tf_quant_finance.black_scholes.asian_option_price

Last updated: 2023-03-16.

tf_quant_finance.black_scholes.asian_option_price#

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Computes the Black Scholes price for a batch of asian options.

tf_quant_finance.black_scholes.asian_option_price(
    *, volatilities, strikes, expiries, spots=None, forwards=None,
    sampling_times=None, past_fixings=None, discount_rates=None,
    dividend_rates=None, discount_factors=None, is_call_options=None,
    is_normal_volatility=False,
    averaging_type=tf_quant_finance.black_scholes.AveragingType.GEOMETRIC,
    averaging_frequency=tf_quant_finance.black_scholes.AveragingFrequency.DISCRETE,
    dtype=None, name=None
)

In Black-Scholes, the marginal distribution of the underlying at each sampling date is lognormal. The product of a sequence of lognormal variables is also lognormal so we can re-express these options as vanilla options with modified parameters and use the vanilla pricer to price them.

TODO(b/261568763): support volatility term structures

Example#

  # Price a batch of 5 seasoned discrete geometric Asian 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
  sampling_times = np.array([[0.5, 0.5, 0.5, 0.5, 0.5],
                             [1.0, 1.0, 1.0, 1.0, 1.0]])
  past_fixings = np.array([[1.0, 2.0, 3.0, 4.0, 5.0]])
  computed_prices = tff.black_scholes.asian_option_price(
      volatilities=volatilities,
      strikes=strikes,
      expiries=expiries,
      forwards=forwards,
      sampling_times=sampling_times,
      past_fixings=past_fixings)
# Expected print output of computed prices:
# [ 0.0, 0.0, 0.52833763, 0.99555802, 1.91452834]

References:#

[1] Haug, E. G., The Complete Guide to Option Pricing Formulas. McGraw-Hill.

Args:#

  • volatilities: Real Tensor of any shape compatible with a batch_shape and and anyy real dtype. The volatilities to expiry of the options to price. Here batch_shape corresponds to a batch of priced options.

  • 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.

  • sampling_times: A real Tensor of same dtype as expiries and shape [n] + batch_shape where n is the number of sampling times for the Asian options Default value: None, which will raise an error for discrete sampling Asian options

  • past_fixings: A real Tensor of same dtype as spots or forwards and shape [n] + batch_shape where n is the number of past fixings that have already been observed. Default value: None, equivalent to no past fixings (ie. unseasoned)

  • 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)). Mutually exclusive with discount_rates. 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.

  • 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.

  • averaging_type: Enum value of AveragingType to select the averaging method for the payoff calculation. Default value: AveragingType.GEOMETRIC

  • averaging_frequency: Enum value of AveragingFrequency to select the averaging type for the payoff calculation (discrete vs continuous) Default value: AveragingFrequency.DISCRETE

  • 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 asian_option_price.

Returns:#

  • option_prices: A Tensor of shape batch_shape and the same dtype as volatilities. The Black Scholes price of the Asian 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.

  • ValueError: If is_normal_volatility is true and option is geometric, or is_normal_volatility is false (ie. lognormal) and option is arithmetic.

  • ValueError: If option is discrete averaging and sampling_dates is None of if last sampling date is later than option expiry date.

  • NotImplementedError: if option is continuous averaging.

  • NotImplementedError: if option is arithmetic.

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