Solving Replication Problems in Complete Market by Orthogonal Series Expansion

We reconsider the replication problem for contingent claims in a complete market under a general framework. Since there are various limitations in the Black–Scholes pricing formula, we propose a new method to obtain an explicit self–financing trading strategy expression for replications of claims in a general model. The main advantage of our method is that we propose using an orthogonal expansion method to derive a closed–form expression for the self–financing strategy that is associated with some general underlying asset processes. As a consequence, a replication strategy is obtained for a European option. Converse to the traditional Black-Scholes theory, we derive a pricing formula for a European option from the proposed replication strategy that is quite different from the Black-Scholes pricing formula. We then provide an implementation procedure and then a numerical example to show how the proposed trading strategy works in practice and then compare with a replication strategy based on the Black-Scholes theory.