Reader question: Standard stochastic-volatility models make volatility evolve through a small Markov state. What changes if market volatility is empirically much rougher at short scales than those diffusions can naturally reproduce?
The rough Bergomi model replaces ordinary Markovian volatility dynamics with a fractional-kernel Gaussian driver. The resulting variance process remembers the path through a singular power-law kernel and can generate the steep short-maturity implied-volatility skew associated with rough-volatility models.
This article owns the rough-volatility simulation and calibration problem: forward variance curve + Hurst exponent + volatility-of-volatility + price/variance correlation + fractional-kernel simulation → variance paths, option prices, implied-volatility surfaces, calibration residuals and numerical-convergence diagnostics.
It does not own Heston calibration, local-volatility extraction, generic Monte Carlo, or volatility forecasting from GARCH. Those are separate mechanisms. The distinct page role here is to explain how a non-Markovian fractional kernel changes the state, the simulation algorithm and the failure tests.
This is public mathematical and computational education. It is not financial advice or a recommendation to trade options or volatility products.
1. What “rough” means
A fractional Brownian motion with Hurst exponent H has increment magnitudes that scale roughly with time interval Δ as:
|ΔWH| ∝ ΔH.
When H = 1/2, the scaling matches ordinary Brownian motion. When H < 1/2, paths are locally more irregular.
The empirical rough-volatility literature reported estimates of H around 0.1 for log-volatility in several datasets. The important idea is not one universal number; it is that H materially below 1/2 changes short-time scaling.
Roughness should not be confused with long memory. A process can be locally rough without possessing classical long-range dependence.
2. The rough Bergomi variance process
A common rough Bergomi specification begins with a deterministic forward-variance curve ξ0(t).
Define the Volterra Gaussian process:
W̃tH = √(2H) ∫0t (t−s)H−1/2dWs.
The instantaneous variance can be written:
Vt = ξ0(t) exp[ηW̃tH − ½η²t2H],
where η controls volatility-of-volatility.
The subtraction of ½η²t2H normalises the lognormal exponential so that its expectation is aligned with the input forward-variance curve under the model setup.
3. The stock-price process
Under a risk-neutral pricing measure, a simplified discounted stock process is:
dSt/St = √VtdBt.
The Brownian driver B is correlated with the Brownian motion W feeding the rough variance process:
d⟨B,W⟩t = ρdt.
A negative ρ creates the familiar leverage effect: negative price shocks tend to coincide with upward volatility shocks.
4. Four core inputs have distinct jobs
- ξ0(t): anchors the forward variance term structure.
- H: controls roughness and short-time scaling.
- η: controls the strength of variance fluctuations.
- ρ: controls price/variance co-movement and strongly influences skew.
These parameters can compensate for one another in calibration. A good fit alone does not prove that each parameter is identified.
5. Why the model is non-Markovian
In Heston, current variance is itself a Markov state: knowing the current state is sufficient for the future conditional law.
In rough Bergomi, the future variance law depends on the accumulated Brownian history through the convolution kernel:
(t−s)H−1/2.
For H < 1/2, the exponent is negative. Recent shocks receive a singularly strong local weight, while older shocks remain embedded through the integral.
That memory makes direct simulation more computationally demanding.
6. The covariance structure must be simulated correctly
The rough driver at two times is jointly Gaussian. A naive algorithm that draws each W̃tH independently destroys the temporal covariance and therefore destroys the model.
One exact-grid approach builds the covariance matrix of the Gaussian vector and applies Cholesky factorisation. For N time nodes, dense factorisation can be expensive and memory-heavy.
Verification invariant: simulated sample covariance should converge toward the theoretical covariance of the Volterra process.
7. Hybrid simulation
The kernel is most difficult near s=t, where the power law is singular. The hybrid scheme of Bennedsen, Lunde and Pakkanen handles that local region more carefully than a plain Riemann discretisation.
Conceptually:
- split the Volterra integral into a near-singular part and a smoother historical part;
- integrate the power-law behaviour accurately close to the singularity;
- approximate the remaining kernel with simpler weighted increments;
- reuse Gaussian increments efficiently across time steps.
The purpose is not merely speed. It reduces discretisation error for the rough kernel at similar computational complexity to cruder schemes.
8. A minimal Monte Carlo pricing algorithm
- Construct the forward-variance curve ξ0(t).
- Choose H, η, ρ.
- Build a time grid.
- Generate correlated Gaussian increments for W and an independent driver.
- Construct W̃H with a validated rough-kernel scheme.
- Compute Vt at each grid node.
- Form the correlated price driver B.
- Evolve the stock under the chosen discretisation.
- Evaluate payoff path by path.
- Discount and average.
- Convert model option prices to implied volatilities for calibration comparison.
- Repeat with finer grids and more paths to separate discretisation error from Monte Carlo error.
9. Inputs and outputs
Inputs can include:
- spot/forward level;
- discount and dividend curves;
- forward variance curve;
- Hurst exponent H;
- volatility-of-volatility η;
- correlation ρ;
- option strikes and maturities;
- time grid;
- path count;
- Gaussian simulation scheme;
- variance-reduction method;
- implied-volatility inversion routine.
Outputs can include:
- variance and stock paths;
- option prices;
- model implied-volatility surface;
- calibration residuals;
- Monte Carlo standard errors;
- time-grid convergence;
- parameter sensitivities;
- forward-variance reproduction checks;
- covariance diagnostics.
10. The forward variance curve is not an optional detail
Rough Bergomi is a forward-variance model. The initial term structure ξ0(t) determines the mean variance level by horizon.
If the input forward-variance curve is wrong, calibration can push η, ρ or H to compensate for a term-structure error that belongs elsewhere.
Diagnostic: freeze rough parameters and perturb the forward-variance curve. If smile parameters move dramatically on recalibration, the decomposition is weakly identified.
11. Calibration is expensive because every objective evaluation may require simulation
A basic least-squares objective can be:
J(H,η,ρ) = Σj wj[σmodel,j − σmarket,j]².
But σmodel may come from Monte Carlo option prices. That means the objective itself is noisy unless common random numbers, sufficiently many paths, or surrogate methods are used.
A noisy objective can create false local minima and unstable gradients.
12. Common random numbers are a calibration control
If each candidate parameter vector uses an unrelated random seed, two objective values differ because of both parameter change and Monte Carlo noise.
Using common random numbers across nearby parameter evaluations makes the objective comparison much smoother.
Falsifier: rerun the calibration under several independent master seeds. Large parameter dispersion relative to pricing tolerances indicates simulation noise is influencing the fit.
13. Evidence polarity
Evidence for confidence includes stable calibration across seeds, small and pattern-free implied-volatility residuals, correct reproduction of the input forward-variance curve, covariance checks for the rough driver, convergence under time-grid refinement, Monte Carlo errors well below calibration errors, plausible parameter stability through adjacent dates, and out-of-sample smile dynamics that are not systematically wrong.
Evidence against confidence includes parameters pinned to constraints, materially different fits under nearby grids, unstable H estimates, strong dependence on one maturity region, option-price residuals with systematic strike patterns, forward-variance mismatch, large seed dependence, or a model that fits static smiles while producing implausible dynamics.
14. Counterexample: “rough” inferred from noisy data
High-frequency volatility estimates contain microstructure noise and measurement error. Statistical procedures can infer artificial roughness if data construction is poor.
Falsifier: estimate roughness across sampling frequencies, volatility estimators and noise corrections. If H changes radically with data treatment, the roughness estimate is not robust.
15. Counterexample: static fit does not prove dynamic fit
Many models can reproduce one option surface after calibration. A model can fit today’s smile and still generate the wrong evolution of skew, forward variance or hedging sensitivities tomorrow.
Falsifier: test out-of-sample smile changes and hedge errors, not only same-day repricing.
16. Counterexample: time grid too coarse
The singular fractional kernel makes near-time behaviour particularly sensitive to discretisation.
A coarse grid can bias option prices while Monte Carlo standard errors look small. Increasing the number of paths will not repair time-discretisation bias.
Falsifier: double time resolution while holding the random experiment comparable. If prices move more than the reported Monte Carlo error, discretisation dominates.
17. Counterexample: Cholesky is exact on a grid but too expensive
Dense covariance factorisation can provide an accurate Gaussian vector on a fixed grid, but cost grows quickly with grid size.
A method can therefore be mathematically clean but operationally unsuitable for calibration loops requiring thousands of repricings.
Alternative: hybrid convolution schemes, FFT acceleration, Markovian multi-factor approximations or precomputed/surrogate pricing.
18. Counterexample: Markovian approximation can erase the very roughness being modelled
A popular acceleration represents the power-law kernel as a finite sum of exponentials. Each exponential creates a Markov factor, turning the non-Markovian model into a higher-dimensional Markov system.
This can be extremely useful, but too few factors distort the kernel.
Falsifier: compare the approximating kernel, covariance, option prices and short-maturity skew against the reference rough model as factor count increases.
19. Counterexample: calibration non-identifiability
H, η and ρ all influence skew shape. With limited strikes and maturities, different parameter triples can produce similar prices.
Falsifier: profile the calibration objective around the optimum. Long flat valleys indicate that point estimates are more precise than the data justify.
20. Diagnostics
- Forward-variance expectation check: verify simulated mean variance against ξ0(t).
- Gaussian covariance check: compare theoretical and empirical rough-driver covariance.
- Time-step convergence: refine the rough-kernel grid.
- Path convergence: increase Monte Carlo samples and track standard errors.
- Seed stability: repeat calibration under independent random seeds.
- Parameter profiling: inspect objective curvature around H,η,ρ.
- Smile residual map: plot errors by strike and maturity.
- Short-maturity skew test: inspect whether the model reproduces observed scaling where roughness matters most.
- Out-of-sample dynamics: test tomorrow’s smile/hedges, not only today’s fit.
- Cross-engine benchmark: compare hybrid, Cholesky and Markovian-approximation prices on a controlled case.
21. Alternatives
Heston stochastic volatility uses a low-dimensional Markov square-root variance process and often has faster transform pricing.
Local volatility can exactly fit a static vanilla surface under idealised construction but generates different dynamics.
Local-stochastic volatility combines local fit flexibility with a stochastic variance factor.
Lifted/Markovian rough approximations replace the fractional kernel with multiple exponential factors.
Jump-volatility models explain short-maturity smiles through discontinuities rather than rough continuous variance.
22. Connections to the surrounding knowledge estate
Heston calibration algorithms provide a Markov stochastic-volatility comparison.
Dupire local-volatility algorithms provide the deterministic state-dependent alternative.
Monte Carlo pricing algorithms supply the general simulation and convergence framework used by rough Bergomi.
Particle-filter algorithms show another computational response when volatility is latent and state estimation is required.
The full lane is indexed at Finance & Banking Algorithms | Applied Mathematics in Real Financial Systems.
23. What would falsify confidence?
Confidence should be withdrawn if simulated rough-driver covariance is wrong; if prices do not converge under time-grid refinement; if calibration is seed-dependent; if parameters are unidentifiable; if the static fit hides persistent dynamic hedge errors; if roughness estimates vanish under better data treatment; or if a simpler model performs equally well out of sample with materially less complexity.
24. Verification and update triggers
Preserve the forward-variance curve, H, η, ρ, grid, path count, random seed policy, simulation scheme, option dataset, weights and implied-volatility solver for every calibration.
Revalidate after material volatility-regime changes, forward-variance curve methodology changes, new short-dated products, simulation-library upgrades, parameter-bound changes, persistent smile residuals, hedge deterioration or a switch between exact rough simulation and Markovian approximations.
25. Primary and high-quality references
- Christian Bayer, Peter Friz and Jim Gatheral, Pricing under Rough Volatility, Quantitative Finance, 2016, introducing and studying the rough Bergomi model.
- Jim Gatheral, Thibault Jaisson and Mathieu Rosenbaum, Volatility Is Rough, Quantitative Finance, 2018.
- Mikkel Bennedsen, Asger Lunde and Mikko S. Pakkanen, Hybrid Scheme for Brownian Semistationary Processes, including numerical application to rough Bergomi simulation.
- Markovian Approximation of the Rough Bergomi Model for Monte Carlo Option Pricing, for finite-factor approximation and convergence diagnostics.
- Federal Reserve Board and OCC, SR 11-7 — Guidance on Model Risk Management, for validation, benchmarking and model limitation principles.
Educational boundary: Rough Bergomi is a sophisticated option-pricing model built on a specific fractional-kernel representation. A good calibration is evidence that the model reproduces selected prices, not proof that market volatility literally follows that process.