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Fit Jump Parameters#

Estimates jump parameters and diffusion volatility from historical returns.

This function separates historical log returns into a "diffusion" component (normal daily movements) and a "jump" component (extreme movements) based on a standard deviation threshold. It then calculates the annualized parameters for a jump-diffusion model like Merton's.

Parameters:

Name Type Description Default
log_returns Series

A pandas Series of daily log returns.

 required
threshold_stds float

The number of standard deviations to use as a threshold for identifying jumps, by default 3.0.

 3.0

Returns:

Type Description
dict

A dictionary containing the estimated parameters: 'sigma', 'lambda', 'mu_j', and 'sigma_j'.
Source code in src/quantfin/calibration/fit_jump_parameters.py 
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def fit_jump_params_from_history(
    log_returns: pd.Series,
    threshold_stds: float = 3.0,
) -> dict:
    """
    Estimates jump parameters and diffusion volatility from historical returns.

    This function separates historical log returns into a "diffusion" component
    (normal daily movements) and a "jump" component (extreme movements) based
    on a standard deviation threshold. It then calculates the annualized
    parameters for a jump-diffusion model like Merton's.

    Parameters
    ----------
    log_returns : pd.Series
        A pandas Series of daily log returns.
    threshold_stds : float, optional
        The number of standard deviations to use as a threshold for identifying
        jumps, by default 3.0.

    Returns
    -------
    dict
        A dictionary containing the estimated parameters: 'sigma', 'lambda',
        'mu_j', and 'sigma_j'.
    """
    print("Fitting jump parameters from historical returns...")

    std_dev = log_returns.std()
    jump_threshold = threshold_stds * std_dev

    diffusion_returns = log_returns[abs(log_returns) < jump_threshold]
    jump_returns = log_returns[abs(log_returns) >= jump_threshold]

    # Annualize daily std dev by multiplying by sqrt(252 trading days)
    sigma_est = diffusion_returns.std() * np.sqrt(252)

    if len(jump_returns) > 2:
        lambda_est = len(jump_returns) / len(log_returns) * 252
        mu_j_est = jump_returns.mean()
        sigma_j_est = jump_returns.std()
    else:
        lambda_est, mu_j_est, sigma_j_est = 0.1, 0.0, 0.0
        print("  -> Warning: Not enough jumps detected. Using default jump parameters.")

    # Use the key 'sigma' to match the parameter name in the Merton/Kou models.
    fitted_params = {
        "sigma": sigma_est,
        "lambda": lambda_est,
        "mu_j": mu_j_est,
        "sigma_j": sigma_j_est,
    }

    formatted = ", ".join(f"{k}: {v:.4f}" for k, v in fitted_params.items())
    print(f"  -> Estimated Historical Params: {{{formatted}}}")
    return fitted_params