*Last updated: 2023-03-16.*
# tf_quant_finance.experimental.american_option_pricing.andersen_lake.andersen_lake
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Computes American option prices using the Andersen-Lake approximation.
```python
tf_quant_finance.experimental.american_option_pricing.andersen_lake.andersen_lake(
*, volatilities, strikes, expiries, spots=None, forwards=None,
discount_rates=None, discount_factors=None, dividend_rates=None,
is_call_options=None, grid_num_points=10, integration_num_points_kronrod=31,
integration_num_points_legendre=32, max_iterations_exercise_boundary=30,
max_depth_kronrod=30, tolerance_exercise_boundary=1e-08,
tolerance_kronrod=1e-08, dtype=None, name=None
)
```
#### Example
```python
volatilities = [0.1, 0.15]
strikes = [3, 2]
expiries = [1, 2]
spots = [8.0, 9.0]
discount_rates = [0.01, 0.02]
dividend_rates = [0.01, 0.02]
is_call_options = [True, False]
grid_num_points = 40
integration_num_points_kronrod = 31
integration_num_points_legendre = 32
max_iterations_exercise_boundary = 500
max_depth_kronrod = 50
tolerance_exercise_boundary = 1e-11
tolerance_kronrod = 1e-11
computed_prices = andersen_lake(
volatilities=volatilities,
strikes=strikes,
expiries=expiries,
spots=spots,
discount_rates=discount_rates,
dividend_rates=dividend_rates,
is_call_options=is_call_options,
grid_num_points=grid_num_points,
integration_num_points_kronrod=integration_num_points_kronrod,
integration_num_points_legendre=integration_num_points_legendre,
max_iterations_exercise_boundary=max_iterations_exercise_boundary,
max_depth_kronrod=max_depth_kronrod,
tolerance_exercise_boundary=tolerance_exercise_boundary,
tolerance_kronrod=tolerance_kronrod
dtype=tf.float64)
# Expected print output of computed prices:
# [4.950249e+00, 7.768513e-14]
```
#### References:
[1] Leif Andersen, Mark Lake and Dimitri Offengenden. High-performance
American option pricing. 2015
https://engineering.nyu.edu/sites/default/files/2019-03/Carr-adjusting-exponential-levy-models.pdf#page=46
#### Args:
* `volatilities`: Real `Tensor` of any real dtype and shape `[num_options]`.
The volatilities to expiry of the options to price.
* `strikes`: A real `Tensor` of the same dtype and same shape as `volatilities`.
The strikes of the options to be priced.
* `expiries`: A real `Tensor` of same dtype and same shape as `volatilities`.
The expiry of each option. The units should be such that
`expiry * volatility**2` is dimensionless.
* `spots`: A real `Tensor` of same shape as `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 same shape as `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 shape and 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`, which maps to `-log(discount_factors) / expiries`
if `discount_factors` is not `None`, or maps to `0` when
`discount_factors` is also `None`.
* `discount_factors`: An optional real `Tensor` of same shape and 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.
Default value: `None`.
* `dividend_rates`: An optional real `Tensor` of same shape and 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.
* `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.
* `grid_num_points`: positive `int`. The number of equidistant points to divide
the values given in `expiries` into in the grid of `tau_grid`.
Default value: 10.
* `integration_num_points_kronrod`: positive `int`. The number of points used in
the Gauss-Kronrod integration approximation method used for
calculating the option prices.
Default value: 31.
* `integration_num_points_legendre`: positive `int`. The number of points used
in the Gauss-Legendre integration approximation method used for
calculating the exercise boundary function used for pricing the options.
Default value: 32.
* `max_iterations_exercise_boundary`: positive `int`. Maximum number of
iterations for calculating the exercise boundary if it doesn't converge
earlier.
Default value: 30.
* `max_depth_kronrod`: positive `int`. Maximum number of iterations for
calculating the Gauss-Kronrod integration approximation.
Default value: 30.
* `tolerance_exercise_boundary`: Positive scalar `Tensor`. The tolerance for the
convergence of calculating the exercise boundary function.
Default value: 1e-8.
* `tolerance_kronrod`: Positive scalar `Tensor`. The tolerance for the
convergence of calculating the Gauss-Kronrod integration approximation.
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 `andersen_lake`.
#### Returns:
`Tensor` of shape `[num_options]`, containing the calculated American option
prices.
#### Raises:
* `ValueError`: (a) If both `forwards` and `spots` are supplied or if neither is supplied.