Pricing Survivor Swaps with Mortality Jumps and Default Risk

考慮隨機跳躍與違約風險下存活交換合約之定價

10.29628/AEP.201006.0001

張傳章(Chuang-Chang Chang)；陳志展(Chih-Chan Chen)；蔡明宏(Min-Hung Tsay)

存活交換合約 ； 死亡率相依衍生性商品 ； 信用風險 ； 王氏轉換 ； 死亡率跳躍 ； Survivor swaps ； Mortality derivatives ； Default risk ； Wang transform ； Mortality jumps

經濟論文

38卷2期（2010 / 06 / 01）

119 - 156

英文

本文利用Cox et al. (2006)所發展之模型並同時納入考慮隨機跳躍的死亡率過程與違約風險來進行存活交換合約之定價。白本文評價結果發現：市場風險價格與死亡率之隨機跳躍頻率皆與存活交換合約價格存在正向關係，且與前者的正向關係亦隨死亡率之隨機跳躍頻率上升而增強。再者我們亦發現在所考慮的各種違約風險程度下市場風險價格與死亡率之隨機跳躍頻率與存活交換合約價格之正向關係依然存在。另一方面，在其他條件不變下本文發現高（低）違約風險隱含相對高（低）的存活交換合約價格。上述結果可以清楚地提供合約發行機構與投資人於發行或購買合約時進行訂定或判斷合約價格所需考量之價格攸關因素。

A survivor swap refers to an agreement for the future exchange of cash flows based upon a mortality-dependent index. Pricing survivor swaps involves the determination of a fixed proportional premium, π, which results in an initial zero value of the swaps for each party. In order to provide a precise valuation of a mortality derivative, such as a survivor swap, an appropriate model is necessary for the forecasting of the mortality rate. The model proposed for use in this study extends the model provided by Cox et al. (2006) by simultaneously considering mortality jumps and default risk. Using this extended model to price survivor swaps in this study, we mainly find from our simulated results, based upon the data collected, that the survivor swap premium has a positive correlation with both the market price of risk and the frequency of jumps. Furthermore, we find that a stronger jump frequency will enhance the effects of the market price of risk on the premium. On the other hand, the survivor swap premium is also found to have a positive correlation with the market price of risk and the frequency of jumps at any level of default risk. However, when the market price of risk and jump frequency levels are both fixed, a higher (lower) default risk will imply a lower (higher) premium.

1. Wang, S.-S. (2006), "Normalized Exponential Tilting: Pricing and Measuring Multivariate Risks," Working Paper, Georgia State University.
2. Madan, D. and H. Unal (2004), "Risk-Neutralizing Statistical Distributions: With an Application to Pricing Reinsurance Contracts on FDIC Losses," FDIC Center for Financial Research Working Paper, No. 2004–01.
3. Blake, D.,Burrows, W.(2001).Survivor Bonds: Helping to Hedge Mortality Risk.Journal of Risk and Insurance,68,339-348.
4. Blake, D.,Cairns, A. J. G.,Dowd, K.(2006).Living with Mortality: Longevity Bonds and Other Mortality-Linked Securities.British Actuarial Journal,12,153-197.
5. Cox, S. H.,Lin, Y.,Wang, S.-S.(2006).Multivariate Exponential Tilting and Pricing Implications for Mortality Securitization.Journal of Risk and Insurance,73,719-736.
6. Dowd, K.(2003).Survivor Bonds: A Comment on Blake and Burrows.Journal of Risk and Insurance,70,339-348.
7. Dowd, K.,Blake, D.,Cairns, A. J. G.,Dawson, P.(2006).Survivor Swaps.Journal of Risk and Insurance,73,1-17.
8. Jarrow, R. A.,Turnbull, S. M.(1995).Pricing Derivatives on Financial Securities Subject to Credit Risk.Journal of Finance,50,53-85.
9. Jarrow, R. A.,Yu, F.(2001).Counterparty Risk and the Pricing of Defaultable Securities.Journal of Finance,56,1765-1798.
10. Lin, Y.,Cox, S. H.(2008).Securitization of Catastrophe Mortality Risks.Insurance: Mathematics and Economics,42,628-637.
11. Lin, Y.,Cox, S. H.(2005).Securitization of Mortality Risks in Life Annuities.Journal of Risk and Insurance,72,227-252.
12. Merton, R.(1976).Option Pricing When Underlying Stock Returns Are Discontinuous.Journal of Financial Economics,3,125-144.
13. Michell, O. S.,Poterba, J. M.,Warshawsky, M. J.,Brown, J. R.(1999).New Evidence on the Money's Worth of Individual Annuities.American Economic Review,89,1299-1318.
14. Vasicek, O.(1977).An Equilibrium Characterization of the Term Structure.Journal of Financial Economics,5,177-188.
15. Wang, S.-S.(2002).A Universal Framework for Pricing Financial and Insurance Risks.ASTIN Bulletin,32,213-234.
16. Wang, S.-S.(2000).A Class of Distortion Operators for Pricing Financial and Insurance Risks.Journal of Risk and Insurance,67,15-36.
17. Wang, S.-S.(1996).Premium Calculation by Transforming the Layer Premium Density.ASTIN Bulletin,26,71-92.

1. 葉仕國、張森林、林丙輝(2016)。台灣衍生性金融商品定價、避險與套利文獻回顧與展望。臺大管理論叢，27(1)，255-304。