Rough volatility, observation noise and approximation. In progress

I’m trying to understand the connection between rough volatility and regularity structures. The starting question is fairly simple: if volatility is very irregular at small scales, what changes in the way I should model it?

I can follow the noise calculation and the interpolation example below. I’m less clear about what Hairer’s framework needs before reconstruction can work.

Starting with the Hurst exponent

Write variance and volatility together: v_t = σ_t², and Y_t = log v_t = 2 log σ_t. Switching from log variance to log volatility divides increments by two and their second moments by four; it changes the amplitude, while leaving the scaling exponent unchanged.

If $Y_t$ is log variance, the sort of scaling relation I have in mind is

\mathbb E[(Y_{t+\Delta}-Y_t)^2]\approx c\Delta^{2H}.

For fractional Brownian motion, this scaling is exact. The Brownian case has $H=1/2$ ; smaller values correspond to rougher paths. Gatheral, Jaisson and Rosenbaum found scaling consistent with $H$ around $0.1$ in the volatility series they examined. [1]

Volatility is estimated from prices, which makes the apparent small-scale irregularity hard to interpret. How much belongs to the underlying process, and how much to the estimate?

I also need to keep local roughness separate from long memory. One concerns nearby increments; the other concerns dependence over long gaps. A fractional model can connect them through the same parameter, which makes it easy to slide from one idea into the other.

The measurement problem seems hard to avoid

Suppose, just as a toy model, that an observed log-variance value is the underlying value plus an error. Let those errors be independent of each other and of the underlying process, with mean zero and variance $\tau^2$ .

If the underlying second moment is $A u^{2H}$ , where $u=\Delta/\Delta_0$ is lag relative to a fixed reference interval, the observed one becomes

m_2^{\mathrm{obs}}(u)=A u^{2H}+2\tau^2.

The extra term doesn’t shrink with the lag. At small lags it can dominate the increment variance and flatten the log-log slope. That would make the apparent $H$ smaller.

Can observation noise make a process look rougher?

Keep the underlying H at 0.5. Change the observation-noise ratio and inspect the local apparent exponent at a chosen lag.

Underlying H0.500

Local apparent exponent0.100

Relative lag u0.0100

Apparent exponentUnderlying H = 0.5

At relative lag 0.0100, noise ratio 0.040 gives local apparent exponent = 0.100. The underlying H remains 0.5. Shorter lags give the constant noise term more weight.

Exact toy relation from this working note: m₂(u) = Au²ᴴ + 2τ². Half its local log-log slope is H × u²ᴴ / (u²ᴴ + 2τ²/A). With H = 0.5, u = 0.01 and noise ratio 0.04, local apparent exponent is 0.1. Assumes independent, equal-variance observation errors, independent of the process. This is a model illustration, not an estimator correction or an empirical study.

Data and code

Toy model input / output	Value
Underlying H	0.5
Relative lag	0.01000
Noise ratio	0.040
Local apparent exponent	0.10000

Interactive values (JSON) Original figure inputs Reproduction script

Static figure and caption

A toy observation model with an underlying Hurst exponent of 0.5. Adding independent observation noise lowers the local apparent exponent, especially at small lags. — The underlying exponent is 0.5 in every curve. Only the amount of observation noise changes. “Noise ratio” means $2\tau^2/A$ ; the local apparent exponent is half the local log-log slope. These are invented inputs, not market estimates.

Averaging windows and dependent estimation errors would change this calculation. It isolates one source of slope flattening, rather than providing a correction for a real volatility estimate.

Cont and Das examine this problem more seriously. [2] Fukasawa, Takabatake and Westphal account for estimation error and still find evidence of rough volatility in their study. [3] So I don’t read the measurement objection as settling the question. It makes me want to understand the estimator before attaching too much meaning to $H$ .

A kernel I can start from

One way to build a rough process is to weight past Brownian increments with a singular kernel:

\begin{aligned} X_t&=\int_0^t K_H(t-s)\,dW_s,\\ K_H(r)&=\sqrt{2H}\,r^{H-1/2},\qquad r>0. \end{aligned}

Here $W$ is Brownian motion and $0<H<1/2$ . The kernel gets large near zero, but its square is integrable. That is enough for this Itô integral, and its variance works out to $t^{2H}$ .

This gives me something concrete to calculate with. It is a Riemann–Liouville-type process with nonstationary increments, so I can’t simply carry over the formulas for fractional Brownian motion.

The price can still be driven by ordinary Brownian motion, with the rough process supplying its volatility. That construction does not require integrating the price against fractional Brownian motion.

Why smoothing it may change the answer

The next object is something like

I_T=\int_0^T f(X_t)\,dW_t.

For adapted $X$ and suitable $f$ , this already makes sense as an Itô integral. My first instinct would be to approximate the paths on a grid and calculate the integral from there. The catch is that the approximation can change what the integral means.

For example, linearly interpolating $W$ uses the Brownian increment across a whole grid cell. Inside that cell, the slope already depends on its endpoint. That is different from using information available at the left endpoint in an Itô sum.

Even the simple case $f(x)=x$ gives a warning. With that interpolation and the corresponding kernel convolution, the uncorrected integral has a mean proportional to $\delta^{H-1/2}$ , where $\delta$ is the grid spacing and the total horizon is fixed. For $H<1/2$ , this grows as the grid gets finer. The intended Itô integral has mean zero.

Deriving the mean for this approximation

Take T = Nδ and interpolate Brownian motion linearly on the uniform grid. On cell [kδ, (k+1)δ], its derivative is ξ^δ_t = ΔW_k/δ. Define X^δ_t = ∫₀^t K_H(t−s)ξ^δ_s ds and J^δ = ∫₀^T X^δ_t ξ^δ_t dt. This is the pathwise f(x)=x calculation used in the download.

Independent Brownian cell increments have variance δ. Thus E[ξ^δ_sξ^δ_t] = 1/δ for s and t in the same cell, and zero in different cells. For t = kδ + a, with 0 ≤ a ≤ δ, only the current cell contributes:

E[X^δ_tξ^δ_t] = (1/δ) ∫_kδ^t √(2H)(t−s)^H−1/2 ds
= √(2H) a^H+1/2 / [δ(H+1/2)].
E[J^δ] = N ∫₀^δ √(2H) a^H+1/2 / [δ(H+1/2)] da
= T √(2H) δ^H−1/2 / [(H+1/2)(H+3/2)].

At H = 1/2 this is T/2. For 0 < H < 1/2 the exponent is negative, so the uncorrected mean diverges as δ tends to zero. The adapted Itô integral ∫X_t dW_t has mean zero. This derives the stated rate for this kernel and interpolation from time zero. Subtracting a mean alone does not prove convergence, or identify the correction for a different nonlinear integrand or approximation scheme; those require the approximation theory in [4].

Bayer, Friz, Gassiat, Martin and Stemper develop the connection to regularity structures and prove renormalised approximation results. [4]

Where regularity structures seem to enter

My current picture is an extension of the idea behind a Taylor expansion. Instead of describing a local object using only powers of $t-s$ , one allows symbols suited to the noise and the equation. Hairer’s introductory notes are the reference I’m using for this. [5]

In the rough-volatility example, $\Xi$ represents white noise and $\mathcal I(\Xi)$ records its convolution with the kernel. Products such as $\mathcal I(\Xi)\Xi$ then appear when expanding the integrand locally. Suppressing small regularity losses, the scaling assigned to this product is roughly

|\mathcal I(\Xi)\Xi|\approx H-\tfrac12.

The bars here denote the assigned regularity, not an absolute value. White noise contributes roughly $-1/2$ and the integrated driver contributes roughly $H$ . For the range of $H$ I’m interested in, their sum is negative. Pointwise multiplication doesn’t generally give a meaning to the product; extra information has to be supplied.

I can see how the Hurst exponent’s scaling enters the local objects of a regularity structure. I still need to work through how the model, changes of base point and reconstruction fit together.

What I’d like to work out next

Follow one corrected integral all the way through, with the approximation and the subtraction written out explicitly.
Put a diffusive volatility model through the same observation procedure as a rough one and see what the estimated exponents look like.
Understand reconstruction well enough to explain the role of each object without relying on the notation.

A PSIM study would start with one asset on the hourly grid, fit on past observations and evaluate forward with the choices fixed. I haven’t run it. First I want to be clearer about what the volatility estimate is measuring.

Reading alongside these notes

Gatheral, Jaisson and Rosenbaum. Volatility is rough (2018).
Cont and Das. Rough volatility: fact or artefact? (journal version, 2024).
Fukasawa, Takabatake and Westphal. Is Volatility Rough? (2019 preprint).
Bayer, Friz, Gassiat, Martin and Stemper. A regularity structure for rough volatility (journal version, 2020).
Hairer. Introduction to regularity structures (2014).

The examples use a small calculation script. It reproduces the local slope and interpolation mean derived above.