Financial models with dependence on the past: a survey

In this paper we will examine some models for the financial markets where the evolution of the prices of the assets depends not only on the current value but also from the values assumed in the past. First we characterise the shape of volatility that a “good” financial model, also depending on past values, must have, and prove the completeness of the market in this general framework. Then, we analyse two models taken from the literature, one with “finite memory” which in general does not give nice numerical results, and one with “infinite memory”, where the model reduces to a Markov system with more state variables than the number of risky assets. Finally, we prove that a generalisation of the results of this last model with finite delay horison is not feasible without using anticipative stochastic calculus.