Heston

The Heston Model is a popular choice for pricing equity and FX options, where capturing the dynamics of volatility is important. It is a type of stochastic volatility model, which assumes that volatility is not constant but instead follows a random process. The Heston model has a semi-closed form solution for European options, enabling fast calibration of its parameters using market option prices.

Model Dynamics

In the Heston model the lognormal stock process \(X_t\) is given by,

\[ dX_t = (\mu - \frac{\nu_t}{2}) dt + \sqrt \nu_t dW_s \]

and the variance follows the process $$ d \nu_t = \kappa (\theta - \nu_t) dt + \xi \sqrt \nu_tdW_t $$

where \(dW_s\) and \(dW_t\) are Wiener processes with correlation \(\rho\).

Dataset

The model specific component in the dataset (HESTON) is a dict with the following parameters:

• ASSET: the name of the asset
• INITIAL_VAR: the initial variance \(\nu_0\)
• LONG_VAR: the long variance \(\theta\)
• VOL_OF_VOL: the volatility of the volatility \(\xi\)
• MEANREV: the mean reversion rate \(\kappa\)
• CORRELATION: the correlation \(\rho\)

Example

from finmc.models.heston import HestonMC

heston_params = {
    "ASSET": "SPX",
    "INITIAL_VAR": 0.015,
    "LONG_VAR": 0.052,
    "VOL_OF_VOL": 0.88,
    "MEANREV": 2.78,
    "CORRELATION": -0.85,
}

dataset = {
    "MC": {"PATHS": 100_000, "TIMESTEP": 1 / 250},
    "BASE": "USD",
    "ASSETS": {
        "USD": ("ZERO_RATES", np.array([[2.0, 0.05]])),
        "SPX": ("FORWARD", np.array([[0.0, 5500], [1.0, 5600]])),
    },
    "HESTON": heston_params
}
model = HestonMC(dataset)
model.advance(1.0)
spots = model.get_value("SPX")

See complete example here