tf_quant_finance.experimental.svi.calibration

Last updated: 2023-03-16.

tf_quant_finance.experimental.svi.calibration#

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Calibrates the SVI model parameters for a batch of volatility skews.

tf_quant_finance.experimental.svi.calibration(
    *, forwards, expiries, strikes, volatilities, weights=None,
    initial_position=None, optimizer_fn=None, tolerance=1e-06, x_tolerance=0,
    f_relative_tolerance=0, maximum_iterations=100, dtype=None, name=None
)

This function optimizes the SVI model parameters to fit the given volatilities at various strikes. The loss function is the L2 norm of the differences in the volatility space.

Each volatility skew in the batch corresponds to a fixed expiry for options on some underlying assets. Optimization is done independently for each skew.

TODO(b/189458981): add flexibility to accept higher rank tensors as inputs.

Example#

This example shows how to calibrate a single skew, loosely based on market prices for GOOG210820C* (GOOG calls with 2021-08-20 expiry) as of 2021-05-27. https://finance.yahoo.com/quote/GOOG/options?p=GOOG&date=1629417600

import numpy as np
import tensorflow as tf
import tf_quant_finance as tff

forwards = np.array([2402.])
expiries = np.array([0.23])
strikes = np.array([[
    1700., 1800., 1900., 2000., 2050., 2100., 2200., 2250., 2350., 2400.,
    2450., 2500., 2550., 2600., 2650., 2700., 2750., 2800., 2850., 2900.,
    2950., 3000.
]])
volatilities = np.array([[
    0.5335, 0.4882, 0.4389, 0.3937, 0.3749, 0.3569, 0.3259, 0.3135, 0.29,
    0.283, 0.2717, 0.2667, 0.2592, 0.2566, 0.2564, 0.2574, 0.2595, 0.2621,
    0.2669, 0.2732, 0.2826, 0.2967
]])

(svi_params, converged, _) = tff.experimental.svi.calibration(
    forwards=forwards,
    expiries=expiries,
    strikes=strikes,
    volatilities=volatilities)

# Expected results are tensors containing (up to numerical tolerance):
# svi_params: [[-0.2978, 0.4212, 0.0415, 0.1282, 0.7436]]
# converged: [True]

Args:#

  • forwards: A rank 1 real Tensor of shape [batch_size]. The forward prices of the underlyig asset for each skew in the batch.

  • expiries: A rank 1 real Tensor of shape [batch_size]. The option expiries for each skew in the batch.

  • strikes: A rank 2 real Tensor of shape [batch_size, num_strikes]. The strike prices of the options.

  • volatilities: A rank 2 real Tensor of shape [batch_size, num_strikes]. The market implied Black-Scholes volatilities to calibrate.

  • weights: An optional rank 2 real Tensor of shape [batch_size, num_strikes]. Used to define the loss function as the weighted L2 norm of the residuals. Default value: None, in which case weights are set to 1.

  • initial_position: A rank 2 real Tensor of shape [batch_size, 5]. Raw SVI parameter tuples (a, b, rho, m, sigma) to be used as the initial values for the optimization. Default value: None, in which case the initial values are estimated heuristically and may lead to slower convergence.

  • optimizer_fn: Optional Python callable which implements the algorithm used to minimize the objective function during calibration. It should have the following interface: result = optimizer_fn(value_and_gradients_function, initial_position, tolerance, max_iterations) value_and_gradients_function is a Python callable that accepts a point as a real Tensor and returns a tuple of Tensors of real dtype containing the value of the function and its gradient at that point. ‘initial_position’ is a real Tensor containing the starting point of the optimization, ‘tolerance’ is a real scalar Tensor for stopping tolerance for the procedure and max_iterations specifies the maximum number of iterations. optimizer_fn should return a namedtuple containing the items: position (a tensor containing the optimal value), converged (a boolean indicating whether the optimize converged according the specified criteria), failed (a boolean indicating if the optimization resulted in a failure), num_iterations (the number of iterations used), and objective_value ( the value of the objective function at the optimal value). The default value for optimizer_fn is None and conjugate gradient algorithm is used.

  • tolerance: Scalar Tensor of real dtype. Specifies the gradient tolerance for the procedure. If the supremum norm of the gradient vector is below this number, the algorithm is stopped. Default value: 1e-6.

  • x_tolerance: Scalar Tensor of real dtype. If the absolute change in the position between one iteration and the next is smaller than this number, the algorithm is stopped. Default value: 1e-6.

  • f_relative_tolerance: Scalar Tensor of real dtype. If the relative change in the objective value between one iteration and the next is smaller than this value, the algorithm is stopped.

  • maximum_iterations: Scalar positive int32 Tensor. The maximum number of iterations during the optimization. Default value: 200.

  • dtype: The default dtype to use when converting values to Tensors. Default value: None, uses the default dtypes inferred by TensorFlow.

  • name: Python string. The name to give to the ops created by this function. Default value: None, maps to the default name svi_skew_calibration.

Returns:#

A Tuple of three elements: (parameters, status, iterations)

  • parameters: a tensor of shape [batch_size, 5] representing raw parameters for the SVI model calibrated with given input Black-Scholes volatilities.

  • status: boolean, whether the optimization algorithm succeeded in finding the optimal point based on the specified convergance criteria.

  • iterations: the number of iterations performed during the optimization.

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