Last updated: 2023-03-16.
tf_quant_finance.rates.swap_curve_fit#
Constructs the zero swap curve using optimization.
tf_quant_finance.rates.swap_curve_fit(
float_leg_start_times, float_leg_end_times, float_leg_daycount_fractions,
fixed_leg_start_times, fixed_leg_end_times, fixed_leg_daycount_fractions,
fixed_leg_cashflows, present_values, initial_curve_rates,
present_values_settlement_times=None, float_leg_discount_rates=None,
float_leg_discount_times=None, fixed_leg_discount_rates=None,
fixed_leg_discount_times=None, optimizer=None, curve_interpolator=None,
instrument_weights=None, curve_tolerance=1e-08, maximum_iterations=50,
dtype=None, name=None
)
A zero swap curve is a function of time which gives the interest rate that
can be used to project forward rates at arbitrary t for the valuation of
interest rate securities.
Suppose we have a set of N Interest Rate Swaps (IRS) S_i with increasing
expiries whose market prices are known.
Suppose also that the ith IRS issues cashflows at times T_{ij} where
1 <= j <= n_i and n_i is the number of cashflows (including expiry)
for the ith swap.
Denote by T_i the time of final payment for the ith swap
(hence T_i = T_{i,n_i}). This function estimates a set of rates r(T_i)
such that when these rates are interpolated to all other cashflow times,
the computed value of the swaps matches the market value of the swaps
(within some tolerance). Rates at intermediate times are interpolated using
the user specified interpolation method (the default interpolation method
is linear interpolation on rates).
Example:#
The following example illustrates the usage by building an implied swap curve from four vanilla (fixed to float) LIBOR swaps.
dtype = np.float64
# Next we will set up LIBOR reset and payment times for four spot starting
# swaps with maturities 1Y, 2Y, 3Y, 4Y. The LIBOR rate spans 6M.
float_leg_start_times = [
np.array([0., 0.5], dtype=dtype),
np.array([0., 0.5, 1., 1.5], dtype=dtype),
np.array([0., 0.5, 1.0, 1.5, 2.0, 2.5], dtype=dtype),
np.array([0., 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5], dtype=dtype)
]
float_leg_end_times = [
np.array([0.5, 1.0], dtype=dtype),
np.array([0.5, 1., 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Next we will set up start and end times for semi-annual fixed coupons.
fixed_leg_start_times = [
np.array([0., 0.5], dtype=dtype),
np.array([0., 0.5, 1., 1.5], dtype=dtype),
np.array([0., 0.5, 1.0, 1.5, 2.0, 2.5], dtype=dtype),
np.array([0., 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5], dtype=dtype)
]
fixed_leg_end_times = [
np.array([0.5, 1.0], dtype=dtype),
np.array([0.5, 1., 1.5, 2.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0], dtype=dtype),
np.array([0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0], dtype=dtype)
]
# Next setup a trivial daycount for floating and fixed legs.
float_leg_daycount = [
np.array([0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5, 0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], dtype=dtype)
]
fixed_leg_daycount = [
np.array([0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5, 0.5, 0.5], dtype=dtype),
np.array([0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5, 0.5], dtype=dtype)
]
fixed_leg_cashflows = [
# 1 year swap with 2.855% semi-annual fixed payments.
np.array([-0.02855, -0.02855], dtype=dtype),
# 2 year swap with 3.097% semi-annual fixed payments.
np.array([-0.03097, -0.03097, -0.03097, -0.03097], dtype=dtype),
# 3 year swap with 3.1% semi-annual fixed payments.
np.array([-0.031, -0.031, -0.031, -0.031, -0.031, -0.031], dtype=dtype),
# 4 year swap with 3.2% semi-annual fixed payments.
np.array([-0.032, -0.032, -0.032, -0.032, -0.032, -0.032, -0.032,
-0.032], dtype=dtype)
]
# The present values of the above IRS.
pvs = np.array([0., 0., 0., 0.], dtype=dtype)
# Initial state of the curve.
initial_curve_rates = np.array([0.01, 0.01, 0.01, 0.01], dtype=dtype)
results = swap_curve_fit(float_leg_start_times, float_leg_end_times,
float_leg_daycount, fixed_leg_start_times,
fixed_leg_end_times, fixed_leg_cashflows,
fixed_leg_daycount, pvs, dtype=dtype,
initial_curve_rates=initial_curve_rates)
#### References:
[1]: Leif B.G. Andersen and Vladimir V. Piterbarg. Interest Rate Modeling,
Volume I: Foundations and Vanilla Models. Chapter 6. 2010.
#### Args:
* <b>`float_leg_start_times`</b>: List of `Tensor`s. Each `Tensor` must be either of
shape `batch_shape + [k_i]` or `[k_i]` and of the same real dtype. `k_i`
may be of different sizes. Each `Tensor` represents the beginning of the
accrual period for the forward rate which determines the floating payment.
Each element in the list belong to a unique swap to be used to build the
curve.
* <b>`float_leg_end_times`</b>: List of `Tensor`s of shapes and `dtype` compatible with
`float_leg_start_times`. Each `Tensor` represents the end of the
accrual period for the forward rate which determines the floating payment.
* <b>`float_leg_daycount_fractions`</b>: List of `Tensor`s of shapes and `dtype`
compatible with `float_leg_start_times`. Each `Tensor` represents the
daycount fraction of the forward rate which determines the floating
payment.
* <b>`fixed_leg_start_times`</b>: List of `Tensor`s. Each `Tensor` must be either of
shape `batch_shape + [n_i]` or `[n_i]` and of the same real dtype.
`n_i` may be of different sizes. All elements must have the same `dtype`
as `float_leg_start_times`. Each `Tensor` represents the beginning of the
accrual period fixed coupon.
* <b>`fixed_leg_end_times`</b>: List of `Tensor`s of shapes and `dtype` compatible with
`fixed_leg_start_times`. All elements must have the same `dtype` as
`fixed_leg_start_times`. Each `Tensor` represents the
end of the accrual period for the fixed coupon.
* <b>`fixed_leg_daycount_fractions`</b>: List of `Tensor`s of shapes and `dtype`
compatible with
`fixed_leg_start_times` Each `Tensor` represents the daycount fraction
applicable for the fixed payment.
* <b>`fixed_leg_cashflows`</b>: List of `Tensor`s of shapes and `dtype` compatible with
`fixed_leg_start_times`. The input contains fixed cashflows at each
coupon payment time including notional (if any). The sign should be
negative (positive) to indicate net outgoing (incoming) cashflow.
* <b>`present_values`</b>: List containing `Tensor`s of the same dtype as
elements of `fixed_leg_cashflows` and of shapes compatible with
`batch_shape`. The length of the list must be the same as the length of
`fixed_leg_cashflows`. The input contains the market price of the
underlying instruments.
* <b>`initial_curve_rates`</b>: A `Tensor` of the `dtype` as `present_values` and of
shape `[batch_shape, num_instruments]` where `num_instruments` is the
length of `float_leg_start_times`. The starting guess for the discount
rates used to initialize the iterative procedure.
* <b>`present_values_settlement_times`</b>: Optional list of `Tensor`s with the shapes
and `dtype` compatible with `present_values` The settlement times for the
present values is the time from now when the instrument is traded to the
time that the purchase price is actually delivered. If not supplied, then
it is assumed that the settlement times are zero for every bond.
Default value: `None` which is equivalent to zero settlement times.
* <b>`float_leg_discount_rates`</b>: Optional list of `Tensor`s with the shapes
and `dtype` compatible with `present_values`. This input contains the
continuously compounded discount rates the will be used
to discount the floating cashflows. This allows the swap curve to
constructed using an independent discount curve (e.g. OIS curve). By
default the cashflows are discounted using the curve that is being
constructed.
* <b>`float_leg_discount_times`</b>: Optional list of `Tensor`s with the shapes
and `dtype` compatible with `present_values`. This input contains the
times corresponding to the rates specified via
the `float_leg_discount_rates`.
* <b>`fixed_leg_discount_rates`</b>: Optional list of `Tensor`s with the shapes
and `dtype` compatible with `present_values`. This input contains the
continuously compounded discount rates the will be used to discount the
fixed cashflows. This allows the swap curve to constructed using an
independent discount curve (e.g. OIS curve). By default the cashflows are
discounted using the curve that is being constructed.
* <b>`fixed_leg_discount_times`</b>: Optional list of `Tensor`s with the shapes
and `dtype` compatible with `present_values`. This input contains the
times corresponding to the rates specified via the
`fixed_leg_discount_rates`.
* <b>`optimizer`</b>: Optional Python callable which implements the algorithm used
to minimize the objective function during calibration. It should have
the following interface: result =
optimizer(value_and_gradients_function, initial_position, tolerance,
max_iterations) `value_and_gradients_function` is a Python callable that
accepts a point as a real `Tensor` and returns a tuple of `Tensor`s of
real dtype containing the value of the function and its gradient at that
point. 'initial_position' is a real `Tensor` containing the starting
point of the optimization, 'tolerance' is a real scalar `Tensor` for
stopping tolerance for the procedure and `max_iterations` specifies the
maximum number of iterations.
`optimizer` should return a namedtuple containing the items: `position`
(a tensor containing the optimal value), `converged` (a boolean
indicating whether the optimize converged according the specified
criteria), `failed` (a boolean indicating if the optimization resulted
in a failure), `num_iterations` (the number of iterations used), and
`objective_value` ( the value of the objective function at the optimal
value). The default value for `optimizer` is None and conjugate
gradient algorithm is used.
Default value: `None` - indicating conjugate gradient minimizer.
* <b>`curve_interpolator`</b>: Optional Python callable used to interpolate the zero
swap rates at cashflow times. It should have the following interface:
yi = curve_interpolator(xi, x, y)
`x`, `y`, 'xi', 'yi' are all `Tensors` of real dtype. `x` and `y` are the
sample points and values (respectively) of the function to be
interpolated. `xi` are the points at which the interpolation is
desired and `yi` are the corresponding interpolated values returned by the
function. The default value for `curve_interpolator` is None in which
case linear interpolation is used.
Default value: `None`. If not supplied, the yields to maturity for the
bonds is used as the initial value.
* <b>`instrument_weights`</b>: Optional 'Tensor' of the same dtype and shape as
`initial_curve_rates`. This input contains the weight of each instrument
in computing the objective function for the conjugate gradient
optimization. By default the weights are set to be the inverse of
maturities.
* <b>`curve_tolerance`</b>: Optional positive scalar `Tensor` of same dtype as
elements of `bond_cashflows`. The absolute tolerance for terminating the
iterations used to fit the rate curve. The iterations are stopped when the
estimated discounts at the expiry times of the bond_cashflows change by a
amount smaller than `discount_tolerance` in an iteration.
Default value: 1e-8.
* <b>`maximum_iterations`</b>: Optional positive integer `Tensor`. The maximum number
of iterations permitted when fitting the curve.
Default value: 50.
* <b>`dtype`</b>: `tf.Dtype`. If supplied the dtype for the (elements of)
`float_leg_start_times`, and `fixed_leg_start_times`.
Default value: None which maps to the default dtype inferred by
TensorFlow.
* <b>`name`</b>: Python str. The name to give to the ops created by this function.
Default value: `None` which maps to 'swap_curve'.
#### Returns:
* <b>`curve_builder_result`</b>: An instance of `SwapCurveBuilderResult` containing the
following attributes.
times: Rank 1 real `Tensor`. Times for the computed discount rates. These
are chosen to be the expiry times of the supplied cashflows.
discount_rates: Rank 1 `Tensor` of the same dtype as `times`.
The inferred discount rates.
discount_factor: Rank 1 `Tensor` of the same dtype as `times`.
The inferred discount factors.
initial_discount_rates: Rank 1 `Tensor` of the same dtype as `times`. The
initial guess for the discount rates.
converged: Scalar boolean `Tensor`. Whether the procedure converged.
The procedure is said to have converged when the maximum absolute
difference in the discount factors from one iteration to the next falls
below the `discount_tolerance`.
failed: Scalar boolean `Tensor`. Whether the procedure failed. Procedure
may fail either because a NaN value was encountered for the discount
rates or the discount factors.
iterations: Scalar int32 `Tensor`. Number of iterations performed.
objective_value: Scalar real `Tensor`. The value of the ibjective function
evaluated using the fitted swap curve.
#### Raises:
* <b>`ValueError`</b>: If the initial state of the curve is not
supplied to the function.