tf_quant_finance.models.hull_white.calibration_from_swaptions

Last updated: 2023-03-16.

tf_quant_finance.models.hull_white.calibration_from_swaptions#

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Calibrates the Hull-White model using European Swaptions.

tf_quant_finance.models.hull_white.calibration_from_swaptions(
    *, prices, expiries, floating_leg_start_times, floating_leg_end_times,
    fixed_leg_payment_times, floating_leg_daycount_fractions,
    fixed_leg_daycount_fractions, fixed_leg_coupon, reference_rate_fn,
    mean_reversion, volatility, notional=None, is_payer_swaption=True,
    use_analytic_pricing=True, num_samples=1, random_type=None, seed=None, skip=0,
    time_step=None, volatility_based_calibration=True, optimizer_fn=None,
    mean_reversion_lower_bound=0.001, mean_reversion_upper_bound=0.5,
    volatility_lower_bound=1e-05, volatility_upper_bound=0.1, tolerance=1e-06,
    maximum_iterations=50, dtype=None, name=None
)

This function estimates the mean-reversion rate and volatility parameters of a Hull-White 1-factor model using a set of European swaption prices as the target. The calibration is performed using least-squares optimization where the loss function minimizes the squared error between the target swaption prices and the model implied swaption prices.

Example#

The example shows how to calibrate a Hull-White model with constant mean reversion rate and constant volatility.

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

dtype = tf.float64

mean_reversion = [0.03]
volatility = [0.01]
expiries = np.array(
    [0.5, 0.5, 1.0, 1.0, 2.0, 2.0, 3.0, 3.0, 4.0, 4.0, 5.0, 5.0, 10., 10.])
float_leg_start_times = np.array([
    [0.5, 1.0, 1.5, 2.0, 2.5, 2.5, 2.5, 2.5, 2.5, 2.5],  # 6M x 2Y  swap
    [0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0],  # 6M x 5Y  swap
    [1.0, 1.5, 2.0, 2.5, 3.0, 3.0, 3.0, 3.0, 3.0, 3.0],  # 1Y x 2Y  swap
    [1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5],  # 1Y x 5Y  swap
    [2.0, 2.5, 3.0, 3.5, 4.0, 4.0, 4.0, 4.0, 4.0, 4.0],  # 2Y x 2Y  swap
    [2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5],  # 2Y x 5Y  swap
    [3.0, 3.5, 4.0, 4.5, 5.0, 5.0, 5.0, 5.0, 5.0, 5.0],  # 3Y x 2Y  swap
    [3.0, 3.5, 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5],  # 3Y x 5Y  swap
    [4.0, 4.5, 5.0, 5.5, 6.0, 6.0, 6.0, 6.0, 6.0, 6.0],  # 4Y x 2Y  swap
    [4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5],  # 4Y x 5Y  swap
    [5.0, 5.5, 6.0, 6.5, 7.0, 7.0, 7.0, 7.0, 7.0, 7.0],  # 5Y x 2Y  swap
    [5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, 9.5],  # 5Y x 5Y  swap
    [10.0, 10.5, 11.0, 11.5, 12.0, 12.0, 12.0, 12.0, 12.0,
     12.0],  # 10Y x 2Y  swap
    [10.0, 10.5, 11.0, 11.5, 12.0, 12.5, 13.0, 13.5, 14.0,
     14.5]  # 10Y x 5Y  swap
])
float_leg_end_times = float_leg_start_times + 0.5
max_maturities = np.array(
    [2.5, 5.5, 3.0, 6.0, 4., 7., 5., 8., 6., 9., 7., 10., 12., 15.])
for i in range(float_leg_end_times.shape[0]):
  float_leg_end_times[i] = np.clip(
      float_leg_end_times[i], 0.0, max_maturities[i])

fixed_leg_payment_times = float_leg_end_times
float_leg_daycount_fractions = (
    float_leg_end_times - float_leg_start_times)
fixed_leg_daycount_fractions = float_leg_daycount_fractions
fixed_leg_coupon = 0.01 * np.ones_like(fixed_leg_payment_times)

zero_rate_fn = lambda x: 0.01 * tf.ones_like(x, dtype=dtype)
prices = tff.models.hull_white.swaption_price(
    expiries=expiries,
    floating_leg_start_times=float_leg_start_times,
    floating_leg_end_times=float_leg_end_times,
    fixed_leg_payment_times=fixed_leg_payment_times,
    floating_leg_daycount_fractions=float_leg_daycount_fractions,
    fixed_leg_daycount_fractions=fixed_leg_daycount_fractions,
    fixed_leg_coupon=fixed_leg_coupon,
    reference_rate_fn=zero_rate_fn,
    notional=100.,
    dim=1,
    mean_reversion=mean_reversion,
    volatility=volatility,
    use_analytic_pricing=True,
    dtype=dtype)

calibrated_parameters = tff.models.hull_white.calibration_from_swaptions(
    prices=prices[:, 0],
    expiries=expiries,
    floating_leg_start_times=float_leg_start_times,
    floating_leg_end_times=float_leg_end_times,
    fixed_leg_payment_times=fixed_leg_payment_times,
    floating_leg_daycount_fractions=float_leg_daycount_fractions,
    fixed_leg_daycount_fractions=fixed_leg_daycount_fractions,
    fixed_leg_coupon=fixed_leg_coupon,
    reference_rate_fn=zero_rate_fn,
    notional=100.,
    mean_reversion=[0.01],  # Initial guess for mean reversion rate
    volatility=[0.005],  # Initial guess for volatility
    maximum_iterations=50,
    dtype=dtype)
# Expected calibrated_parameters.mean_reversion.values(): [0.03]
# Expected calibrated_parameters.volatility.values(): [0.01]

Args:#

  • prices: A rank 1 real Tensor. The prices of swaptions used for calibration.

  • expiries: A real Tensor of same shape and dtype as prices. The time to expiration of the swaptions.

  • floating_leg_start_times: A real Tensor of the same dtype as expiries. The times when accrual begins for each payment in the floating leg. The shape of this input should be expiries.shape + [m] where m denotes the number of floating payments in each leg.

  • floating_leg_end_times: A real Tensor of the same dtype as expiries. The times when accrual ends for each payment in the floating leg. The shape of this input should be expiries.shape + [m] where m denotes the number of floating payments in each leg.

  • fixed_leg_payment_times: A real Tensor of the same dtype as expiries. The payment times for each payment in the fixed leg. The shape of this input should be expiries.shape + [n] where n denotes the number of fixed payments in each leg.

  • floating_leg_daycount_fractions: A real Tensor of the same dtype and compatible shape as floating_leg_start_times. The daycount fractions for each payment in the floating leg.

  • fixed_leg_daycount_fractions: A real Tensor of the same dtype and compatible shape as fixed_leg_payment_times. The daycount fractions for each payment in the fixed leg.

  • fixed_leg_coupon: A real Tensor of the same dtype and compatible shape as fixed_leg_payment_times. The fixed rate for each payment in the fixed leg.

  • reference_rate_fn: A Python callable that accepts expiry time as a real Tensor and returns a Tensor of either shape input_shape or input_shape. Returns the continuously compounded zero rate at the present time for the input expiry time.

  • mean_reversion: A real positive scalar Tensor or a Python callable. The callable can be one of the following: (a) A left-continuous piecewise constant object (e.g., tff.math.piecewise.PiecewiseConstantFunc) that has a property is_piecewise_constant set to True. In this case the object should have a method jump_locations(self) that returns a Tensor of shape [num_jumps]. The return value of mean_reversion(t) should return a Tensor of shape t.shape, t is a rank 1 Tensor of the same dtype as the output. See example in the class docstring. (b) A callable that accepts scalars (stands for time t) and returns a scalar Tensor of the same dtype as strikes. Corresponds to the mean reversion rate.

  • volatility: A real positive Tensor of the same dtype as mean_reversion or a callable with the same specs as above. Corresponds to the long run price variance.

  • notional: An optional Tensor of same dtype and compatible shape as strikesspecifying the notional amount for the underlying swap. Default value: None in which case the notional is set to 1.

  • is_payer_swaption: A boolean Tensor of a shape compatible with expiries. Indicates whether the prices correspond to payer (if True) or receiver (if False) swaption. If not supplied, payer swaptions are assumed.

  • use_analytic_pricing: A Python boolean specifying if swaption pricing is done analytically during calibration. Analytic valuation is only supported for constant mean_reversion and piecewise constant volatility. If the input is False, then valuation using Monte-Carlo simulations is performed. Default value: The default value is True.

  • num_samples: Positive scalar int32 Tensor. The number of simulation paths during Monte-Carlo valuation of swaptions. This input is ignored during analytic valuation. Default value: The default value is 1.

  • random_type: Enum value of RandomType. The type of (quasi)-random number generator to use to generate the simulation paths. This input is relevant only for Monte-Carlo valuation and ignored during analytic valuation. Default value: None which maps to the standard pseudo-random numbers.

  • seed: Seed for the random number generator. The seed is only relevant if random_type is one of [STATELESS, PSEUDO, HALTON_RANDOMIZED, PSEUDO_ANTITHETIC,   STATELESS_ANTITHETIC]. For PSEUDO, PSEUDO_ANTITHETIC and HALTON_RANDOMIZED the seed should be an Python integer. For STATELESS and STATELESS_ANTITHETIC must be supplied as an integer Tensor of shape [2]. This input is relevant only for Monte-Carlo valuation and ignored during analytic valuation. Default value: None which means no seed is set.

  • skip: int32 0-d Tensor. The number of initial points of the Sobol or Halton sequence to skip. Used only when random_type is ‘SOBOL’, ‘HALTON’, or ‘HALTON_RANDOMIZED’, otherwise ignored. Default value: 0.

  • time_step: Scalar real Tensor. Maximal distance between time grid points in Euler scheme. Relevant when Euler scheme is used for simulation. This input is ignored during analytic valuation. Default value: None.

  • volatility_based_calibration: An optional Python boolean specifying whether calibration is performed using swaption implied volatilities. If the input is True, then the swaption prices are first converted to normal implied volatilities and calibration is performed by minimizing the error between input implied volatilities and model implied volatilities. Default value: True.

  • 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.

  • mean_reversion_lower_bound: An optional scalar Tensor specifying the lower limit of mean reversion rate during calibration. Default value: 0.001.

  • mean_reversion_upper_bound: An optional scalar Tensor specifying the upper limit of mean reversion rate during calibration. Default value: 0.5.

  • volatility_lower_bound: An optional scalar Tensor specifying the lower limit of Hull White volatility during calibration. Default value: 0.00001 (0.1 basis points).

  • volatility_upper_bound: An optional scalar Tensor specifying the upper limit of Hull White volatility during calibration. Default value: 0.1.

  • tolerance: Scalar Tensor of real dtype. The absolute tolerance for terminating the iterations. Default value: 1e-6.

  • maximum_iterations: Scalar positive int32 Tensor. The maximum number of iterations during the optimization. Default value: 50.

  • dtype: The default dtype to use when converting values to Tensors. Default value: None which means that default dtypes inferred by TensorFlow are used.

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

Returns:#

A Tuple of three elements:

  • The first element is an instance of CalibrationResult whose parameters are calibrated to the input swaption prices.

  • A Tensor of optimization status for each batch element (whether the optimization algorithm has found the optimal point based on the specified convergance criteria).

  • A Tensor containing the number of iterations performed by the optimization algorithm.

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