Long Term Risk: A Martingale Approach

We extend long-term factorization of the pricing kernel due to Alvarez and Jermann (2005) in discrete time ergodic environments and Hansen and Scheinkman (2009) in continuous ergodic Markovian environments to general semimartingale environments, without assuming the Markov property. An easy to verify sufficient condition is given that guarantees convergence in semimartingale topology of trading strategies that invest in T-maturity pure discount bonds to the long bond and convergence in total variation of T-forward measures to the long forward measure. We explicitly construct long-term factorization in Heath-Jarrow-Morton (1992) models and decompose the market price of Brownian risk into the volatility of the long bond plus an additional risk premium defining the permanent martingale component in the long-term factorization. When Markovian and ergodicity assumptions are added, we recover Hansen and Scheinkman (2009) results linking the long-term factorization with the principal eigenfunction of the pricing operator, and explicitly identify their distorted probability measure with the long forward measure.