{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "bb795a90-6814-49e8-ad0b-a3c354f1ec8f",
   "metadata": {},
   "source": [
    "# Another Example of an Exogenous Variable (SABR's Beta)\n",
    "\n",
    "The development of **exogenous variables** in *rateslib* came with attempting to capture sensitivity to *recovery risk* for a credit name with *CDSs*.\n",
    "But, their scope is completely general and they are easily used.\n",
    "\n",
    "Here we will give a second example of capturing sentivity to the **beta** parameter in a SABR model for *FXOptions*.\n",
    "The **beta** parameter is usually characterised as representing how normal or log-normal the underlying price process is; zero for fully normal and one for fully log-normal with a ranged value representing a blend.\n",
    "\n",
    "We can plot the variation in a *SabrSmile* below for differing values.\n",
    "\n",
    "First we create the *FX Forwards* market consistent with some of the other *FX Option* cookbooks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "846e5f66-2f30-4666-b6eb-e69ae445da06",
   "metadata": {},
   "outputs": [],
   "source": [
    "from rateslib import *\n",
    "from pandas import DataFrame\n",
    "\n",
    "eur = Curve({dt(2009, 5, 3): 1.0, dt(2011, 5, 10): 1.0})\n",
    "usd = Curve({dt(2009, 5, 3): 1.0, dt(2011, 5, 10): 1.0})\n",
    "fxf = FXForwards(\n",
    "    fx_rates=FXRates({\"eurusd\": 1.34664}, settlement=dt(2009, 5, 5)),\n",
    "    fx_curves={\"eureur\": eur, \"usdusd\": usd, \"eurusd\": eur},\n",
    ")\n",
    "fx_solver = Solver(\n",
    "    curves=[eur, usd],\n",
    "    instruments=[\n",
    "        Value(dt(2009, 5, 4), curves=eur, metric=\"cc_zero_rate\"),\n",
    "        Value(dt(2009, 5, 4), curves=usd, metric=\"cc_zero_rate\")\n",
    "    ],\n",
    "    s=[1.00, 0.4759550366220911],\n",
    "    fx=fxf,\n",
    ")"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5450450e-97a3-4f74-90df-823c623c7c63",
   "metadata": {},
   "source": [
    "Then build and calibrate an *FXSabrSmile*."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "252f3e84-7143-4326-bc36-557745227625",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxs = FXSabrSmile(\n",
    "    nodes={\"alpha\": 0.04, \"beta\": Variable(0.9, [\"beta\"]), \"rho\": 0.00, \"nu\": 0.01},\n",
    "    eval_date=dt(2009, 5, 3), expiry=dt(2010, 5, 3), pair=\"eurusd\",\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "b055bd0e-fea0-452d-8725-6989f6001a59",
   "metadata": {},
   "outputs": [],
   "source": [
    "solver = Solver(\n",
    "    pre_solvers=[fx_solver],\n",
    "    curves=[fxs],\n",
    "    instruments=[\n",
    "        FXCall(expiry=dt(2010, 5, 3), pair=\"eurusd\", strike=\"25d\", curves=[None, eur, None, usd], vol=fxs, metric=\"vol\"),\n",
    "        FXCall(expiry=dt(2010, 5, 3), pair=\"eurusd\", strike=\"atm_delta\", curves=[None, eur, None, usd], vol=fxs, metric=\"vol\"),\n",
    "        FXCall(expiry=dt(2010, 5, 3), pair=\"eurusd\", strike=\"75d\", curves=[None, eur, None, usd], vol=fxs, metric=\"vol\"),\n",
    "    ],\n",
    "    s=[10.0, 9.0, 9.9],\n",
    "    fx=fxf,\n",
    "    id=\"options\",\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "a7a53062-f2b5-4a01-a97a-bd90391953fa",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxs.plot(f=fxf)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f5f5a0ae-8282-4667-b45f-591824589fc8",
   "metadata": {},
   "source": [
    "## Beta sensitivity\n",
    "\n",
    "What is the sensitivity if **beta** changes on this *Smile*? We can evaluate it numerically for a specific *Option*.\n",
    "This numeric method involves a finite difference approach, shifting it up and down and revaluing the *Option*."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "85b08e49-81fb-49fa-b740-f6743f0381b2",
   "metadata": {},
   "source": [
    "### Numerically: Finite Difference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "deefaa37-f516-4540-b430-628bf8b04d13",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxc = FXCall(expiry=dt(2010, 5, 3), pair=\"eurusd\", strike=1.40, curves=[None, eur, None, usd], vol=fxs, premium=26710)\n",
    "base_npv = fxc.npv(solver=solver)\n",
    "base_npv"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "97bd04e6-0080-4d6d-91ff-071bab56e489",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxs.update_node(\"beta\", 0.91)\n",
    "solver.iterate()\n",
    "(fxc.npv(solver=solver) - base_npv) * 100.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "8ae3ea49-ef24-4e17-bbe3-4c05c7287dda",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxs.update_node(\"beta\", 0.90)  # reset\n",
    "solver.iterate()\n",
    "fxs.update_node(\"beta\", 0.89)\n",
    "solver.iterate()\n",
    "(fxc.npv(solver=solver) - base_npv) * 100.0"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24d7c40a-9487-4774-a726-48d14993a4ae",
   "metadata": {},
   "source": [
    "### Rateslib AD: ``exo_delta``"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "19d77bee-7db9-438a-981c-4e97899eb8e0",
   "metadata": {},
   "source": [
    "The result of the finite difference approach is that if **beta** is increased by one unit the *Option* will lose in value by c.77.6USD, whilst if it is decreased by one unit the *Option* will gain in value by c.77.6USD.\n",
    "\n",
    "This **sensitivity** can be expressly calculated using ``exo_delta``. We have injected *Variable* sensitivity into the calculation process and all of the internal calculations are configured to recognise this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "73c45a04-5387-49bd-9bf5-42a45b016650",
   "metadata": {},
   "outputs": [],
   "source": [
    "fxs.update_node(\"beta\", Variable(0.9, [\"beta\"]))\n",
    "solver.iterate()\n",
    "fxc.exo_delta(solver=solver, vars=[\"beta\"])"
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.12.0"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}