On pricing and hedging in financial markets with long-range dependence

Abstract

We study a mixed financial market with risky asset governed by both the standard Brownian motion and the fractional Brownian motion with Hurst index \({H\in(\frac12, 1)}\). We use representations of Hitsuda and Cheridito for the mixed Brownian and fractional Brownian process and present the solution of the problem of efficient hedging for \({H\in(\frac34, 1)}\). To solve the problem for \({H\in(\frac12, 1)}\) and to avoid some computational difficulties, we introduce the approximate incomplete semimartingale market, and the solution of the approximate problem of efficient hedging is considered. Then we pass to the limit and observe the asymptotic behavior of the solution of the efficient hedging problem.