Rough Volatility
A first pass. Incomplete by design.
Part of the track Derived, Borrowed, Learned
The finding
Mandelbrot argued from the 1960s that price series have a roughness the random-walk toolkit does not capture. The claim was eventually tested on volatility rather than prices. Gatheral, Jaisson, and Rosenbaum's "Volatility is rough" (Quantitative Finance, 2018) estimated the Hurst exponent of log realized volatility for a range of equity indices and found H of roughly 0.1, consistent across assets and time periods. A Brownian path has H = 0.5; volatility is much rougher.
Why it matters for models
- Classical stochastic-volatility models (Heston and relatives) drive volatility with a Brownian motion and so assume H = 0.5. They struggle to fit the steep short-maturity skew observed in option markets.
- Rough volatility models (rough Bergomi, rough Heston) replace that driver with fractional Brownian motion at H ≈ 0.1 and fit the term structure of implied volatility with few parameters.
- At H ≈ 0.1 the driving path is not a semimartingale, so the mathematics needed is that of Rough Path TheoryNoteSeedRough Path TheoryLyons (1998): a rough path alone does not determine the integrals along it, so enrich the path with its iterated integrals. Once you do, differential equations driven by paths of any roughness become well defined, with no probability required., not Itô.
Why it belongs in the essay
It is the measured confirmation of the intuition Mandelbrot gave the author: markets are measurable, but not with integer tools. See Hurst Exponent and Fractal DimensionNoteSeedHurst Exponent and Fractal DimensionTwo measurements of roughness that are one measurement: for a self-affine path, D = 2 − H. A random walk has H = 0.5 and D = 1.5; anything else is a path of another roughness. for the roughness measure and From Smooth to JaggedEssayFrom Smooth to JaggedHow mathematics grew a new tool each time reality posed a problem the existing math could not solve, from Taylor series to the Fourier transform, stochastic calculus, fractal geometry, and rough paths, ending with a personal encounter with Benoit Mandelbrot and its bearing on reading markets. for the argument.
What this note does not yet cover
- The microstructural explanation (Hawkes processes, order-flow clustering) for why volatility should be rough.
- Estimation pitfalls: H estimates depend on the realized-volatility proxy and sampling frequency.
- Practical adoption in trading desks versus academic literature.
Sources
- Gatheral, J., Jaisson, T., & Rosenbaum, M. (2018). "Volatility is rough." Quantitative Finance, 18(6), 933-949.
Review log
- 2026-09-08: seeded.
Appears in
Essays that draw on this note.
- From Prediction to AgencyA draft essay on frontier language models: each stage in their development answered a limit hit by the previous one, and each answer changed what a model is. It ends with the question the companion essays set up: whether a learned model of human behavior could become a substrate for markets, and why reflexivity puts a ceiling on that.
- From Smooth to JaggedHow mathematics grew a new tool each time reality posed a problem the existing math could not solve, from Taylor series to the Fourier transform, stochastic calculus, fractal geometry, and rough paths, ending with a personal encounter with Benoit Mandelbrot and its bearing on reading markets.
Links out
- Rough Path TheoryLyons (1998): a rough path alone does not determine the integrals along it, so enrich the path with its iterated integrals. Once you do, differential equations driven by paths of any roughness become well defined, with no probability required.
- Hurst Exponent and Fractal DimensionTwo measurements of roughness that are one measurement: for a self-affine path, D = 2 − H. A random walk has H = 0.5 and D = 1.5; anything else is a path of another roughness.
- From Smooth to JaggedHow mathematics grew a new tool each time reality posed a problem the existing math could not solve, from Taylor series to the Fourier transform, stochastic calculus, fractal geometry, and rough paths, ending with a personal encounter with Benoit Mandelbrot and its bearing on reading markets.
Referenced by
Other notes and pages that link here.
- Hurst Exponent and Fractal DimensionTwo measurements of roughness that are one measurement: for a self-affine path, D = 2 − H. A random walk has H = 0.5 and D = 1.5; anything else is a path of another roughness.
- ReflexivityA model of a market changes the market. Soros's reflexivity, the Lucas critique, and Goodhart's law are three statements of the same limit, and it is the one way markets are permanently harder than chemistry: molecules do not read the paper.
- Rough Path TheoryLyons (1998): a rough path alone does not determine the integrals along it, so enrich the path with its iterated integrals. Once you do, differential equations driven by paths of any roughness become well defined, with no probability required.
Related
- Brownian Motion and Itô CalculusA path continuous everywhere and differentiable nowhere broke ordinary calculus. Itô's lemma repaired it with one extra term, tuned exactly to Brownian roughness.