tf_quant_finance.black_scholes.approximations.adesi_whaley

Last updated: 2023-03-16.

tf_quant_finance.black_scholes.approximations.adesi_whaley#

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Computes American option prices using the Baron-Adesi Whaley approximation.

tf_quant_finance.black_scholes.approximations.adesi_whaley(
    *, volatilities, strikes, expiries, spots=None, forwards=None,
    discount_rates=None, dividend_rates=None, discount_factors=None,
    is_call_options=None, max_iterations=100, tolerance=1e-08, dtype=None, name=None
)

Example#

spots = [80.0, 90.0, 100.0, 110.0, 120.0]
strikes = [100.0, 100.0, 100.0, 100.0, 100.0]
volatilities = [0.2, 0.2, 0.2, 0.2, 0.2]
expiries = 0.25
dividends = 0.12
discount_rates = 0.08
computed_prices = adesi_whaley(
    volatilities=volatilities,
    strikes=strikes,
    expiries=expiries,
    discount_rates=discount_rates,
    dividend_rates=dividends,
    spots=spots,
    dtype=tf.float64)
# Expected print output of computed prices:
# [0.03, 0.59, 3.52, 10.31, 20.0]

References:#

[1] Baron-Adesi, Whaley, Efficient Analytic Approximation of American Option Values, The Journal of Finance, Vol XLII, No. 2, June 1987 https://deriscope.com/docs/Barone_Adesi_Whaley_1987.pdf

Args:#

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

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

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

  • spots: A real Tensor of any shape that broadcasts to the shape of the 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 any shape that broadcasts to the shape of 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 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, equivalent to r = 0 and discount factors = 1 when discount_factors also not given.

  • dividend_rates: An optional real Tensor of same 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.

  • discount_factors: An optional real Tensor of same 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. At most one of discount_rates and discount_factors can be supplied. Default value: None, which maps to -log(discount_factors) / expiries

  • 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.

  • max_iterations: positive int. The maximum number of iterations of Newton’s root finding method to find the critical spot price above and below which the pricing formula is different. Default value: 100

  • tolerance: Positive scalar Tensor. As with max_iterations, used with the Newton root finder to find the critical spot price. The root finder will judge an element to have converged if |f(x_n) - a| is less than tolerance (where f is the target function as defined in [1] and x_n is the estimated critical value), or if x_n becomes nan. When an element is judged to have converged it will no longer be updated. If all elements converge before max_iterations is reached then the root finder will return early. 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 adesi_whaley.

Returns:#

A 3-tuple containing the following items in order: (a) option_prices: A Tensor of the same shape as forwards. The Black Scholes price of the options. (b) converged: A boolean Tensor of the same shape as option_prices above. Indicates whether the corresponding adesi-whaley approximation has converged to within tolerance. (c) failed: A boolean Tensor of the same shape as option_prices above. Indicates whether the corresponding options price is NaN or not a finite number. Note that converged being True implies that failed will be false. However, it may happen that converged is False but failed is not True. This indicates the search did not converge in the permitted number of iterations but may converge if the iterations are increased.

Raises:#

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

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