Where the Smile Comes From#

If Black and Scholes were right, every option on a stock would imply the same volatility. Markets disagree: implied volatility is higher for low strikes than for high ones, a skew, and higher at both ends than in the middle, a smile. This tutorial builds a skew from a model in financialkit.stochastic_volatility, reads it back through the implied-volatility tools in financialkit.options, recovers the risk-neutral distribution it encodes, and fits the parametrization traders actually quote.

1. Implied volatility inverts the formula#

Implied volatility is the volatility that makes Black-Scholes reproduce a price. Run a Black-Scholes price back through it and the input comes back, at every strike:

import numpy as np
import financialkit as fk

S, T, r = 100.0, 0.5, 0.03
K = np.linspace(70.0, 130.0, 13)
flat = fk.options.black_scholes_price(S, K, T, r, 0.2)
print(np.round(fk.options.implied_volatility(flat, S, K, T, r), 6))
# [0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2]

2. Stochastic volatility skews the smile#

In Heston’s model the variance follows its own mean-reverting process, correlated with the stock. A negative correlation, which is what equity markets show, means volatility rises when the stock falls. Price options under Heston by Fourier inversion, then ask what volatility Black-Scholes would need:

heston = fk.stochastic_volatility.HestonModel(v0=0.04, kappa=2.0, theta=0.04, xi=0.6, rho=-0.7)
prices = fk.stochastic_volatility.heston_price(heston, S, K, T, r)
smile = fk.options.implied_vol_smile(prices, K, S, T, r)
for k, v in zip(K[::3], smile.vols[::3]):
    print(f"K = {k:5.1f}: implied vol {100 * v:.2f}%")
# K =  70.0: implied vol 29.39%
# K =  85.0: implied vol 24.06%
# K = 100.0: implied vol 18.39%
# K = 115.0: implied vol 14.22%
# K = 130.0: implied vol 14.28%

The long-run volatility is 20% everywhere, yet a 70 strike implies almost 30% and a 115 strike barely 14%; at 130 the curve turns up again, so the skew carries a smile. Before trusting a Fourier price, check it independently: simulate Heston paths with Andersen’s QE scheme and price the at-the-money call by Monte Carlo:

mc = fk.stochastic_volatility.heston_mc_price(heston, S, 100.0, T, r, n_steps=100, n_paths=40_000, rng=5)
print(f"Fourier: {prices[6]:.4f}, Monte Carlo: {mc.estimate:.4f} +/- {mc.std_error:.4f}")
# Fourier: 5.9237, Monte Carlo: 5.9039 +/- 0.0339

3. The distribution behind the prices#

Breeden and Litzenberger showed that the second derivative of call prices in the strike is the discounted risk-neutral density. Differencing a fine strip of Heston prices recovers it:

from scipy.integrate import trapezoid

strikes = np.linspace(40.0, 200.0, 801)
density = fk.options.breeden_litzenberger_density(strikes, fk.stochastic_volatility.heston_price(heston, S, strikes, T, r), T, r)
x, p = density.strikes, density.density
mean = trapezoid(x * p, x)
sd = np.sqrt(trapezoid((x - mean) ** 2 * p, x))
print(f"total probability {trapezoid(p, x):.4f}")
# total probability 0.9994
print(f"mean {mean:.2f} against the forward {S * np.exp(r * T):.2f}")
# mean 101.49 against the forward 101.51
print(f"skewness {trapezoid((x - mean) ** 3 * p, x) / sd**3:.2f}")
# skewness -0.89

The density integrates to one and is centred on the forward, as the absence of arbitrage requires. Its negative skewness is the skew: a fat left tail makes low-strike puts expensive, so their implied volatility is high. A lognormal with the same volatility would have positive skewness.

4. Jumps smile too#

Stochastic volatility is not the only explanation. Merton added occasional jumps to the stock, and downward jumps also fatten the left tail:

merton = fk.stochastic_volatility.MertonJumpModel(sigma=0.15, lam=0.5, mu_j=-0.1, delta=0.1)
print(np.round(100 * merton.implied_vol(S, K[::3], T, r), 2))
# [22.5  19.5  17.44 16.6  16.33]

The two models leave different fingerprints across maturities, though. Jumps make the smile steep for short expiries and flat for long ones, while stochastic volatility does the opposite, which is why the Bates model combines them.

5. What traders quote: SABR#

Desks rarely quote a full model. They fit a smile parametrization, and Hagan’s SABR formula is the standard in rates and FX. Calibrating it to the Heston smile above:

forward = S * np.exp(r * T)
sabr = fk.stochastic_volatility.sabr_calibrate(forward, K, T, smile.vols, beta=0.5)
print(f"rho {sabr.rho:.3f}, nu {sabr.nu:.3f}, RMSE {1e4 * sabr.rmse:.0f} bp of vol")
# rho -0.646, nu 0.824, RMSE 35 bp of vol

SABR’s correlation rho and vol of vol nu play the same roles as Heston’s rho and xi, and the fit is close: three parameters summarise the whole smile.