The authors study partial regularity of viscosity solutions for a class of Kolmogorov equations. They consider a hedging problem for the derivative of a risky asset whose volatility, as well as claim, may depend on the past history of the asset. Related Kolmogorov equations are thus associated to stochastic delay problems. These Kolmogorov equations are linear second-order partial differential equations in an infinite dimensional Hilbert space with a drift term which contains an unbounded operator and a second-order term which only depends on a finite dimensional component of the Hilbert space. The proposed strategy for proving partial regularity of the value function is the following. Stochastic differential equations with smoothed out coefficients are considered and the unbounded operator is replaced by its Yosida approximations. The corresponding value functions with smoothed out payoff function are then studied. The new value functions satisfy their associated Kolmogorov equations. The the authors prove that their finite dimensional sections are viscosity solutions of certain linear finite-dimensional parabolic equations for which \(C^{1+\alpha}\) estimates are established. Passing to the limit with the approximations, these estimates are preserved, giving \(C^{1+\alpha}\) partial regularity for finite dimensional sections of the original value function.