Recovery of time dependent volatility coefficient by linearization

Victor Isakov

doi:10.3934/eect.2014.3.119

Article Contents

2014, Volume 3, Issue 1: 119-134. Doi: 10.3934/eect.2014.3.119

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Recovery of time dependent volatility coefficient by linearization

Victor Isakov^1,

1.
Department of Mathematics, Statistics, and Physics, Wichita State University, Wichita, KS 67260-0033

Received: August 31, 2013

Revised: November 30, 2013

Published: February 2014

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Abstract

We study the problem of reconstruction of special time dependent local volatility from market prices of options with different strikes at two expiration times. For a general diffusion process we apply the linearization technique and we conclude that the option price can be obtained as the sum of the Black-Scholes formula and of an operator $W$ which is linear in perturbation of volatility. We further simplify the linearized inverse problem and obtain unique solvability result in basic functional spaces. By using the Laplace transform in time we simplify the kernels of integral operators for $W$ and we obtain uniqueness and stability results for volatility under natural condition of smallness of the spacial interval where one prescribes the (market) data. We propose a numerical algorithm based on our analysis of the linearized problem.

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Mathematics Subject Classification: Primary: 35R30; Secondary: 92G20.

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