Risk Preference Based Option Pricing in a Continuous Time Fractional Brownian Market

After the success of risk-neutral valuation in the Markovian models by Black, Scholes and Merton (see Black and Scholes (1973), Merton (1973)), it was aspired to extend the famous option pricing formula for use in a fractional context. In the course of the last few years however, it turned out that no arbitrage pricing methods could not be sensibly applied within the fractional market model (see e.g. Rogers (1997) and Bjork and Hult (2005)). In this chapter which further develops a preceding joint article of Rostek and Schöbel (2006), we look at a market where randomness is driven by fractional Brownian motion. While the price process evolves continuously, we will assume that a single investor faces certain restrictions concerning the speed of his transactions. In the sense of Cheridito (2003), we introduce a minimal amount of time—which can be arbitrarily small—between two consecutive transactions by one and the same investor. The idea behind this is that the great multitude of investors in the market ensures that there are other transactions in-between. In other words, this assumption restricts a single investor from being as fast as the market as a whole which evolves continuously. This restriction is sufficient to exclude arbitrage in the fractional Brownian market (see Cheridito (2003)). Note that, as we give up on continuous tradability, the well-known no arbitrage pricing approach based on dynamical hedging arguments is unsuitable within this modified framework. This problem is solved the most naturally by introducing risk preferences (see Brennan (1979)). Concerning these risk preferences, the market has to satisfy a basic equilibrium condition which we will investigate. In the special case of risk-neutral investors, the option pricing problem will prove to be the calculation of the discounted conditional mean of the relevant payoff.