The price of granularity and fractional finance

The purpose of this paper is to assess the risk premium a fractional financial lognormal (Black-Scholes or BS) process relative to a non-fractional and complete financial pricing model. While fractional Brownian BS models based on the Duncan and Wicks calculus were shown to define a no arbitrage financial model, this paper claim is that this martingale need not be the pricing martingale. There may be many martingales corresponding to a no-arbitrage financial model. In this vein, the intent of this paper are twofold. On the one hand, provide a definition of the risk premium implied by the discount rate applied to future fractional returns as well as justify this premium compared to a non-fractional financial model. To do so, an insurance rationale for the risk implied by the financial asset volatility is used when defining a no-arbitrage risk neutral probability measure. A “granular risk premium”, is then defined relative to the model granularity. On the other, the paper highlights the effects of a model granularity and its Hurst index on financial risk management. In particular, we argue that that such an index is defined by both the granularity underlying the model as well as the effects of mixed models (combining both diffusion and jump processes) on relative range and volatility variations (and thus the measurement of the Hurst index). 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.