Hostname: page-component-77f85d65b8-6c7dr Total loading time: 0 Render date: 2026-03-28T14:44:02.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Super-replication of life-contingent options under the Black–Scholes framework

Part of: Mathematical finance Stochastic analysis

Published online by Cambridge University Press:  05 April 2024

Ze-An Ng ,
You-Beng Koh Open the ORCID record for You-Beng Koh [Opens in a new window] ,
Tee-How Loo  and
Hailiang Yang
Ze-An Ng*
Affiliation:
Universiti Malaya
You-Beng Koh*
Affiliation:
Universiti Malaya
Tee-How Loo*
Affiliation:
Universiti Malaya
Hailiang Yang*
Affiliation:
Xi’an Jiaotong-Liverpool University
*
*Postal address: Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia.
*Postal address: Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia.
*Postal address: Institute of Mathematical Sciences, Faculty of Science, Universiti Malaya, 50603, Kuala Lumpur, Malaysia.
*****Postal address: Department of Financial and Actuarial Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, China. Email: hailiang.yang@xjtlu.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the super-replication problem for a class of exotic options known as life-contingent options within the framework of the Black–Scholes market model. The option is allowed to be exercised if the death of the option holder occurs before the expiry date, otherwise there is a compensation payoff at the expiry date. We show that there exists a minimal super-replication portfolio and determine the associated initial investment. We then give a characterisation of when replication of the option is possible. Finally, we give an example of an explicit super-replicating hedge for a simple life-contingent option.

Keywords

Exotic optionsoption pricingoption hedgingstochastic analysis

MSC classification

Primary: 91G20: Derivative securities
Secondary: 60H30: Applications of stochastic analysis (to PDE, etc.)

Information

Type
Original Article
Information
Journal of Applied Probability , Volume 61 , Issue 4 , December 2024 , pp. 1263 - 1277
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Elliott, R. J., and Kopp, P. E. (2005). Mathematics of Financial Markets, 2nd edn. Springer, New York.Google Scholar
Eyraud-Loisel, A., and Royer-Carenz, M. (2010). BSDEs with random terminal time under enlarged filtration. American-style options hedging by an insider. Random Operators Stoch. Equat. 18, 141–163.CrossRefGoogle Scholar
Gerber, H. U., Shiu, E. S. W., and Yang, H. (2012). Valuing equity-linked death benefits and other contingent options: A discounted density approach. Insurance Math. Econom. 51, 73–92.CrossRefGoogle Scholar
Gerber, H. U., Shiu, E. S. W., and Yang, H. (2013). Valuing equity-linked death benefits in jump diffusion models. Insurance Math. Econom. 53, 615–623.CrossRefGoogle Scholar
Gerber, H. U., Shiu, E. S. W., and Yang, H. (2015). Geometric stopping of a random walk and its applications to valuing equity-linked death benefits. Insurance Math. Econom. 64, 313–325.CrossRefGoogle Scholar
Kelani, A., and Quittard-Pinon, F. (2017). Pricing and hedging variable annuities in a Lévy market: A risk management perspective. J. Risk Insurance 84, 209–238.CrossRefGoogle Scholar
Leao, D. Jr, Fragoso, M., and Ruffino, P. (2004). Regular conditional probability, disintegration of probability and Radon spaces. Proyecciones 23, 15–19.Google Scholar
Pascucci, A. (2011). PDE and Martingale Methods in Option Pricing. Springer, New York.CrossRefGoogle Scholar
Wang, W., Shen, Y., Qian, L., and Yang, Z. (2022). Hedging strategy for unit-linked life insurance contracts with self-exciting jump clustering. J. Industrial Management Optimization 18, 2369–2399.CrossRefGoogle Scholar
Wang, Y. (2009). Quantile hedging for guaranteed minimum death benefits. Insurance Math. Econom. 45, 449–458.CrossRefGoogle Scholar
Yong, J., and Zhou, X. Y. (1999). Stochastic Controls: Hamiltonian Systems and HJB Equations. Springer, New York.CrossRefGoogle Scholar
Zhang, J. (2017). Backward Stochastic Differential Equations. Springer, New York.CrossRefGoogle Scholar
Zhang, Z., Yong, Y., and Yu, W. (2020). Valuing equity-linked death benefits in general exponential Lévy models. J. Comput. Appl. Math. 365, 112–377.CrossRefGoogle Scholar