Formulate the following problem for the accrual barrier option as a Black–Scholes partial differential equation with appropriate final and boundary conditions:

The option has barriers at levels S_u and S_d, above and below the initial asset price, respectively. If the asset touches either barrier before expiry then the option knocks out with an immediate payoff of ϕ(T − t). Otherwise, at expiry the option has a payoff of max(S − E, 0).