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Calculus I / Infinite Sequences and Series

Applications of Taylor Polynomials: Error Bounds and Finance Approximations

Taylor polynomials turn "functions are hard" into "polynomials are easy". This lesson applies them to two families of problem: (1) approximating functions and integrals — evaluating sin⁡(0.1)\sin(0.1), e0.5e^{0.5}, or ∫01e−x2 dx\int_0^1 e^{-x^2}\,dx to any desired precision using Taylor's Inequality to control the error; and (2) finance approximations — the duration-convexity bond pricing model is a second-order Taylor approximation, and the "small-σ\sigma" expansion of option prices is a Taylor expansion in volatility. These are not decorative analogies: traders compute duration and convexity from the Taylor formula, and approximation errors (ignored higher-order terms) create real P&L risk when rate moves are large.
Learning objectives. By the end of this note you should be able to:
  • use Tn(x)T_n(x) to approximate f(x)f(x) and bound the error with Taylor's Inequality;
  • use ∫Tn(x) dx\int T_n(x)\,dx to approximate ∫f(x) dx\int f(x)\,dx for non-elementary functions;
  • interpret duration and convexity as the first two Taylor coefficients of bond price in yield;
  • apply the small-vol expansion of a call option's price.

In this lesson

  1. Approximating functions with TnT_n
  2. Approximating integrals with TnT_n
  3. Finance application: duration and convexity
  4. Finance application: small-vol expansion

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