Equity

Local Volatility Model

class rivapy.models.LocalVol(vol_param, x_strikes: array, time_grid: array, call_prices: ndarray = None, local_vol_grid: ndarray = None)

Bases: object

Local Volatility Class

Parameters:

• vol_param – a grid or a parametrisation of the volatility

• x_strikes (np.array) – strikes

• time_grid (np.array) – time_grid

• call_param (np.ndarray, optional) – A grid of call prices. Not compatible with vol_param. Defaults to None.

apply_mc_step(x: ndarray, t0: float, t1: float, rnd: ndarray, inplace: bool = True)

Apply a MC-Euler step for the LV Model for n different paths.

Parameters:

• x (np.ndarray) – 2-d array containing the start values for the spot and variance. The first column contains the spot, the second the variance values.

• t0 ([type]) – [description]

• t1 ([type]) – [description]

• rnd ([type]) – [description]

static compute_local_var(vol_param, x_strikes: array, time_grid: array, call_param: ndarray = None, min_lv=0.01, max_lv=1.5)

Calculate the local variance from vol_param or call_param for x_strikes on time_grid

Parameters:

• vol_param – a grid or a parametrisation of the volatility

• x_strikes (np.array) – strikes

• time_grid (np.array) – time_grid

• call_param (np.ndarray, optional) – A grid of call prices. Not compatible with vol_param. Defaults to None.

Returns:

local volatility surface on the grid

Heston Model

class rivapy.models.HestonModel(long_run_variance, mean_reversion_speed, vol_of_vol, initial_variance, correlation)

Bases: object

_summary_

Parameters:

• long_run_variance (_type_) – _description_

• mean_reversion_speed (_type_) – _description_

• vol_of_vol (_type_) – _description_

• initial_variance (_type_) – _description_

• correlation (_type_) – _description_

apply_mc_step(x: ndarray, t0: float, t1: float, rnd: ndarray, inplace: bool = True, slv: ndarray = None)

Apply a MC-Euler step for the Heston Model for n different paths.

Parameters:

• x (np.ndarray) – 2-d array containing the start values for the spot and variance. The first column contains the spot, the second the variance values.

• t0 (float) – The current time.

• t1 (float) – The next timestep to be computed.

• rnd (np.ndarray) – Two-dimensional array of shape (n_sims,2) containing the normal random numbers. Each row of the array is used to compute the correlated random numbers for the respective simulation.

• slv (np.ndarray) – Stochastic local variance (for each path) to be multiplied with the heston variance. This is used by the StochasticVolatilityModel and can be ignored.

call_price(s0: float, v0: float, K: ndarray | float, ttm: ndarray | float) → ndarray | float

Computes a call price for the Heston model via integration over characteristic function.

Parameters:

• s0 (float) – current spot

• v0 (float) – current variance

• K (float) – strike

• ttm (float) – time to maturity

feller_condition()

Return True if the model parameter fulfill the Feller condition ..:

Returns:

True->Feller condition is fullfilled

Return type:

bool

get_initial_value() → ndarray

Return the initial value (x0, v0)

Returns:

Initial value.

Return type:

np.ndarray

Scott Chesney

Scott Chesney Model is a stochastic volatility model of the form

\[dS = e^y S dW_S\]

\[dy = \kappa (\theta-y)dt \alpha dW_y\]

\[E[dW_S\dot dW_y] = \rho dt\]

class rivapy.models.ScottChesneyModel(kappa: float, theta: float, alpha: float, correlation: float, y0: float)

Bases: object

Scott-Chesney Model Generates a timeseries according to

\[dS = e^y S dW_S\]

\[dy = `{\kappa}` (`{ heta}`-y)dt `{lpha}` dW_y\]

\[E[dW_s\dot dW_y] = \rho dt\]

Parameters:

• kappa (float) – speed of mean reversion

• theta (float) – mean reversion level

• alpha (float) – vol of (log)vol

• correlation (float) – correlation between (log)vol and spot

• y0 – start value (float): (log) vol

apply_mc_step(x: ndarray, t0: float, t1: float, rnd: ndarray, inplace: bool = True, slv: ndarray = None)

Apply a MC-Euler step for the Scott-Chesney Model for n different paths.

Parameters:

• x (np.ndarray) – 2-d array containing the start values for the spot and variance. The first column contains the spot, the second the variance values.

• t0 ([type]) – [description]

• t1 ([type]) – [description]

• rnd ([type]) – [description]

• slv (np.ndarray) – Stochastic local variance (for each path) to be multiplied with the heston variance. This is used by the StochasticVolatilityModel.

Stochastic Local Volatility Model

class rivapy.models.StochasticLocalVol(stoch_vol_model)

Bases: object

Stochastic Local Volatility model

Parameters:

stochastic_vol_model (StochasticVolModel) – The underlying stochastic vol model

apply_mc_step(x: ndarray, t0: float, t1: float, rnd: ndarray, inplace: bool = True)

Apply a MC-Euler step for the Heston Local Vol Model for n different paths.

Parameters:

• x (np.ndarray) – 2-d array containing the start values for the spot and variance. The first column contains the spot, the second the variance values.

• t0 ([type]) – [description]

• t1 ([type]) – [description]

• rnd ([type]) – [description]

calibrate_MC(vol_param, x_strikes: ndarray, time_grid: ndarray, n_sims, local_var: ndarray = None, call_prices: ndarray = None)

Calibrate the Heston Local Volatility Model using kernel regression.

This method calibrates the local volatility part of the Heston Model given a volatility parametrization so that the respective implied volatilities from the given vol parametrization are reproduced by the Heston Local Volatility model. The calibration is based on kernel regression as described in Applied Machine Learning for Stochastic Local Volatility Calibration.

Parameters:

• vol_param ([type]) – [description]

• x_strikes (np.array) – [description]

• time_grid (np.array) – [description]

• n_sims ([type]) – [description]

• local_var (np.ndarray, optional) – [description]. Defaults to None.

• call_prices (np.ndarray, optional) – Defaults to None.

get_initial_value() → ndarray

Return the initial value (x0, v0)

Returns:

Initial value.

Return type:

np.ndarray