3.12.6 Advanced Extensions

Basic Prerequisites: 05 · Failure Modes & Practice and 02 · The rBergomi Model.

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1. Intuition & Practical Objective

rBergomi is the minimal rough model. This page maps the ladder of extensions, each fixing a specific shortcoming:

rBergomi limitation	extension	what it buys
no characteristic function, non-Markovian, MC-only pricing	rough Heston (El Euch–Rosenbaum)	an affine-in-$\mathbb{E}[\int v]$ fractional Riccati system ⇒ semi-explicit Fourier pricing
simulation is slow (singular kernel, many steps)	multifactor Markovian lifts (rational approximations, small-$\epsilon$ OU towers)	near-Markovian proxies with standard MC/PDE machinery
VIX/realized-variance joint fit to SPX is hard	VIX-aware calibration (Jacquier–Martini–Muguruza)	closed-form VIX futures ⇒ joint SPX/VIX calibration
H≈0.1 lacks a microstructural explanation	Hawkes foundations (Jaisson–Rosenbaum; Bacry–Muzy)	rough vol as the scaling limit of nearly-unstable heavy-tailed Hawkes flows

The practical objective: know which extension is the right answer for which problem, what it costs (parameters, tractability, identifiability), and which are desk-ready versus research-frontier. The one thing to internalise: every extension buys tractability at the price of either more parameters or an approximation error, and the empirical anchor is always the same - the H≈0.1 roughness and the $\psi (T)\propto T^{H-\frac{1}{2}}$ skew.

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2. Mathematical Ground Truth & Derivations

2.1 Rough Heston (El Euch–Rosenbaum 2019)

Replace the Heston variance SDE’s $dW$ with a fractional/Volterra driver:

$$v_t=v_0+\frac{1}{\Gamma (\alpha )}{\int }_0^t(t-s{)}^{\alpha -1}\kappa (\theta -v_s)ds+\frac{\nu }{\Gamma (\alpha )}{\int }_0^t(t-s{)}^{\alpha -1}\sqrt{v_s}d{W}_s,\alpha =H+\frac{1}{2},$$

with $0<\alpha <\frac{1}{2}$ (the rough regime). The key tractability: the fractional Riccati system has a closed form, so the log-price characteristic function is

$${\phi }_t(u)=\exp \left(\mathbb{E}\right[{\int }_0^t{v}_{s}ds]\Psi (u,t)+\dots ),$$

where $\Psi$ solves a fractional Riccati ODE (a Mittag-Leffler-type function). This restores a Fourier-pricing route that rBergomi lacks. Short-time ATM skew again behaves as $T^{\alpha -\frac{1}{2}}=T^H$, and the model is richer: it reproduces both the rough skew and, via the $\mathbb{E}[\int v]$ leverage, the variance-swap/VIX structure.

2.2 Multifactor Markovian lifts

The singular Volterra kernel $(t-u{)}^{\alpha -1}$ can be approximated by a sum of exponentials, yielding a finite-dimensional Markovian representation of the fractional driver:

$$(t-u{)}^{\alpha -1}\approx \sum _{j=1}^M{w}_j{e}^{-{\kappa }_j(t-u)},$$

so the fractional integral ${\int }_0^t(t-u{)}^{\alpha -1}{X}_{u}d{W}_u$ becomes a sum of $M$ OU-like factors. This is the “rational approximation” programme (Harang–Langrené; Bayer–Friz; Abi Jaber–El Euch): $M$ factors of spectral accuracy capture the short-time skew to arbitrary precision, and then standard MC/PDE/Fourier machinery applies. Cost: $M$ extra state variables and a delicate small-$t$ blowup to preserve.

2.3 VIX in rough vol (Jacquier–Martini–Muguruza 2018)

VIX is (a scaling of) the 30-day variance-swap variance $\frac{1}{\tau }{\int }_t^{t+\tau }{\xi }_t(u)du$. Because ${\xi }_t(u)$ is a lognormal forward-variance, VIX futures are $\mathbb{E}[\sqrt{\cdot }]$ - a concave functional, so VIX futures trade below the VS strike by Jensen (the same convexity gap as in the Markovian two-factor model of SV · 06 Advanced Extensions). In rough vol this gap is a closed form for the futures price, which is what makes joint SPX/VIX calibration feasible without nested simulation.

2.4 Hawkes microstructural foundations (Jaisson–Rosenbaum 2016)

Rough vol has a bottom-up explanation: a nearly-unstable Hawkes process (a self-exciting point process whose branching ratio approaches 1 from below) converges, under a heavy-tailed jump distribution and in the right rescaling, to a fractional process with $H<1/2$. In Jaisson–Rosenbaum, the order flow of price jumps is a Hawkes process; its cumulative intensity converges to a rough fractional diffusion - so the $H\approx 0.1$ roughness is emergent from microstructure rather than an exogenous assumption. This is the deepest of the extensions: it turns rough vol from a fit into a scaling limit.

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3. Computational Implementation - the short-end blowup that a Markovian lift cannot fake

We demonstrate the structural obstruction at the heart of the lift programme: the rough Volterra kernel $(x{)}^{H-\frac{1}{2}}$ diverges as $x\to 0$, while any finite sum of exponentials (the Markovian representation) is bounded there. This is precisely why a naive finite-factor approximation fails at the short end, and why the rigorous lifts add a dedicated small-time factor. Stdlib only.

Reading the output. The boundedness test is the honest heart of the lift problem: as $x\to 0$, the rough kernel $x^{H-\frac{1}{2}}$ grows without bound ($1.00\to 63.10$ as $x:1\to 10^{-5}$), while the finite exponential tower is trapped below its finite weight-sum ($5$ for $M=5$, $20$ for $M=20$). The tower can match the kernel at intermediate lags but is structurally incapable of the short-end divergence that carries the ATMF skew $\psi (T)\propto T^{H-\frac{1}{2}}$. That is exactly why the rigorous Markovian lifts (Harang–Langrené, Abi Jaber–El Euch) wrap a dedicated small-time factor around the exponential tower. The second block confirms the structural consistency: rough Heston and rBergomi share the same short-time skew exponent $H-\frac{1}{2}$ by construction - the roughness is carried by the single parameter $H$, whichever model you price with.

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4. Failure Modes & First-Principles Breakdowns

1. Using a naive exponential lift for the short end. As §3 shows, a plain sum of exponentials cannot reproduce the divergent $x^{H-\frac{1}{2}}$ kernel near $x=0$ (error $\to 100\%$). A correct lift needs a small-time blow-up factor; otherwise the short-dated skew is silently capped and you’ve reverted to a Markovian model.
2. Rough Heston is affine in $\int v$, not in $v$. Its tractability rests on the fractional Riccati structure of the integrated variance; mistaking it for standard Heston’s affine-in-$v$ characteristic function breaks the derivation.
3. Joint SPX/VIX calibration is not free. It requires the VIX futures closed form; without it (i.e. nested simulation) the joint fit is numerically prohibitive. Adopt the closed-form route (JMM 2018) or accept the cost.
4. Hawkes foundations are a scaling limit, not the model. The microstructural derivation is asymptotic (nearly-unstable branching, heavy tails, specific rescaling); it motivates and explains $H<1/2$ but does not hand you a calibrated pricing model. Don’t present the micro-foundation as a pricing alternative.
5. H remains the empirical anchor. Every extension is calibrated to the same $H\approx 0.1$ (variogram) and $\psi (T)\propto T^{H-\frac{1}{2}}$ (skew). If an extension needs a different $H$ to fit, that is a red flag, not a free parameter.
6. More parameters always fit better. The rough-Heston and lift programmes add parameters; the relevant test is whether they change a dynamic quantity (VIX term structure, SSR, forward skew) in the direction the market says - otherwise they are unidentifiable noise (the same lesson as SV · 06 Advanced Extensions).

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5. References

• El Euch & Rosenbaum (2019), The characteristic function of rough Heston models, Mathematical Finance 29(1), 3–38
• Jaisson & Rosenbaum (2016), Rough fractional diffusions as scaling limits of nearly unstable heavy tailed Hawkes processes, Annals of Applied Probability 26(5), 2860–2882
• Jacquier, Martini & Muguruza (2018), On VIX futures in the rough Bergomi model, Quantitative Finance 18(1), 45–61
• Abi Jaber & El Euch (2019) and Harang & Langrené (2019), multifactor/rational Markovian approximations of the Volterra kernel
• Bayer, Friz & Gatheral (2016), Pricing under rough volatility, Quantitative Finance 16(6), 887–904
• Bacry & Muzy (2014), Hawkes model for price and trades high-frequency dynamics
• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough, Quantitative Finance 18(6), 933–949

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · Index Hub
• Sibling / back-references: SV · 06 Advanced Extensions (the launchpad that first flagged rough vol) · Heston & SABR (flat note) · VS · 06 Advanced Extensions
• Forward: Calibration & Market Practice (fitting rough vol to SPX/VIX) · ES) (fat tails, model risk) · Stochastic Calculus & Itô (fractional / Volterra calculus)
• Base: Probability & Measure Theory (point processes, Hawkes) · Numerical Methods (MC, approximation theory)