Abstract
Numerous kinds of uncertainties may affect an economy, e.g. economic, political, and environmental ones. We model the aggregate impact by the uncertainties on an economy and its associated financial market by randomised mixtures of Levy processes. We assume that market participants observe the randomised mixtures only through best estimates based on noisy market information. The concept of incomplete information introduces an element of stochastic filtering theory in constructing what we term “filtered Esscher martingales”. We make use of this family of martingales to develop pricing kernel models. Examples of bond price models are examined, and we show that the choice of the random mixture has a significant effect on the model dynamics and the types of movements observed in the associated yield curves. Parameter sensitivity is analysed and option price processes are derived. We extend the class of pricing kernel models by considering a weighted heat kernel approach, and develop models driven by mixtures of Markov processes.
Highlights
We develop interest rate models that offer consistent dynamics in the short, medium, and long term
The question arises as to how one can create interest rate models, which are sensitive to market changes over both short and long time intervals, so that they remain useful for the pricing of securities of various tenors
We propose a family of pricing kernel models which may generate interest rate dynamics sufficiently flexible to allow for diverse behaviour over short, medium and long periods of time
Summary
We develop interest rate models that offer consistent dynamics in the short, medium, and long term. One can imagine an investor holding a portfolio of securities maturing over various periods of time, perhaps spanning several years Another situation requiring interest rate models that are valid over short and long terms, is where illiquid long-term fixed-income assets need to be replicated with (rolled-over) liquid shorter-term derivatives. When designing interest rate models that are sensitive to the states an economy may take, subject to its response to the underlying uncertainty factors, one may wonder a) how many stochastic factor processes ought to be considered, and b) what is the combination, or mixture, of factor processes determining the dynamics of an economy and its associated financial market. The set of stochastic processes chosen to model an economy’s response to uncertainty, the particular mixture of those, and the pricing kernel model jointly characterize the dynamics of the derived interest rate model. We conclude the paper by introducing randomised weighted heat kernel models, along the lines of Akahori et al (2014) and Akahori and Macrina (2012), which extend the class of pricing kernels developed in the first part of this paper
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