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

thesis
posted on 07.12.2012, 00:00 by Miao Xu
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.

Knessl, Charles

## Department

Mathematics, Statistics, and Computer Science

## Degree Grantor

University of Illinois at Chicago

Doctoral

## Committee Member

Nicholls, David Yang, Jie Abramov, Rafael Sclove, Stanley

2011-08

en

07/12/2012

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