*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.brownian_bridge_double
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Computes probability of not touching the barriers for a 1D Brownian Bridge.
```python
tf_quant_finance.black_scholes.brownian_bridge_double(
*, x_start, x_end, variance, upper_barrier, lower_barrier, n_cutoff=3,
dtype=None, name=None
)
```
The Brownian bridge starts at `x_start`, ends at `x_end` and has a variance
`variance`. The no-touch probabilities are calculated assuming that `x_start`
and `x_end` are within the barriers 'lower_barrier' and 'upper_barrier'.
This can be used in Monte Carlo pricing for adjusting probability of
touching the barriers from discrete case to continuous case.
Typically in practice, the tensors `x_start`, `x_end` and `variance` should be
of rank 2 (with time steps and paths being the 2 dimensions).
#### Example
```python
x_start = np.asarray([[4.5, 4.5, 4.5], [4.5, 4.6, 4.7]])
x_end = np.asarray([[5.0, 4.9, 4.8], [4.8, 4.9, 5.0]])
variance = np.asarray([[0.1, 0.2, 0.1], [0.3, 0.1, 0.2]])
upper_barrier = 5.1
lower_barrier = 4.4
no_touch_proba = brownian_bridge_double(
x_start=x_start,
x_end=x_end,
variance=variance,
upper_barrier=upper_barrier,
lower_barrier=lower_barrier,
n_cutoff=3,
)
# Expected print output of no_touch_proba:
#[[0.45842169 0.21510919 0.52704599]
#[0.09394963 0.73302813 0.22595022]]
```
#### References
[1] Emmanuel Gobet. Advanced Monte Carlo methods for barrier and related
exotic options.
https://papers.ssrn.com/sol3/papers.cfm?abstract_id=1265669
#### Args:
* `x_start`: A real `Tensor` of any shape and dtype.
* `x_end`: A real `Tensor` of the same dtype and compatible shape as
`x_start`.
* `variance`: A real `Tensor` of the same dtype and compatible shape as
`x_start`.
* `upper_barrier`: A scalar `Tensor` of the same dtype as `x_start`. Stands for
the upper boundary for the Brownian Bridge.
* `lower_barrier`: A scalar `Tensor` of the same dtype as `x_start`. Stands for
lower the boundary for the Brownian Bridge.
* `n_cutoff`: A positive scalar int32 `Tensor`. This controls when to cutoff
the sum which would otherwise have an infinite number of terms.
Default value: 3.
* `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
`brownian_bridge_double`.
#### Returns:
A `Tensor` of the same shape as the input data which is the probability
of not touching the upper and lower barrier.