fat_tails_and_their_impact_on_option_prices1

ee929a79-ba6d-4fdc-bb0a-d222eb041759.py

# define normal pdf
def norm_pdf(x, sigma=1):
    return (1 / np.sqrt(2*np.pi)) * (1 / sigma) * np.exp(-0.5 * (x/sigma)**2)

# define x's we want to compute pdf for
xs = np.linspace(-4, 4, 8*200 + 1)
# define epsilsons
eps = [0, 0.1, 0.2, 0.5]
# create pdfs
perturbs = []
for e in eps[::-1]:
    p = [0.5 * (norm_pdf(x, 1 + e) + norm_pdf(x, 1 - e)) for x in xs]
    perturbs.append([xs, p])

# plot
fig, ax = plt.subplots(figsize=(25,10))

for dist, e in zip(perturbs, eps[::-1]):
    ax.plot(dist[0], dist[1], label="e = {}".format(e))

ax.set_title("PDFs when averaging normal distributions", fontsize=24)
ax.set_xlabel("x", fontsize=20)
ax.set_ylabel("Density", fontsize=20)
ax.legend(fontsize=14);