Longevity risk constitutes an important risk factor for insurance companies and pension plans. For its analysis, but also for evaluating mortality-contingent structured financial products, modeling approaches allowing for uncertainties in mortality projections are needed. One model class that has attracted interest in applied research as well as among practitioners are forward mortality models, which are defined based on forecasts of survival probabilities as can be found in generation life tables and infer dynamics on the entire age/term structure, or forward surface, of mortality. However, thus far, there has been little guidance on identifying suitable specifications and their properties. The current paper provides a detailed analysis of forward mortality models driven by a finite-dimensional Brownian motion. In particular, after discussing basic properties, we present an infinite-dimensional formulation, and we examine the existence of finite-dimensional realizations for time-homogeneous deterministic volatility models, which are shown to possess important advantages for practical applications.


  1. stochastic mortality
  2. HJM framework
  3. Musiela parametrization
  4. translation semigroups
  5. finite-dimensional realization

MSC codes

  1. 91G80
  2. 62P05
  3. 60H15

Get full access to this article

View all available purchase options and get full access to this article.


P. Barrieu, H. Bensusan, N. El Karoui, C. Hillairet, S. Loisel, C. Ravanelli, and Y. Salhi (2012), Understanding, modelling and managing longevity risk: Key issues and main challenges, Scand. Actuar. J., 2012/3, pp. 203--231.
D. Bauer (2008), Stochastic Mortality Modeling and Securitization of Mortality Risk, IFA-Verlag, Ulm, Germany.
D. Bauer, M. Börger, and J. Ruŭshapeß (2010), On the Pricing of Longevity-Linked Securities, Insurance Math. Econom., 42, pp. 139--149.
E. Biffis (2005), Affine processes for dynamic mortality and actuarial valuation, Insurance Math. Econom., 37, pp. 443--468.
E. Biffis, M. Denuit, and P. Devolder (2010), Stochastic mortality under measure changes, Scand. Actuar. J., 2010/4, pp. 284--311.
T. Björk (1999), Arbitrage Theory in Continuous Time, Oxford University Press, Oxford, UK.
T. Björk (2003), On the Geometry of Interest Rate Models, Lecture Notes in Math. 1847, Springer, Berlin.
T. Björk and B. J. Christensen (1997), Interest rate dynamics and consistent forward rate curves, Math. Finance, 9, pp. 323--348.
T. Björk and A. Gombani (1999), Minimal realizations of interest rate models, Finance Stoch., 3, pp. 413--432.
T. Björk and L. Svensson (2001), On the existence of finite-dimensional realizations for nonlinear forward rate models, Math. Finance, 11, pp. 205--243.
D. Blake, A. J. G. Cairns, and K. Dowd (2008), The birth of the life market, Alternative Investments Quarterly, Fourth Quarter.
H. Booth (2006), Demographic forecasting: $1980$ to $2005$ in review, Internat. J. Forecasting, 22, pp. 547--581.
A. J. G. Cairns, D. Blake, and K. Dowd (2006a), Pricing death: Frameworks for the valuation and securitization of mortality risk, Astin Bull., 36, pp. 79--120.
A. J. G. Cairns, D. Blake, and K. Dowd (2006b), A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration, J. Risk Insur., 73, pp. 687--718.
R. Carmona (2007), HJM: A unified approach to dynamic models for fixed income, credit and equity markets, in Paris-Princeton Lectures on Mathematical Finance 2004, Lecture Notes in Math. 1919, Springer, Berlin, pp. 1--50.
R. Carmona and M. Tehranchi (2006), Interest Rate Models: An Infinite Dimensional Stochastic Analysis Perspective, Springer Finance, Springer, Berlin.
G. Da Prato and J. Zabczyk (1992), Stochastic Equations in Infinite Dimensions, Encyclopedia Math. Appl. 44, Cambridge University Press, Cambridge, UK.
M. Dahl (2004), Stochastic mortality in life insurance: Market reserves and mortality-linked insurance contracts, Insurance Math. Econom., 35, pp. 113--136.
M. Dahl and T. Mø ller (2006), Valuation and hedging of life insurance liabilities with systematic mortality risk, Insurance Math. Econom., 39, pp. 193--217.
P. Dawson, K. Dowd, A. J. G. Cairns, and D. Blake (2010), Survivor derivatives: A consistent pricing framework, J. Risk Insur., 77, pp. 579--596.
F. Delbaen and W. Schachermayer (1994), A general version of the fundamental theorem of asset pricing, Math. Ann., 300, pp. 463--520.
J. Duchassing and F. Suter (2009), Longevity: A “simple" stochastic modelling of mortality, Presentation at the 2009 LIFE Colloquium on behalf of Partner RE.
D. Duffie, D. Filipović, and W. Schachermayer (2003), Affine processes and applications in finance, Ann. Appl. Probab., 13, pp. 984--1053.
D. Duffie, J. Pan, and K. Singleton (2000), Transform analysis and asset pricing for affine jump-diffusions, Econometrica, 68, pp. 1343--1376.
E. Eberlein and S. Raible (1999), Term structure models driven by general Lévy processes, Math. Finance, 9, pp. 31--54.
K,-J. Engel and R. Nagel (2000), One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, New York.
D. Filipović (2001), Consistency Problems for Heath-Jarrow-Morton Interest Rate Models, Lecture Notes in Math. 1760, Springer, Berlin.
D. Filipović and J. Teichmann (2002), On Finite-Dimensional Term Structure Models, Working paper, Princeton University, Princeton, NJ.
M. Harrison and D. Kreps (1979), Martingales and arbitrage in multiperiod security markets, J. Econom. Theory, 20, pp. 381--408.
D. Heath, R. Jarrow, and R. Morton (1992), Bond pricing and the term structure of interest rates: A new methodology for contingent claim valuation, Econometrica, 60, pp. 77--105.
D. Lando (1998), On Cox processes and credit risky securities, Rev. Derivatives Research, 2, pp. 99--120.
R. D. Lee and L. R. Carter (1992), Modeling and forecasting U.S. mortality, J. Amer. Statist. Assoc., 87, pp. 659--671.
M. Ludkovski and E. Bayraktar (2009), Relative hedging of systematic mortality risk, N. Am. Actuar. J., 13, pp. 106--140.
M. A. Milevsky and S. D. Promislow (2001), Mortality derivatives and the option to annuitize, Insurance Math. Econom., 29, pp. 299--318.
K. R. Miltersen and S. A. Persson (2005), Is Mortality Dead? Stochastic Force of Mortality Determined by No Arbitrage, Working paper, Norwegian School of Economics and Business Administration, Bergen, Norway.
M. Musiela (1993), Stochastic PDEs and term structure models, Journées Internationales de Finance, IGR-AFFI, La Baule.
R. Norberg (2010), Forward mortality and other vital rates---are they the way forward?, Insurance Math. Econom., 47, pp. 105--112.
D. Schrager (2006), Affine stochastic mortality, Insurance Math. Econom., 38, pp. 81--97.
T. Vargiolu (1999), Invariant measures for the Musiela equation with deterministic diffusion term, Finance Stoch., 3, pp. 483--492.
N. Zhu and D. Bauer (2011a), Coherent Modeling of the Risk in Mortality Projections: A Semi-parametric Approach, Working paper, Georgia State University, Atlanta, GA.
N. Zhu and D. Bauer (2011b), Applications of forward mortality factor models in life insurance practice, Geneva Papers on Risk and Insurance -- Issues and Practice, 36, pp. 567--594.

Information & Authors


Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 639 - 666
ISSN (online): 1945-497X


Submitted: 14 December 2010
Accepted: 28 June 2012
Published online: 11 October 2012


  1. stochastic mortality
  2. HJM framework
  3. Musiela parametrization
  4. translation semigroups
  5. finite-dimensional realization

MSC codes

  1. 91G80
  2. 62P05
  3. 60H15



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.