A penalty-based method from reconstructing smooth local volatility surface from American options

Kai Zhang; Kok Lay Teo

doi:10.3934/jimo.2015.11.631

Article Contents

2015, Volume 11, Issue 2: 631-644. Doi: 10.3934/jimo.2015.11.631

This issue Previous Article A family of extragradient methods for solving equilibrium problems Next Article Neural network smoothing approximation method for stochastic variational inequality problems

A penalty-based method from reconstructing smooth local volatility surface from American options

Kai Zhang^1, and
Kok Lay Teo^2,

1.
China Center for Special Economic Zone Research, Shenzhen University, 3688 Nanhai Ave., Shenzhen, 518060
2.
Department of Mathematics and Statistics, Curtin University, Perth, Western Australia, WA 6845

Received: November 2013

Revised: May 2014

Published: April 2015

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Abstract

This paper is devoted to develop a robust penalty-based method of reconstructing smooth local volatility surface from the observed American option prices. This reconstruction problem is posed as an inverse problem: given a finite set of observed American option prices, find a local volatility function such that the theoretical option prices matches the observed ones optimally with respect to a prescribed performance criterion. The theoretical American option prices are governed by a set of partial differential complementarity problems (PDCP). We propose a penalty-based numerical method for the solution of the PDCP. Typically, the reconstruction problem is ill-posed and a bicubic spline regularization technique is thus proposed to overcome this difficulty. We apply a gradient-based optimization algorithm to solve this nonlinear optimization problem, where the Jacobian of the cost function is computed via finite difference approximation. Two numerical experiments: a synthetic American put option example and a real market American put option example, are performed to show the robustness and effectiveness of the proposed method to reconstructing the unknown volatility surface.

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Mathematics Subject Classification: Primary: 91G60 , 91G20; Secondary: 90C33.

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