Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order

Schlögl, E

Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order

Schlögl, E

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Publication Type:: Journal Article
Citation:: Journal of Economic Dynamics and Control, 2013, 37 (3), pp. 611 - 632
Issue Date:: 2013-03-01

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Field	Value	Language
dc.contributor.author	Schlögl, E https://orcid.org/0000-0001-8898-6879	en_US
dc.date.issued	2013-03-01	en_US
dc.identifier.citation	Journal of Economic Dynamics and Control, 2013, 37 (3), pp. 611 - 632	en_US
dc.identifier.issn	0165-1889	en_US
dc.identifier.uri	http://hdl.handle.net/10453/23442
dc.description.abstract	If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner. © 2012 Elsevier B.V.	en_US
dc.relation.ispartof	Journal of Economic Dynamics and Control	en_US
dc.relation.isbasedon	10.1016/j.jedc.2012.10.001	en_US
dc.subject.classification	Economics	en_US
dc.title	Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order	en_US
dc.type	Journal Article
utslib.citation.volume	3	en_US
utslib.citation.volume	37	en_US
utslib.for	1502 Banking, Finance and Investment	en_US
utslib.for	1402 Applied Economics	en_US
utslib.for	1401 Economic Theory	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Science
pubs.organisational-group	/University of Technology Sydney/Faculty of Science/School of Mathematical and Physical Sciences
pubs.organisational-group	/University of Technology Sydney/Strength - QFRC - Quantitative Finance Research Centre
utslib.copyright.status	open_access
pubs.issue	3	en_US
pubs.publication-status	Published	en_US
pubs.volume	37	en_US

Abstract:

If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner. © 2012 Elsevier B.V.

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http://hdl.handle.net/10453/23442