# -*- coding: utf-8 -*-
"""
Spyder Editor

This is a temporary script file.
"""

import numpy as np
import matplotlib.pyplot as plt

T=1
N=255
dt=T/N
M=1000

S0=100
K=110
mu=0.15
sigma=0.2
r=0.05

#construction mouvement brownien par somme cummulative 
dW= np.sqrt(dt)*np.random.randn(N,M)
W= np.zeros((N+1,M))
W[0,:]=0
W[1:,:]=np.cumsum(dW,axis=0)

t=np.linspace(0,T,N+1,endpoint=True)
#plot
plt.figure('brownian')
plt.plot(W)

# construction du processus de prix S: dS/S = mu*dt + sigma* dWt
S= S0*np.exp( (mu-sigma**2/2)*t[:,None] + sigma*W)
#plot
plt.plot(t,S)

S_risque_neutre= S0*np.exp( (r-sigma**2/2)*t[:,None] + sigma*W)
prix_option= np.exp(-r*T)*np.mean(
    np.maximum(S_risque_neutre[-1,:]-K,0))