We propose a nonparametric local volatility Cheyette model and apply it to pricing interest rate swaptions. Concretely, given market prices of swaptions, we show how to construct a unique diffusion process consistent with these prices. We then link the resulting local volatility to the dynamics of the entire interest rate curve. The model preserves completeness and allows consistent pricing of illiquid, out-of-the-money and exotic interest rate products. The model is relatively straightforward to implement and calibrate and less involved than stochastic volatility approaches.