*Last updated: 2023-03-16.*
# tf_quant_finance.models.hjm.swaption_price
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Calculates the price of European swaptions using the HJM model.
```python
tf_quant_finance.models.hjm.swaption_price(
*, expiries, fixed_leg_payment_times, fixed_leg_daycount_fractions,
fixed_leg_coupon, reference_rate_fn, num_hjm_factors, mean_reversion,
volatility, times=None, time_step=None, num_time_steps=None, curve_times=None,
corr_matrix=None, notional=None, is_payer_swaption=None,
valuation_method=tf_quant_finance.models.ValuationMethod.MONTE_CARLO,
num_samples=1, random_type=None, seed=None, skip=0,
time_step_finite_difference=None, num_time_steps_finite_difference=None,
num_grid_points_finite_difference=101, dtype=None, name=None
)
```
A European Swaption is a contract that gives the holder an option to enter a
swap contract at a future date at a prespecified fixed rate. A swaption that
grants the holder the right to pay fixed rate and receive floating rate is
called a payer swaption while the swaption that grants the holder the right to
receive fixed and pay floating payments is called the receiver swaption.
Typically the start date (or the inception date) of the swap coincides with
the expiry of the swaption. Mid-curve swaptions are currently not supported
(b/160061740).
This implementation uses the HJM model to numerically value European
swaptions. For more information on the formulation of the HJM model, see
quasi_gaussian_hjm.py.
#### Example
````python
import numpy as np
import tensorflow as tf
import tf_quant_finance as tff
dtype = tf.float64
# Price 1y x 1y swaption with quarterly payments using Monte Carlo
# simulations.
expiries = np.array([1.0])
fixed_leg_payment_times = np.array([1.25, 1.5, 1.75, 2.0])
fixed_leg_daycount_fractions = 0.25 * np.ones_like(fixed_leg_payment_times)
fixed_leg_coupon = 0.011 * np.ones_like(fixed_leg_payment_times)
zero_rate_fn = lambda x: 0.01 * tf.ones_like(x, dtype=dtype)
mean_reversion = [0.03]
volatility = [0.02]
price = tff.models.hjm.swaption_price(
expiries=expiries,
fixed_leg_payment_times=fixed_leg_payment_times,
fixed_leg_daycount_fractions=fixed_leg_daycount_fractions,
fixed_leg_coupon=fixed_leg_coupon,
reference_rate_fn=zero_rate_fn,
notional=100.,
num_hjm_factors=1,
mean_reversion=mean_reversion,
volatility=volatility,
valuation_method=tff.model.ValuationMethod.MONTE_CARLO,
num_samples=500000,
time_step=0.1,
random_type=tff.math.random.RandomType.STATELESS_ANTITHETIC,
seed=[1, 2])
# Expected value: [[0.716]]
````
#### References:
[1]: D. Brigo, F. Mercurio. Interest Rate Models-Theory and Practice.
Second Edition. 2007. Section 6.7, page 237.
#### Args:
* `expiries`: A real `Tensor` of any shape and dtype. The time to expiration of
the swaptions. The shape of this input along with the batch shape of the
HJM model determines the number (and shape) of swaptions to be priced and
the shape of the output. If the batch shape of HJM models is
`model_batch_shape`, then the leading dimensions of `expiries` must be
broadcastable to `model_batch_shape`. For example, if the rank of
`model_batch_shape` is `n` and the rank of `expiries.shape` is `m`, then
`m>=n` and the leading `n` dimensions of `expiries.shape` must be
broadcastable to `model_batch_shape`.
* `fixed_leg_payment_times`: A real `Tensor` of the same dtype as `expiries`.
The payment times for each payment in the fixed leg. The shape of this
input should be `expiries.shape + [n]` where `n` denotes the number of
fixed payments in each leg. The `fixed_leg_payment_times` should be
greater-than or equal-to the corresponding expiries.
* `fixed_leg_daycount_fractions`: A real `Tensor` of the same dtype and
compatible shape as `fixed_leg_payment_times`. The daycount fractions for
each payment in the fixed leg.
* `fixed_leg_coupon`: A real `Tensor` of the same dtype and compatible shape as
`fixed_leg_payment_times`. The fixed rate for each payment in the fixed
leg.
* `reference_rate_fn`: A Python callable that accepts expiry time as a real
`Tensor` and returns a `Tensor` of shape
`model_batch_shape + input_shape`. Returns the continuously compounded
zero rate at the present time for the input expiry time.
* `num_hjm_factors`: A Python scalar which corresponds to the number of factors
in the batch of HJM models to be used for pricing.
* `mean_reversion`: A real positive `Tensor` of shape
`model_batch_shape + [num_hjm_factors]`.
Corresponds to the mean reversion rate of each factor in the batch.
* `volatility`: A real positive `Tensor` of the same `dtype` and shape as
`mean_reversion` or a callable with the following properties:
(a) The callable should accept a scalar `Tensor` `t` and a `Tensor`
`r(t)` of shape `batch_shape + [num_samples]` and returns a `Tensor` of
shape compatible with `batch_shape + [num_samples, dim]`. The variable
`t` stands for time and `r(t)` is the short rate at time `t`. The
function returns instantaneous volatility `sigma(t) = sigma(t, r(t))`.
When `volatility` is specified as a real `Tensor`, each factor is
assumed to have a constant instantaneous volatility and the model is
effectively a Gaussian HJM model.
Corresponds to the instantaneous volatility of each factor.
* `times`: An optional rank 1 `Tensor` of increasing positive real values. The
times at which Monte Carlo simulations are performed. Relevant when
swaption valuation is done using Monte Calro simulations.
Default value: `None` in which case simulation times are computed based
on either `time_step` or `num_time_steps` inputs.
* `time_step`: Optional scalar real `Tensor`. Maximal distance between time
grid points in Euler scheme. Relevant when Euler scheme is used for
simulation. This input or `num_time_steps` are required when valuation
method is Monte Carlo.
Default Value: `None`.
* `num_time_steps`: An optional scalar integer `Tensor` - a total number of
time steps during Monte Carlo simulations. The maximal distance betwen
points in grid is bounded by
`times[-1] / (num_time_steps - times.shape[0])`.
Either this or `time_step` should be supplied when the valuation method
is Monte Carlo.
Default value: `None`.
* `curve_times`: An optional rank 1 `Tensor` of positive and increasing real
values. The maturities at which spot discount curve is computed during
simulations.
Default value: `None` in which case `curve_times` is computed based on
swaption expiries and `fixed_leg_payments_times` inputs.
* `corr_matrix`: A `Tensor` of shape `[num_hjm_factors, num_hjm_factors]` and
the same `dtype` as `mean_reversion`. Specifies the correlation between
HJM factors.
Default value: `None` in which case the factors are assumed to be
uncorrelated.
* `notional`: An optional `Tensor` of same dtype and compatible shape as
`strikes`specifying the notional amount for the underlying swaps.
Default value: None in which case the notional is set to 1.
* `is_payer_swaption`: A boolean `Tensor` of a shape compatible with `expiries`.
Indicates whether the swaption is a payer (if True) or a receiver (if
False) swaption. If not supplied, payer swaptions are assumed.
* `valuation_method`: An enum of type `ValuationMethod` specifying
the method to be used for swaption valuation. Currently the valuation is
supported using `MONTE_CARLO` and `FINITE_DIFFERENCE` methods. Valuation
using finite difference is only supported for Gaussian HJM models, i.e.
for models with constant mean-reversion rate and time-dependent
volatility.
Default value: ValuationMethod.MONTE_CARLO, in which case
swaption valuation is done using Monte Carlo simulations.
* `num_samples`: Positive scalar `int32` `Tensor`. The number of simulation
paths during Monte-Carlo valuation. This input is ignored during analytic
valuation.
Default value: The default value is 1.
* `random_type`: Enum value of `RandomType`. The type of (quasi)-random number
generator to use to generate the simulation paths. This input is relevant
only for Monte-Carlo valuation and ignored during analytic valuation.
Default value: `None` which maps to the standard pseudo-random numbers.
* `seed`: Seed for the random number generator. The seed is only relevant if
`random_type` is one of `[STATELESS, PSEUDO, HALTON_RANDOMIZED,
PSEUDO_ANTITHETIC, STATELESS_ANTITHETIC]`. For `PSEUDO`,
`PSEUDO_ANTITHETIC` and `HALTON_RANDOMIZED` the seed should be an Python
integer. For `STATELESS` and `STATELESS_ANTITHETIC` must be supplied as
an integer `Tensor` of shape `[2]`. This input is relevant only for
Monte-Carlo valuation and ignored during analytic valuation.
Default value: `None` which means no seed is set.
* `skip`: `int32` 0-d `Tensor`. The number of initial points of the Sobol or
Halton sequence to skip. Used only when `random_type` is 'SOBOL',
'HALTON', or 'HALTON_RANDOMIZED', otherwise ignored.
Default value: `0`.
* `time_step_finite_difference`: Optional scalar real `Tensor`. Spacing between
time grid points in finite difference discretization. This input is only
relevant for valuation using finite difference.
Default value: `None`. If `num_time_steps_finite_difference` is also
unspecified then a `time_step` corresponding to 100 discretization steps
is used.
* `num_time_steps_finite_difference`: Optional scalar real `Tensor`. Number of
time grid points in finite difference discretization. This input is only
relevant for valuation using finite difference.
Default value: `None`. If `time_step_finite_difference` is also
unspecified, then 100 time steps are used.
* `num_grid_points_finite_difference`: Optional scalar real `Tensor`. Number of
spatial grid points per dimension. Currently, we construct an uniform grid
for spatial discretization. This input is only relevant for valuation
using finite difference.
Default value: 101.
* `dtype`: The default dtype to use when converting values to `Tensor`s.
Default value: `None` which means that default dtypes inferred by
TensorFlow are used.
* `name`: Python string. The name to give to the ops created by this function.
Default value: `None` which maps to the default name `hjm_swaption_price`.
#### Returns:
A `Tensor` of real dtype and shape derived from `model_batch_shape` and
expiries.shape containing the computed swaption prices. The shape of the
output is as follows:
* If the `model_batch_shape` is [], then the shape of the output is
expiries.shape
* Otherwise, the shape of the output is
`model_batch_shape + expiries.shape[model_batch_shape.rank:]`
For swaptions that have reset in the past (expiries<0), the function sets
the corresponding option prices to 0.0.