# Real-world forward rate dynamics with affine realizations

## TL;DR

This paper extends the understanding of affine realizations in interest rate models driven by Lévy processes from the risk-neutral to the real-world measure, revealing conditions under which such models are feasible.

## Contribution

It demonstrates that affine realizations valid under risk-neutral measures also hold under the real-world measure for Wiener and certain jump processes, with restrictions for infinite activity jumps.

## Key findings

- Affine realizations transfer from risk-neutral to real-world measures for Wiener-driven models.
- Models with finite activity jumps also admit affine realizations under the real-world measure.
- Infinite activity jumps impose strict restrictions on the market price of risk, often requiring it to be constant.

## Abstract

We investigate the existence of affine realizations for L\'{e}vy driven interest rate term structure models under the real-world probability measure, which so far has only been studied under an assumed risk-neutral probability measure. For models driven by Wiener processes, all results obtained under the risk-neutral approach concerning the existence of affine realizations are transferred to the general case. A similar result holds true for models driven by compound Poisson processes with finite jump size distributions. However, in the presence of jumps with infinite activity we obtain severe restrictions on the structure of the market price of risk; typically, it must even be constant.