Geometric Brownian Motion (GBM) is the fundamental stochastic process used to model asset prices in continuous time. It serves as the foundation for pricing models such as the Black–Scholes–Merton framework. However, some critics argue that the assumptions of GBM are unrealistic, and researchers continue to explore ways to modify and improve the model.
Reference [1] proposed an extension based on fractional Brownian motion (FBM), which incorporates trend fractal dimensions (FTD)—distinguishing between upward (D⁺) and downward (D⁻) dimensions—combined with momentum lifecycle theory.
The authors developed a pricing framework for futures under this setup. Because FBM is not a semimartingale in the classical sense, they adjusted the drift of the log-price process to reconcile fractal dynamics with approximate arbitrage-free pricing.
Afterward, they constructed a futures pricing model and designed an arbitrage strategy based on the futures–cash basis. The strategy operates as follows:
- Rule 1: Execute a positive arbitrage (sell futures, buy spot/ETF) when the basis series enters the low reversal phase, as identified by the conditions on D⁺ and D⁻.
- Rule 2: Close the positive arbitrage position (buy futures, sell spot/ETF) when the basis series enters the high reversal phase, or, depending on market rules and strategy design, open a negative arbitrage position.
The authors pointed out,
… a dynamic arbitrage strategy engineered from fractal principles significantly outperforms conventional static-threshold methods in both profitability and risk control. By leveraging trend fractal dimensions (D+ and D−) and momentum lifecycle logic, our strategy adapted to the evolving structure of the basis. The backtest results are compelling: the fractal strategy delivered a net total return of 12.71%, substantially higher than the 7.06% from the traditional strategy. Critically, it achieved a positive after-cost Sharpe ratio of 0.32, indicating genuine risk-adjusted profitability, whereas the traditional strategy’s Sharpe ratio was negative (−0.61), rendering it economically unviable. This superior performance was further validated in a severe stress test during the 2015 market crash. While a buy-and-hold strategy collapsed and the traditional arbitrage strategy incurred significant losses (−5.82%), the fractal strategy demonstrated exceptional resilience, limiting its loss to a near breakeven at −0.83% and maintaining a much smaller drawdown. This robustness in extreme conditions underscores its practical value for capital preservation.
In brief, the proposed futures pricing model performed well, improving the basis arbitrage strategy’s risk-adjusted return.
This research constitutes another interesting extension of the GBM model, particularly as it incorporates the trend behavior of the underlying asset. It would be interesting to see this approach applied to options.
Let us know what you think in the comments below or in the discussion forum.
References
[1] Xu Wu and Yi Xiong, A fractal market perspective on improving futures pricing and optimizing cash-and-carry arbitrage strategies, Quantitative Finance and Economics, Volume 9, Issue 4, 713–744.
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