tf_quant_finance.black_scholes.binary_price

Last updated: 2023-03-16.

tf_quant_finance.black_scholes.binary_price#

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Computes the Black Scholes price for a batch of binary call or put options.

tf_quant_finance.black_scholes.binary_price(
    *, volatilities, strikes, expiries, spots=None, forwards=None,
    discount_rates=None, dividend_rates=None, discount_factors=None,
    is_call_options=None, is_normal_volatility=False, dtype=None, name=None
)

The binary call (resp. put) option priced here is that which pays off a unit of cash if the underlying asset has a value greater (resp. smaller) than the strike price at expiry. Hence the binary option price is the discounted probability that the asset will end up higher (resp. lower) than the strike price at expiry.

Example#

  # Price a batch of 5 binary call options.
  volatilities = np.array([0.0001, 102.0, 2.0, 0.1, 0.4])
  forwards = np.array([1.0, 2.0, 3.0, 4.0, 5.0])
  # Strikes will automatically be broadcasted to shape [5].
  strikes = np.array([3.0])
  # Expiries will be broadcast to shape [5], i.e. each option has strike=3
  # and expiry = 1.
  expiries = 1.0
  computed_prices = tff.black_scholes.binary_price(
      volatilities=volatilities,
      strikes=strikes,
      expiries=expiries,
      forwards=forwards)
# Expected print output of prices:
# [0.         0.         0.15865525 0.99764937 0.85927418]

References:#

[1] Hull, John C., Options, Futures and Other Derivatives. Pearson, 2018. [2] Wikipedia contributors. Binary option. Available at: https://en.wikipedia.org/w/index.php?title=Binary_option

Args:#

  • volatilities: Real Tensor of any shape and 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 and of the shape that broadcasts with volatilities. Default value: None, equivalent to q = 0.

  • 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)). If None, 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, equivalent to discount factors = 1.

  • 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.

  • is_normal_volatility: An optional Python boolean specifying whether the volatilities correspond to lognormal Black volatility (if False) or normal Black volatility (if True). Default value: False, which corresponds to lognormal volatility.

  • 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 (float32).

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

Returns:#

  • binary_prices: A Tensor of the same shape as forwards. The Black Scholes price of the binary options.

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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