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Analysis of nonlinear Black Scholes models


Please use this identifier to cite or link to this item: http://hdl.handle.net/1928/11130

Analysis of nonlinear Black Scholes models

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Title: Analysis of nonlinear Black Scholes models
Author: Qiu, Yan
Advisor(s): Lorenz, Jens
Committee Member(s): Embid, Pedro
Sauer, Christine
Pereyra, Cristina
Department: University of New Mexico. Dept. of Mathematics and Statistics
Subject: Black Scholes Model
Nonlinear PDE
Moving Boundary Problem
LC Subject(s): Derivative securities--Prices--Mathematical models.
Nonlinear partial differential operators.
Degree Level: Doctoral
Abstract: The Black Scholes equation is a fundamental model for derivative pricing. Modifying its assumptions will lead to more realistic but mathematically more complicated models. This dissertation consists of analytical and numerical studies about one particular type of nonlinear Black Scholes models, whose nonlinearity lies in the highest spatial derivative 1 with discontinuous coefficient function. First we smooth out the discontinuous term and focus only on the nonlinearity. We consider the case where the volatility is a smooth function and present some basic existence and uniqueness results. To study the discontinuity we simplify the problem by discretizing the Partial Differential Equation PDE only in time and consider the evolution in a given tiny time step from initial data. We perform convergence and perturbation analysis to the Ordinary Differential Equation (ODE) with discontinuous coefficient and obtain some insight of how the curves, where the discontinuity occurs, evolve in the space-time plane for the PDE. Last we obtain numerical results for the nonlinear PDE in the setting of a moving boundary problem.
Graduation Date: July 2010
URI: http://hdl.handle.net/1928/11130

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