tf_quant_finance.black_scholes.brownian_bridge_single

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.

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#

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.

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