In calibration, one example takes, for each instrument, the squared difference between the model price and the market price, multiplies it by a weight, and sums the results12. The smaller this value, the better the model reproduces market prices.

Consider a hypothetical example in which the market implied volatilities are 18%, 20%, and 23% and the model has a flat volatility of 20%. The differences (model minus market) are 2, 0, and −3 percentage points. With all weights equal to 1, the objective function is 22+02+322^2 + 0^2 + 3^2, which is 13.

The choice of weights changes the parameters that minimize it. Depending on its shape, an objective function can have several minimum points, and the search then depends on where it starts.

Notes

  1. Rama Cont and Peter Tankov, "Calibration of jump-diffusion option-pricing models: a robust non-parametric approach" (second version, September 2002), section 3.1 (page 13), equation (30). It minimizes a weighted sum of squares. ↩

  2. Cui, del Baño Rollin, and Germano, "Full and fast calibration of the Heston stochastic volatility model" (second version, May 2016), section 3 (page 5), equations (4) and (5). It minimizes one half of the sum of squared residuals. ↩