Calibration and Parameter Estimation in Models › Calibration and Parameter Estimation in Models

A $1 pricing error on a $100 option and a $0.10 error on a $10 option are equal in percentage terms, but raw-price squared error gives the first one a hundred times the influence. Calibration starts with choices like this: the objective determines which discrepancies the optimizer tries hardest to reduce. We will compare objectives, fit synthetic option and yield-curve data, and use held-out errors and parameter diagnostics to assess the results.

Throughout this book, we've developed sophisticated pricing models: the Black-Scholes framework for options, short-rate models like Vasicek and Hull-White for interest rate derivatives, and GARCH models for volatility dynamics. Each of these models contains parameters that must be determined from data. The volatility parameter $\sigma$ in Black-Scholes, the mean reversion speed $\kappa$ in the Vasicek model, and the persistence coefficient in GARCH: none of these are given to us by theory. They must be estimated or calibrated.

Statistical estimation typically uses historical observations to fit a model of past and future real-world dynamics, often through maximum likelihood or method of moments. Calibration to derivative prices instead fits current market quotes and generally identifies parameters under a pricing measure, often denoted $Q$. These prices reflect uncertainty together with risk premia and market conditions; pricing parameters are not automatically forecasts under the physical measure $P$. Combining the two requires a model of their relationship, rather than interchanging the fitted parameters.

This chapter develops the theory and practice of calibration. We'll formalize the calibration problem, explore different objective functions and optimization techniques, and work through practical examples. We'll also address the dangers of overfitting: the tendency to fit noise rather than signal when models have too many free parameters relative to the information content of the data.