3.5.6 Advanced Extensions

Basic Prerequisites: 04 · SV Dynamics and 05 · Failure Modes & Practice.

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1. Intuition & Practical Objective

Heston and SABR are the entry points, not the destination. Each of their failures points at a specific extension, and this page maps the ladder:

failure of Heston/SABR	extension	what it buys
cannot fit the short-dated skew (which rises faster than any time-homogeneous SV allows)	jumps (SVJ / SVJJ / Lévy)	the $-2{\mu }_J$ additive skew term at $\tau \to 0$
cannot fit a general variance-swap term structure and a power-law vol-of-vol term structure	forward-variance (multi-factor OU) models	exact VS calibration by construction; term-structure control
calibrated vanillas do not pin the dynamics	Bergomi–Guyon (04)	an inventory of which functionals matter
local vol fits statics but kills forward skew; Heston fits statics but not the short end	local-stochastic vol (LSV)	both - if the model is “usable”
the short-dated skew behaves as $T^{H-\frac{1}{2}}$ with $H\approx 0.1$, not $T^{1/2}$	rough volatility	the empirical short-end scaling

The practical objective: know what each extension actually fixes, what it costs (parameters, simulation complexity, identifiability), and which are desk-standard versus frontier. The one thing to internalise: extensions are bought to fix dynamics, and the price is paid in identifiability.

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2. Mathematical Ground Truth & Derivations

2.1 Jumps: SVJ and SVJJ (Gatheral ch 5)

Add a compound Poisson jump to the spot (Merton), keeping Heston variance - the SVJ model:

$$dS=\mu Sdt+\sqrt{v}Sd{Z}_1+(e^{\alpha +\delta Z}-1)Sdq,Z\sim N(0,1),dv=-\lambda (v-\bar{v})dt+\eta \sqrt{v}d{Z}_2,$$

with log-jump size $\sim \mathcal{N}(\alpha ,{\delta }^2)$. The characteristic function factorises (Gatheral ch 5):

$${\phi }_T(u)=e^{C(u,T)\bar{v}+D(u,T)v_0}{e}^{\psi (u)T},\psi (u)=-{\lambda }_{J}iu(e^{\alpha +{\delta }^2/2}-1)+{\lambda }_J(e^{iu\alpha -u^2{\delta }^2/2}-1),$$

so the Heston $C,D$ are untouched and the jump part is a separate, cheap term. Two facts decide how to fit it:

• The short-dated variance skew is exactly additive: ${\partial }_k{\sigma }_{BS}^2{|}_0\to \rho \eta /2-2{\mu }_J$ with compensator ${\mu }_J={\lambda }_J\mathbb{E}[J]$ (Gatheral 7.8). Fit Heston on the long end (which it handles well) and the jump parameters on the short end (which it does not).
• Jumps in volatility (SVJJ, Matytsin) do not help the short-dated skew at all. Because the short skew depends only on the jump compensator, identical in SVJ and SVJJ (after the jump an ATM option is out-of-the-money with zero time value whatever the vol does). Vol-jumps do reduce the vol-of-vol needed for longer-dated skew - at the cost of three more parameters and a harder fit. Gatheral’s verdict: SVJ beats SVJJ on real SPX data (Table 5.5: $v_0=0.0158,\bar{v}=0.0439,\eta =0.3038,\rho =-0.6974,\lambda =0.5394,{\lambda }_J=0.1308,\delta =0.0967,\alpha =-0.1151$ reproduces the surface where Heston cannot).
• Drawback: after a price jump, volatility stays fixed (jumps and vol are uncorrelated), inconsistent with the observation that implied vols jump when spot jumps. And jumps destroy the static-hedging/completeness arguments that make diffusive models tractable: with uncertain jump size there is no replicating hedge - options stop being redundant (Gatheral §5.1).
• Reading (Bergomi ch 10): a Lévy/jump component is best understood not as a dynamics but as an embedded stress-test reserve policy - extra theta, a cost-of-capital levy ($\lambda =\beta \mu$), or a minimum return on stress-limit usage. “Using a diffusive process for pricing does not mean we assume securities behave as diffusions.”

2.2 Forward-variance models: the desk-grade fix (Bergomi ch 7)

Model the curve $\{{\xi }_t^T\}$ directly. The pricing equation (7.4) plus the cardinal rule $\sigma (t,S,\xi {)}^2={\xi }_t^t$ gives a driftless, exactly-calibratable-to-VS system, with the $N$-factor Markov representation

$$d{\xi }_t^T=\omega {\alpha }_w{\xi }_t^T\sum _i{w}_i{e}^{-k_i(T-t)}d{W}_t^i,{\xi }_t^T={\xi }_0^T\exp (\omega \sum _i{w}_i{e}^{-k_i(T-t)}{X}_t^i-\frac{{\omega }^2}{2}\sum _{ij}{w}_i{w}_j{e}^{-(k_i+k_j)(T-t)}\mathbb{E}[X_t^i{X}_t^j]),$$

with $\omega =2\nu$, ${\alpha }_w=1/\sqrt{{\sum }_{ij}{w}_i{w}_j{\rho }_{ij}}$, and the exact OU transitions (7.15–7.18)

$$X_{{\tau }_{n+1}}^i=e^{-k_i\delta \tau }{X}_{{\tau }_n}^i+\delta X^i,\mathbb{E}[\delta X^i\delta X^j]={\rho }_{ij}\frac{1-e^{-(k_i+k_j)\delta \tau }}{k_i+k_j},\mathbb{E}[\delta W^S\delta X^i]={\rho }_{iS}\frac{1-e^{-k_i\delta \tau }}{k_i}.$$

Because the transition law is exact, no time discretisation bias enters for variance-only payoffs (realized variance, variance swaps, VIX) - a decisive practical advantage over Heston simulation. The stationary vol-of-vol and correlation structures are

$${\nu }_T(t)=\nu {\alpha }_w\sqrt{{\sum }_{ij}{w}_i{w}_j{\rho }_{ij}I(k_i(T-t))I(k_j(T-t))},I(x)=\frac{1-e^{-x}}{x},$$ $$\omega (T-t)=2\nu {\alpha }_w\sqrt{{\sum }_{ij}{w}_i{w}_j{\rho }_{ij}{e}^{-(k_i+k_j)(T-t)}},{\rho }_t({\xi }^T,{\xi }^{T^{\prime }})=\frac{\sum w_i{w}_j{\rho }_{ij}{e}^{-k_i(T-t)}{e}^{-k_j(T^{\prime }-t)}}{\sqrt{\cdots }},$$

and the break-even covariances (7.27a,b) that close the pricing equation. VIX is (a scaling of) the variance-swap variance for $T=30$d; VIX futures are ${\mathbb{E}}_t[\sqrt{\cdot }]$ in this model, priced by two-dimensional quadrature over $(X^1,X^2)$. Vega hedging caveat (Bergomi §7.3.3): the $N$-factor structure does not justify hedging only $N$ variance swaps - deltas must immunise against all deformations $\delta {\xi }^T$; the factors only fix the rank of the break-even covariance matrix.

2.3 Local-stochastic volatility (LSV) - and why “usable” is a real constraint

LSV multiplies the local-vol component onto the stochastic variance (Bergomi 12.1–12.2):

$${\sigma }_t=\sqrt{{\zeta }_t^t}\sigma (t,S_t),d{S}_t=(r-q)S_{t}dt+\sigma (t,S_t)\sqrt{{\zeta }_t^t}{S}_{t}d{W}^S,d{\zeta }_t^T=2\nu {\alpha }_{\theta }{\zeta }_t^T[(1-\theta )e^{-k_1(T-t)}d{W}^1+\theta e^{-k_2(T-t)}d{W}^2]$$

with the mixed covariance $\mu (t,u,\xi )=2\nu {\xi }^u\sqrt{{\xi }^t}{\alpha }_{\theta }[{\rho }_{S{X}^1}(1-\theta )e^{-k_1(u-t)}+{\rho }_{S{X}^2}\theta e^{-k_2(u-t)}]$ (12.4/8.50 - note the radical covers only ${\xi }^t$). The appeal is obvious: the local part fits the statics exactly, the stochastic part restores forward skew. The warning is equally clear (Bergomi §12.2.2): the LSV pricing equation is not derived from a replication argument - it is the forward-variance equation with the ansatz $\sqrt{{\zeta }_t^t}\to \sqrt{{\zeta }_t^t}\sigma (t,S_t)$. Whether its solution is a price (i.e. whether the hedge P&L has the usual gamma/theta form) must be verified a posteriori: “most local-stochastic volatility models are not usable models.” The characterisation of usable LSV, the decomposition of the ATMF skew into its components, and the resulting SSR/vol-of-vol dynamics are Bergomi §12.3–12.4 - a separate deep-read from the corpus used here.

2.4 Rough volatility

Empirically, log realized volatility has Hurst exponent $H\approx 0.1$ (Gatheral–Jaisson–Rosenbaum 2018) - far rougher than a diffusion ($H=\frac{1}{2}$), with a power-law autocorrelation of log-vol. Replacing the OU factors by a fractional Brownian motion, ${\sigma }_t=\exp (X_t)$ with $X$ fBm of Hurst $H$, gives a short-dated ATM skew that scales as

$$\frac{\partial {\sigma }_{BS}}{\partial k}\sim T^{H-\frac{1}{2}},$$

i.e. much steeper than any Markovian SV model permits - precisely the short-end failure flagged in §04/§05, and the same power-law family as Bergomi’s benchmark ${\nu }_T(t)={\sigma }_0({\tau }_0/(T-t){)}^{\alpha }$, $\alpha \approx 0.4$ (note $\alpha \leftrightarrow \frac{1}{2}-H$ in spirit). The cost is that the model is non-Markovian, harder to simulate, and harder to calibrate jointly to SPX and VIX. This is a forward pointer beyond the corpus files used for this folder.

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3. Computational Implementation - the two-factor model, simulated exactly

We verify the two structural facts that make forward-variance models desk-usable: (i) the closed-form $\chi (t,T)$ used in the exponent equals the exact integral ${\int }_{T-t}^T{\eta }^2(u)du$ (so ${\xi }_t^T$ is a genuine martingale), (ii) the exact OU simulation reproduces $\mathbb{E}[{\xi }_t^T]={\xi }_0^T$ and the instantaneous vol-of-vol ${\omega }^2{\eta }^2(T)$, and (iii) the model implies a VIX-versus-variance-swap convexity gap of the right sign. Stdlib only.

Reading the output.

• (1) The exponent’s $\chi (t,T)$ is exactly ${\int }_{T-t}^T{\eta }^2(u)du$ - agreement to $10^{-15}$ across four $(t,T)$ pairs. This is why ${\xi }_t^T$ is a martingale: the driftless lognormal ansatz needs $\mathrm{Var}[\omega x_t^T]={\omega }^2\chi$ to cancel in the expectation, and the identity provides exactly that.
• (2) The martingale property holds in simulation. $\mathbb{E}[{\xi }_1^T]$ is $0.040078$, $0.039755$, $0.039902$ for $T=1,2,5$ against the initial curve $0.04$ - all within Monte Carlo error at 40 000 paths. And the instantaneous vol-of-vol check is sharp: at $t=0.01$ the simulated $\mathrm{Var}[\log {\xi }_t^T]/t=0.6638$ against the analytic ${\omega }^2{\eta }^2(T)=0.6593$ ($0.7\%$, the MC error of a variance estimate on 40 000 paths). The $t=1$ rows are not this limit (the window $[T-t,T]$ is long), and are shown only to display the maturity profile: forward variance for far maturities is much better behaved than for near ones.
• (3) VIX versus variance swap. $\mathbb{E}[\sqrt{V}]=0.1685$ against $\sqrt{\mathbb{E}[V]}=0.2057$ - a $(16.85,20.57)$ vol-point pair, i.e. the model-implied VIX-futures/VS-vol gap, with the correct sign ($\mathbb{E}[\sqrt{\cdot }]<\sqrt{\mathbb{E}[\cdot ]}$ by Jensen). The magnitude here is dictated by Set II’s deliberately high $\nu =174\%$ (chosen to fit the power-law benchmark over 0.25–5y, §04); with realistic index $\nu$ the gap is on the order of one vol point. The mechanism - VIX is a concave (square-root) functional of the variance curve, so VIX futures trade below the variance-swap strike - is the reason the two are separate products with separate hedges.

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4. Failure Modes & First-Principles Breakdowns

1. Adding jumps to fix the long end. Jumps contribute $-2{\mu }_J$ to the short-dated variance skew and nothing structural to the term structure. Using SVJ to repair the long end over-fits the short end and mis-attributes the risk (Gatheral ch 5).
2. Buying vol-jumps (SVJJ) expecting a better short end. They contribute nothing at $\tau \to 0$ (the skew depends only on the jump compensator) while adding three parameters and a harder fit. SVJ fits the SPX surface with fewer parameters.
3. Using an LSV model without the usability check. LSV’s pricing equation is an ansatz, not a replication result; the hedge P&L must be verified. “Most LSV models are not usable models” (Bergomi §12.2.2).
4. Hedging a forward-variance model with only $N$ variance swaps. The Markov structure fixes the rank of the break-even covariance matrix; deltas must immunise against all deformations $\delta {\xi }^T$ (Bergomi §7.3.3).
5. Treating ${\xi }_0^T$ as a calibration output. In forward-variance models the variance curve is a market input from the (more liquid) variance-swap market. Calibrating it to vanillas discards the best-quality information in the market.
6. Ignoring the dividend/rate effect on variance swaps. The log-contract replication needs a discrete-dividend correction (Bergomi 5.44–5.47), and index VS vols and ATMF vols differ through rate vol too: ${\hat{\sigma }}_{VS,T}={\hat{\sigma }}_T-\frac{\rho }{2}{\sigma }_{r}T$ (5.54–5.55) - one vol point for 5y EUR rates at 5bp/day.
7. Jumping to rough vol without the infrastructure. Rough vol fixes the short-end scaling but is non-Markovian: pricing needs fractional-kernel simulation or the multifactor Markov approximation, and joint SPX/VIX calibration is materially harder. Adopt it for a reason (the $T^{H-1/2}$ skew), not for fashion.
8. Extending a model because it fits better. More parameters always fit better; the relevant test is whether the extension changes a dynamic quantity (SSR, ${\nu }_T$, forward skew) in the direction the market says. Otherwise it is a stress reserve (jumps) or unidentifiable noise (extra factors).

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Gatheral, Jaisson & Rosenbaum (2018), Volatility is rough
• Hagan et al. (2002) for SABR (03); Duffie–Pan–Singleton (2000) for the affine class; Matytsin (1999) for SVJJ; Andersen–Andreasen (2000) and Bakshi–Cao–Chen (1997) for the empirical jump fits cited by Gatheral Table 5.4.
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 05 · Failure Modes & Practice · 04 · SV Dynamics · Index Hub
• Sibling / back-references: VS · 06 Advanced Extensions · Heston & SABR (flat note) · VS · 05 Failure Modes
• Forward: Interest-Rate & Term-Structure Models (SABR / LMM) · Quantitative Risk (stress reserves and model risk)
• Base: Stochastic Calculus & Itô · Probability & Measure Theory · Econometrics & Time Series