Rough Volatility Models
Classical stochastic-volatility models assume that variance is driven by a diffusion with relatively smooth paths. High-frequency realized-volatility studies instead often find very irregular, nearly non-differentiable behavior. Rough volatility models encode that evidence by giving log variance a fractional regularity parameter H below one half, commonly near 0.1. The resulting short-time implied-volatility skew can match equity-index observations more naturally than standard Markovian models.
“Rough” describes path regularity, not merely a volatile market. It is a statistical property of the modeled variance process. Its appeal does not eliminate the hard parts: estimating H from noisy data, simulating a memory process, and proving that a better cross-sectional fit improves risk management.
From Heston to rough Bergomi
In the Heston model, variance is Markovian: the next state depends on the current state. In a rough model, the state depends on a weighted history. A stylized Volterra Gaussian driver is:
W_t^H = integral_0^t K(t-s) dW_s
K(u) = u^(H - 1/2) / Gamma(H + 1/2)
For H < 1/2, the kernel is singular near zero, giving rough paths and long memory in the representation. In rough Bergomi, forward variance is lognormal with such a driver, and the asset return has correlated Brownian noise.
| Parameter | Role | Observable influence |
|---|---|---|
H | roughness exponent | short-time skew scaling |
eta | vol-of-vol amplitude | smile curvature |
rho | spot/vol correlation | skew direction |
| forward variance curve | expected future variance | ATM term structure |
Under broad rough-volatility settings, the at-the-money skew has an approximate short-maturity power law proportional to T^(H - 1/2). A low H thus allows steep short-dated skews without forcing extreme instantaneous correlation or vol-of-vol.
Estimating roughness without fooling yourself
Log realized variance at fine sampling intervals is contaminated by bid-ask bounce, discretization error, jumps, and intraday seasonality. A naïve log-log regression of increment moments can therefore estimate the microstructure process rather than volatility roughness.
| Estimator risk | Why it matters | Mitigation |
|---|---|---|
| Microstructure noise | inflates short-scale variation | pre-averaging, coarser grids |
| Intraday seasonality | creates deterministic scale effects | deseasonalize first |
| Jump contamination | violates continuous-path scaling | robust estimators or jump filter |
| Limited range | slope is sensitive to selected scales | report scale sensitivity |
Estimate on multiple liquid instruments, dates, and sampling choices. Bootstrap by blocks to preserve serial dependence. A narrow confidence interval from overlapping intraday observations is not evidence of precision. It often reflects an invalid independence assumption.
Forward variance is the natural state
For option pricing, forward variance xit(u) = Et[v_u] is more useful than a single spot variance. It connects directly to the term structure and can be initialized from a cleaned implied-volatility surface. The calibration problem then fits selected liquid quotes after matching the forward curve.
def short_maturity_skew(T, H, eta, rho, scale):
# diagnostic approximation, not a full pricing engine
return scale * rho * eta * T ** (H - 0.5)
Hs = np.linspace(0.03, 0.45, 100)
candidate_skews = [short_maturity_skew(5/252, h, eta=.8, rho=-.7, scale=.2)
for h in Hs]
Never use this approximation as a calibration substitute outside its asymptotic regime. Quoted maturities, discrete dividends, rates, and bid-ask uncertainty all matter. For pricing, use a validated hybrid scheme, Markovian lift, or transform method appropriate to the exact model.
Simulation and computational choices
The direct convolution simulation of W^H costs quadratically in time steps. Hybrid schemes approximate the kernel near its singularity accurately; convolution methods use FFT acceleration; multi-factor Markovian lifts approximate the power-law kernel by a sum of exponentials. The lift is especially useful for risk systems because it recovers a finite-dimensional state.
| Method | Advantage | Trade-off |
|---|---|---|
| Hybrid scheme | accurate near zero | implementation complexity |
| FFT convolution | fast long paths | grid and wraparound care |
| Markovian lift | filters and hedges naturally | approximation factors |
| Plain Euler convolution | simple prototype | slow and biased near kernel |
Check convergence jointly in time step, number of paths, and kernel approximation. A stable option price does not guarantee stable Greeks; hedging requires the latter.
Model risk and trading relevance
Roughness is often presented as a structural explanation of the volatility surface. That is not equivalent to an alpha signal. Parameters can move with data construction, calibration weights, and market regime. Daily re-fitting can generate large model P&L and hedge turnover even when quotes move little.
Compare rough-volatility hedges to Heston stochastic volatility using an honest hedging experiment: freeze parameters at a stated clock, rebalance at executable prices, and attribute transaction costs, vega, and parameter re-estimation. Test static-arbitrage quality across the complete surface, not just fitted points. A rough model may be preferable for short-dated index options while adding no value for longer maturities or single names.
Key takeaways
- Rough volatility models use
H < 1/2to represent highly irregular variance paths. - Their principal empirical attraction is realistic short-maturity skew scaling.
- Roughness estimates are vulnerable to microstructure noise, jumps, and scale selection.
- Judge a model by robust calibration and out-of-sample hedge performance, not elegance.