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Long Term Risk: A Martingale Approach

2014/11/12 by Likuan Qin, Vadim Linetsky, Qin, Likuan +1 · 2 citations
Economics, Econometrics and Finance · Social Sciences · #Economic theories and models #FOS: Economics and business #General Economics (econ.GN) #Insurance, Mortality, Demography, Risk Management #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1411.3078

openalex publication_date 2014/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper extends the long-term factorization of the stochastic discount factor introduced and studied by Alvarez and Jermann (2005) in discretetime ergodic environments and by Hansen and Scheinkman (2009) and Hansen (2012) in Markovian environments to general semimartingale environments. The transitory component discounts at the stochastic rate of return on the long bond and is factorized into discounting at the long-term yield and a positive semimartingale that extends the principal eigenfunction of Hansen and Scheinkman (2009) to the semimartingale setting. The permanent component is a martingale that accomplishes a change of probabilities to the long forward measure, the limit of T-forward measures. The change of probabilities from the data generating to the long forward measure absorbs the long-term risk-return trade-off and interprets the latter as the long-term risk-neutral measure.

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