Uniform asymptotic expansions for pricing European options
type: Item
instance of: scholarly article
title: Uniform asymptotic expansions for pricing European options
zbMATH Open document ID: 1087.60045
published in: Applied Mathematics and Optimization
publication date: 2006-03-02T00:00:00Z
review text: A fast varying stochastic volatility \({\sigma}(t,{\omega})=f(y^{\varepsilon}(t))\) which plays the role of the constant volatility in the Black-Scholes' SDE is considered to describe the risky asset price, where \(y^{\varepsilon}(t)\) satisfies the following SDE: \[ dy^{\varepsilon}(t)=\frac{1}{\varepsilon}m(y^{\varepsilon}(t))dt +\frac{1}{\sqrt{\varepsilon}}{\beta}(y^{\varepsilon}(t))d \tilde{z^{\varepsilon}} (t) \] driven by \(\tilde{z^{\varepsilon}}(t)\), a ``Brownian motion on a circle'' of circumference \textit{a}. Let \(P^{\varepsilon}(t,x,y)\) be the option price at \(t<T\) of a payoff at maturity time \textit{T} of the stock with the present value \textit{x} and a current value of the volatility \textit{y}. Under conditions of smoothness of the coefficient and a compact support restriction of the payoff function, it is proven that \(P^{\varepsilon}(t,x,y)\) has an asymptotic expansion of \(P_0(t,x,y)+ {\varepsilon}^{1/2}P_1(t,x,y)+{\varepsilon}P_2(t,x,y) +{\varepsilon}Q_2((T-t)/\varepsilon,x,y)\) unifomly in \(t<T,x,y\) with an error of \(O({\varepsilon}^{3/2})\).
reviewed by: Gong Guanglu
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 91G20
Mathematics Subject Classification ID: 34E20
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 5009600
zbMATH Keywords: Singular perturbation
zbMATH Keywords: Two-time scale
zbMATH Keywords: European option
zbMATH Keywords: diffusion
author: R. Z. Khas'minskiĭ
author: G. Yin
MaRDI profile type: Publication
full work available at URL: s00245-005-0833-2
OpenAlex ID: W1991813615
DOI: 10.1007/S00245-005-0833-2
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description: scientific article; zbMATH DE number 5009600