tf_quant_finance.experimental.american_option_pricing.andersen_lake.andersen_lake

Last updated: 2023-03-16.

tf_quant_finance.experimental.american_option_pricing.andersen_lake.andersen_lake#

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Computes American option prices using the Andersen-Lake approximation.

tf_quant_finance.experimental.american_option_pricing.andersen_lake.andersen_lake(
    *, volatilities, strikes, expiries, spots=None, forwards=None,
    discount_rates=None, discount_factors=None, dividend_rates=None,
    is_call_options=None, grid_num_points=10, integration_num_points_kronrod=31,
    integration_num_points_legendre=32, max_iterations_exercise_boundary=30,
    max_depth_kronrod=30, tolerance_exercise_boundary=1e-08,
    tolerance_kronrod=1e-08, dtype=None, name=None
)

Example#

volatilities = [0.1, 0.15]
strikes = [3, 2]
expiries = [1, 2]
spots = [8.0, 9.0]
discount_rates = [0.01, 0.02]
dividend_rates = [0.01, 0.02]
is_call_options = [True, False]
grid_num_points = 40
integration_num_points_kronrod = 31
integration_num_points_legendre = 32
max_iterations_exercise_boundary = 500
max_depth_kronrod = 50
tolerance_exercise_boundary = 1e-11
tolerance_kronrod = 1e-11
computed_prices = andersen_lake(
    volatilities=volatilities,
    strikes=strikes,
    expiries=expiries,
    spots=spots,
    discount_rates=discount_rates,
    dividend_rates=dividend_rates,
    is_call_options=is_call_options,
    grid_num_points=grid_num_points,
    integration_num_points_kronrod=integration_num_points_kronrod,
    integration_num_points_legendre=integration_num_points_legendre,
    max_iterations_exercise_boundary=max_iterations_exercise_boundary,
    max_depth_kronrod=max_depth_kronrod,
    tolerance_exercise_boundary=tolerance_exercise_boundary,
    tolerance_kronrod=tolerance_kronrod
    dtype=tf.float64)
# Expected print output of computed prices:
# [4.950249e+00, 7.768513e-14]

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=46

Args:#

  • volatilities: Real Tensor of any real dtype and shape [num_options]. The volatilities to expiry of the options to price.

  • strikes: A real Tensor of the same dtype and same shape as volatilities. The strikes of the options to be priced.

  • expiries: A real Tensor of same dtype and same shape as volatilities. The expiry of each option. The units should be such that expiry * volatility**2 is dimensionless.

  • spots: A real Tensor of same shape as volatilities. The current spot price of the underlying. Either this argument or the forwards (but not both) must be supplied.

  • forwards: A real Tensor of same shape as volatilities. The forwards to maturity. Either this argument or the spots must be supplied but both must not be supplied.

  • discount_rates: An optional real Tensor of same shape and dtype as the volatilities. If not None, discount factors are calculated as e^(-rT), where r are the discount rates, or risk free rates. Default value: None, which maps to -log(discount_factors) / expiries if discount_factors is not None, or maps to 0 when discount_factors is also None.

  • discount_factors: An optional real Tensor of same shape and dtype as the volatilities. If not None, these are the discount factors to expiry (i.e. e^(-rT)). Mutually exclusive with discount_rate. If neither is given, no discounting is applied (i.e. the undiscounted option price is returned). If spots is supplied and discount_factors is not None then this is also used to compute the forwards to expiry. Default value: None.

  • dividend_rates: An optional real Tensor of same shape and dtype as the volatilities. The continuous dividend rate on the underliers. May be negative (to indicate costs of holding the underlier). Default value: None, equivalent to zero dividends.

  • is_call_options: A boolean Tensor of a shape compatible with volatilities. Indicates whether the option is a call (if True) or a put (if False). If not supplied, call options are assumed.

  • grid_num_points: positive int. The number of equidistant points to divide the values given in expiries into in the grid of tau_grid. Default value: 10.

  • integration_num_points_kronrod: positive int. The number of points used in the Gauss-Kronrod integration approximation method used for calculating the option prices. Default value: 31.

  • integration_num_points_legendre: positive int. The number of points used in the Gauss-Legendre integration approximation method used for calculating the exercise boundary function used for pricing the options. Default value: 32.

  • max_iterations_exercise_boundary: positive int. Maximum number of iterations for calculating the exercise boundary if it doesn’t converge earlier. Default value: 30.

  • max_depth_kronrod: positive int. Maximum number of iterations for calculating the Gauss-Kronrod integration approximation. Default value: 30.

  • tolerance_exercise_boundary: Positive scalar Tensor. The tolerance for the convergence of calculating the exercise boundary function. Default value: 1e-8.

  • tolerance_kronrod: Positive scalar Tensor. The tolerance for the convergence of calculating the Gauss-Kronrod integration approximation. Default value: 1e-8.

  • dtype: Optional tf.DType. If supplied, the dtype to be used for conversion of any supplied non-Tensor arguments to Tensor. Default value: None which maps to the default dtype inferred by TensorFlow.

  • name: str. The name for the ops created by this function. Default value: None which is mapped to the default name andersen_lake.

Returns:#

Tensor of shape [num_options], containing the calculated American option prices.

Raises:#

  • ValueError: (a) If both forwards and spots are supplied or if neither is supplied.

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