This document contains Python code to simulate call option prices using Monte Carlo methods. It generates random trajectories for underlying asset prices, calculates the payoffs at expiration for each trajectory, discounts the payoffs back to the present, and takes the average to estimate the call option price. Key steps include generating random variables, applying the Black-Scholes formula to the trajectories over a time period, and computing the average discounted payoff.

1. from math import sqrt
from random import gauss
from numpy import ones, random,exp,average,maximum, zeros,max
Traj=1000000
S=100*ones( (1,Traj) )
vol=(random.normal(0,1,Traj))
v = 0.10
r = 0.001
T = 0.5
K=105*ones( (1,Traj) )
St=S* exp((r - 0.5 * v**2) * T + v * sqrt(T) * vol)
payoff=maximum(St-K,zeros( (1,Traj) ))
callprice=payoff*exp(-r*T)
print(average(callprice))