tf_quant_finance.black_scholes.variance_swap_fair_strike

Last updated: 2023-03-16.

tf_quant_finance.black_scholes.variance_swap_fair_strike#

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Calculates the fair value strike for a variance swap contract.

tf_quant_finance.black_scholes.variance_swap_fair_strike(
    put_strikes, put_volatilities, call_strikes, call_volatilities, expiries,
    discount_rates, spots, reference_strikes, validate_args=False, dtype=None,
    name=None
)

This implements the approach in Appendix A of Demeterfi et al (1999), where a variance swap is defined as a forward contract on the square of annualized realized volatility (though the approach assumes continuous sampling). The variance swap payoff is, then:

notional * (realized_volatility^2 - variance_strike)

The method calculates the weight of each European option required to approximately replicate such a payoff using the discrete range of strike prices and implied volatilities of European options traded on the market. The fair value variance_strike is that which is expected to produce zero payoff.

Example#

dtype = tf.float64
call_strikes = tf.constant([[100, 105, 110, 115], [1000, 1100, 1200, 1300]],
  dtype=dtype)
call_vols = 0.2 * tf.ones((2, 4), dtype=dtype)
put_strikes = tf.constant([[100, 95, 90, 85], [1000, 900, 800, 700]],
  dtype=dtype)
put_vols = 0.2 * tf.ones((2, 4), dtype=dtype)
reference_strikes = tf.constant([100.0, 1000.0], dtype=dtype)
expiries = tf.constant([0.25, 0.25], dtype=dtype)
discount_rates = tf.constant([0.05, 0.05], dtype=dtype)
variance_swap_price(
  put_strikes,
  put_vols,
  call_strikes,
  put_vols,
  expiries,
  discount_rates,
  reference_strikes,
  reference_strikes,
  dtype=tf.float64)
# [0.03825004, 0.04659269]

References#

[1] Demeterfi, K., Derman, E., Kamal, M. and Zou, J., 1999. More Than You Ever Wanted To Know About Volatility Swaps. Goldman Sachs Quantitative Strategies Research Notes.

Args:#

  • put_strikes: A real Tensor of shape batch_shape + [num_put_strikes] containing the strike values of traded puts. This must be supplied in descending order, and its elements should be less than or equal to the reference_strike.

  • put_volatilities: A real Tensor of shape batch_shape + [num_put_strikes] containing the market volatility for each strike in `put_strikes. The final value is unused.

  • call_strikes: A real Tensor of shape batch_shape + [num_call_strikes] containing the strike values of traded calls. This must be supplied in ascending order, and its elements should be greater than or equal to the reference_strike.

  • call_volatilities: A real Tensor of shape batch_shape + [num_call_strikes] containing the market volatility for each strike in call_strikes. The final value is unused.

  • expiries: A real Tensor of shape compatible with batch_shape containing the time to expiries of the contracts.

  • discount_rates: A real Tensor of shape compatible with batch_shape containing the discount rate to be applied.

  • spots: A real Tensor of shape compatible with batch_shape containing the current spot price of the asset.

  • reference_strikes: A real Tensor of shape compatible with batch_shape containing an arbitrary value demarcating the atm boundary between liquid calls and puts. Typically either the spot price or the (common) first value of put_strikes or call_strikes.

  • validate_args: Python bool. When True, input Tensors are checked for validity. The checks verify the the matching length of strikes and volatilties. When False invalid inputs may silently render incorrect outputs, yet runtime performance will be improved. Default value: False.

  • dtype: tf.Dtype. If supplied the dtype for the input and output Tensors. Default value: None, leading to the default value inferred by Tensorflow.

  • name: Python str. The name to give to the ops created by this function. Default value: None which maps to ‘variance_swap_price’.

Returns:#

A Tensor of shape batch_shape containing the fair value of variance for each item in the batch. Note this is on the decimal rather than square percentage scale.

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