*Last updated: 2023-03-16.*
# tf_quant_finance.models.hull_white.cap_floor_price
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Calculates the prices of interest rate Caps/Floors using Hull-White model.
```python
tf_quant_finance.models.hull_white.cap_floor_price(
*, strikes, expiries, maturities, daycount_fractions, reference_rate_fn,
mean_reversion, volatility, notional=1.0, is_cap=True,
use_analytic_pricing=True, num_samples=1, random_type=None, seed=None, skip=0,
time_step=None, dtype=None, name=None
)
```
An interest Cap (or Floor) is a portfolio of call (or put) options where the
underlying for the individual options are successive forward rates. The
individual options comprising a Cap are called Caplets and the corresponding
options comprising a Floor are called Floorlets. For example, a
caplet on forward rate `F(T_i, T_{i+1})` has the following payoff at time
`T_{i_1}`:
```None
caplet payoff = tau_i * max[F(T_i, T_{i+1}) - X, 0]
```
where where `X` is the strake rate and `tau_i` is the daycount fraction. The
caplet payoff (at `T_{i+1}`) can be expressed as the following at `T_i`:
```None
caplet_payoff = (1.0 + tau_i * X) *
max[1.0 / (1 + tau_i * X) - P(T_i, T_{t+1}), 0]
```
where `P(T_i, T_{i+1})` is the price at `T_i` of a zero coupon bond with
maturity `T_{i+1}. Thus, a caplet can be priced as a put option on zero
coupon bond [1].
#### References
[1]: D. Brigo, F. Mercurio. Interest Rate Models-Theory and Practice.
Second Edition. 2007.
#### Example
The example shows how value a batch containing spot starting 1-year and
2-year Caps and with quarterly frequency.
````python
import numpy as np
import tensorflow as tf
import tf_quant_finance as tff
dtype = tf.float64
reference_rate_fn = lambda x: 0.01 * tf.ones_like(x, dtype=dtype)
expiries = np.array([[0.0, 0.25, 0.5, 0.75, 1.0, 1.0, 1.0, 1.0],
[0.0, 0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75]])
maturities = np.array([[0.25, 0.5, 0.75, 1.0, 1.0, 1.0, 1.0, 1.0],
[0.25, 0.5, 0.75, 1.0, 1.25, 1.5, 1.75, 2.0]])
strikes = 0.01 * np.ones_like(expiries)
daycount_fractions = np.array([[0.25, 0.25, 0.25, 0.25, 0.0, 0.0, 0.0, 0.0],
[0.0, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25, 0.25]])
price = tff.models.hull_white.cap_floor_price(
strikes=strikes,
expiries=expiries,
maturities=maturities,
daycount_fractions=daycount_fractions,
notional=1.0e6,
mean_reversion=[0.03],
volatility=[0.02],
reference_rate_fn=reference_rate_fn,
use_analytic_pricing=True,
dtype=dtype)
# Expected value: [[0.4072088281493774], [1.3031872853339002]]
````
#### Args:
* `strikes`: A real `Tensor` of any shape and dtype. The strike rate of the
caplets or floorlets. The shape of this input determines the number
(and shape) of the options to be priced and the shape of the output. For
an N-dimensional input `Tensor`, the first N-1 dimensions correspond to
the batch dimension, i.e., the distinct caps and floors and the last
dimension correspond to the caplets or floorlets contained with an
instrument.
* `expiries`: A real `Tensor` of the same dtype and compatible shape as
`strikes`. The reset time of each caplet (or floorlet).
* `maturities`: A real `Tensor` of the same dtype and compatible shape as
`strikes`. The maturity time of each caplet (or floorlet) and also the
time at which payment is made.
* `daycount_fractions`: A real `Tensor` of the same dtype and compatible shape
as `strikes`. The daycount fractions associated with the underlying
forward rates.
* `reference_rate_fn`: A Python callable that accepts expiry time as a real
`Tensor` and returns a `Tensor` of the same shape as the input. Returns
the continuously compounded zero rate at the present time for the input
expiry time.
* `mean_reversion`: A real positive scalar `Tensor` or a Python callable. The
callable can be one of the following:
(a) A left-continuous piecewise constant object (e.g.,
`tff.math.piecewise.PiecewiseConstantFunc`) that has a property
`is_piecewise_constant` set to `True`. In this case the object should
have a method `jump_locations(self)` that returns a `Tensor` of shape
`[num_jumps]`. The return value of `mean_reversion(t)` should return a
`Tensor` of shape `t.shape`, `t` is a rank 1 `Tensor` of the same `dtype`
as the output. See example in the class docstring.
(b) A callable that accepts scalars (stands for time `t`) and returns a
scalar `Tensor` of the same `dtype` as `strikes`.
Corresponds to the mean reversion rate.
* `volatility`: A real positive `Tensor` of the same `dtype` as
`mean_reversion` or a callable with the same specs as above.
Corresponds to the long run price variance.
* `notional`: An optional `Tensor` of same dtype and compatible shape as
`strikes`specifying the notional amount for the cap (or floor).
Default value: None in which case the notional is set to 1.
* `is_cap`: A boolean `Tensor` of a shape compatible with `strikes`. Indicates
whether the option is a Cap (if True) or a Floor (if False). If not
supplied, Caps are assumed.
* `use_analytic_pricing`: A Python boolean specifying if analytic valuation
should be performed. Analytic valuation is only supported for constant
`mean_reversion` and piecewise constant `volatility`. If the input is
`False`, then valuation using Monte-Carlo simulations is performed.
* `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`: Scalar real `Tensor`. Maximal distance between time grid points
in Euler scheme. Relevant when Euler scheme is used for simulation. This
input is ignored during analytic valuation.
Default value: `None`.
* `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 class.
Default value: `None` which maps to the default name
`hw_cap_floor_price`.
#### Returns:
A `Tensor` of real dtype and shape strikes.shape[:-1] containing
the computed option prices. For caplets that have reset in the past
(expiries<0), the function sets the corresponding caplet prices to 0.0.