*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.implied_vol
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Finds the implied volatilities of options under the Black Scholes model.
```python
tf_quant_finance.black_scholes.implied_vol(
*, prices, strikes, expiries, spots=None, forwards=None, discount_factors=None,
is_call_options=None,
method=tf_quant_finance.black_scholes.ImpliedVolMethod.NEWTON, underlying_distri
bution=tf_quant_finance.black_scholes.ImpliedVolUnderlyingDistribution.LOG_NORMA
L, validate_args=False, dtype=None, name=None, **kwargs
)
```
#### Examples
```python
import numpy as np
import tf_quant_finance as tff
forwards = np.array([1.0, 1.0, 1.0, 1.0])
strikes = np.array([1.0, 2.0, 1.0, 0.5])
expiries = np.array([1.0, 2.0, 1.0, 3.0])
discount_factors = np.array([0.95, 0.9, 0.95, 0.8])
is_call_options = np.array([True, True, False, False])
volatilities = np.array([0.5, 0.3, 0.2, 1.0])
prices = tff.black_scholes.option_price(
volatilities=volatilities,
strikes=strikes,
expiries=expiries,
forwards=forwards,
discount_factors=discount_factors,
is_call_options=is_call_options)
implied_vols = tff.black_scholes.implied_vol(
prices=prices,
strikes=strikes,
expiries=expiries,
forwards=forwards,
discount_factors=discount_factors,
is_call_options=is_call_options)
# Expected output:
# [0.5, 0.3, 0.2, 1.0]
#### Args:
* `prices`: A real `Tensor` of any shape. The prices of the options whose
implied vol is to be calculated.
* `strikes`: A real `Tensor` of the same dtype as `prices` and a shape that
broadcasts with `prices`. The strikes of the options.
* `expiries`: A real `Tensor` of the same dtype as `prices` and a shape that
broadcasts with `prices`. The expiry for 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
`prices`. The current spot price of the underlying. Either this argument
or the `forwards` (but not both) must be supplied.
Default value: None.
* `forwards`: A real `Tensor` of any shape that broadcasts to the shape of
`prices`. The forwards to maturity. Either this argument or the `spots`
must be supplied but both must not be supplied.
Default value: None.
* `discount_factors`: An optional real `Tensor` of same dtype as the `prices`.
If not None, these are the discount factors to expiry (i.e. e^(-rT)). If
None, no discounting is applied (i.e. it is assumed that the undiscounted
option prices are provided ). 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 `prices`.
Indicates whether the option is a call (if True) or a put (if False). If
not supplied, call options are assumed.
Default value: None.
* `method`: Enum value of ImpliedVolMethod to select the algorithm to use to
infer the implied volatility.
Default value: ImpliedVolMethod.NEWTON
* `underlying_distribution`: Enum value of ImpliedVolUnderlyingDistribution to
select the distribution of the underlying.
Default value: UnderlyingDistribution.LOG_NORMAL
* `validate_args`: A Python bool. If True, indicates that arguments should be
checked for correctness before performing the computation. The checks
performed are: (1) Forwards and strikes are positive. (2) The prices
satisfy the arbitrage bounds (i.e. for call options, checks the
inequality `max(F-K, 0) <= Price <= F` and for put options, checks that
`max(K-F, 0) <= Price <= K`.). (3) Checks that the prices are not too
close to the bounds. It is numerically unstable to compute the implied
vols from options too far in the money or out of the money.
Default value: False.
* `dtype`: `tf.Dtype` to use when converting arguments to `Tensor`s. If not
supplied, the default TensorFlow conversion will take place. Note that
this argument does not do any casting for `Tensor`s or numpy arrays.
Default value: None.
* `name`: (Optional) Python str. The name prefixed to the ops created by this
function. If not supplied, the default name 'implied_vol' is used.
Default value: None
* `**kwargs`: Any other keyword arguments to be passed to the specific
implementation. (See black_scholes.implied_vol_approx and
black_scholes.implied_vol_newton for details).
#### Returns:
* `implied_vols`: A `Tensor` of the same dtype as `prices` and shape as the
common broadcasted shape of `(prices, spots/forwards, strikes, expiries)`.
The implied volatilities as inferred by the chosen method.
#### Raises:
* `ValueError`: If both `forwards` and `spots` are supplied or if neither is
supplied. Or, if `underlying_distribution` is
`UnderlyingDistribution.NORMAL` when `method` is `FAST_APPROX`.