Nonlinear Integral-Differential evolution equation arising in option pricing when including transaction costs: A viscosity solution approach

Resumen

Since the Black-Scholes paper (1973) presented a formula for pricing options, there has been an increasing interest in problems arising in Financial Mathematics and in particular in derivatives pricing. The standard approach to this problem leads to the study of parabolic equations. One of the classic assumptions of the Black- Scholes model solution (1973) is that the investor’s portfolio revalues continuously. This dynamic implies transaction costs, due to the buying/selling of stocks for maintaining the portfolio’s equilibrium. Black- Scholes models that include proportional transaction costs have been studied by many authors Leland (1985).In this work we suppose that transaction costs behave as a non-increasing positive linear function, h(x) =a-bx, (a, b>0) which depends on the stock trading needed for hedging the portfolio that replicates the contingent claim. Moreover, if the underlying asset follows a jump-diffusion process, Merton (1993), we obtain an integro-differential evolution problem, with boundary value conditions, extending the paper Amster, Averbuj, Mariani, Rial (2005). Under adequate conditions, we extend their results, including a jump process, and we propose the existence of  an unique convex solution to the corresponding evolution Dirichlet problem.