A note on the use of fractional Brownian motion for financial modeling
Highlights
► Models with Brownian motion: no arbitrage, continuous tradability, pricing formula ► This finding is irrespective of the integration calculus (Itô vs Stratonovich). ► With fractional Brownian motion: continuous tradability = existence of arbitrage ► FBM-models ignoring arbitrage lead to a wrong pricing formula. ► This finding is irrespective of the integration calculus (Wick vs Stratonovich).
Introduction
Fractional Brownian motion was introduced by Mandelbrot and van Ness (1968). Being extensions of classical Brownian motion, both models of randomness still have some key properties in common, most importantly, they are Gaussian. For all Hurst parameters , there are however also important differences: while Brownian motion has independent increments, the increments of fractional Brownian are serially correlated. Thereby, new information has a persisting influence on the process, which implies a certain level of predictability: In contrast to the classical Brownian case, the historical trajectory of the process does matter when forecasting its future evolution.
Our analysis will be based on a continuous time market setup with two assets. We introduce a riskless asset At followingand, referring to the definition of fractional Brownian motion, a risky asset St by means of a geometric fractional Brownian motion:
The parameters for the riskless interest rate r as well as for the drift μ and the volatility σ of the stock price process are constant. The differential Eq. (2) can be interpreted in different ways depending on the chosen stochastic integration calculus. Throughout this paper we will focus on pathwise integration on one hand and Wick-based integration on the other hand.
The remainder of the paper is organized as follows: In the next section, we briefly recall the debate concerning the problem of arbitrage that is inherent to markets driven by fractional Brownian motion. We also look at the recent literature focusing on a market microstructure perspective toward fractional Brownian motion. Keeping these results in mind, we use in Section 3 a fractional analog to the work of Sethi and Lehoczky (1981) to show that fractional Brownian motion and continuous tradability are incompatible. In Section 4, we then show how the conceptual discussion can be brought to a more applied basis showing that the correct handling of this incompatibility leads to results that are more intuitive. In Section 5 we summarize our main findings and revisit them based on the insights gained by the market microstructure perspective.
Section snippets
Characteristics of financial models with fBM
Within the last two years, a number of articles have been published choosing fractional Brownian motion as an underlying diffusive process (e.g. Gu et al., 2012, Meng and Wang, 2010, Xiao et al., 2010). The references used therein seem to show that some of the insights of the first part of the last decade have been buried in oblivion. This may be due to the fact, that the discussion then had become rather technical. Still, a clear statement whether to use fractional Brownian motion as a model
FBM and continuous trading are incompatible
For decreasing values of ϵ Cheridito's mixed model approaches the fractional Brownian motion market. However, with ϵ → 0, the volatility of the option σϵ also vanishes and hence randomness disappears. Cheridito (2001b) argues that as soon as the Brownian noise component disappears the market participants can (i) exploit the predictability of the fractional Brownian motion, (ii) find an appropriate trading strategy and (iii) eliminate randomness.
This strange behavior in the limiting case above is
Example: closed-form solutions for European call options
In this section, we substantiate the conceptual discussion above by switching to the more applied level of closed-form option pricing formulas. The following comparison of two different approaches taken from the finance literature serves to show that a thorough choice of the model framework helps to remedy counter-intuitive findings with respect to fractional Brownian markets.
In his still commonly cited paper, Necula (2002) disregards the incompatibility of continuous tradability and fractional
Conclusion
Evidently, fractional Brownian motion comes along with some characteristics that do not fit the assumption of continuous tradability. However, if single investors do not act as fast as the market and show some kind of delay in the sequence of their transactions instead, arbitrage opportunities disappear. The link to the market microstructure perspective completes this picture: it is just this delay—whether it is on purpose (in the form of investor's inertia) or whether it is just a physical
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