Abstract
In this paper, we study term structure movements in the spirit of Heath et al. (Econometrica 60(1):77–105, 1992) under volatility uncertainty. We model the instantaneous forward rate as a diffusion process driven by a G-Brownian motion. The G-Brownian motion represents the uncertainty about the volatility. Within this framework, we derive a sufficient condition for the absence of arbitrage, known as the drift condition. In contrast to the traditional model, the drift condition consists of several equations and several market prices, termed market price of risk and market prices of uncertainty, respectively. The drift condition is still consistent with the classical one if there is no volatility uncertainty. Similar to the traditional model, the risk-neutral dynamics of the forward rate are completely determined by its diffusion term. The drift condition allows to construct arbitrage-free term structure models that are completely robust with respect to the volatility. In particular, we obtain robust versions of classical term structure models.
Highlights
Most approaches to volatility modeling in mathematical finance are subject to model uncertainty, termed volatility uncertainty, which can be tamed by making models robust with respect to the volatility
In the traditional HJM model, the absence of arbitrage on the related bond market is ensured by the HJM drift condition, which assumes the existence of a market price of risk and characterizes the drift of the forward rate in terms of its diffusion coefficient
The drift condition fully characterizes the risk-neutral dynamics of the forward rate in terms of its diffusion term
Summary
Most approaches to volatility modeling in mathematical finance are subject to model uncertainty, termed volatility uncertainty, which can be tamed by making models robust with respect to the volatility. Similar to the traditional HJM model, the main result of the present work is a drift condition which implies that the related bond market is arbitrage-free. Due to the main result of the present work, we are able to obtain arbitrage-free term structure models in the presence of volatility uncertainty by considering specific examples. The present work is the first in the literature on robust finance that provides a general HJM framework for arbitrage-free term structure modeling in the presence of volatility uncertainty. We start from a probabilistic setting similar to the one of Denis and Martini [11], since it is a natural approach to represent volatility uncertainty from an economic point of view, and use the results of Denis, Hu, and Peng [10] to acquire all results from the calculus of G-Brownian motion. Section B in the Appendix provides a sufficient condition for the discounted bonds to be well-posed
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
