumnVector mid_band = args(0).column_vector_value () ;
double sum ;

double up_1 ;
double low_1 ;

double up_2 ;
double low_2 ;
double sqrt_2 = sqrt(2.0) ; // pre-calculated for speed optimisation

double up_3 ;
double low_3 ;
double sqrt_3 = sqrt(3.0) ; // pre-calculated for speed optimisation

double up_4 ;
double low_4 ;
double sqrt_4 = sqrt(4.0) ; // pre-calculated for speed optimisation

double up_5 ;
double low_5 ;
double sqrt_5 = sqrt(5.0) ; // pre-calculated for speed optimisation

double up_6 ;
double low_6 ;
double sqrt_6 = sqrt(6.0) ; // pre-calculated for speed optimisation

double up_7 ;
double low_7 ;
double sqrt_7 = sqrt(7.0) ; // pre-calculated for speed optimisation

double up_8 ;
double low_8 ;
double sqrt_8 = sqrt(8.0) ; // pre-calculated for speed optimisation

double up_9 ;
double low_9 ;
double sqrt_9 = sqrt(9.0) ; // pre-calculated for speed optimisation

double up_10 ;
double low_10 ;
double sqrt_10 = sqrt(10.0) ; // pre-calculated for speed optimisation

double up_11 ;
double low_11 ;
double sqrt_11 = sqrt(11.0) ; // pre-calculated for speed optimisation

double up_12 ;
double low_12 ;
double sqrt_12 = sqrt(12.0) ; // pre-calculated for speed optimisation

double up_13 ;
double low_13 ;
double sqrt_13 = sqrt(13.0) ; // pre-calculated for speed optimisation

double up_14 ;
double low_14 ;
double sqrt_14 = sqrt(14.0) ; // pre-calculated for speed optimisation

double up_15 ;
double low_15 ;
double sqrt_15 = sqrt(15.0) ; // pre-calculated for speed optimisation

double up_16 ;
double low_16 ;
double sqrt_16 = sqrt(16.0) ; // pre-calculated for speed optimisation

double up_17 ;
double low_17 ;
double sqrt_17 = sqrt(17.0) ; // pre-calculated for speed optimisation

double up_18 ;
double low_18 ;
double sqrt_18 = sqrt(18.0) ; // pre-calculated for speed optimisation

double up_19 ;
double low_19 ;
double sqrt_19 = sqrt(19.0) ; // pre-calculated for speed optimisation

double up_20 ;
double low_20 ;
double sqrt_20 = sqrt(20.0) ; // pre-calculated for speed optimisation

double up_21 ;
double low_21 ;
double sqrt_21 = sqrt(21.0) ; // pre-calculated for speed optimisation

double up_22 ;
double low_22 ;
double sqrt_22 = sqrt(22.0) ; // pre-calculated for speed optimisation

double up_23 ;
double low_23 ;
double sqrt_23 = sqrt(23.0) ; // pre-calculated for speed optimisation

double up_24 ;
double low_24 ;
double sqrt_24 = sqrt(24.0) ; // pre-calculated for speed optimisation

double up_25 ;
double low_25 ;
double sqrt_25 = sqrt(25.0) ; // pre-calculated for speed optimisation

for ( octave_idx_type ii (1) ; ii < 25 ; ii++ ) // initialising loop 
    {
    abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;
    }
    
for ( octave_idx_type ii (25) ; ii < args(0).length () ; ii++ ) // main loop
    {
      
    abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;

    sum = abs_log_price_diff(ii) ;
    up_1 = exp( log( price(ii-1) ) + sum ) ;
    low_1 = exp( log( price(ii-1) ) - sum ) ;

    sum += abs_log_price_diff(ii-1) ;
    up_2 = exp( log( price(ii-2) ) + (sum/2.0) * sqrt_2 ) ;
    low_2 = exp( log( price(ii-2) ) - (sum/2.0) * sqrt_2 ) ;

    sum += abs_log_price_diff(ii-2) ;
    up_3 = exp( log( price(ii-3) ) + (sum/3.0) * sqrt_3 ) ;
    low_3 = exp( log( price(ii-3) ) - (sum/3.0) * sqrt_3 ) ;
    
    sum += abs_log_price_diff(ii-3) ;
    up_4 = exp( log( price(ii-4) ) + (sum/4.0) * sqrt_4 ) ;
    low_4 = exp( log( price(ii-4) ) - (sum/4.0) * sqrt_4 ) ;

    sum += abs_log_price_diff(ii-4) ;
    up_5 = exp( log( price(ii-5) ) + (sum/5.0) * sqrt_5 ) ;
    low_5 = exp( log( price(ii-5) ) - (sum/5.0) * sqrt_5 ) ;    

    sum += abs_log_price_diff(ii-5) ;
    up_6 = exp( log( price(ii-6) ) + (sum/6.0) * sqrt_6 ) ;
    low_6 = exp( log( price(ii-6) ) - (sum/6.0) * sqrt_6 ) ;
    
    sum += abs_log_price_diff(ii-6) ;
    up_7 = exp( log( price(ii-7) ) + (sum/7.0) * sqrt_7 ) ;
    low_7 = exp( log( price(ii-7) ) - (sum/7.0) * sqrt_7 ) ;
    
    sum += abs_log_price_diff(ii-7) ;
    up_8 = exp( log( price(ii-8) ) + (sum/8.0) * sqrt_8 ) ;
    low_8 = exp( log( price(ii-8) ) - (sum/8.0) * sqrt_8 ) ;
    
    sum += abs_log_price_diff(ii-8) ;
    up_9 = exp( log( price(ii-9) ) + (sum/9.0) * sqrt_9 ) ;
    low_9 = exp( log( price(ii-9) ) - (sum/9.0) * sqrt_9 ) ;
    
    sum += abs_log_price_diff(ii-9) ;
    up_10 = exp( log( price(ii-10) ) + (sum/10.0) * sqrt_10 ) ;
    low_10 = exp( log( price(ii-10) ) - (sum/10.0) * sqrt_10 ) ;
    
    sum += abs_log_price_diff(ii-10) ;
    up_11 = exp( log( price(ii-11) ) + (sum/11.0) * sqrt_11 ) ;
    low_11 = exp( log( price(ii-11) ) - (sum/11.0) * sqrt_11 ) ;
    
    sum += abs_log_price_diff(ii-11) ;
    up_12 = exp( log( price(ii-12) ) + (sum/12.0) * sqrt_12 ) ;
    low_12 = exp( log( price(ii-12) ) - (sum/12.0) * sqrt_12 ) ;
    
    sum += abs_log_price_diff(ii-12) ;
    up_13 = exp( log( price(ii-13) ) + (sum/13.0) * sqrt_13 ) ;
    low_13 = exp( log( price(ii-13) ) - (sum/13.0) * sqrt_13 ) ;
    
    sum += abs_log_price_diff(ii-13) ;
    up_14 = exp( log( price(ii-14) ) + (sum/14.0) * sqrt_14 ) ;
    low_14 = exp( log( price(ii-14) ) - (sum/14.0) * sqrt_14 ) ;
    
    sum += abs_log_price_diff(ii-14) ;
    up_15 = exp( log( price(ii-15) ) + (sum/15.0) * sqrt_15 ) ;
    low_15 = exp( log( price(ii-15) ) - (sum/15.0) * sqrt_15 ) ;
    
    sum += abs_log_price_diff(ii-15) ;
    up_16 = exp( log( price(ii-16) ) + (sum/16.0) * sqrt_16 ) ;
    low_16 = exp( log( price(ii-16) ) - (sum/16.0) * sqrt_16 ) ;
    
    sum += abs_log_price_diff(ii-16) ;
    up_17 = exp( log( price(ii-17) ) + (sum/17.0) * sqrt_17 ) ;
    low_17 = exp( log( price(ii-17) ) - (sum/17.0) * sqrt_17 ) ;
    
    sum += abs_log_price_diff(ii-17) ;
    up_18 = exp( log( price(ii-18) ) + (sum/18.0) * sqrt_18 ) ;
    low_18 = exp( log( price(ii-18) ) - (sum/18.0) * sqrt_18 ) ;
    
    sum += abs_log_price_diff(ii-18) ;
    up_19 = exp( log( price(ii-19) ) + (sum/19.0) * sqrt_19 ) ;
    low_19 = exp( log( price(ii-19) ) - (sum/19.0) * sqrt_19 ) ;
    
    sum += abs_log_price_diff(ii-19) ;
    up_20 = exp( log( price(ii-20) ) + (sum/20.0) * sqrt_20 ) ;
    low_20 = exp( log( price(ii-20) ) - (sum/20.0) * sqrt_20 ) ;
    
    sum += abs_log_price_diff(ii-20) ;
    up_21 = exp( log( price(ii-21) ) + (sum/21.0) * sqrt_21 ) ;
    low_21 = exp( log( price(ii-21) ) - (sum/21.0) * sqrt_21 ) ;
    
    sum += abs_log_price_diff(ii-21) ;
    up_22 = exp( log( price(ii-22) ) + (sum/22.0) * sqrt_22 ) ;
    low_22 = exp( log( price(ii-22) ) - (sum/22.0) * sqrt_22 ) ;
    
    sum += abs_log_price_diff(ii-22) ;
    up_23 = exp( log( price(ii-23) ) + (sum/23.0) * sqrt_23 ) ;
    low_23 = exp( log( price(ii-23) ) - (sum/23.0) * sqrt_23 ) ;
    
    sum += abs_log_price_diff(ii-23) ;
    up_24 = exp( log( price(ii-24) ) + (sum/24.0) * sqrt_24 ) ;
    low_24 = exp( log( price(ii-24) ) - (sum/24.0) * sqrt_24 ) ;
    
    sum += abs_log_price_diff(ii-24) ;
    up_25 = exp( log( price(ii-25) ) + (sum/25.0) * sqrt_25 ) ;
    low_25 = exp( log( price(ii-25) ) - (sum/25.0) * sqrt_25 ) ;
    
    upper_band(ii) = (up_1+up_2+up_3+up_4+up_5+up_6+up_7+up_8+up_9+up_10+up_11+up_12+up_13+up_14+up_15+up_16+up_17+up_18+up_19+up_20+up_21+up_22+up_23+up_24+up_25)/25.0 ;
    lower_band(ii) = (low_1+low_2+low_3+low_4+low_5+low_6+low_7+low_8+low_9+low_10+low_11+low_12+low_13+low_14+low_15+low_16+low_17+low_18+low_19+low_20+low_21+low_22+low_23+up_24+low_25)/25.0 ;
    
    // round the upper_band up to the nearest tick
    upper_band(ii) = ceil( upper_band(ii)/tick_size ) * tick_size ;
    
    // round the lower_band down to the nearest tick
    lower_band(ii) = floor( lower_band(ii)/tick_size ) * tick_size ;
    
    mid_band(ii) = ( upper_band(ii) + lower_band(ii) ) / 2.0 ;
    
    } // end of main loop   

    retval_list(2) = mid_band ;
    retval_list(1) = lower_band ;
    retval_list(0) = upper_band ;

return retval_list ;

} // end of function

This next code box contains code for an adjustable look back length, the length being a user input to the function. This code wraps the above unrolled loop into a nested loop.

DEFUN_DLD ( brownian_bands_adjustable, args, nargout, 
"-*- texinfo -*-\n\
@deftypefn {Function File} {} brownian_bands_adjustable (@var{price,lookback,tick_size})\n\
This function takes a price vector input, a value for the lookback\n\
length and a value for the tick size of the instrument and outputs\n\
upper and lower bands based on the concept of Brownian motion. The bands\n\
are the average of lookback period bands from 1 to lookback length.\n\
@end deftypefn" )

{
octave_value_list retval_list ;
int nargin = args.length () ;

// check the input arguments
if ( nargin != 3 )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
   
if ( args(1).length () != 1 ) // lookback length argument
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
   
double lookback = args(1).double_value() ;   

if ( args(0).length () < lookback ) // check length of price vector input
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
   
if ( args(2).length () != 1 ) // tick size
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }   
   
if ( error_state )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
// end of input checking   

ColumnVector price = args(0).column_vector_value () ;
double tick_size = args(2).double_value() ;
ColumnVector abs_log_price_diff = args(0).column_vector_value () ;
ColumnVector upper_band = args(0).column_vector_value () ;
ColumnVector lower_band = args(0).column_vector_value () ;
ColumnVector mid_band = args(0).column_vector_value () ;
double sum ;
double up_bb ;
double low_bb ;
int jj ;

for ( octave_idx_type ii (1) ; ii < lookback ; ii++ ) // initialising loop 
    {
    abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;
    }
    
for ( octave_idx_type ii (lookback) ; ii < args(0).length () ; ii++ ) // main loop
    {    
    // initialise calculation values  
    sum = 0.0 ;
    up_bb = 0.0 ;
    low_bb = 0.0 ;

    for ( jj = 1 ; jj < lookback + 1 ; jj++ ) // nested jj loop
        {
   
 abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;
        sum += abs_log_price_diff(ii-jj+1) ;
        up_bb += exp( log( price(ii-jj) ) + (sum/double(jj)) * sqrt(double(jj)) ) ;
        low_bb += exp( log( price(ii-jj) ) - (sum/double(jj)) * sqrt(double(jj)) ) ;  

 } // end of nested jj loop  
    
    upper_band(ii) = up_bb / lookback ;
    lower_band(ii) = low_bb / lookback ;
    
    // round the upper_band up to the nearest tick
    upper_band(ii) = ceil( upper_band(ii)/tick_size ) * tick_size ;
    
    // round the lower_band down to the nearest tick
    lower_band(ii) = floor( lower_band(ii)/tick_size ) * tick_size ;
    
    mid_band(ii) = ( upper_band(ii) + lower_band(ii) ) / 2.0 ;
    
    } // end of main loop   

    retval_list(2) = mid_band ;
    retval_list(1) = lower_band ;
    retval_list(0) = upper_band ;

return retval_list ;

} // end of function

Finally, this code is the adaptive look back version where the length of the look back is contained in a vector which is also a user input.

DEFUN_DLD ( brownian_bands_adaptive, args, nargout, 
"-*- texinfo -*-\n\
@deftypefn {Function File} {} brownian_bands_adaptive (@var{price,period,tick_size})\n\
This function takes price and period vector inputs & a value for the tick\n\
size of the instrument and outputs upper and lower bands based on\n\
the concept of Brownian motion. The bands are adaptive averages of look\n\
back lengths of 1 to period bars inclusive.\n\
@end deftypefn" )

{
octave_value_list retval_list ;
int nargin = args.length () ;

// check the input arguments
if ( nargin != 3 )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
   
if ( args(0).length () < 50 ) // check length of price vector input
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }   
   
if ( args(1).length () != args(0).length () ) // lookback period length argument
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
      
if ( args(2).length () != 1 ) // tick size
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }   
   
if ( error_state )
   {
   error ( "Invalid arguments. Type help to see usage." ) ;
   return retval_list ;
   }
// end of input checking   

ColumnVector price = args(0).column_vector_value () ;
ColumnVector period = args(1).column_vector_value () ; 
double tick_size = args(2).double_value() ;
ColumnVector abs_log_price_diff = args(0).column_vector_value () ;
ColumnVector upper_band = args(0).column_vector_value () ;
ColumnVector lower_band = args(0).column_vector_value () ;
ColumnVector mid_band = args(0).column_vector_value () ;
double sum ;
double up_bb ;
double low_bb ;
int jj ;

for ( octave_idx_type ii (1) ; ii < 50 ; ii++ ) // initialising loop 
    {
    abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;
    }
    
for ( octave_idx_type ii (50) ; ii < args(0).length () ; ii++ ) // main loop
    {    
    // initialise calculation values  
    sum = 0.0 ;
    up_bb = 0.0 ;
    low_bb = 0.0 ;

    for ( jj = 1 ; jj < period(ii) + 1 ; jj++ ) // nested jj loop
        {
   
 abs_log_price_diff(ii) = fabs( log( price(ii) ) - log( price(ii-1) ) ) ;
        sum += abs_log_price_diff(ii-jj+1) ;
        up_bb += exp( log( price(ii-jj) ) + (sum/double(jj)) * sqrt(double(jj)) ) ;
        low_bb += exp( log( price(ii-jj) ) - (sum/double(jj)) * sqrt(double(jj)) ) ;  

 } // end of nested jj loop  
    
    upper_band(ii) = up_bb / period(ii) ;
    lower_band(ii) = low_bb / period(ii) ;
    
    // round the upper_band up to the nearest tick
    upper_band(ii) = ceil( upper_band(ii)/tick_size ) * tick_size ;
    
    // round the lower_band down to the nearest tick
    lower_band(ii) = floor( lower_band(ii)/tick_size ) * tick_size ;
    
    mid_band(ii) = ( upper_band(ii) + lower_band(ii) ) / 2.0 ;
    
    } // end of main loop   

    retval_list(2) = mid_band ;
    retval_list(1) = lower_band ;
    retval_list(0) = upper_band ;

return retval_list ;

} // end of function

Due to formatting issues on Blogger none of the code boxes contain the required header statements, which should be:-

#include octave/oct.h
#include octave/dColVector.h
#include math.h

with the customary < before "octave" and "math" and > after ".h"