近年來，壽險公司在過去販售的保險商品開始面臨著重大的長壽風險。長壽風險的緣由來自於近代醫療保健和藥學的進步，人類平均壽命延長，導致壽險理賠金額高於發行時的預期而虧損。因此將此類風險轉移至資本市場的長壽型衍生性商品便孕育而生，相較於傳統的金融商品普遍以利率做折現，多數長壽衍生性商品的模型中包含死亡率與利率兩種隨機過程，導致訂價困難，包括死亡率在不同年齡層之間存在差異而產生的世代效應， 以及受重大災害引發的死亡率驟升，例如2020年全球爆發的Covid-19疫情，使全球死亡趨勢呈現跳躍的巨災風險。本篇模型同時考慮上述兩項難題，建立更加貼近現實情形的隨機死亡率模型，分別納入解決世代效應的多世代仿射過程，以及捕捉巨災死亡風險的雙指數跳躍擴散過程。最後將模型結合無套利Nelson-Siegel利率期限結構應用於長壽債券選擇權，數值結果顯示相較於原始長壽債券選擇權，採用本文死亡率模型對選擇權價值有顯著的正向影響。

Life insurance companies have recently been widely exposed to longevity risk, as the number of years people can live is longer than expected due to improvements in healthcare or medicine. Life insurers have also become aware of catastrophic mortality risks, such as COVID- 19 that occurred globally in 2020-2022, leading to a positive jump in the global mortality trend. While, it is obvious that survival probability varies by cohort, if people are born in the same year and in the same region, then their survival curves will be similar. Therefore, financial securitization like longevity-linked securities and derivatives, provides an important approach to transfer these risks from insurers to the capital market. This research extends the affine death rate model presented in the literature that does not consider catastrophic jump risk. When a mortality jump event occurs, our model applies a double exponential jump diffusion process to simulate catastrophic jumps in mortality trends. We also use our mortality rate model and an arbitrage-free Nelson-Siegel term structure to derive the formula of European options on survival bonds and show that catastrophic mortality jump and cohort effect significantly impact the option value.

Andersen, T., and Gestsson, M. (2020). Annuitization and aggregate mortality risk, Journal of Risk and Insurance, 88(1):79-99.
Andreev, K., and Vaupel, J. (2005). Patterns of mortality improvement over age and time in developed countries: Estimation, presentation, and implications for mortality forecasting. In Paper presented PAA annual meetings, Philadelphia.
Bahl, R. K. (2015). Price Bounds for the Swiss Re Mortality Bond 2003. University of Edinburgh. https://www.actuaries.org.uk/system/files/field/document/R%20Bahl%20PARTY2015.pdf.
Bahl, R., and Sabanis, S. (2021). Model-independent price bounds for Catastrophic Mortality Bonds, Insurance: Mathematics and Economics, 96: 276-291.
Bauer, D., Borger, M., and Russ, J. (2010). On the pricing of longevity-linked securities, Insurance: Mathematics and Economics, 46(1): 139-149.
Bauer, D., and Russ, J. (2006). Pricing longevity bonds using implied survival probabilities, presented Annual Meeting of the American Risk and Insurance Association.
Biffis, E. (2005). Affine processes for dynamic mortality and actuarial valuations, Insurance: Mathematics and Economics, 37:443-468.
Blake, D., Cairns, A., Dowd, K., and MacMinn, R. (2006). Longevity bonds: financial engineering, valuation, and hedging, The Journal of Risk and Insurance, 73(4): 647-672.
Blake, D., Boardman, T., and Cairns, A. (2010). The case for longevity bonds, Center for Retirement Research, Issues in Brief.
Blake, D., and Burrows, W. (2001). Survivor bonds: helping to hedge mortality risk, Journal of Risk and Insurance, 68(2): 339-348.
Blake, D., and Cairns, A. J. (2021). Longevity risk and capital markets: The 2019-20 update,
Insurance: Mathematics and Economics, 99:395-439.
Blackburn, C., and Sherris, M. (2013). Consistent dynamic affine mortality models for longevity risk applications, Insurance: Mathematics and Economics, 53:64-73.
Bravo, J. M., and Nunes, J. P. V. (2021). Pricing longevity derivatives via Fourier transforms, Insurance: Mathematics and Economics, 96:81-97.
Brouhns, N., Denuit, M., and Vermunt, J. (2002). A Poisson log-bilinear regression approach to the construction of projected lifetables, Insurance: Mathematics and Economics, 31:373-393.
Chan, W.S., Li, J.S.-H., and Li, J. (2014). The CBD mortality indexes: modeling and applications, North American Actuarial Journal, 18:38-58.
Chen, H., and Cox, S. H. (2009). Modeling mortality with jumps: applications to mortality securitization, Journal of Risk and Insurance, 76(3):727-751.
Christensen, J. H. E., Diebold, F. X., and Rudebusch, G. D. (2011). The affine arbitrage-free class of Nelson-Siegel term structure models, Journal of Econometrics, 164(2011):4-20.
Coughlan, G., Epstein, D., Sinha, A., and Honig, P. (2007). q-Forwards: Derivatives for transferring longevity and mortality risk, Technical Report, J.P. Morgan.
Cox, S. H., Lin, Y., and Pedersen, H. (2010). Mortality risk modeling: application to insurance securitization, Insurance: Mathematics and Economics, 46: 242-253.
Dahl, M., and Moller, T. (2006). Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance: Mathematics and Economics, 39(2): 193-217.
Deng, Y., Brockett P., and MacMinn, R. (2012). Longevity/Mortality risk modeling and securities pricing, Journal of Risk and Insurance, 79(3): 697-721.
Diebold, F. X., and Li, C. (2005). Forecasting the term structure of government bond yields, Journal of Econometrics, 130(2006): 337-364
Dowd, K., Blake, D., Cairns, A. J., and Dawson P. (2006). Survivor swaps, Journal of Risk and Insurance, 73(1): 1-17.
Gallop, A. (2008). Mortality projections in the United Kingdom. In Presented at the Living to 100 and Beyond Symposium, Orlando, Fla.
Jevtic, P., Luciano, E., and Vigna, E. (2013). Mortality surface by means of continuous time cohort models. Insurance: Mathematics and Economics, 53:122-133.
Kou, S. G., and Wang, H. (2004). Option pricing under a double exponential jump diffusion model, Management Science, 50(9): 1178-1192.
Lee, R. D., and Carter, L. R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87:659-671.
Leung, M., Man, C. F., and O’Hare, C. (2018). A comparative study of pricing approaches for longevity instruments. Insurance: Mathematics and Economics, 82:95-116.
Li, J. S., Li, J., Balasooriya, U., and Zhou, K. Q. (2013). Constructing out-of-the-money longevity hedges using parametric mortality indexes. North American Actuarial Journal, 1-32.
Li, H., Liu, H., Tang, Q., and Yuan, Z. (2023). Pricing extreme mortality risk in the wake of the COVID-19 pandemic. Insurance: Mathematics and Economics, 108:84-106.
Lin, Y., and Cox, S. H. (2008). Securitization of catastrophe mortality risks, Insurance: Mathematics and Economics, 42: 628-637.
Lippi, D., Bianucci, R., and Donell, S. (2020). Gender medicine: its historical roots. Postgraduate medical journal, 96(1138), 480–486.
Menoncin, F., and Regis, L. (2020). Optimal life-cycle labour supply, consumption, and investment: The role of longevity-linked assets. Journal of Banking and Finance, 120.
Renshaw, A., and Haberman, S. (2006). A cohort-based extension to the Lee-Carter model for mortality reduction factors. Insurance: Mathematics and Economics, 38:556-570.
Shen, Y., and Siu, T. K. (2013). Longevity bond pricing under stochastic interest rate and mortality with regime-switching, Insurance: Mathematics and Economics, 52:114-123.
Toole, J. (2007). May, Potential impact of pandemic influenza on the U.S. life insurance Industry, Research report, Society of Actuaries.
Xu, Y., Sherris, M., and Ziveyi, J. (2015). The application of affine processes in multi-cohort mortality models, Working Paper, UNSW Business School Research Paper.
Xu, Y., Sherris, M., and Ziveyi, J. (2019). Market price of longevity risk for a multi-cohort mortality model with application to longevity bond option pricing, Journal of Risk and Insurance, 20(20): 1-25.
Zanjani, G. (2002). Pricing and capital allocation in catastrophe insurance. Journal of Financial Economics, 65: 283-305.