Real-world forward rate dynamics with affine realizations
Eckhard Platen, Stefan Tappe
TL;DR
This work analyzes when Lévy-driven HJMM forward-rate dynamics admit affine realizations under the real-world probability measure within the Benchmark Approach. A central result states that an affine realization exists iff the risk-neutral HJMM has an affine realization and the finite-dimensional condition $\dim U_{\Psi,\gamma} < \infty$ holds, where $U_{\Psi,\gamma}$ aggregates jump-related market-risk terms. In the Wiener-only case, the real-world and risk-neutral criteria coincide, enabling a direct transfer of known results; with infinite-activity jumps, strong restrictions on the market price of risk typically arise, often forcing it to be constant. The paper provides explicit sufficiency conditions (e.g., finite-jump, constant volatility structures, or quasi-exponential volatilities) and necessary conditions that illuminate the delicate balance between real-world dynamics and affine tractability, guiding practical modeling choices under the Benchmark Approach.
Abstract
We investigate the existence of affine realizations for Lévy 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.