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A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation

We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula’s theorem, which states that a truncated model may exhibit poor statistical properties, we show...

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Detalles Bibliográficos
Autores principales: Popescu, Dan M., Lipan, Ovidiu
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Public Library of Science 2015
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4308111/
https://www.ncbi.nlm.nih.gov/pubmed/25625856
http://dx.doi.org/10.1371/journal.pone.0116752
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author Popescu, Dan M.
Lipan, Ovidiu
author_facet Popescu, Dan M.
Lipan, Ovidiu
author_sort Popescu, Dan M.
collection PubMed
description We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula’s theorem, which states that a truncated model may exhibit poor statistical properties, we show that for a third-order Kramers-Moyal truncation model of an option’s and its underlier’s price, important properties emerge: (i) the option price can be written in a closed analytical form that involves the Airy function, (ii) the price is a positive function for positive skewness in the distribution, (iii) for negative skewness, the price becomes negative only for price values that are close to zero. Moreover, using third-order noise in option valuation reveals additional properties: (iv) the inconsistencies between two popular option pricing approaches (using a “delta-hedged” portfolio and using an option replicating portfolio) that are otherwise equivalent up to the second moment, (v) the ability to develop a measure R of how accurately an option can be replicated by a mixture of the underlying stocks and cash, (vi) further limitations of second-order models revealed by introducing third-order noise.
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spelling pubmed-43081112015-02-06 A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation Popescu, Dan M. Lipan, Ovidiu PLoS One Research Article We propose the use of the Kramers-Moyal expansion in the analysis of third-order noise. In particular, we show how the approach can be applied in the theoretical study of option valuation. Despite Pawula’s theorem, which states that a truncated model may exhibit poor statistical properties, we show that for a third-order Kramers-Moyal truncation model of an option’s and its underlier’s price, important properties emerge: (i) the option price can be written in a closed analytical form that involves the Airy function, (ii) the price is a positive function for positive skewness in the distribution, (iii) for negative skewness, the price becomes negative only for price values that are close to zero. Moreover, using third-order noise in option valuation reveals additional properties: (iv) the inconsistencies between two popular option pricing approaches (using a “delta-hedged” portfolio and using an option replicating portfolio) that are otherwise equivalent up to the second moment, (v) the ability to develop a measure R of how accurately an option can be replicated by a mixture of the underlying stocks and cash, (vi) further limitations of second-order models revealed by introducing third-order noise. Public Library of Science 2015-01-27 /pmc/articles/PMC4308111/ /pubmed/25625856 http://dx.doi.org/10.1371/journal.pone.0116752 Text en © 2015 Popescu, Lipan http://creativecommons.org/licenses/by/4.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are properly credited.
spellingShingle Research Article
Popescu, Dan M.
Lipan, Ovidiu
A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title_full A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title_fullStr A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title_full_unstemmed A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title_short A Kramers-Moyal Approach to the Analysis of Third-Order Noise with Applications in Option Valuation
title_sort kramers-moyal approach to the analysis of third-order noise with applications in option valuation
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4308111/
https://www.ncbi.nlm.nih.gov/pubmed/25625856
http://dx.doi.org/10.1371/journal.pone.0116752
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