Stochastic volatility Gaussian Heath-Jarrow-Morton models

Abstract

This paper extends the class of deterministic volatility Heath-Jarrow-Morton models to a Markov chain stochastic volatility framework allowing for jump discontinuities and a variety of deformations of the term structure of forward rate volatilities. Analytical solutions for the dynamics of the volatility term structure are obtained. Semimartingale decompositions of the interest rates under a spot and forward martingale measures are identified. Stochastic volatility versions of the continuous time Ho-Lee and Hull-White extended Vasicek models are obtained. Introducing a regime shift in volatility that is an exponential function of time to maturity leads to a Vasicek dynamics with regime switching coefficients of the short rate.term structure of interest rates, Heath-Jarrow-Morton model, stochastic volatility, continuous time Markov chains, piecewise-deterministic Markov processes,

Pricing credit-risky bonds and spread options modelling credit-spread term structures with two-dimensional Markov-modulated jump-diffusion

“…Heath et al (1992) proposed a forward interest rates model, known as the Heath-Jarrow-Morton (HJM) model. The HJM framework provides extra flexibility in specifying uncertainty, which allows for differences in risk across the maturity spectrum (Valchev 2004). Therefore, we assumed an instantaneous forward rate based on the HJM framework.…”

Pricing variance swaps under stochastic volatility and stochastic interest rate

“…Valchev [105] where only Gaussian models were focused and no finite-dimensional issue was considered. In [32], the volatility was specified as function of continuous-time finite…”