Ultra Long Run Term Structure Models

The ultra long run (ULR) discount rates are key inputs for valuing pension funds, life annuities, or firms with long run investments due to low carbon transition. However the corresponding zero-coupon bonds are only actively traded up to a last liquid point (LLP) such as 20 years, say. This leads to “extrapolating” the term structures beyond this LLP. The current extrapo- lation approaches have at least two drawbacks: i) they are not compatible with the theoretical literature, in particular not arbitrage free; ii) they fix a deterministic ultimate forward rate, that is they set to zero the risk on the ULR rate.This paper proposes a solution to these issues by considering sequences of arbitrage free markets. It explains how such sequences can be justified in a context effect and how to implement them. This fills the gap between the current practice in Solvency II and the theoretical literature