Recovery of time dependent volatility coefficient by linearization

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Authors
Isakov, Victor
Advisors
Issue Date
2014-03
Type
Article
Keywords
Inverse problems , Parabolic equations , Integral transforms , Inverse option pricing , Derivatives securities
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Organizational Units
Journal Issue
Citation
Isakov, Victor, 1947-. 2014. Recovery of time dependent volatility coefficient by linearization. Evolution Equations and Control Theory (EECT), vol. 3:no. 1:pp 119-134
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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Publisher
American Institute of Mathematical Sciences
Journal
Book Title
Series
Evolution Equations and Control Theory (EECT);v.3:no.1
PubMed ID
DOI
ISSN
2163-2480
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