Implied Cap Volatility Curves and Swaption Surfaces: Calibrating Short-Rate Models

A short-rate model calibrated only to today's yield curve — exact initial fit, all $\vartheta(t)$ determined — still has free parameters: mean-reversion speed $a$ and volatility $\sigma$. The market tells you how to pin these down through the cap and swaption volatility structures. But the calibration reveals a fundamental tension: no one-factor short-rate model can simultaneously fit both the humped shape of the cap vol curve and the two-dimensional swaption volatility matrix. Understanding why models fail — and in what specific way each model's vol structure bends or flattens under calibration pressure — is what tells a rates trader which model to reach for and which pricing artifacts to anticipate. This note builds the conceptual and practical framework: how to extract model-implied cap vols from any short-rate model, what shapes different models can and cannot produce, and what the calibrated results on Euro market data actually looked like.