*Last updated: 2023-03-16.*
# tf_quant_finance.models.sabr.approximations.implied_volatility
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Computes the implied volatility under the SABR model.
```python
tf_quant_finance.models.sabr.approximations.implied_volatility(
*, strikes, expiries, forwards, alpha, beta, volvol, rho, shift=0.0, volatility_
type=tf_quant_finance.models.sabr.approximations.SabrImpliedVolatilityType.LOGNO
RMAL, approximation_type=tf_quant_finance.models.sabr.approximations.SabrApproxi
mationType.HAGAN, dtype=None, name=None
)
```
The SABR model specifies the risk neutral dynamics of the underlying as the
following set of stochastic differential equations:
```
dF = sigma F^beta dW_1
dsigma = volvol sigma dW_2
dW1 dW2 = rho dt
F(0) = f
sigma(0) = alpha
```
where F(t) represents the value of the forward price as a function of time,
and sigma(t) is the volatility.
Here, we implement an approximate solution as proposed by Hagan [1], and back
out the equivalent implied volatility that would've been obtained under either
the normal model or the Black model.
#### Example
```python
import tf_quant_finance as tff
import tensorflow as tf
equiv_vol = tff.models.sabr.approximations.implied_volatility(
strikes=np.array([106.0, 11.0]),
expiries=np.array([17.0 / 365.0, 400.0 / 365.0]),
forwards=np.array([120.0, 20.0]),
alpha=1.63,
beta=0.6,
rho=0.00002,
volvol=3.3,
dtype=tf.float64)
# Expected: [0.33284656705268817, 1.9828728139982792]
# Running this inside a unit test passes:
# equiv_vol = self.evaluate(equiv_vol)
# self.assertAllClose(equiv_vol, 0.33284656705268817)
```
#### References
[1] Hagan et al, Managing Smile Risk, Wilmott (2002), 1:84-108
#### Args:
* `strikes`: Real `Tensor` of arbitrary shape, specifying the strike prices.
Values must be strictly positive.
* `expiries`: Real `Tensor` of shape compatible with that of `strikes`,
specifying the corresponding time-to-expiries of the options. Values must
be strictly positive.
* `forwards`: Real `Tensor` of shape compatible with that of `strikes`,
specifying the observed forward prices of the underlying. Values must be
strictly positive.
* `alpha`: Real `Tensor` of shape compatible with that of `strikes`, specifying
the initial values of the stochastic volatility. Values must be strictly
positive.
* `beta`: Real `Tensor` of shape compatible with that of `strikes`, specifying
the model exponent `beta`. Values must satisfy 0 <= `beta` <= 1.
* `volvol`: Real `Tensor` of shape compatible with that of `strikes`,
specifying the model vol-vol multipliers. Values of `volvol` must be
non-negative.
* `rho`: Real `Tensor` of shape compatible with that of `strikes`, specifying
the correlation factors between the Wiener processes modeling the forward
and the volatility. Values must satisfy -1 < `rho` < 1.
* `shift`: Optional `Tensor` of shape compatible with that of `strkies`,
specifying the shift parameter(s). In the shifted model, the process
modeling the forward is modified as: dF = sigma * (F + shift) ^ beta * dW.
With this modification, negative forward rates are valid as long as
F > -shift.
Default value: 0.0
* `volatility_type`: Either SabrImpliedVolatility.NORMAL or LOGNORMAL.
Default value: `LOGNORMAL`.
* `approximation_type`: Instance of `SabrApproxmationScheme`.
Default value: `HAGAN`.
* `dtype`: Optional: `tf.DType`. If supplied, the dtype to be used for
converting values to `Tensor`s.
Default value: `None`, which means that the default dtypes inferred from
`strikes` is used.
* `name`: str. The name for the ops created by this function.
Default value: 'sabr_approx_implied_volatility'.
#### Returns:
A real `Tensor` of the same shape as `strikes`, containing the
corresponding equivalent implied volatilities.