Abstract
This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional Markov processes, we present the main characteristics of bonds and short-term rates in this setting. Their dynamics under risk neutral and forward measures are studied. Finally, bond options are valued with a discretization scheme and a discrete Fourier’s transform.
Highlights
The interest rate depends on its path through the integral of past occurrences weighted by an exponential decaying function, called the memory kernel
Under the assumption of normality, parameters are estimated by log-likelihood maximization whereas the goodness of fit is measured with the Akaike Information Criterion (AIC)
This article contributes to the literature on interest rate models by proposing an alternative to standard mean reverting processes
Summary
Eberlein and Raible (1999) were among the first to propose a short-term rate model driven by Lévy processes. In their setting, the short-term rate is ruled by Ornstein–Uhlenbeck processes reverting in an exponential manner to a mean level. The short-term rate is ruled by Ornstein–Uhlenbeck processes reverting in an exponential manner to a mean level Within this framework, Eberlein and Kluge (2005) derived analytical formulae for the prices of caps and floors using bilateral Laplace transforms. A wide majority of short-term rate models are driven by Markov mean reverting processes In this setting, the interest rate depends on its path through the integral of past occurrences weighted by an exponential decaying function, called the memory kernel.
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