.. _delta-doc:

.. ipython:: python
   :suppress:

   from rateslib.fx import *
   from datetime import datetime as dt

*****************
Delta Risk
*****************

Using Solvers
-------------

The :class:`~rateslib.solver.Solver` serves two purposes:

- **solving curves** relative to **calibrating instruments** and market rates,
- obtaining **risk sensitivities** to those **calibrating instruments**.

The mathematical processes involved here are technical and better explained in the
supplementary material (TODO link). Essentially a mapping is created between
the **fundamental derivatives** obtained through AD and
the **calibrating instrument rates**.
The :class:`~rateslib.solver.Solver` stores and uses this mapping to create the
:meth:`~rateslib.instruments.Sensitivities.delta`.

.. ipython:: python

   usd_curve = Curve(
       nodes={
           dt(2022, 1, 1): 1.0,
           dt(2022, 2, 1): 1.0,
           dt(2022, 4, 1): 1.0,
           dt(2023, 1, 1): 1.0,
        },
       id="sofr",
   )
   instruments = [
       IRS(dt(2022, 1, 1), "1m", "A", curves="sofr"),
       IRS(dt(2022, 1, 1), "3m", "A", curves="sofr"),
       IRS(dt(2022, 1, 1), "1y", "A", curves="sofr"),
   ]
   usd_solver = Solver(
       curves=[usd_curve],
       id="usd_sofr",
       instruments=instruments,
       s=[2.5, 3.25, 4.0],
       instrument_labels=["1m", "3m", "1y"],
   )
   irs = IRS(
       effective=dt(2022, 1, 1),
       termination="6m",
       frequency="A",
       currency="usd",
   )
   irs.curves = "sofr"
   irs.delta(solver=usd_solver)

A typical scenario in which FX exposures are created (if the instrument is not
multi-currency) is when ``base`` is set to something other than local currency.

.. ipython:: python

   fxr = FXRates({"eurusd": 1.1})
   irs.fixed_rate = 6.0  # create a negative NPV of approx -11.2k USD
   irs.delta(solver=usd_solver, base="eur", fx=fxr)

The NPV of the :class:`~rateslib.instruments.IRS` in EUR here is approximately -10.2k.
If the EURUSD exchange rate increases by 1000 pips to 1.20, then the EUR NPV increases
to only about -9.3k, meaning a gain of about 900 EUR. The FX sensitivity of about
0.9 EUR/pip is visible in the delta exposure dataframe.