python and derivatives pricing tutorial – brownian motion

In pricing models (black-scholes, local vol, libor market model) the source of uncertainty is Brownian motion. how to simulate it in python? python download

first, lets simulate dWt , we know it’s a gaussian variable with mean 0 and deviation $$ \sqrt {dt} $$
google “python gaussian” and we get to numpy gaussian
so dWt is just np.random.randn()*math.sqrt(dt)

knowing dWt how to we get to Wt at time t? $$ W_T=\int_0^T {dW_t} $$ and integral is just a sum , so $$ W_T = \sum {dW_t}$$ , so final function to calculate W_T is:

 
import numpy as np
dt=0.01
def W(T):
  sum=0
  N=T/dt
  for i in xrange(int(N)):
    sum+=np.random.randn()*np.sqrt(dt)
  return sum

now check calculate properties of brownian motion:

1) mean should be zero

 
T=1.0
nSim=10000
mean=0
for i in xrange(nSim):
  mean+=W(T)
mean/=nSim  
print mean

2) variance should be T

 
T=3.0
nSim=10000
mean=0
for i in xrange(nSim):
  wt=W(T)
  mean+=wt*wt
mean/=nSim  
print mean

now lets calculate probability $$ P( W(1)<0 and W(2) < 0) $$

nSim=100000
proba=0
for i in xrange(nSim):
  w1=W(1)
  w2=w1+W(1)
  if w1<0 and w2<0:
    proba+=1.
proba/=nSim  
print proba