Affiliations: [a] Department of Finance and Risk Engineering, New York University School of Engineering, New York, USA | [b] Graduate School of Business, Concordia University, Montreal, Canada | [c] Department of Mathematics, Universite du Quebec, Montreal, Canada
Correspondence:
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Address for correspondence: Charles S. Tapiero, Department of Finance and Risk Engineering, New York University School of Engineering, New York, USA. E-mail: cst262@nyu.edu.
Abstract: 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.
