Description: The series expansion of a probability density function, known to actuaries by Esscher's name and to statisticians as the saddlepoint approximation, typically gets integrated to approximate probability values under the density, with a location parameter in the expansion chosen to optimize convergence of the integrated series. We review the general form, the accuracy and the derivation of the Esscher approximation. Then we propose using related series expansions for the derivative of a probability density function to approximate the value of the point of maximum likelihood for the density. The key is to make the value of the location parameter an unknown and assume it to be the point of maximum likelihood, which leads to a tractable optimization problem to find its value.

Description: Prior work indicates that a regime-switching stochastic model with randomized regime parameters creates a more plausible set of extreme paths than do the usual stochastic interest rate models. Generalizing the Black-Karasinski model by randomizing the mean reversion target provides an example. Better to understand such models, as well as to calibrate their parameters without trial and error runs of a stochastic model, this paper uses an asymptotic expansion to estimate the moments of the integrated stochastic process over time. Mention is made of a family of related asymptotic expansions that could be of wider interest.

Description: The behavior of extreme paths in the usual stochastic interest rate models is not nearly so plausible as the behavior of their expected values and variances. This paper proposes a new class of models that deliver more plausible extreme paths while preserving the usual expected value and variance behavior. Along the way we derive the closed form solution for the traditional mean-reverting lognormal process, including the drift compensation.