A transaction cost convergence result for general hedging strategies

This paper focuses on the valuation of financial derivatives with transaction costs. These financial products derive their value from other financial products, which are usually modeled as diffusion processes and called ‘underlyings’. The prices of such financial derivatives are generally written as the expectation of a function of the underlying process and can therefore be written as solutions of Partial Differential Equations. If there are transaction costs in the market, we prove that the price of a derivative converges towards the solution of a non-linear Partial Differential Equation as these transaction costs go to zero and the frequency of their payment goes to infinity. Our result generalizes those of Leland and Henrotte. It holds for derivatives that are functionals of the underlying process rather than just functions (so-called ‘path-dependent’). It also holds if other derivatives are used in lieu of underlyings, and for derivatives whose value is not supposed to be a convex function of the underlying's price.