open-source-modelling / bisection_alpha_python

Simple bisection method that finds the optimal parameter α for the Smith & Wilson algorithm.

🐍 Bisection for Smith-Wilson parameter 🐍

This repository has an implementation for a simple bisection method that finds the optimal parameter α for the Smith & Wilson algorithm often used in insurance to interpolate/extrapolate rates or yields.

The implementation is based on Technical documentation of the Methodology to derive EIOPA's risk-free interest rate term structure and Wiki on Bisection method.

Problem

Before using the Smith & Wilson algorithm, the user needs to provide the convergence speed parameter α. This parameter needs to be calibrated primarily so that that the extrapolated result matches the desired long-term behaviour.

Solution

By transforming the minimization problem at the point of convergence into a problem of finding a root of the shifted function g(α) - τ, this repository implements a simple bisection algorithm to find the optimal α.

Input

• The minimum allowed value of the convergence speed parameter α.
• The maximum allowed value of the convergence speed parameter α.
• Maturities of bonds, observed on the market and provided as output.
• Zero-coupon rates, for which the user wishes to calibrate the algorithm. Each rate belongs to an observable zero-coupon bond with a known maturity.
• The ultimate forward rate towards which the user wishes the resulting curve to converge.
• Allowed difference between the given ultimate forward rate and the resulting curve.
• The numeric precision of the calculation. Higher the precision, more accurate the estimation of the root.
• The maximum number of iterations allowed. This is to prevent an infinite loop in case the method does not converge to a solution.

Output

• Optimal value of the parameter α (if the bisection method converged).

Note that to be consistent with EIOPA's recommendations, the lower bound of the interval should be set to 0.05.

Getting started

import numpy as np
from SWCalibrate import SWCalibrate as SWCalibrate
from SWExtrapolate import SWExtrapolate as SWExtrapolate
from bisection_alpha import Galfa as Galfa
from bisection_alpha import BisectionAlpha as BisectionAlpha

# Maturities of bonds observed on the market
M_Obs = np.transpose(np.array([1, 2, 4, 5, 6, 7]))

# Yields observed on the market
r_Obs = np.transpose(np.array([0.01, 0.02, 0.03, 0.032, 0.035, 0.04]))

# Ultimate forward rate
ufr = 0.04
# Numeric precision of the optimisation
Precision = 0.0000000001

# Targeted distance between the extrapolated curve and the ultimate forward rate at the convergence point
Tau = 0.0001 # 1 basis point

# Examples of a call to Galfa and BisectionAlpha
print("Example in the documentation for Galfa: "+ str(Galfa(M_Obs, r_Obs, ufr, 0.15, Tau)))
print("Example in the documentation for BisectionAlpha: "+ str(BisectionAlpha(0.05, 0.5, M_Obs, r_Obs, ufr, Tau, Precision, 1000)))

Note that this implementation use functions SWCalibrate and SWExtrapolate from the Smith & Wilson implementation. They are duplicated to this repository for completeness. If there are any inconsistencies or suggestions, raise an issue or contact us directly.