`skfolio.moments`.ImpliedCovariance#

class skfolio.moments.ImpliedCovariance(prior_covariance_estimator=None, annualized_factor=252.0, window_size=20, linear_regressor=None, volatility_risk_premium_adj=None, nearest=True, higham=False, higham_max_iteration=100)[source]#

Implied Covariance estimator.

For each asset, the implied volatility time series is used to estimate the realised volatility using the non-overlapping log-transformed OLS model [6]:

\[\ln(RV_{t}) = \alpha + \beta_{1} \ln(IV_{t-1}) + \beta_{2} \ln(RV_{t-1}) + \epsilon\]

with $\alpha$, $\beta_{1}$ and $\beta_{2}$ the intercept and coefficients to estimate, $RV$ the realised volatility, and $IV$ the implied volatility. The training set uses non-overlapping data of sample size window_size to avoid possible regression errors caused by auto-correlation. The logarithmic transformation of volatilities is used for its better finite sample properties and distribution, which is closer to normality, less skewed and leptokurtic [6].

Alternatively, if volatility_risk_premium_adj is provided, the realised volatility is estimated using:

\[RV_{t} = \frac{IV_{t-1}}{VRPA}\]

with $VRPA$ the volatility risk premium adjustment.

The final step is the reconstruction of the covariance matrix from the correlation and estimated realised volatilities $D$:

\[\Sigma = D \ Corr \ D\]

With $Corr$, the correlation matrix computed from the prior covariance estimator. The default is the EmpiricalCovariance. It can be changed to any covariance estimator using prior_covariance_estimator.

Parameters:

prior_covariance_estimatorBaseCovariance, optional

Covariance estimator to estimate the covariance matrix used for the correlation estimates prior the volatilities update. The default (None) is to use EmpiricalCovariance.

annualized_factorfloat, default=252

Annualized factor (AF) used to covert the implied volatilities into the same frequency as the returns using $\frac{IV}{\sqrt{AF}}$. The default is 252 which corresponds to daily returns and implied volatility expressed in p.a.

window_sizeint, default=20

Window size used to construct the non-overlapping training set of realised volatilities and implied volatilities used in the regression. The default is 20 observations.

linear_regressorBaseEstimator, optional

Estimator of the linear regression used to estimate the realised volatilities from the implied volatilities. The default is to use the scikit-learn OLS estimator LinearRegression.

volatility_risk_premium_adjfloat | dict[str, float] | array-like of shape (n_assets, ), optional

If provided, instead of using the regression model, the realised volatilities are estimated using:

\[RV_{t} = \frac{IV_{t-1}}{VRPA}\]

with $VRPA$ the volatility risk premium adjustment.

If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset $VRPA$) and the input X of the fit method must be a DataFrame with the assets names in columns.

nearestbool, default=True

If this is set to True, the covariance is replaced by the nearest covariance matrix that is positive definite and with a Cholesky decomposition than can be computed. The variance is left unchanged. A covariance matrix that is not positive definite often occurs in high dimensional problems. It can be due to multicollinearity, floating-point inaccuracies, or when the number of observations is smaller than the number of assets. For more details, see cov_nearest. The default is True.

highambool, default=False

If this is set to True, the Higham & Nick (2002) algorithm is used to find the nearest PD covariance, otherwise the eigenvalues are clipped to a threshold above zeros (1e-13). The default is False and use the clipping method as the Higham & Nick algorithm can be slow for large datasets.

higham_max_iterationint, default=100

Maximum number of iteration of the Higham & Nick (2002) algorithm. The default value is 100.

Attributes:

covariance_ndarray of shape (n_assets, n_assets): Estimated covariance matrix.
prior_covariance_estimator_BaseEstimator: Fitted prior covariance estimator.
pred_realised_vols_ndarray of shape (n_assets,): The predicted realised volatilities
linear_regressors_list[BaseEstimator]: The fitted linear regressions.
coefs_ndarray of shape (n_assets, 2): The coefficients of the log transformed regression model for each asset.
intercepts_ndarray of shape (n_assets,): The intercepts of the log transformed regression model for each asset.
n_features_in_int: Number of assets seen during fit.
feature_names_in_ndarray of shape (n_features_in_,): Names of assets seen during fit. Defined only when returns has assets names that are all strings.

References

[1]

“New evidence on the implied-realized volatility relation”. Christensen & Hansen (2002).

[2]

“The relation between implied and realized volatility”. Christensen & Prabhala (2002).

[3]

“Can implied volatility predict returns on the carry trade?”. Egbers & Swinkels (2015).

[4]

“Volatility and correlation forecasting”. Egbers & Swinkels (2015).

[5]

“Volatility and correlation forecasting”. Andersen, Bollerslev, Christoffersen & Diebol (2006).

[6] (1,2)

“How Well Does Implied Volatility Predict Future Stock Index Returns and Volatility? : A Study of Option-Implied Volatility Derived from OMXS30 Index Options”. Sara Vikberg & Julia Björkman (2020).

Methods

`fit`(X[, y, implied_vol])	Fit the implied covariance estimator.
`get_metadata_routing`()	Get metadata routing of this object.
`get_params`([deep])	Get parameters for this estimator.
`set_fit_request`(*[, implied_vol])	Request metadata passed to the `fit` method.
`set_params`(**params)	Set the parameters of this estimator.

fit(X, y=None, implied_vol=None, **fit_params)[source]#

Fit the implied covariance estimator.

Parameters:

Xarray-like of shape (n_observations, n_assets): Price returns of the assets.
yIgnored: Not used, present for API consistency by convention.
implied_volarray-like of shape (n_observations, n_assets): Implied volatilities of the assets.
**fit_paramsdict: Parameters to pass to the underlying estimators. Only available if enable_metadata_routing=True, which can be set by using sklearn.set_config(enable_metadata_routing=True). See Metadata Routing User Guide for more details.

Returns:

selfImpliedCovariance: Fitted estimator.

get_metadata_routing()[source]#

Get metadata routing of this object.

Please check User Guide on how the routing mechanism works.

Returns:

routingMetadataRequest: A MetadataRequest encapsulating routing information.

get_params(deep=True)#

Get parameters for this estimator.

Parameters:

deepbool, default=True: If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns:

paramsdict: Parameter names mapped to their values.

set_fit_request(*, implied_vol='$UNCHANGED$')#

Request metadata passed to the fit method.

Note that this method is only relevant if enable_metadata_routing=True (see sklearn.set_config). Please see User Guide on how the routing mechanism works.

The options for each parameter are:

True: metadata is requested, and passed to fit if provided. The request is ignored if metadata is not provided.
False: metadata is not requested and the meta-estimator will not pass it to fit.
None: metadata is not requested, and the meta-estimator will raise an error if the user provides it.
str: metadata should be passed to the meta-estimator with this given alias instead of the original name.

The default (sklearn.utils.metadata_routing.UNCHANGED) retains the existing request. This allows you to change the request for some parameters and not others.

Added in version 1.3.

Note

This method is only relevant if this estimator is used as a sub-estimator of a meta-estimator, e.g. used inside a Pipeline. Otherwise it has no effect.

Parameters:

implied_volstr, True, False, or None, default=sklearn.utils.metadata_routing.UNCHANGED: Metadata routing for implied_vol parameter in fit.

Returns:

selfobject: The updated object.

set_params(**params)#

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as Pipeline). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters:

**paramsdict: Estimator parameters.

Returns:

selfestimator instance: Estimator instance.

skfolio.moments.ImpliedCovariance#

`skfolio.moments`.ImpliedCovariance#