We study an infinite-dimensional Black-Scholes-Barenblatt equation which is a Hamilton-Jacobi-Bellman equation that is related to option pricing in the Musiela model of interest rate dynamics. We prove the existence and uniqueness of viscosity solutions of the Black-Scholes-Barenblatt equation and discuss their stochastic optimal control interpretation. We also show that in some cases the solution can be locally uniformly approximated by solutions of suitable finite-dimensional Hamilton-Jacobi-Bellman equations.