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Rough Bergomi Model

Rough Bergomi explains something that has puzzled traders for years: why are short-dated smiles so steep? The answer turns out to be that vol paths in real markets are far more jagged than classical models assume. When you measure the "roughness" of actual realized vol in BTC, ETH, or the S&P 500, you find it is much rougher than anything Heston or SABR can produce.

This model is not used for real-time surface fitting -- it is too slow. Its value is theoretical: it tells you why vol surfaces look the way they do, and gives you the right intuition when fitting practical models like SVI to short-dated crypto options. The implied volatility patterns it explains are visible across every liquid options market.

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The roughness insight

Measured across equities, FX, and crypto, vol paths are far more jagged than standard models assume. This roughness naturally produces the steep short-dated skew observed in markets -- no jumps or extreme parameters needed.

Interactive: Roughness and Skew​

Use the slider below to see both effects of the roughness parameter (H) in action. The left panel shows how lower H produces jaggier, more irregular paths. The right panel shows how that roughness translates to steeper short-dated skew.

Rough Paths Explorer

H (Hurst exponent)0.10
Rougher (jaggier paths, steeper skew)Smoother (standard Brownian motion)
Path Roughness
H=0.1 (rough)H=0.3H=0.5 (Brownian)H=0.10 (slider)Time stepsPath value
ATM Skew vs Maturity (log-log)
1d7d30d90d1yTime to expiry|ATM skew|T^(-0.5) classicalT^(-0.4) H=0.1T^(-0.4) H=0.10

Drag the slider to change H. Lower H produces jaggier paths (left) and steeper short-dated skew (right). At H=0.5, the path is standard Brownian motion and skew follows the classical T^(-0.5) decay.

What "Rough" Means​

Classical models like Heston give vol smooth, gently meandering paths -- like a river. Rough Bergomi gives vol jagged, coastline-like paths. This is not a modeling choice -- it is what the data shows when you measure real vol paths at high frequency.

The roughness is controlled by a single number: the Hurst parameter H. Lower H = rougher paths = steeper short-dated skew.

H value
Path character
What it means for skew
0.1 (observed)
Extremely rough, spiky, coastline-like
Very steep short-dated skew. Matches BTC/ETH markets.
0.3
Moderately rough, noticeable jitter
Moderate short-dated skew. Steeper than classical but less than observed.
0.5 (classical)
Standard Brownian motion -- smooth-looking
Classical skew. Too steep at very short dates, not steep enough at medium dates.
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H near 0.1 is a fact, not a choice

Researchers find H near 0.1 whether they measure S&P 500, individual stocks, BTC, or ETH. The data itself says vol paths are rough. The model is built on what the data shows.

The ATM skew power law​

The roughness parameter H controls how ATM skew decays from short to long expiries. With H near 0.1, short-dated skew is steep and it flattens out as you go longer. This single parameter explains the entire term structure of skew from 1 day to 1 year -- in both crypto and equities.

Classical models (Heston, SABR) systematically get this wrong: they overpredict skew at 1 day and underpredict it at 30 days. Rough Bergomi with H near 0.1 threads the needle. The Black-Scholes framework cannot capture this power-law behavior at all.

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Steep short-dated skew explained

Rough Bergomi explains why short-dated skew is so steep. It is a theoretical insight, not a production tool.

Parameters​

Three free parameters, plus the forward variance curve from market data.

Parameter
Typical value
What it does
H (Hurst)
0.07 - 0.12
Roughness of vol paths. Lower = rougher = steeper short-dated skew
eta (vol of vol)
1.5 - 3.0
How much vol fluctuates. Controls smile width and butterfly level
rho (correlation)
-0.7 to -0.9
Spot-vol correlation. Negative = put skew (standard)

Strengths and Limitations​

Strength
What it means for you
Matches observed skew scaling
A single parameter (H) explains how skew decays from short to long expiries. Works for crypto and equities.
Explains steep short-dated smiles
Classical models need extreme parameters or added jumps. Rough Bergomi produces steep short-dated skew naturally.
Empirically grounded
H near 0.1 is measured from real data, not chosen for convenience.
Limitation
What it means for you
No pricing formula
Every price requires Monte Carlo simulation. Orders of magnitude slower than SABR or SVI.
Path-dependent (remembers its history)
You cannot write a PDE for option prices. No simple numerical solver. Greeks like delta and vega must be computed by simulation.
Fitting takes minutes to hours
Each candidate parameter set requires a full Monte Carlo run. Compare to milliseconds for SVI.
Not practical for real-time use
Production vol surfaces need to update in milliseconds. Rough Bergomi is too slow.

Comparison with Classical Models​

Property
Rough Bergomi
Heston
SABR
Skew scaling
Correct (H-based power law)
Wrong (too steep at short dates)
Wrong (same issue)
Pricing speed
Monte Carlo only (slow)
Semi-analytic (fast)
Formula (fastest)
Fitting speed
Minutes to hours
Seconds
Milliseconds
Short-dated smile
Excellent
Poor without jumps
Moderate
Best for
Theoretical insight, skew research
Equity exotics, structured products
Rates, FX, crypto smile fitting

Why It Matters for Crypto​

ℹ️
A lens, not a production tool

Rough Bergomi is like Black-Scholes -- not the model you run in production, but the framework that gives you the right language and intuition.

It explains why crypto smiles look the way they do. BTC and ETH vol surfaces have steep short-dated skews. Rough Bergomi says: this steepness is the natural consequence of rough vol paths, which is what the data shows.

It gives you the right prior for SVI fitting. If you are fitting SVI to sparse short-dated data, rough vol tells you the skew should be steep. The power law gives you a quantitative expectation for how skew should evolve across expiries. Useful when data is thin. At each strike, the expected implied vol follows from the roughness of the underlying variance process.

It frames the research frontier. Deep-learning fitting of rough vol models, hybrid rough-local vol, and rough Heston variants may eventually be fast enough for real-time use. Understanding the framework now means you will recognize these tools when they arrive. Concepts like delta hedging and vega exposure remain the same, but their computation becomes much harder under rough dynamics. The challenge is computing these Greeks without calendar arbitrage violations when stitching simulated slices together, something OTM wings are especially sensitive to.

Equation Explorer​

Convert between implied vol, total variance, log-moneyness, and option prices.

Equation Explorer

w = σ2 × Ttotal variance = IV2 × time
%
The implied volatility
days
Calendar days to expiration
Total Variance (w)
0.022225
Annualized Variance (σ²)
0.2704
Round-trip IV
52.00%
Total variance is what SVI and other models fit. It scales with time, so a 50% vol for 30 days has less total variance than 50% vol for 90 days.

Self-Check​

Test your understanding before moving on.

Q: Why does Rough Bergomi produce steeper short-dated skew than Heston or SABR, without needing extreme parameters?
Q: If Rough Bergomi is theoretically superior, why isn't it used for real-time vol surface fitting?
Q: A trader notices that BTC 1-day implied vol skew is much steeper than 30-day skew. How does rough volatility explain this?
Q: How can the rough vol insight help you when fitting SVI to sparse short-dated crypto data?

💡 Tip: Try answering each question yourself before revealing the answer.

Building mathematical intuition​

Learn Rough Bergomi from scratchInteractive lesson · no prerequisites

This lesson starts with the rough-volatility insight, then explains the Hurst parameter, the variance process, and why roughness naturally steepens the short end of the smile.

See also:

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