A discrete stochastic model for investment with an application to the transaction costs case

This work consists of two parts. In the first one, we study a model where the assets are investment opportunities, which are completely described by their cash-flows. Those cash-flows follow some binomial processes and have the following property called stationarity: it is possible to initiate them at any time and in any state of the world at the same condition. In such a model, we prove that the absence of arbitrage condition implies the existence of a discount rate and a particular probability measure such that the expected value of the net present value of each investment is non-positive if there are short-sales constraints and equal to zero otherwise. This extends the works of Cantor-Lippman who studied a deterministic setup. In the second part, we apply this result to a financial model in the spirit of Cox-Ross-Rubinstein Cox, but where there are transaction costs on the assets. This model appears to be stationary. At the equilibrium, the Cox-Ross-Rubinstein's price of a European option is always included between its buying and its selling price. Moreover, if there is transaction cost only on the underlying asset, the option price will be equal to the Cox-Ross-Rubinstein's price. Those results give more information than the results of Jouini-Kallal Jouini, which where working in a finite horizon model.