*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.approximations.bjerksund_stensland
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Computes prices of a batch of American options using Bjerksund-Stensland.
```python
tf_quant_finance.black_scholes.approximations.bjerksund_stensland(
*, volatilities, strikes, expiries, spots=None, forwards=None,
discount_rates=None, dividend_rates=None, discount_factors=None,
is_call_options=None, modified_boundary=True, dtype=None, name=None
)
```
#### Example
```python
import tf_quant_finance as tff
# Price a batch of 5 american call options.
volatilities = [0.2, 0.2, 0.2, 0.2, 0.2]
forwards = [80.0, 90.0, 100.0, 110.0, 120.0]
# Strikes will automatically be broadcasted to shape [5].
strikes = np.array([100.0])
# Expiries will be broadcast to shape [5], i.e. each option has strike=100
# and expiry = 0.25.
expiries = 0.25
discount_rates = 0.08
dividend_rates = 0.12
computed_prices = tff.black_scholes.approximations.bjerksund_stensland(
volatilities=volatilities,
strikes=strikes,
expiries=expiries,
discount_rates=discount_rates,
dividend_rates=dividend_rates,
forwards=forwards,
is_call_options=True
modified_boundary=True)
# Expected print output of computed prices:
# [ 0.03931201 0.70745419 4.01937524 11.31429842 21.20602005]
```
#### References:
[1] Bjerksund, P. and Stensland G., Closed Form Valuation of American Options,
2002
https://core.ac.uk/download/pdf/30824897.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` and of the shape that broadcasts with `volatilities`.
If not `None`, discount factors are calculated as e^(-rT),
where r are the discount rates, or risk free rates. At most one of
discount_rates and discount_factors can be supplied.
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 and cost_of_carry.
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 e^(-rT) calculated from
discount_rates.
* `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.
* `modified_boundary`: Python `bool`. Indicates whether the Bjerksund-Stensland
1993 algorithm (single boundary) if False or Bjerksund-Stensland 2002
algorithm (modified boundary) if True, is to be used.
* `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 `option_price`.
#### Returns:
A `Tensor` of the same shape as `forwards`.
#### Raises:
* `ValueError`: If both `forwards` and `spots` are supplied or if neither is
supplied.
* `ValueError`: If both `discount_rates` and `discount_factors` is supplied.