Abstract
We consider a family of mixed processes given as the sum of a fractional Brownian motion with Hurst parameter $H\in(3/4,1)$ and a multiple of an independent standard Brownian motion, the family being indexed by the scaling factor in front of the Brownian motion. We analyze the underlying markets with methods from large financial markets. More precisely, we show the existence of a strong asymptotic arbitrage (defined as in Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)]) when the scaling factor converges to zero. We apply a result of Kabanov and Kramkov [Finance Stoch. 2(2), 143--172 (1998)] that characterizes the notion of strong asymptotic arbitrage in terms of the entire asymptotic separation of two sequences of probability measures. The main part of the paper consists of proving the entire separation and is based on a dichotomy result for sequences of Gaussian measures and the concept of relative entropy.
Highlights
Empirical studies of financial time series led to the conclusion that the log-return increments exhibit long-range dependence. This fact supports the idea of modelling the randomness of a risky asset using a fractional Brownian motion with Hurst parameter H > 1/2
A Black–Scholes type model in which the randomness of the risky asset is driven by a mixed fractional Brownian motion is arbitrage free and complete
We focus on the notion of strong asymptotic arbitrage (SAA) introduced by Kabanov and Kramkov in [9] as the possibility of getting arbitrarily rich with probability arbitrarily close to one by taking a vanishing risk
Summary
Empirical studies of financial time series led to the conclusion that the log-return increments exhibit long-range dependence This fact supports the idea of modelling the randomness of a risky asset using a fractional Brownian motion with Hurst parameter H > 1/2. Our proof follows using tightness arguments for the sequence of Radon– Nikodym derivatives of the objective probability measures with respect to equivalent martingale measures and the fact that two sequences of Gaussian measures are either mutually contiguous or entirely separable. The latter is known in the literature as the equivalence/singularity dichotomy for sequences of Gaussian processes, see [5]. We end our work with Appendix A in which we recall the definition of relative entropy and an equivalent characterization in terms of the Radon– Nikodym derivative
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