tf_quant_finance.experimental.american_option_pricing.andersen_lake.exercise_boundary.boundary_denominator

Last updated: 2023-03-16.

tf_quant_finance.experimental.american_option_pricing.andersen_lake.exercise_boundary.boundary_denominator#

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Calculates the denominator of the exercise boundary function of an American option.

tf_quant_finance.experimental.american_option_pricing.andersen_lake.exercise_boundary.boundary_denominator(
    tau_grid, b, k, r, q, sigma, integration_num_points=32, dtype=None
)

Calculates the denominator part of the calculation required to get the exercise boundary function of an American option. This corresponds to D in formula (3.8) in the paper [1].

References#

[1] Leif Andersen, Mark Lake and Dimitri Offengenden. High-performance American option pricing. 2015 https://engineering.nyu.edu/sites/default/files/2019-03/Carr-adjusting-exponential-levy-models.pdf#page=54

Example#

  dtype = tf.float64
  tau_grid = tf.constant([[0., 0.5, 1.], [0., 1., 2.],  [0., 3., 6.]],
                         dtype=dtype)
  k = tf.constant([1, 2, 2], dtype=dtype)
  r = tf.constant([0.01, 0.02, 0.04], dtype=dtype)
  q = tf.constant([0.01, 0.02, 0.0], dtype=dtype)
  sigma = tf.constant([0.1, 0.15, 0.05], dtype=dtype)
  k_exp = k[:, tf.newaxis, tf.newaxis]
  r_exp = r[:, tf.newaxis, tf.newaxis]
  q_exp = q[:, tf.newaxis, tf.newaxis]
  epsilon = machine_eps(dtype)
  def b_0(tau_grid_exp):
    one = tf.constant(1.0, dtype=dtype)
    return tf.ones_like(tau_grid_exp) * k_exp_exp * tf.where(
        tf.math.abs(q_exp_exp) < epsilon, one,
        tf.math.minimum(one, r_exp_exp / q_exp_exp))
  integration_num_points = 32
  boundary_denominator(tau_grid, b_0, k, r, q, sigma, integration_num_points)
  # Returns a tensor of shape [3, 3].

Args:#

  • tau_grid: Grid of values of shape [num_options, grid_num_points] indicating the time left until option maturity.

  • b: Represents the exercise boundary function for the option. Receives Tensor of rank tau_grid.rank + 1 and returns a Tensor of same shape.

  • k: Same dtype as tau_grid with shape num_options representing the strike price of the option.

  • r: Same shape and dtype as k representing the annualized risk-free interest rate, continuously compounded.

  • q: Same shape and dtype as k representing the dividend rate.

  • sigma: Same shape and dtype as k representing the volatility of the option’s returns.

  • integration_num_points: The number of points used in the integration approximation method. Default value: 32.

  • dtype: If supplied, the dtype for all input tensors. Result will have the same dtype. Default value: None which maps to dtype of tau_grid.

Returns:#

Tensor of shape [num_options, grid_num_points], containing a partial result for calculating the exercise boundary.

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