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

Bernard Wong

doi:10.1155/2009/215817

Abstract

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 long horizon asset-liability management as they allow the modelling of long term stock returns with heavy tail ergodic diffusions, with tractable, time homogeneous dynamics, and which moreover admit a complete financial market, leading to unique pricing and hedging strategies. Unfortunately the standard specifications of these models in 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.

Highlights

Ever since the fundamental work of Black and Scholes, there has been extensive work in the literature on alternative stock price models
The financial market under these models will be complete, and the valuation of options and guarantees can be performed without requiring extra assumptions regarding the market price of risk
In this paper we investigated the arbitrage-free property of the class of stock price models where the local martingale component is based on an ergodic diffusion with a specified stationary distribution

Summary

Introduction

Ever since the fundamental work of Black and Scholes, there has been extensive work in the literature on alternative stock price models. Extensive references can be found in Cont and Tankov 1 and Fouque et al 2 These models and tools have proven invaluable for long term asset liability management, in particular with applications to insurance and pensions. In this paper we investigate the arbitrage-free property of the class of Brownian based stock price models where the local martingale component of the log stock returns is assumed to be an ergodic diffusion. This class of models was first investigated by Bibby and Sørensen 14 and Rydberg 15, 16 “BSR” who reported good fit of their models to financial data.

Local Martingales Based on Ergodic Diffusions

Financial Market

Arbitrage Opportunities under the Standard Specification

Arbitrage-Free Specification

Stock Markets with Stochastic Term Structure of Interest Rates

Conclusions