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Home  >  Transasctions of NAMP VOL2

28. An Application of Sturm-Liouville Equation to the Solution of the Black–Scholes equation with Transaction cost and Portfolio Risk Measures by Bright O. Osu, Chidinma Olunkwa, Anthony C. Akpanta and Chisom Onwuegbula Trans. of NAMP Vol 2, pp 307 – 312
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In this research work, we shall obtain the analytical solution of non-linear Black-Sholes equation with transaction cost measure and portfolio risk measure. We will reduce the Black-Scholes Partial Differential Equation (PDE) to a form of the Sturm–Liouville equation and obtain solutions given different boundary conditions. In each case we obtain a series solution which is a sequence of special functions. Furthermore we discuss the dynamic stability of equilibrium of the solution.It is observed that the equilibrium is dynamically stable if and only if the portfolio value (or the time path) u(x,t) is convergent under the condition that the eigenvalue and minimized total riskh_i<0.

Keywords: Black-Scholes PDE ,Eigenvalue problems,Transaction cost measure,Portfolio risk measure Sturm- Liouville Boundary Value Problem.