Arbitrage-Free Valuation of Interest Rate Securities under Forward Curves with Stochastic Speed & Acceleration

Abstract

Arbitrage-free models for valuing interest rate securities posit that stochastic changes in spot or forward interest rates (forward rate "speed") follow a diffusion process. This paper extends the Heath, Jarrow and Morton (1992, HJM) framework by allowing diffusive shocks to both the "speed" and "acceleration" of forward rates. The arbitrage-free restriction on forward rates is identified and involves volatilities of the speed and acceleration dynamics and their correlation. Although the extended forward rates remain in the diffusive framework and evolve continuously, they may exhibit large changes over short intervals (as with jumps) due to stochastic acceleration. Comparisons of bond prices show that the proposed model generates more complex and intricate shapes for the restricted forward curve with the same number of stochastic factors and volatility.

Keywords: HJM, forward rates, arbitrage-free restriction, volatility estimation

JEL Classification: G13, E43, C1