Journal article
Valuing equity-linked death benefits and other contingent options: A discounted density approach
Insurance, mathematics & economics, Vol.51(1), pp.73-92
07/2012
DOI: 10.1016/j.insmatheco.2012.03.001
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. ► The paper presents a discounted probability density function approach to value equity-linked death benefits, such as Guaranteed Minimum Death Benefits in variable annuities. ► Closed-form formulas are obtained for various contingent options, for dynamic fund protection, and for dynamic withdrawal benefits. ► Closed-form formulas are found for the contingent call and put options when there is a fixed expiry date. From these, results for De Moivre’s law are obtained as limits. ► The paper also discusses equity-linked death benefit reserves and investment strategies for maintaining such reserves. ► The results in this paper are not restricted to valuing equity-linked benefits payable at the moment of death. The method is applicable to equity-linked benefits payable at the occurrence of a catastrophe and other cases.
Details
- Title: Subtitle
- Valuing equity-linked death benefits and other contingent options: A discounted density approach
- Creators
- Hans U Gerber - Faculty of Business and Economics, University of Lausanne, CH-1015 Lausanne, SwitzerlandElias S.W Shiu - Department of Statistics and Actuarial Science, The University of Iowa, Iowa City, IA 52242-1409, USAHailiang Yang - Department of Statistics and Actuarial Science, The University of Hong Kong, Pokfulam Road, Hong Kong
- Resource Type
- Journal article
- Publication Details
- Insurance, mathematics & economics, Vol.51(1), pp.73-92
- DOI
- 10.1016/j.insmatheco.2012.03.001
- ISSN
- 0167-6687
- eISSN
- 1873-5959
- Publisher
- Elsevier B.V
- Language
- English
- Date published
- 07/2012
- Academic Unit
- Statistics and Actuarial Science
- Record Identifier
- 9983985954602771
Metrics
25 Record Views