*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.approximations.adesi_whaley
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Computes American option prices using the Baron-Adesi Whaley approximation.
```python
tf_quant_finance.black_scholes.approximations.adesi_whaley(
*, volatilities, strikes, expiries, spots=None, forwards=None,
discount_rates=None, dividend_rates=None, discount_factors=None,
is_call_options=None, max_iterations=100, tolerance=1e-08, dtype=None, name=None
)
```
#### Example
```python
spots = [80.0, 90.0, 100.0, 110.0, 120.0]
strikes = [100.0, 100.0, 100.0, 100.0, 100.0]
volatilities = [0.2, 0.2, 0.2, 0.2, 0.2]
expiries = 0.25
dividends = 0.12
discount_rates = 0.08
computed_prices = adesi_whaley(
volatilities=volatilities,
strikes=strikes,
expiries=expiries,
discount_rates=discount_rates,
dividend_rates=dividends,
spots=spots,
dtype=tf.float64)
# Expected print output of computed prices:
# [0.03, 0.59, 3.52, 10.31, 20.0]
```
#### References:
[1] Baron-Adesi, Whaley, Efficient Analytic Approximation of American Option
Values, The Journal of Finance, Vol XLII, No. 2, June 1987
https://deriscope.com/docs/Barone_Adesi_Whaley_1987.pdf
#### Args:
* `volatilities`: Real `Tensor` of any shape and real dtype. The volatilities to
expiry of the options to price.
* `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.
* `discount_rates`: An optional real `Tensor` of same dtype as the
`volatilities`. If not `None`, discount factors are calculated as e^(-rT),
where r are the discount rates, or risk free rates.
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`. The continuous dividend rate on the underliers. May be
negative (to indicate costs of holding the underlier).
Default value: `None`, equivalent to zero dividends.
* `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_rate. 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 `-log(discount_factors) / expiries`
* `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.
* `max_iterations`: positive `int`. The maximum number of iterations of Newton's
root finding method to find the critical spot price above and below which
the pricing formula is different.
Default value: 100
* `tolerance`: Positive scalar `Tensor`. As with `max_iterations`, used with the
Newton root finder to find the critical spot price. The root finder will
judge an element to have converged if `|f(x_n) - a|` is less than
`tolerance` (where `f` is the target function as defined in [1] and `x_n`
is the estimated critical value), or if `x_n` becomes `nan`. When an
element is judged to have converged it will no longer be updated. If all
elements converge before `max_iterations` is reached then the root finder
will return early.
Default value: 1e-8
* `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 `adesi_whaley`.
#### Returns:
A 3-tuple containing the following items in order:
(a) option_prices: A `Tensor` of the same shape as `forwards`. The Black
Scholes price of the options.
(b) converged: A boolean `Tensor` of the same shape as `option_prices`
above. Indicates whether the corresponding adesi-whaley approximation
has converged to within tolerance.
(c) failed: A boolean `Tensor` of the same shape as `option_prices`
above. Indicates whether the corresponding options price is NaN or not
a finite number. Note that converged being True implies that failed
will be false. However, it may happen that converged is False but
failed is not True. This indicates the search did not converge in the
permitted number of iterations but may converge if the iterations are
increased.
#### Raises:
* `ValueError`: (a) If both `forwards` and `spots` are supplied or if neither is supplied.