tf_quant_finance.black_scholes.approximations.bjerksund_stensland

Last updated: 2023-03-16.

tf_quant_finance.black_scholes.approximations.bjerksund_stensland#

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Computes prices of a batch of American options using Bjerksund-Stensland.

tf_quant_finance.black_scholes.approximations.bjerksund_stensland(
    *, volatilities, strikes, expiries, spots=None, forwards=None,
    discount_rates=None, dividend_rates=None, discount_factors=None,
    is_call_options=None, modified_boundary=True, dtype=None, name=None
)

Example#

  import tf_quant_finance as tff
  # Price a batch of 5 american call options.
  volatilities = [0.2, 0.2, 0.2, 0.2, 0.2]
  forwards = [80.0, 90.0, 100.0, 110.0, 120.0]
  # Strikes will automatically be broadcasted to shape [5].
  strikes = np.array([100.0])
  # Expiries will be broadcast to shape [5], i.e. each option has strike=100
  # and expiry = 0.25.
  expiries = 0.25
  discount_rates = 0.08
  dividend_rates = 0.12
  computed_prices = tff.black_scholes.approximations.bjerksund_stensland(
      volatilities=volatilities,
      strikes=strikes,
      expiries=expiries,
      discount_rates=discount_rates,
      dividend_rates=dividend_rates,
      forwards=forwards,
      is_call_options=True
      modified_boundary=True)
# Expected print output of computed prices:
# [ 0.03931201  0.70745419  4.01937524 11.31429842 21.20602005]

References:#

[1] Bjerksund, P. and Stensland G., Closed Form Valuation of American Options, 2002 https://core.ac.uk/download/pdf/30824897.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 and of the shape that broadcasts with volatilities. If not None, discount factors are calculated as e^(-rT), where r are the discount rates, or risk free rates. At most one of discount_rates and discount_factors can be supplied. 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 and cost_of_carry. 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 e^(-rT) calculated from discount_rates.

  • 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.

  • modified_boundary: Python bool. Indicates whether the Bjerksund-Stensland 1993 algorithm (single boundary) if False or Bjerksund-Stensland 2002 algorithm (modified boundary) if True, is to be used.

  • 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 option_price.

Returns:#

A Tensor of the same shape as forwards.

Raises:#

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

  • ValueError: If both discount_rates and discount_factors is supplied.

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