Mandelbrot and van Ness (1968) suggested fractional Brownian motion as a
parsimonious model for the dynamics of financial price data, which allows
for dependence between returns over time. Starting with Rogers (1997) there
is an ongoing dispute on the proper usage of fractional Brownian motion in
option pricing theory. Problems arise because fractional Brownian motion is
not a semimartingale and therefore “no arbitrage pricing” cannot be applied.
While this is consensus, the consequences are not as clear. The orthodox
interpretation is simply that fractional Brownian motion is an inadequate
candidate for a price process. However, as shown by Cheridito (2003) any
theoretical arbitrage opportunities disappear by assuming that market participants
cannot react instantaneously.