Calibration reverses the direction of pricing: it solves for the parameters from market prices, which mathematics calls an inverse problem1. Prices of actively traded options are set by the market, so they serve as inputs, and the calibrated model prices exotic options and other instruments that have no market price2.

Consider a hypothetical example in which the implied volatilities at strikes 90, 100, and 110 are 18%, 20%, and 23%. Fitting a model with one flat volatility by minimizing the three squared differences with equal weights gives a volatility of 20.33%. The three prices cannot be matched at once, so an error remains.

The fitted parameters depend on the input instruments and on how the error is measured. Even if every input price is reproduced, nothing guarantees that the model gives correct prices for other instruments.

Notes

  1. Rama Cont and Peter Tankov, "Calibration of jump-diffusion option-pricing models: a robust non-parametric approach" (second version, September 2002), section 1 (page 3), "Calibration Problem 1". ↩

  2. Rama Cont, "Model uncertainty and its impact on the pricing of derivative instruments" (published in Mathematical Finance in 2006; this is the pre-publication version), section 3.2 (page 11). ↩