Duplicating Contingent Claims by the Lagrange Method

The problem of investing y (0) dollars at time 0 to duplicate a contingent claim is formulated as a dynamic optimization problem and solved by the Lagrange method. As an example, the well‐known formula of Black and Scholes on option pricing is derived. If the function defining dy ( t ) is concave in y ( t ), owing to costs of trading in incomplete markets, there is economy of scale in producing many claims simultaneously, thus explaining the profitability of institutions in providing such financial services.