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.

Ever since the fundamental work of Black and Scholes, there has been extensive work in the literature on alternative stock price models. In a continuous time setting, these include, for example, jump diffusions, Levy processes, stochastic volatility models, and regime switching models. Extensive references can be found in Cont and Tankov [

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 [

A significant drawback of the ergodic diffusion approach however was also noted by BSR (in particular, Bibby and Sørensen [

Following on from the previous discussion in literature, in this paper we provide a detailed proof that any ergodic diffusion process used as a stock return model, and as specified in literature, will admit arbitrage. We further analyze in detail the cause for these arbitrage opportunities and consequently propose a modification that is arbitrage-free. This modification once again opens up the application of ergodic diffusion models to problems in insurance and finance.

The outline of this paper is as follows. In Section

Let

Consider an interval

Using standard diffusion theory (cf. Karlin and Taylor [

It is worth noting that the existence of a stationary distribution for (

Note that the above is not the only method of constructing a Brownian-based model with a specified distribution. There are two alternative approaches. The first alternative (Bibby et al. [

For the financial market we consider a probability space

An extension of our framework to include stochastic interest rates can be found in the Section

Portfolios are formed by holding an amount of

Ergodic diffusion-based stock price models were first considered by Bibby and Sørensen [

As

Bibby and Sørensen [

The financial market with stock price modelled by (

By Ito's lemma the process

The martingale property of

Note that

For calculation purposes it is convenient to consider, under the measure

As this is a complete market we can identify a specific arbitrage portfolio and strategy using the techniques of Levental and Skorohod [

In the previous section we have shown that the model specification (

As

As

The financial market with stock model (

Note firstly that we have, by the assumptions on

From Ito's lemma the discounted stock price process

Under the unique equivalent local martingale measure

In Sections

To allow for imperfect correlation between stock and interest rates we now consider a probability space

Interest rate variability will be introduced via the second Brownian motion

Under the above setup the savings account

The financial market with stock model (

The discounted stock price process

Consider now the nonnegative

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. The dynamics of these models are time homogeneous and, as it is based on Brownian motion, tractable. The financial market under these models will be complete, and hence the valuation of options and guarantees can be performed without requiring extra assumptions regarding the market price of risk. In this paper we provided a detailed proof that any ergodic diffusion process used as a stock return model, and as specified in the existing literature, will admit arbitrage in general. We further analyzed the technical cause for these arbitrage opportunities and consequently constructed a modification that is arbitrage-free. This arbitrage free property is shown to be true in financial markets both with and without stochastic interest rates. Our modification once again opens up the application of ergodic diffusion models to problems in insurance and finance.