Fractional Brownian Motion in Finance: Simulation and Model Limits
Summary
The document points to references on fractional Brownian motion, including material on its numerical simulation and its potential use in financial modeling. Its central modeling feature is long-range dependence, which can make it relevant to financial time series. The suggested simulation reference is identified as a resource rather than explained in the text, so the document does not give an algorithm or implementation details.
The cited discussion raises a major limitation: fractional Brownian motion combined with continuous tradability can create arbitrage concerns. It argues that these processes are incompatible under that assumption, and suggests that a market-microstructure perspective entails dynamic market incompleteness. It also leaves open whether newer fractional models are economically reasonable. These points make the document a pointer to theoretical debate, not a validated forecasting or pricing method.
Key ideas
- Fractional Brownian motion can model long-range dependence in time series.
- The document points to a reference on numerical simulation but does not describe a simulation procedure.
- Continuous tradability is presented as incompatible with fractional Brownian motion because of arbitrage concerns.
- Market microstructure and dynamic incompleteness are relevant to assessing such models.
- The economic suitability of newer fractional models remains unsettled in the cited discussion.
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# Fractional Brownian motion references # Fractional Brownian motion references Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you. ## Answer by Theodore (score 6, accepted) https://quant.stackexchange.com/a/42428 Sure. - A note on the use of fractional Brownian motion for financial modeling > Abstract In the second part of the past decade, the usage of fractional Brownian motion for financial models was stuck. The favorable time-series properties of fractional Brownian motion exhibiting long-range dependence came along with an apparently insuperable shortcoming: the existence of arbitrage. Within the last two years, several new models using fractional Brownian motion have been published. However, still the problem remains unsolved whether such models are reasonable choices from an economic perspective. In this article, we take on a straightforward mathematical argument in order to clarify when and why fractional Brownian motion is suited for economic modeling: We provide a fractional analog to the work of Sethi and Lehoczky (1981) thereby confirming that fractional Brownian motion and continuous tradability are incompatible. In the light of a market microstructure perspective to fractional Brownian motion, it becomes clear that the correct usage of fractional Brownian motion inherently implies dynamic market incompleteness. - Also, from Columbia: Simulation of fractional Brownian Motion (this one is a pdf you can just access online). also see https://sci-hub.tw/10.1016/j.econmod.2012.09.003 to access the first paper
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