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Article: Valuing equity-linked death benefits and other contingent options: A discounted density approach
Title | Valuing equity-linked death benefits and other contingent options: A discounted density approach |
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Authors | |
Keywords | Discounted Density Equity-Linked Death Benefits Exponential Stopping Minimum Guaranteed Death Benefits Option Pricing Variable Annuities |
Issue Date | 2012 |
Publisher | Elsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/ime |
Citation | Insurance: Mathematics and Economics, 2012, v. 51 n. 1, p. 73-92 How to Cite? |
Abstract | Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment turns out to be the payoff of an option, we call the contract for the payment a (life) contingent option. Because each time-until-death distribution can be approximated by a combination of exponential distributions, the analysis is made for the case where the time until death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The time-until-death random variable is assumed to be independent of the stock price process which is a geometric Brownian motion. Our key tool is a discounted joint density function. A substantial series of closed-form formulas is obtained, for the contingent call and put options, for lookback options, for barrier options, for dynamic fund protection, and for dynamic withdrawal benefits. In a section on several stocks, the method of Esscher transforms proves to be useful for finding among others an explicit result for valuing contingent Margrabe options or exchange options. For the case where the contracts have a finite expiry date, closed-form formulas are found for the contingent call and put options. From these, results for De Moivre's law are obtained as limits. We also discuss equity-linked death benefit reserves and investment strategies for maintaining such reserves. The elasticity of the reserve with respect to the stock price plays an important role. Whereas in the most important applications the stopping time is the time of death, it could be different in other applications, for example, the time of the next catastrophe. © 2012 Elsevier B.V. |
Persistent Identifier | http://hdl.handle.net/10722/172493 |
ISSN | 2021 Impact Factor: 2.168 2020 SCImago Journal Rankings: 1.139 |
ISI Accession Number ID | |
References |
DC Field | Value | Language |
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dc.contributor.author | Gerber, HU | en_US |
dc.contributor.author | Shiu, ESW | en_US |
dc.contributor.author | Yang, H | en_US |
dc.date.accessioned | 2012-10-30T06:22:47Z | - |
dc.date.available | 2012-10-30T06:22:47Z | - |
dc.date.issued | 2012 | en_US |
dc.identifier.citation | Insurance: Mathematics and Economics, 2012, v. 51 n. 1, p. 73-92 | en_US |
dc.identifier.issn | 0167-6687 | en_US |
dc.identifier.uri | http://hdl.handle.net/10722/172493 | - |
dc.description.abstract | Motivated by the Guaranteed Minimum Death Benefits in various deferred annuities, we investigate the calculation of the expected discounted value of a payment at the time of death. The payment depends on the price of a stock at that time and possibly also on the history of the stock price. If the payment turns out to be the payoff of an option, we call the contract for the payment a (life) contingent option. Because each time-until-death distribution can be approximated by a combination of exponential distributions, the analysis is made for the case where the time until death is exponentially distributed, i.e., under the assumption of a constant force of mortality. The time-until-death random variable is assumed to be independent of the stock price process which is a geometric Brownian motion. Our key tool is a discounted joint density function. A substantial series of closed-form formulas is obtained, for the contingent call and put options, for lookback options, for barrier options, for dynamic fund protection, and for dynamic withdrawal benefits. In a section on several stocks, the method of Esscher transforms proves to be useful for finding among others an explicit result for valuing contingent Margrabe options or exchange options. For the case where the contracts have a finite expiry date, closed-form formulas are found for the contingent call and put options. From these, results for De Moivre's law are obtained as limits. We also discuss equity-linked death benefit reserves and investment strategies for maintaining such reserves. The elasticity of the reserve with respect to the stock price plays an important role. Whereas in the most important applications the stopping time is the time of death, it could be different in other applications, for example, the time of the next catastrophe. © 2012 Elsevier B.V. | en_US |
dc.language | eng | en_US |
dc.publisher | Elsevier BV. The Journal's web site is located at http://www.elsevier.com/locate/ime | en_US |
dc.relation.ispartof | Insurance: Mathematics and Economics | en_US |
dc.rights | NOTICE: this is the author’s version of a work that was accepted for publication in Insurance: Mathematics And Economics. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Insurance: Mathematics And Economics, 2012, v. 51 n. 1, p. 73-92. DOI: 10.1016/j.insmatheco.2012.03.001 | - |
dc.rights | This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. | - |
dc.subject | Discounted Density | en_US |
dc.subject | Equity-Linked Death Benefits | en_US |
dc.subject | Exponential Stopping | en_US |
dc.subject | Minimum Guaranteed Death Benefits | en_US |
dc.subject | Option Pricing | en_US |
dc.subject | Variable Annuities | en_US |
dc.title | Valuing equity-linked death benefits and other contingent options: A discounted density approach | en_US |
dc.type | Article | en_US |
dc.identifier.email | Yang, H: hlyang@hku.hk | en_US |
dc.identifier.authority | Yang, H=rp00826 | en_US |
dc.description.nature | postprint | en_US |
dc.identifier.doi | 10.1016/j.insmatheco.2012.03.001 | en_US |
dc.identifier.scopus | eid_2-s2.0-84859339556 | en_US |
dc.identifier.hkuros | 218762 | - |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-84859339556&selection=ref&src=s&origin=recordpage | en_US |
dc.identifier.volume | 51 | en_US |
dc.identifier.issue | 1 | en_US |
dc.identifier.spage | 73 | en_US |
dc.identifier.epage | 92 | en_US |
dc.identifier.eissn | 1873-5959 | - |
dc.identifier.isi | WOS:000305259800008 | - |
dc.publisher.place | Netherlands | en_US |
dc.identifier.scopusauthorid | Gerber, HU=7202185517 | en_US |
dc.identifier.scopusauthorid | Shiu, ESW=6603568601 | en_US |
dc.identifier.scopusauthorid | Yang, H=7406559537 | en_US |
dc.identifier.citeulike | 10487941 | - |
dc.identifier.issnl | 0167-6687 | - |