Why Quadratic Variation Matters in Brownian Financial Models

Summary

The document gives a concise reason quadratic variation matters in quantitative finance: Brownian motion has nonzero quadratic variation, whereas a continuously differentiable path has zero quadratic variation. Because Brownian motion is a central model for the evolution of financial prices and returns, this property is foundational to stochastic calculus used in finance.

The discussion is introductory and does not develop rough path theory, pathwise integration, or a trading application. It offers no empirical evidence or comparison of models. Its practical takeaway is limited but important: stochastic price models based on Brownian motion have path behavior that differs mathematically from smooth curves, and quadratic variation captures part of that distinction. Readers seeking a fuller account of rough paths or their uses in pricing and risk analysis would need additional material.

Key ideas

• Brownian motion has nonzero quadratic variation.
• Continuously differentiable paths have zero quadratic variation.
• Brownian motion is widely used to model financial price and return evolution.
• Quadratic variation is therefore a foundational concept in stochastic finance.

Tags

# Why are quadratic variation and rough paths so important in quantitative finance?


# Why are quadratic variation and rough paths so important in quantitative finance?












I am new to quant finance - come from a mathematics background. I am starting stochastic calculus and have been particularly interested in some papers pathwise integration and rough calculus in general. At this time I struggle to see its applications in the world of quant finance? Does its utility come from the fact that we can then also model "rough paths" rather than just smooth ones?

## Answer by roz (score 2)

https://quant.stackexchange.com/a/51594

Because Brownian motion has non zero quadratic variation (as opposed to continuous differentiable function which have 0). Quadratic variation is a defining characteristic of Brownian motion and Brownian motion is central to financial models of the evolution of prices/returns over time.

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