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

Calibration as an Inverse Problem

Pricing maps parameters to quotes: θ↦p(θ)\boldsymbol\theta\mapsto\mathbf{p}(\boldsymbol\theta). Calibration inverts that map — recover θ\boldsymbol\theta from observed pmkt\mathbf{p}_{\mathrm{mkt}}. When many parameter vectors match the same quotes, the inverse is ill-posed: solutions jump under tiny quote changes, echoing ill-conditioned linear solves. This lesson explains calibration using tools already in the programme: least squares, root-finding, and regularisation as deliberate stabilisation. A small implied-volatility and a discount-factor example are enough.
Learning objectives. By the end of this note you should be able to:
  • distinguish the forward (pricing) map from the inverse (calibration) problem;
  • formulate calibration as a root or a nonlinear least-squares problem;
  • explain non-uniqueness / ill-posedness and why regularisation restores stability;
  • diagnose a toy discount-factor or implied-vol fit using conditioning language.

In this lesson

  1. Forward maps and inverses
  2. Scalar calibration: implied volatility
  3. Vector calibration: discount factors
  4. Ill-posedness and regularisation

Before you start

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