1. Define parameters

K = 1.0 sigma = 0.2 T = 1 x = np.linspace(-0.5, 0.5, 100) tau = np.linspace(0.0001, T, 100) X, TAU = np.meshgrid(x, tau)

def black_scholes(x, tau):

   d_plus = (1 / (sigma * np.sqrt(tau))) * (x + 0.5 * sigma**2 * tau + 0.5 * sigma**2 * tau)
   d_minus = (1 / (sigma * np.sqrt(tau))) * (x + 0.5 * sigma**2 * tau - 0.5 * sigma**2 * tau)
   u = K * (np.exp(x + 0.5 * sigma**2 * tau) * norm.cdf(d_plus) - norm.cdf(d_minus))
   return u

fig = plt.figure(figsize=(10, 10)) ax = fig.add_subplot(projection='3d')

K = 1.0 sigma = 0.05 num_walks = 10 starting_points = [-0.2, -0.1, 0, 0.1, 0.2] num_starting_points = len(starting_points) num_points = 1000 t_values = np.linspace(1e-4, 1, num_points)

brownian_walks = np.zeros((num_starting_points, num_walks, num_points)) for i in range(num_starting_points):

   for j in range(num_walks):
       dt = 1 / num_points
       dW = np.random.normal(0, np.sqrt(dt), num_points)
       brownian_walks[i, j] = np.cumsum(dW) * sigma + starting_points[i]

for i in range(num_starting_points):

   for j in range(num_walks):
       B_t = brownian_walks[i,j]
       S_t = np.exp(B_t)-1
       S_t = S_t[::-1]
       ax.plot(T-t_values, S_t, black_scholes(S_t, t_values), color='w', alpha=0.2)

ax.plot_surface(T-TAU, X, black_scholes(X, TAU), cmap='viridis')