High-Frequency Exponential-Utility Maximization under Fractional Brownian Motion

Abstract

We study utility maximization for high-frequency trading in fractional Brownian motion models. We first consider exponential utility in the discretized arithmetic model. Using spectral methods for stationary Gaussian sequences, we derive the asymptotic growth rate of the optimal certainty equivalent and the scaling limit of the optimal positions. We then consider power and logarithmic utility in a positive-price model with positive wealth and investment fractions in $[0,1]$. In this case, the logarithm of the optimal certainty equivalent grows at rate $n^{1-H}$ for $H\ne 1/2$, with a coefficient determined by the one-sided prediction error of fractional Gaussian noise. We also give a simple binary strategy which attains this leading rate, without claiming exact finite-horizon optimality.