The Evolution of Interest‐Rate Models: From the Yield Curve to the Swaption Cube

Interest‐rate modelling is often taught as a catalogue of competing stochastic equations, obscuring why models were created and why modern sell‐side desks use several simultaneously. This survey reorganises the field around five layers of a pricing architecture: curve construction; arbitrage‐free dynamics; volatility‐smile representation; collateral, multi‐curve and risk‐free‐rate conventions; and numerical implementation. Within this framework, the progression from Vasicek and Cox–Ingersoll–Ross to Hull–White, HJM, the LIBOR market model, SABR and post‐LIBOR forward market models is interpreted as a sequence of responses to new products, quotation practices, computational constraints and market institutions—not as a tournament won by one equation. We derive the core equations, compare the principal model families, connect calibration choices to hedging instruments, and map models to major rates products and XVA applications. Particular attention is paid to risks concealed by exact vanilla calibration, including correlation, conditional curve dynamics, future smile behavior and numerical error. An open Python/C++ companion repository provides reproducible illustrations for selected stages. The resulting framework offers a systematic introduction for new quantitative analysts and a practical reference for researchers, traders, model developers and validators.