Bates: jumps and stochastic volatility

David Bates (1996), studying Deutsche mark options, found that neither jumps nor stochastic volatility alone fit the smile at every maturity. Diffusive stochastic volatility needs time to build up a skew, so Heston’s short-dated smiles are too flat; jumps act at once but their effect washes out with maturity. Putting Merton’s lognormal jumps on top of Heston gives both: the characteristic function is simply the product of the two.

phi_Bates = phi_Heston * phi_jumps: True

import matplotlib.pyplot as plt
import numpy as np

from financialkit.stochastic_volatility import BatesModel, HestonModel, MertonJumpModel
from financialkit.stochastic_volatility.visualizers import plot_smile

S, r = 100.0, 0.02
heston = HestonModel(0.03, 2.0, 0.03, 0.4, -0.6)
bates = BatesModel(0.03, 2.0, 0.03, 0.4, -0.6, lam=0.3, mu_j=-0.12, delta=0.08)

u = np.linspace(-10, 10, 5)
print(
    "phi_Bates = phi_Heston * phi_jumps:",
    np.allclose(bates.char_func(u, 1.0), heston.char_func(u, 1.0) * MertonJumpModel(0.0, 0.3, -0.12, 0.08).char_func(u, 1.0)),
)

fig, axes = plt.subplots(1, 3, figsize=(12, 3.8), sharey=True)
for ax, T in zip(axes, (0.08, 0.5, 3.0), strict=True):
    F = S * np.exp(r * T)
    K = F * np.exp(np.linspace(-0.3, 0.2, 21) * np.sqrt(max(T, 0.25)))
    plot_smile(K, heston.implied_vol(S, K, T, r), forward=F, ax=ax, label="Heston")
    plot_smile(K, bates.implied_vol(S, K, T, r), forward=F, ax=ax, label="Bates")
    ax.set_title(f"T = {T}")
fig.tight_layout()

plt.show()