No-Arbitrage Theorems in Markets Driven by the Fractional Brownian Motion


Starting from the interest of modeling financial products through the usage of non-Markovian processes a possibility is the utilization of the so called fractional Brownian motion. However, since this process is not a semimartingale, the classic mathematical modeling of continuous time finance ensures that there exists some possibility of capital gain without risk. This work presents two theorems about the inexistence of arbitrage possibilities in models driven by the fractional Brownian motion. The adicional hypothesis of the theorems (in relation to the classical modeling) do not turn them more unreal, on the contrary, because they are, in one of the cases, the requirement of a minimum waiting time between two consecutive transactions and, in the other case, the inclusion of transaction costs proportional to the asset’s value. It also presents some estimators for the long memory parameter that can be used to corroborate the hypothesis on the existence of this phenomenon in financial assets pricing.