[Paper Review] The Hull-White Model under Knightian Uncertainty about the Volatility

Julian Hölzermann|arXiv (Cornell University)|Aug 10, 2018

Stochastic processes and financial applicationsEconomics, Econometrics and Finance3 citations

TL;DR

This paper develops an arbitrage-free short rate model under Knightian uncertainty about volatility by employing a set of priors that induce G-Brownian motion. It characterizes the entire term structure without arbitrage, showing that zero-coupon bond pricing must be redefined under this framework to ensure consistency with uncertainty-averse preferences.

ABSTRACT

We construct an arbitrage-free short rate model under Knightian uncertainty about the volatility. The uncertainty is represented by a set of priors, which naturally leads to a G-Brownian motion. Within this framework, it is shown how to characterize the whole term structure without admitting arbitrage. The pricing of zero-coupon bonds in such a setting differs substantially from traditional models, since the prices need to be chosen in a different way in order to exclude arbitrage.

Motivation & Objective

• To model interest rate dynamics under Knightian uncertainty regarding volatility, where uncertainty is represented by a set of priors.
• To develop a term structure model that remains arbitrage-free despite ambiguity in volatility parameters.
• To reformulate zero-coupon bond pricing under Knightian uncertainty to exclude arbitrage opportunities.
• To characterize the full yield curve under volatility ambiguity using a non-linear expectation framework.
• To extend traditional interest rate models by incorporating model uncertainty via G-Brownian motion.

Proposed method

• Represent Knightian uncertainty about volatility through a set of probability measures (priors), leading to a G-expectation framework.
• Model the short rate dynamics using a G-Brownian motion to capture volatility ambiguity in the diffusion coefficient.
• Derive the bond pricing equation under the G-Brownian motion by solving a fully nonlinear PDE arising from the G-expectation.
• Ensure the absence of arbitrage by requiring that the bond price process satisfies the dynamic consistency condition under the G-Girsanov transformation.
• Characterize the term structure as the solution to a fully nonlinear parabolic PDE under the G-Brownian motion framework.
• Use the viscosity solution approach to establish the existence and uniqueness of the bond price under the G-framework.

Experimental results

Research questions

• RQ1 How can a short rate model be constructed under Knightian uncertainty about volatility while preserving no-arbitrage conditions?
• RQ2 What is the impact of volatility ambiguity on the term structure of interest rates in a no-arbitrage setting?
• RQ3 How does the pricing of zero-coupon bonds differ under Knightian uncertainty compared to classical models?
• RQ4 What mathematical framework is required to represent volatility uncertainty in interest rate models without introducing arbitrage?
• RQ5 Can the full yield curve be consistently characterized under a G-Brownian motion framework with Knightian uncertainty?

Key findings

• The model constructs a term structure under Knightian uncertainty by embedding volatility ambiguity into the diffusion coefficient via a set of priors.
• The use of G-Brownian motion allows for a non-linear expectation framework that naturally accommodates volatility uncertainty.
• Zero-coupon bond prices must be determined through a fully nonlinear PDE rather than a linear PDE, reflecting the impact of model uncertainty.
• Arbitrage-freeness is preserved by ensuring the bond price process is a martingale under the G-Girsanov measure.
• The entire yield curve is characterized as the solution to a fully nonlinear parabolic PDE, which generalizes the classical Hull-White model.
• The framework provides a robust pricing mechanism that accounts for ambiguity in volatility without requiring a single probability measure.