Nonlinear Integral-Differential evolution equation arising in option pricing when including transaction costs: A viscosity solution approach
Keywords:
Lei do Petróleo, Poder normativo, Norma-quadro.Abstract
Since Black and Scholes´s paper (1973) presents a formula to pricing option, there has been an increasing interest on problems arising in Financial Mathematics and in particular on derivatives pricing. The standard approach to this problem leads to the study of equations of parabolic type.
One of the classic assumptions with the Black- Scholes models resolution (1973) is that the investor's portfolio revalue in a continuous form. This dynamic implies transaction costs, due to the buy/sell of necessary stocks to maintain the portfolio's equilibrium. Black- Scholes models which include proportional transaction costs were studied by many authors (Leland, 1985).
In this work we suppose that transaction costs behave as a no increasing lineal function, h(x) =a-bx, (a, b>0), depending on the trading stocks need to hedge the portfolio that replicates the contingent claim.
Moreover, if the underlying asset follows a jump-diffusion process, (Merton, 1993), we obtain an Evolution Integro-differential problem, with value boundaries conditions, extending the paper (Amster, Averbuj, 2005)
Under adequate conditions, we propose a unique convex solution to the corresponding evolution Dirichlet problem
Key words: Poisson-diffusion process, transaction costs, nonlinear partial differential equations, viscosity solution, Perron´s method
Code:C02,G13
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