3.12.5 Failure Modes & Practice

Basic Prerequisites: 04 · Hurst Estimation & Simulation and 02 · The rBergomi Model.

---

1. Intuition & Practical Objective

Everything up to here was mathematics; this page is about what actually goes wrong with rough-vol models on a desk, and why. Four symptom classes, each traceable to a first principle:

1. The simulation breaks. The Volterra kernel $(t-u{)}^{H-\frac{1}{2}}$ is singular at $u\to t$; naive Euler diverges. The hybrid scheme exists to fix exactly this.
2. The empirics are misread. H<1/2 is anti-persistence, not long memory; classical long-memory estimators certify long memory on rough data that has none (GJR §4). This misdiagnosis has historically driven the wrong model choice.
3. Calibration is ill-posed and non-Markovian. Three parameters but no characteristic function and no PDE; pricing is Monte Carlo or a Markovian lift, and $H$ and $\eta$ enter the short-time skew only through $\eta \sqrt{2H}$.
4. Model risk is concentrated at the short end. The whole reason to adopt rough vol is the short-dated skew; but the short end is also where estimation error is largest and where the Markovian-vs-rough gap is biggest.

The practical objective: be able to state exactly why naive simulation fails and how the hybrid scheme fixes it; know that “rough” and “long memory” are opposites; size the Markovian-vs-rough skew gap; and know the checks (martingale, positivity) that police any implementation.

---

2. Mathematical Ground Truth & Derivations

2.1 The singular kernel and the hybrid fix

The driver $W_t^{\alpha }=\sqrt{2\alpha +1}{\int }_0^t(t-u{)}^{\alpha }d{W}_u$ with $\alpha =H-\frac{1}{2}\in (-\frac{1}{2},0)$ has a kernel that diverges as $u\to t^{-}$. A naive Euler discretisation $\sqrt{2\alpha +1}{\sum }_j(t_i-u_j{)}^{\alpha }\Delta W_j$ is not a consistent estimator of the Itô integral: the singular proximal cell contributes a term whose discretisation error does not vanish as the grid refines unless handled separately. The hybrid scheme (BLP 2017) splits off the proximal integral and treats it as a Gaussian of variance ${\Delta }^{2\alpha +1}/(2\alpha +1)$ - which is what makes $\mathrm{Var}[W_t^{\alpha }]\to t^{2H}$ (verified in 04 · Simulation).

2.2 Rough is not long memory (GJR §4)

For $H<\frac{1}{2}$, fBm increments are anti-correlated and the autocovariance is negative at every nonzero lag (${\rho }_1=\frac{1}{2}(2^{2H}-2)<0$). Classical long memory means $H>\frac{1}{2}$ (positive, non-integrable autocovariance). GJR’s §4 is the sharp empirical statement: run a long-memory test (R/S, fractional differencing with $d=0.4$) on a well-calibrated RFSV path and on real SPX vol - both return “long memory”, with matching statistics. The apparent long memory of volatility is spurious, an artefact of anti-persistence at short scales. Roughness and long memory are not only different - they are opposite regimes of H.

2.3 Why Markovian models are structurally wrong at the short end

For a one-factor OU (Heston-like) vol-of-vol kernel $g(u-t)=e^{-\kappa (u-t)}$, the Bergomi–Guyon skew functional gives

$$S_T^{OU}=\frac{1}{2{\hat{\sigma }}_T^3{T}^2}{C}^{x\xi },C^{x\xi }\propto {\int }_0^T{\int }_t^T{e}^{-\kappa (u-t)}dudt=\frac{\kappa T-1+e^{-\kappa T}}{{\kappa }^2},$$

so $S_T^{OU}\to \frac{1}{2}$ (bounded) as $T\to 0$. For the rough kernel $(u-t{)}^{H-\frac{1}{2}}$, $S_T\propto T^{H-\frac{1}{2}}\to \infty$. This is the structural failure: no finite-factor Markovian model can produce an ATMF skew that blows up at short maturity, yet the SPX data demand $\psi (T)\propto T^{-0.44}$. The comparison is quantified in §3.

2.4 The short-time skew and the H/η identification

From 02:

$$\psi (T)=\frac{\rho \eta \sqrt{2H}}{2(H+\frac{1}{2})(H+\frac{3}{2})}{T}^{H-\frac{1}{2}}.$$

The parameters enter as products: $\rho \eta \sqrt{2H}$ (level) and $H-\frac{1}{2}$ (exponent). From one short-time skew you can read the exponent $H$ (the slope) but only the product $\rho \eta \sqrt{2H}$ (the level) - you cannot separate $\eta$ from $\rho$ without the wings or additional maturities, and $\rho$ must come from the spot/vol correlation. Three parameters, but the short-time vanillas see them as ~2 independent directions.

---

3. Computational Implementation - MC pricing, the Markovian cap, and the exponent

We (i) run a full Monte Carlo of the rBergomi model and extract the ATMF skew at several maturities to confirm the power-law exponent empirically; (ii) compare the rough-kernel skew to the Markovian OU skew to size the short-end gap. Stdlib only.

What the output establishes.

• (i) The full MC pricing confirms the power-law skew. Running the actual rBergomi model (hybrid scheme for the vol driver + correlated spot) and inverting implied vol gives a negative ATMF skew ($\rho <0$) whose log-log slope is $-0.392$ vs the theory $-0.360$ - agreement within Monte Carlo error (10 000 paths). The power-law short-dated skew is not an artefact of the Bergomi–Guyon expansion; it is present in the raw simulated prices.
• (ii) The Markovian gap is unbounded as T→0. The rough-kernel skew $S_T$ grows without bound ($0.95\to 11.45$ as $T:1\to 0.001$) while the OU (Heston-like) skew caps at $0.4992$ (the $\frac{1}{2}$ limit of §2.3). No finite-factor Markovian model can produce the SPX ${\tau }^{-0.44}$ short-end behaviour - this is the quantitative case for rough vol.

---

4. Failure Modes & First-Principles Breakdowns

1. Using naive Euler on the singular kernel. Divergent, biased, no Brownian-limit consistency. Use the hybrid scheme (or the exact Hosking for fBm-increment tasks). Check $\mathrm{Var}[W_t^{\alpha }]=t^{2H}$ and $\mathbb{E}[v_t]={\xi }_0(t)$ after any scheme change.
2. Reading “rough” as “long memory”. They are opposite regimes of H. Long-memory estimators (R/S, fractional differencing) return long memory on anti-persistent rough data - GJR §4 shows the same “verdict” on real SPX and on a simulated RFSV path. Diagnose with the variogram exponent $2H$, not R/S.
3. Estimating H from too short a sample. The variogram slope is biased on short series (see §04’s downward bias at 400×150); GJR use thousands of days across 21 indices. A weekly H is noise.
4. Trying to separate H from η on a single skew slice. The short-time skew sees $\eta \sqrt{2H}$ and the exponent $H-\frac{1}{2}$; $\eta$ and $\rho$ are not individually pinned by vanillas. Use the variogram of log-vol (for H) + the wing/multi-maturity data (for η, ρ).
5. Believing the Bergomi–Guyon expansion globally. $\psi (T)\propto T^{H-\frac{1}{2}}$ is the order-1 ATMF slope; the truncated density can misbehave at extreme strikes (Lee’s moment formula governs the wings). Use it near the money and for its coefficients, like any SV expansion.
6. Ignoring the non-Markovian pricing cost. No characteristic function, no PDE - pricing is MC (slow, needs variance reduction; see McCrickerd–Pakkanen) or a Markovian lift (rough Heston, El Euch–Rosenbaum). The adoption decision must price in this computational tax.

---

5. References

• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949
• Bennedsen, Lunde & Pakkanen (2017), Hybrid scheme for Brownian semistationary processes, Finance and Stochastics 21(4), 931–965
• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904
• Bergomi (2016), Stochastic Volatility Modeling, ch 8 (Bergomi–Guyon expansion and its validity)
• McCrickerd & Pakkanen (2018), Turbocharging Monte Carlo pricing for the rough Bergomi model
• Fukasawa (2017), Short-time at-the-money skew and rough fractional volatility

---

6. Connected Graph Bridges

• Back: 04 · Hurst Estimation & Simulation · 02 · The rBergomi Model
• Forward: 06 · Advanced Extensions · Index Hub
• Cross-links: SABR · 05 Failure Modes (Feller, discretisation, model risk - the Markovian analogue) · ES) (fat tails) · Foundations · Monte Carlo (variance reduction for rBergomi pricing)