Volume 15, Number 1, Spring 2007
AN ITERATIVE ALGORITHM FOR
EVALUATING APPROXIMATIONS TO THE
OPTIMAL EXERCISE BOUNDARY FOR A
NONLINEAR BLACKSCHOLES EQUATION
DANIEL SEVCOVIC
Abstract. The purpose of this paper is to analyze and
compute the early exercise boundary for a class of nonlinear
BlackScholes equations with a nonlinear volatility which can
be a function of the second derivative of the option price itself.
A motivation for studying the nonlinear BlackScholes equation
with a nonlinear volatility arises from option pricing models
taking into account, e.g., nontrivial transaction costs, investor's
preferences, feedback and illiquid markets effects and risk from
a volatile (unprotected) portfolio. We present a new method
how to transform the free boundary problem for the early exercise boundary position into a solution of a time depending
nonlinear parabolic equation defined on a fixed domain. We
furthermore propose an iterative numerical scheme that can be
used to find an approximation of the free boundary. We present
results of numerical approximation of the early exercise boundary for various types of nonlinear BlackScholes equations and
we discuss dependence of the free boundary on various model
parameters.
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