On Modelling Long Term Stock Returns with Ergodic Diffusion Processes : Arbitrage and Arbitrage-Free Specifications

We investigate the arbitrage-free property of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution. These models are particularly useful for insurer asset-liability management as they allow the modelling of long term stock returns with heavy tailed ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market. Unfortunately the standard specification of these models in the literature admit arbitrage opportunities. We investigate in detail the features of the existing model specifications which create these arbitrage opportunities, and consequently construct a modification that is arbitrage free