The price of granularity and fractional finance

The purpose of this paper is to assess the risk premium of a fractional financial lognormal (Black-Scholes) process relative to a non-fractional and complete financial markets pricing model. The intents of this paper are two-fold. On the one hand, provide a definition of the risk premium implied by the discount rate applied to future fractional returns relative to that of a non-fractional financial (and complete market) model. To do so, an insurance rationale is used to define a no-arbitrage risk neutral probability measure. On the other, highlight the effects of a model granularity and its Hurst index on financial risk models and their implications to risk management. In particular, we argue that fractional Brownian motion (BM) does not define a normal probability distribution but a fractional volatility model. To present simply the ideas underlying this paper, we price an elementary fractional risk free bond and its risk premium relative to a known spot interest rate. Similarly, the Black-Scholes no arbitrage model is presented in both its non-fractional conventional form and in its fractional framework. The granularity risk premium is then calculated.