Conclusion

In the following chapter, we readdressed ourselves to the continuous time process of fractional Brownian motion. Stimulated by the insights of the binomial setting, we investigated in Chap. 4 a financial market setting consisting of a riskless asset as well as a risky one driven by a geometric fractional Brownian motion. We first recalled the debate of the history: The first theoretical results by Delbaen and Schachermayer (1994) as well as by Rogers (1997) had suggested a categorical rejection of fractional market models for reasons of arbitrage. Though the introduction of Wick–Itô calculus had inspired some promising results (Hu and Øksendal (2003) and Elliott and van der Hoek (2003)), Bjork and Hult (2005) had shown that their implied assumptions were economically meaningless. We then worked out that as long as the possibility of continuous tradability exists, the predictability of fractional Brownian motion always eliminates randomness from option pricing. We stated a fractional inversion of the work of Sethi and Lehoczky (1981): They had shown for the classical case that—if applied sensibly—both Stratonovich and Itô calculus lead to the Black–Scholes pricing formula for a European call option. Contrariwise, we showed that the correct usage and interpretation of both pathwise and Wick–Itô calculus does not lead to an option pricing formula la Black–Scholes but to a formula where the price is nothing but the maximum of the discounted inner value and zero. On the one hand, our considerations provided another example concerning the importance of the correct interpretation of integration concepts as disregarding them leads to results like those by Hu and Øksendal (2003). On the other hand, we drew the following conclusion: In order to ensure a reasonable pricing and absence of arbitrage in the fractional Brownian market model, we had to restrict trading strategies to be non-continuous. This was the logical counterpart to the phenomenon we had observed in the discrete time setting. Furthermore, it was perfectly in line with similar findings received by Cheridito (2003).