Fractional Brownian Motion and Arbitrage in Black–Scholes Models

Summary

The document asks how fractional Brownian motion (fBm) might be interpreted when it replaces ordinary Brownian motion in a Black–Scholes framework. Its answer focuses less on an economic interpretation and more on a mathematical consequence: fBm is generally not a semimartingale, a property that underpins many standard results in continuous-time finance.

The response cites established mathematical finance results and several papers showing that Black–Scholes models driven by fBm permit arbitrage under a number of formulations. This is a caution against substituting fBm into the familiar model while assuming its usual no-arbitrage conclusions remain valid. The brief exchange does not explain a financial mechanism represented by fBm, specify the trading assumptions behind each arbitrage result, or compare alternative fractional models. Therefore, it supports a narrow lesson about model consistency and arbitrage risk, rather than a complete interpretation or practical modeling guide. Any application would need to state the exact process and trading framework being assumed.

Key ideas

• Fractional Brownian motion is generally not a semimartingale.
• Replacing Brownian motion with fractional Brownian motion in Black–Scholes changes foundational model properties.
• The cited literature reports arbitrage in several Black–Scholes formulations driven by fractional Brownian motion.
• The exchange does not provide an economic interpretation of fBm or detail the assumptions behind the cited arbitrage results.

Tags

# Soft: Interpretation Fractional BM in finance


# Soft: Interpretation Fractional BM in finance












Suppose we are in the BS framework. If we replace the Brownian Motion with a more general fractional Brownian motion therein, how can it be interpreted?

That is what is a financial interpretation of fractional brownian motion, what can it be understood to represent?

## Answer by user381975 (score 3, accepted)

https://quant.stackexchange.com/a/31583

You can replace BM with FBM.

However, in finance, fBm is not a semi-martingale, the general results of mathematical finance in Delbaen and Schachermayer (1994) already imply that it allows a certain kind of arbitrage.

For example, Rogers (1997); Sottinen (2001); Cheridito et al. (2003); Bender and Elliott (2004); Bj¨ork and Hult (2005) have shown that the Black-Scholes model driven by fBm allows arbitrage in a number of ways.

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