This paper presents a new approximation formula for pricing swaptions and cap/floors under the LIBOR market model of interest rates (LMM) with the local and affine-type stochastic volatility. In particular, two approximation methods are applied in pricing, one of which is so called “drift freezing” that fixes parts of the underlying stochastic processes at their initial values. Another approximation is based on an asymptotic expansion approach. An advantage of our method is that those approximations can be applied in a uniform manner to a general class of local-stochastic volatility models of interest rates. To demonstrate effectiveness of our method, the paper takes CEV-Heston LMM and Quadratic-Heston LMM as examples; it confirms sufficient flexibility of the models for calibration in a cpalet market and enough accuracies of the approximation method for numerical evaluation of swaption values under the models.