*Last updated: 2023-03-16.*
# tf_quant_finance.black_scholes.brownian_bridge_single
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Computes proba of not touching the barrier for a 1D Brownian Bridge.
```python
tf_quant_finance.black_scholes.brownian_bridge_single(
*, x_start, x_end, variance, barrier, 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 the same side of the barrier (either both above or both
below).
This can be used in Monte Carlo pricing for adjusting probability of
touching the barrier from discrete case to continuous case.
Typically in practise, the tensors `x_start`, `x_end` and `variance` should be
bi-dimensional (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]])
barrier = 5.1
no_touch_proba = brownian_bridge_single(
x_start=x_start,
x_end=x_end,
variance=variance,
barrier=barrier)
# Expected print output of no_touch_proba:
# [[0.69880579 0.69880579 0.97267628]
# [0.69880579 0.86466472 0.32967995]]
```
#### 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`.
* `barrier`: A scalar `Tensor` of the same dtype as `x_start`. Stands for the
boundary for the Brownian Bridge.
* `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_single`.
#### Returns:
A `Tensor` of the same shape as the input data which is the probability
of not touching the barrier.