3.12.2 The rBergomi Model

Basic Prerequisites: 01 · From Zero and SV · 04 Stochastic-Vol Dynamics.

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1. Intuition & Practical Objective

Bayer, Friz & Gatheral (2016) ask: what is the minimal pricing model that inherits the empirical roughness of log-vol without losing tractability? The answer is rBergomi: a forward-variance model whose variance is lognormal and driven by a Riemann–Liouville Volterra process $W^{\alpha }$ with $\alpha =H-\frac{1}{2}<0$ - the fractional driver with the right covariance.

Three things to internalise:

1. It is a forward-variance model, like Bergomi’s $n$-factor model - but with a power-law kernel. The variance is $v_t={\xi }_0(t)\exp (\eta W_t^{\alpha }-\frac{{\eta }^2}{2}{t}^{2H})$, which keeps $\mathbb{E}[v_t]={\xi }_0(t)$ (the forward-variance curve stays a market input). What makes it rough is replacing the exponential OU kernel with $(t-u{)}^{H-\frac{1}{2}}$.
2. The exponent $\alpha =H-\frac{1}{2}$ is the whole design. Because $W_t^{\alpha }=\sqrt{2\alpha +1}{\int }_0^t(t-u{)}^{\alpha }d{W}_u$ has $\mathrm{Var}[W_t^{\alpha }]=t^{2\alpha +1}=t^{2H}$, the log-variance inherits the fBm increment law ${\nu }^2{\Delta }^{2H}$ - the exact GJR scaling. Roughness is built in by the kernel exponent, not added as an afterthought.
3. The skew falls out of the Bergomi–Guyon machinery. The ATMF skew of any forward-variance model is $S_T=\frac{\rho \eta \sqrt{2H}}{2(H+\frac{1}{2})(H+\frac{3}{2})}{T}^{H-\frac{1}{2}}$ at order one (verified in §3) - a power law in $T$ with exponent $H-\frac{1}{2}\approx -0.36$, precisely the SPX ${\tau }^{-0.44}$ regime, and blowing up as $T\to 0$ in a way no Markovian model can.

The practical objective: be able to write the rBergomi SDE, understand why $E[v_t]={\xi }_0(t)$ is the martingale check, know the three parameters $(H,\eta ,\rho )$, and reproduce the power-law ATMF skew from the Bergomi–Guyon expansion.

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2. Mathematical Ground Truth & Derivations

2.1 The model

Under $\mathbb{Q}$, with $S_0$ scaled to 1 and the forward-variance curve ${\xi }_0(u)$ given by the market:

$$S_t=\mathcal{E}\left({\int }_0^t\sqrt{v_u}d\right(\rho W_u^1+\sqrt{1-{\rho }^2}{W}_u^2){)}_t,v_t={\xi }_0(t)\exp (\eta W_t^{\alpha }-\frac{{\eta }^2}{2}{t}^{2\alpha +1}),$$ $$W_t^{\alpha }=\sqrt{2\alpha +1}{\int }_0^t(t-u{)}^{\alpha }d{W}_u^1,\alpha =H-\frac{1}{2}\in (-\frac{1}{2},0),$$

where $W^1,W^2$ are independent Brownian motions and $\mathcal{E}(\cdot )$ is the stochastic exponential. The Volterra process $W^{\alpha }$ is a centred, $(\alpha +\frac{1}{2}-\epsilon )$-Hölder, non-semimartingale Gaussian process with $\mathrm{Var}[W_t^{\alpha }]=t^{2\alpha +1}=t^{2H}$ - “rough” in the precise sense of having $H<\frac{1}{2}$.

2.2 Why $E[v_t]={\xi }_0(t)$ (the martingale check)

Since $W_t^{\alpha }$ is Gaussian with variance $t^{2H}$,

$$\mathbb{E}\left[e^{\eta W_t^{\alpha }}\right]=e^{\frac{{\eta }^2}{2}{t}^{2H}},$$

so $\mathbb{E}[v_t]={\xi }_0(t)e^{-\frac{{\eta }^2}{2}{t}^{2H}}\cdot e^{\frac{{\eta }^2}{2}{t}^{2H}}={\xi }_0(t)$. The drift term $-\frac{{\eta }^2}{2}{t}^{2H}$ is exactly the lognormal compensator - get its sign or the exponent $t^{2H}$ (i.e. the mapping $\alpha =H-\frac{1}{2}$) wrong and the martingale fails. This is the first check on any implementation, verified in §3.

2.3 The ATMF skew from Bergomi–Guyon

For a forward-variance model, the order-1 ATMF skew is (Bergomi ch 8, 04 · SV Dynamics)

$$S_T:={\frac{\partial {\sigma }_{BS}}{\partial k}|}_{k=0}=\frac{C^{x\xi }(T)}{2{\hat{\sigma }}_T^3{T}^2},C^{x\xi }(T)={\int }_0^{T}dt{\int }_t^{T}du\frac{\mathbb{E}[d{x}_{t}d{\xi }_t(u)]}{dt}.$$

For rBergomi the spot/forward-variance covariance is $\frac{\mathbb{E}[d{x}_{t}d{\xi }_t(u)]}{dt}=\rho \sqrt{{\xi }_t^t}{\xi }_t(u)\eta \sqrt{2H}(u-t{)}^{H-\frac{1}{2}}$; on a flat forward-variance curve ${\xi }_0={\sigma }_0^2$ this gives

$$C^{x\xi }(T)=\rho \eta \sqrt{2H}{\sigma }_0^3{\int }_0^{T}dt{\int }_t^{T}du(u-t{)}^{H-\frac{1}{2}}=\rho \eta \sqrt{2H}{\sigma }_0^3\frac{T^{H+\frac{3}{2}}}{(H+\frac{1}{2})(H+\frac{3}{2})},$$ $$\boxed{\psi (T)=|S_T|=\frac{\rho \eta \sqrt{2H}}{2(H+\frac{1}{2})(H+\frac{3}{2})}{T}^{H-\frac{1}{2}}}$$

The double integral is evaluated exactly (§3): ${\int }_0^T{\int }_t^T(u-t{)}^{H-\frac{1}{2}}dudt=\frac{T^{H+3/2}}{(H+1/2)(H+3/2)}$. The result is the power-law skew $\psi (T)\propto T^{H-\frac{1}{2}}$ with exponent $H-\frac{1}{2}=-0.36$ for $H=0.14$ - the empirically observed ${\tau }^{-0.44}$ regime, and qualitatively different from the Markovian cap (§05).

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3. Computational Implementation - the martingale check and the skew power law

We (i) verify the two defining identities of the hybrid-scheme rBergomi - $\mathrm{Var}[W_t^{\alpha }]=t^{2H}$ and $\mathbb{E}[v_t]={\xi }_0(t)$ - and (ii) verify the Bergomi–Guyon skew functional: the double integral equals its closed form, and the resulting $\psi (T)$ scales as $T^{H-\frac{1}{2}}$. Stdlib only.

What the output establishes.

• (i) The hybrid scheme reproduces both defining identities. $\mathrm{Var}[W_t^{\alpha }]$ tracks $t^{2H}$ (within ~3%, the discretisation error of the singular kernel) and - with a small $\eta$ so the estimator is well-conditioned - $\mathbb{E}[v_t]={\xi }_0(t)$ to within $0.55\%$. With $\eta =1.9$ the martingale still holds exactly in law; the naive MC estimate just becomes heavy-tailed (05 · Failure Modes).
• (ii) The closed-form double integral is right to 5 decimals, and $\psi (T)=|S_T|\propto T^{H-\frac{1}{2}}$ with OLS slope $-0.360$ exactly equal to $H-\frac{1}{2}$. The constant of proportionality $\frac{\rho \eta \sqrt{2H}}{2(H+\frac{1}{2})(H+\frac{3}{2})}$ equals $0.4310$ - so $\psi (T)=0.431T^{-0.36}$ for these parameters. A Monte Carlo pricing of the actual skew in §05 confirms the exponent ($-0.392$ with MC noise).

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4. Failure Modes & First-Principles Breakdowns

1. Wrong sign on the compensator. The $-\frac{{\eta }^2}{2}{t}^{2H}$ term is load-bearing: flip it and $\mathbb{E}[v_t]={\xi }_0(t)e^{{\eta }^2{t}^{2H}}\ne {\xi }_0(t)$, silently mispricing everything. Always check the martingale (§3(i)).
2. Confusing $\alpha$ and $H$. The model is written with $\alpha =H-\frac{1}{2}$; the variance exponent is $t^{2\alpha +1}=t^{2H}$. Papers use both - a factor of 2 in the exponent of $t$ is an easy, catastrophic slip.
3. The singular kernel breaks naive Euler. $(t-u{)}^{\alpha }$ diverges at $u\to t$; a plain Euler discretisation is biased and has no Brownian-limit consistency. The hybrid scheme’s proximal-term handling is not cosmetic (04 · Simulation).
4. $\rho ,\eta$ sign and the skew orientation. With equity $\rho <0$ the skew is negative; the formula is $\psi (T)=\frac{\rho \eta \sqrt{2H}}{\cdots }{T}^{H-\frac{1}{2}}$, so a wrong $\rho$ sign flips the skew. Calibrate $\rho$ from the leverage effect / spot-vol covariance, not from H.
5. The order-1 expansion is a near-the-money tool. $\psi (T)\propto T^{H-\frac{1}{2}}$ is the ATMF slope at order one; using it to extrapolate the whole smile (the wings) is outside its validity - Lee’s moment formula still governs extreme strikes (03 · SABR & Asymptotics).

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5. References

• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904
• 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
• Bergomi (2016), Stochastic Volatility Modeling, ch 7–8
• Fukasawa (2017), Short-time at-the-money skew and rough fractional volatility

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6. Connected Graph Bridges

• Back: 01 · From Zero · SV · 04 SV Dynamics (forward-variance framework + Bergomi–Guyon)
• Forward: 04 · Hurst Estimation & Simulation · Index Hub
• Theory: Stochastic Calculus & Itô (Volterra integrals, non-semimartingales) · 02 · The Heston Model (the affine/CF baseline rBergomi abandons) · SV · 06 Advanced Extensions (the frontier flag pointing here)