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Linear Algebra, Optimisation and Numerical Methods / Numerical methods and calibration

Root-Finding and Numerical Integration

A bond's price is a smooth function of yield; markets quote the price and desks solve for the yield. An option expectation is an integral against a density that may lack an elementary antiderivative. This note joins bisection and Newton for scalar roots with one quadrature rule (trapezoid), and lands both on finance readings: yield from price, and an expectation as an integral.
Learning objectives. By the end of this note you should be able to:
  • run bisection given a sign-changing bracket;
  • derive and apply the Newton update xn+1=xn−f(xn)/f′(xn)x_{n+1}=x_n-f(x_n)/f'(x_n);
  • approximate a definite integral by the trapezoid rule and quote its O(n−2)O(n^{-2}) error order;
  • formulate bond yield as a root and an expectation as a numerical integral.

In this lesson

  1. Root-finding: bisection and Newton
  2. Numerical integration

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