code: def _jamshidian_decomposition(
      hw_model, expiries, maturities, coefficients, dtype, name=None
  ):
      """Jamshidian decomposition for European swaption valuation.
  
      Jamshidian decomposition is a widely used technique for the valuation of
      European swaptions (and options on coupon bearing bonds) when the underlying
      models for the term structure are short rate models (such as Hull-White
      model). The method transforms the swaption valuation to the valuation of a
      portfolio of call (put) options on zero-coupon bonds.
  
      Consider the following swaption payoff(assuming unit notional) at the
      exipration time (under a single curve valuation):
  
      ```None
      payoff = max(1 - P(T0, TN, r(T0)) - sum_1^N tau_i * X_i * P(T0, Ti, r(T0)), 0)
             = max(1 - sum_0^N alpha_i * P(T0, Ti, r(T0)), 0)
      ```
  
      where `T0` denotes the swaption expiry, P(T0, Ti, r(T0)) denotes the price
      of the zero coupon bond at `T0` with maturity `Ti` and `r(T0)` is the short
      rate at time `T0`. If `r*` (or breakeven short rate) is the solution of the
      following equation:
  
      ```None
      1 - sum_0^N alpha_i * P(T0, Ti, r*) = 0            (1)
      ```
  
      Then the swaption payoff can be expressed as the following (Ref. [1]):
  
      ```None
      payoff = sum_1^N alpha_i max(P(T0, Ti, r*) - P(T0, Ti), 0)
      ```
      where in the above formulation the swaption payoff is the same as that of
      a portfolio of bond options with strikes `P(T0, Ti, r*)`.
  
      The function accepts relevant inputs for the above computation and returns
      the strikes of the bond options computed using the Jamshidian decomposition.
  
      #### References:
        [1]: Leif B. G. Andersen and Vladimir V. Piterbarg. Interest Rate Modeling.
        Volume II: Term Structure Models. Chapter 10.
  
  
      Args:
        hw_model: An instance of `VectorHullWhiteModel`. The model used for the
          valuation.
        expiries: A real `Tensor` of shape `batch_shape + [n]`, where `n` denotes
          the number of payments in the fixed leg of the underlying swaps. The time
          to expiration of the swaptions.
        maturities: A real `Tensor` of the same shape and dtype as `expiries`. The
          payment times for fixed payments of the underlying swaptions.
        coefficients: A real `Tensor` of the same shape and dtype as `expiries`
        dtype: The default dtype to use when converting values to `Tensor`s.
        name: Python string. The name to give to the ops created by this function.
          Default value: `None` which maps to the default name
          `jamshidian_decomposition`.
  
      Returns:
        A real `Tensor` of shape `expiries.shape + [1]` containing the forward bond
        prices computed at the breakeven short rate using the Jamshidian
        decomposition.
      """
  
      with tf.name_scope(name):
          # Shape ` batch_shape + [n, 1]`
          coefficients = tf.expand_dims(coefficients, axis=-1)
  
          def _zero_fun(x):
              # Get P(t0, t, r(t0)).
              # batch_shape + [n, 1]
              p_t0_t = hw_model.discount_bond_price(
                  x, expiries, maturities
              )
              # return_value.shape = batch_shape + [1, 1]
              return_value = tf.reduce_sum(
                  coefficients * p_t0_t, axis=-2, keepdims=True
              ) + [1.0]
              return return_value
  
          # batch_shape + [1, 1]
          swap_shape = expiries.shape.as_list()[:-1] + [1] + [1]
          lower_bound = -1 * tf.ones(swap_shape, dtype=dtype)
          upper_bound = 1 * tf.ones(swap_shape, dtype=dtype)
          # Solve Eq.(1)
          brent_results = brent.brentq(
              _zero_fun, lower_bound, upper_bound
          )
          breakeven_short_rate = brent_results.estimated_root
          # Shape `batch_shape + [n, 1]`
          return hw_model.discount_bond_price(
              breakeven_short_rate, expiries, maturities
          )
  
imports: from typing import Callable, Union
  import numpy
  import tensorflow.compat.v2
  from tf_quant_finance import types
  from tf_quant_finance.math import pde
  from tf_quant_finance.math import random
  from tf_quant_finance.math.interpolation import linear
  from tf_quant_finance.math.root_search import brent
  from tf_quant_finance.models import utils
  from tf_quant_finance.models.hjm import swaption_util
  from tf_quant_finance.models.hull_white import one_factor
  from tf_quant_finance.models.hull_white import zero_coupon_bond_option
  from tf_quant_finance.models.longstaff_schwartz import lsm
label_desc: Use image node_id 3 for calling a global function with example usage: _jamshidian_decomposition(hw_model, expiries, maturities, coefficients, dtype, name) and returns: hw_model, return_value