The aim of this work is to advocate the use of multifractional Brownian motion (mBm) as a relevant model in financial mathematics. Multifractional Brownian motion is an extension of fractional Brownian motion where the Hurst parameter is allowed to vary in time. This enables the possibility to accommodate for non-stationary local regularity, and to decouple it from long-range dependence properties. While we believe that mBm is potentially useful in a variety of settings, we focus here on a multifractional stochastic volatility Hull & White model that is an extension of the model studied in Long memory continuous-time stochastic volatility models from F.Comte and E.Renault. Using the stochastic calculus with respect to mBm developed in White Noise-based Stochastic Calculus with respect to multifractional Brownian motion (from J.Lebovits and J.Lévy-Véhel), we solve the corresponding stochastic differential equations. Since the solutions are of course not explicit, we take advantage of recently developed numerical techniques, namely functional quantization-based cubature methods, to get accurate approximations. This allows us to test the behavior of our model (as well as the one of F.Comte and E.Renault) with respect to its parameters, and in particular its ability to explain the smile effect of implied volatility. An advantage of our model is that it is able to both fit smiles at different maturities, and to take into account volatility persistence in a more precise way than in Long memory continuous-time stochastic volatility models.