tf_quant_finance.black_scholes.brownian_bridge_double

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.

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#

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.

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