# Asymptotic Methods applied to an American Option under a CEV Process

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 Title: Asymptotic Methods applied to an American Option under a CEV Process Author(s): Xu, Miao Advisor(s): Knessl, Charles Contributor(s): Nicholls, David; Yang, Jie; Abramov, Rafael; Sclove, Stanley Department / Program: Mathematics, Statistics, and Computer Science Graduate Major: Applied Mathematics Degree Granting Institution: University of Illinois at Chicago Degree: PhD, Doctor of Philosophy Genre: Doctoral Subject(s): Asymptotic Methods Partial Differential Equations Mathematical Finance Analysis Abstract: We consider an American put option under the Constant Elasticity of Variance (CEV) process. This corresponds to a free boundary problem for a partial differential equation (PDE). We show that this free boundary satisfies a nonlinear integral equation, and analyze it in the limit of small $\rho$ = $2r/ \sigma^2$, where $r$ is the interest rate and $\sigma$ is the volatility. We find that the free boundary behaves differently for five ranges of time to expiry. We then analyze option price $P(S,t)$, as a function of the asset price $S$ and time to expiry $t$. We obtain the asymptotic expansion of $P$ as $\rho \rightarrow 0$, first via an integral equation formulation, and then using the PDE satisfied by $P$, and analyzing it by perturbation theory and matched asymptotic expansions. Issue Date: 2012-12-07 Genre: thesis URI: http://hdl.handle.net/10027/8921 Rights Information: Copyright 2011 Miao Xu Date Available in INDIGO: 2012-12-072014-04-15 Date Deposited: 2011-08

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