People who are not knowledgeable in mathematics like to think that a definition of trend is as straightforward as it gets.
For example I skimmed through one of the major popular books on trendfollowing and it is full of babble about trends.
The problem is that trends are a very perplexing thing. All the confusion arise from the disregard of the random nature of the process.
One should start any observation with the statement: asset prices are a random process. That is to say that one part of the outcome will always be unknown. It doesn't mean that everything will be unknown. The standard recipe for random proceccess is the brownian motion.
The following picture shows a simulation of a brownian motion with mu = 1 and sigma = 0.3 and sigma = 0.9. Both markets trend, which simply means that mu is not zero. However the ratio of mu to sigma is a measure for strength of the trends. Trends are
not binary, they have a spektrum. In fact they are multidimensional: they can be meanreverting, autocorrelating, correlating, cointegrating, etc. However often sophisticated stochastic calculus misses some very basic things.
Chan writes in his excellent book:
For example I skimmed through one of the major popular books on trendfollowing and it is full of babble about trends.
The problem is that trends are a very perplexing thing. All the confusion arise from the disregard of the random nature of the process.
One should start any observation with the statement: asset prices are a random process. That is to say that one part of the outcome will always be unknown. It doesn't mean that everything will be unknown. The standard recipe for random proceccess is the brownian motion.
The following picture shows a simulation of a brownian motion with mu = 1 and sigma = 0.3 and sigma = 0.9. Both markets trend, which simply means that mu is not zero. However the ratio of mu to sigma is a measure for strength of the trends. Trends are
not binary, they have a spektrum. In fact they are multidimensional: they can be meanreverting, autocorrelating, correlating, cointegrating, etc. However often sophisticated stochastic calculus misses some very basic things.
Chan writes in his excellent book:
Code:
'''
N_sim: number of simulations
T: horizon
dt: step length in years
sigma: volatility per year
mu: drift terms (moving average or long-term mean for stock returns)
S0: initial stock price
'''
from numpy.random import standard_normal
from numpy import array, zeros, sqrt, shape
from pylab import *
S0 = 10.222
T =1
dt =0.0002
#sigma = 0.4
#mu = 4 #1
N_Sim = 5
Steps=round(T/dt); #Steps in years
def plot_brown(sigma,mu):
title('Simulations %d Steps %d Sigma %.6f Mu %.6f S0 %.6f' % (int(N_Sim), int(Steps), sigma, mu, S0))
S = zeros([N_Sim, Steps], dtype=float)
x = range(0, int(Steps), 1)
for j in range(0, N_Sim, 1):
S[j,0]= S0
for i in x[:-1]:
S[j,i+1]=S[j,i]+S[j,i]*(mu-0.5*pow(sigma,2))*dt+sigma*S[j,i]*sqrt(dt)*standard_normal();
plot(x, S[j])
if __name__=='__main__':
subplot(211)
plot_brown(0.3,1)
subplot(212)
plot_brown(0.9,1)
xlabel('steps')
ylabel('stock price')
show()