A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics

Garces, LPDM; Cheang, GHL

A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics

Garces, LPDM Cheang, GHL

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Publication Type:: Working Paper
Citation:: arXiv, 2020
Issue Date:: 2020-02-24

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Field	Value	Language
dc.contributor.author	Garces, LPDM
dc.contributor.author	Cheang, GHL
dc.date.accessioned	2023-04-07T12:12:23Z
dc.date.available	2023-04-07T12:12:23Z
dc.date.issued	2020-02-24
dc.identifier.citation	arXiv, 2020
dc.identifier.uri	http://hdl.handle.net/10453/169382
dc.description.abstract	We price European and American exchange options where the underlying asset prices are modelled using a Merton (1976) jump-diffusion with a common Heston (1993) stochastic volatility process. Pricing is performed under an equivalent martingale measure obtained by setting the second asset yield process as the numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice for the numeraire reduces the exchange option pricing problem, a two-dimensional problem, to pricing a call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratio process and the instantaneous variance process is then determined from the corresponding Kolmogorov backward equation via integral transforms. We then determine integral representations for the European exchange option price and the early exercise premium and state a linked system of integral equations that characterizes the American exchange option price and the associated early exercise boundary. Properties of the early exercise boundary near maturity are also discussed.
dc.language	en
dc.relation.ispartof	arXiv
dc.rights	info:eu-repo/semantics/restrictedAccess
dc.title	A Put-Call Transformation of the Exchange Option Problem under Stochastic Volatility and Jump Diffusion Dynamics
dc.type	Working Paper
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
utslib.copyright.status	recently_added	*
dc.date.updated	2023-04-07T12:12:21Z

Abstract:

We price European and American exchange options where the underlying asset prices are modelled using a Merton (1976) jump-diffusion with a common Heston (1993) stochastic volatility process. Pricing is performed under an equivalent martingale measure obtained by setting the second asset yield process as the numeraire asset, as suggested by Bjerskund and Stensland (1993). Such a choice for the numeraire reduces the exchange option pricing problem, a two-dimensional problem, to pricing a call option written on the ratio of the yield processes of the two assets, a one-dimensional problem. The joint transition density function of the asset yield ratio process and the instantaneous variance process is then determined from the corresponding Kolmogorov backward equation via integral transforms. We then determine integral representations for the European exchange option price and the early exercise premium and state a linked system of integral equations that characterizes the American exchange option price and the associated early exercise boundary. Properties of the early exercise boundary near maturity are also discussed.

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http://hdl.handle.net/10453/169382