*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.variance_swap_fair_strike
View source
Calculates the fair value strike for a variance swap contract.
```python
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
```python
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 `Tensor`s 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 `Tensor`s.
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.