3.5.4 Stochastic-Vol Dynamics

Basic Prerequisites: 02 · The Heston Model and 03 · SABR & Asymptotics.

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1. Intuition & Practical Objective

Here is the uncomfortable theorem of this whole pillar (Gatheral §7.8): all stochastic-volatility-with-jumps models generate essentially the same surface shape. Any of them, with appropriate parameters, will fit today’s vanilla smile. So the static fit cannot select a model; it is a constraint, not a test. This page is about what actually does select one: the dynamics.

Two questions, answerable by two different objects:

question	object	what it measures
Statics - what is today’s shape?	skew level ${\partial }_k{\sigma }_{BS}^2{|}_0=\frac{\rho \eta }{2}\beta (v_0)$ (+ jumps)	the spot/vol covariance, model-independently
Dynamics - how does the surface move?	skew stickiness ratio $R_T$ and the vol-of-vol term structure ${\nu }_T(t)$	how implied vol co-moves with spot, and how vol-of-vol decays with maturity

Everything in this folder that a practitioner cares about lives in the second row. Three concrete claims drive the page:

1. Heston is structurally Type I. Its ATMF skew decays as $1/T$ and its skew stickiness ratio tends to $1$. Equivalently, its skew is hard-wired inversely proportional to the vol level, ${\mathcal{S}}_T\propto 1/{\hat{\sigma }}_{F_{T}T}$ - which reality does not show (Bergomi §6). Real markets are Type II with ${\mathcal{S}}_T\propto T^{-1/2}$.
2. Heston’s vol-of-vol term structure is a fixed one-time-scale shape, $\propto (1-e^{-k(T-t)})/(k(T-t))$, which cannot match the empirical power law ${\nu }_T\propto T^{-0.4}$ over a wide maturity range. Forward-variance (Bergomi) models with a couple of OU factors can - that is their entire reason for existing.
3. The Bergomi–Guyon expansion makes the smile computable in terms of three dimensionless numbers. To second order in vol-of-vol, any SV model’s smile is $\hat{\sigma }(K,T)={\hat{\sigma }}_{F_{T}T}+S_{T}k+\frac{C_T}{2}{k}^2$, with the coefficients given by $C^{x\xi },C^{\xi \xi },D$ - integrals of the spot/vol and vol/vol covariance functions. Same recipe, any model.

The practical objective: know that vanilla smiles barely constrain the exotic book, know which dynamic quantity to measure to discriminate, and know the two settings (vol-of-vol level and term structure) that decide the price of forward-skew products.

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2. Mathematical Ground Truth & Derivations

2.1 Heston as a one-factor forward-variance model (Bergomi ch 6)

Bergomi rewrites Heston in terms of the variance curve ${\xi }_t^T={\mathbb{E}}_t[v_T]$:

$${\xi }_t^T=\bar{v}+e^{-\lambda (T-t)}(v_t-\bar{v}),{\hat{\sigma }}_T^2(t)=\bar{v}+\frac{1-e^{-\lambda (T-t)}}{\lambda (T-t)}(v_t-\bar{v}),$$

and as a forward-variance SDE $d{\xi }_t^T=\eta e^{-\lambda (T-t)}\sqrt{{\xi }_t^t}d{Z}_t$ - driftless, which is why variance-swap implied vols stay fixed under perturbations. The price of this elegance is a hard constraint on the initial curve,

$$\frac{d{\xi }_0^T}{dT}=-\lambda ({\xi }_0^T-\bar{v})(\text{6.2}),$$

so Heston cannot fit a general variance-swap term structure and an ATM term structure simultaneously - the curve has one free parameter ($\bar{v}$) and one exponent ($\lambda$). Its vol-of-vol term structure for a flat VS curve is fixed (6.9):

$$\mathrm{vol}({\hat{\sigma }}_T)\propto \frac{1-e^{-\lambda (T-t)}}{\lambda (T-t)}\text{(short }T:\to 1;\text{long }T:\propto 1/(T-t)).$$

Heston’s ATMF skew at order one in $\eta$ (6.17b, flat curve 6.20):

$${\mathcal{S}}_T=\frac{1}{{\hat{\sigma }}_T^3{T}^2}\frac{\rho \eta }{2}{\int }_0^T{V}_{\tau }\frac{1-e^{-\lambda (T-\tau )}}{\lambda }d\tau \stackrel{\text{flat}}{\to }\frac{\rho \eta }{2\sqrt{\bar{v}}}\frac{\lambda T+e^{-\lambda T}-1}{(\lambda T{)}^2},$$

with the two exact limits

$${\mathcal{S}}_T\to \frac{\rho \eta }{4\sqrt{\bar{v}}}(T\to 0)\text{and}{\mathcal{S}}_T\to \frac{\rho \eta }{2\sqrt{\bar{v}}}\frac{1}{\lambda T}(T\to \infty ).$$

The $1/T$ long-maturity decay is Heston’s signature - and the reason it is a Type I model.

2.2 Skew stickiness and the Type I / Type II classification (Bergomi ch 9)

The skew stickiness ratio measures the co-movement of implied vol with spot, normalised by the skew,

$$R_T=\frac{1}{{\mathcal{S}}_T}\frac{\mathbb{E}\left[d\ln Sd{\hat{\sigma }}_{F_T(S)T}\right]}{\mathbb{E}[(d\ln S{)}^2]}$$

($R_T=1$ sticky-strike, $R_T=0$ sticky-delta). At lowest non-trivial order, for a time-homogeneous model with covariance kernel $\mu$ on a flat curve,

$$R_T=\frac{{\int }_0^T\mu (t)dt}{{\int }_0^T(1-t/T)\mu (t)dt}.$$

For monotone-decaying $\mu$ this yields the model-independent range

$$\boxed{R_T\in [1,2]}\text{with}{R}_0=2\text{(the same short-end limit as local volatility).}$$

Classifying models by the decay exponent $\mu (t)\propto t^{-\gamma }$:

$$\text{Type I }(\gamma >1):{\mathcal{S}}_T\propto \frac{1}{T},R_{\infty }=1;\text{Type II }(\gamma <1):{\mathcal{S}}_T\propto T^{-\gamma },R_{\infty }=2-\gamma ;{\mathcal{S}}_T\propto T^{-(2-R_{\infty })}.$$

Heston ($\mu$ exponential) is Type I, $R_T\to 1$; local volatility gives $R_T\to (2-\gamma )/(1-\gamma )$ (e.g. $3$ for $\gamma =\frac{1}{2}$) - i.e. LV and SV start at the same $R_0=2$ and go in opposite directions. The data say Type II ($\gamma \approx \frac{1}{2}$, $R_{\infty }\approx 1.5$), so neither pure LV nor Heston is right, and the two-factor model is built to be Type II over a practical range:

$${\mathcal{S}}_T=\frac{\omega }{2}\sum _i{w}_i{\rho }_{iS}\frac{k_{i}T-1+e^{-k_{i}T}}{(k_{i}T{)}^2},R_T=\frac{{\sum }_i{w}_i{\rho }_{iS}(1-e^{-k_{i}T})/(k_{i}T)}{{\sum }_i{w}_i{\rho }_{iS}(k_{i}T-(1-e^{-k_{i}T}))/(k_{i}T{)}^2}\text{(9.16a,b)}.$$

Pricing consequence (Bergomi §9): for a long spot/vol cross-gamma book, local volatility is the conservative (higher-$R$) choice; for a short cross-gamma book, SV is. Same smile, opposite prices - that is the whole point of this chapter.

2.3 The Bergomi–Guyon expansion (Bergomi ch 8)

Write the pricing equation as ${\partial }_{t}P+H_{t}P=0$ with $H_t=H_t^0+\epsilon W_t^1+{\epsilon }^2{W}_t^2$, where $\epsilon$ scales the vol-of-vol ($\mu \to \epsilon \mu$, $\nu \to {\epsilon }^2\nu$, then $\epsilon =1$). Because forward variances are driftless, VS implied vols are unchanged by $\epsilon$ - so the expansion perturbs the smile without shifting the level. The price expansion is (8.18)

$$P=[1+\epsilon \frac{C_0^{x\xi }}{2}({\partial }_x^3-{\partial }_x^2)+{\epsilon }^2(\frac{C_0^{\xi \xi }}{8}({\partial }_x^2-{\partial }_x{)}^2+\frac{(C_0^{x\xi }{)}^2}{8}({\partial }_x^3-{\partial }_x{)}^2+\frac{D_0}{2}({\partial }_x^3-{\partial }_x{)}^2)]P_0,$$

with the three dimensionless model-dependent constants

$$C_t^{x\xi }={\int }_t^{T}d\tau {\int }_{\tau }^{T}du\mu (\tau ,u)={\int }_t^T(T-\tau )⟨d\ln S_{\tau }d{\hat{\sigma }}_T^2(\tau )⟩,$$ $$C_t^{\xi \xi }={\int }_t^{T}d\tau {\int }_{\tau }^{T}du{\int }_{\tau }^{T}d{u}^{\prime }\nu (\tau ,u,u^{\prime })={\int }_t^T(T-\tau {)}^2⟨d{\hat{\sigma }}_T^2(\tau )d{\hat{\sigma }}_T^2(\tau )⟩,$$ $$D_t={\int }_t^{T}d\tau {\int }_{\tau }^{T}du\mu (\tau ,u)\frac{\delta C_{\tau }^{x\xi }}{\delta {\xi }^u},$$

so the whole smile at order 2 is a function of $(C^{x\xi },C^{\xi \xi },D)$ - the skew functional, the vol-of-vol functional, and the term-structure functional. The implied-vol expansion (8.20–8.21) is

$$\hat{\sigma }(K,T)={\hat{\sigma }}_{F_{T}T}+S_T\ln (K/F_T)+\frac{C_T}{2}{\ln }^2(K/F_T)+O({\epsilon }^3),$$ $$S_T={\hat{\sigma }}_T\frac{\epsilon C^{x\xi }}{2Q^2}\{1+O(\epsilon )\},Q={\hat{\sigma }}_T^{2}T,{\hat{\sigma }}_{F_{T}T}={\hat{\sigma }}_T+\frac{Q}{2}{S}_T.$$

The order-1 skew is exactly the local-volatility formula (2.89). With $\mu$ the instantaneous spot/VS-vol covariance,

$$S_T=\frac{1}{2{\hat{\sigma }}_T^{3}T}{\int }_0^T\frac{T-\tau }{T}⟨d\ln S_{\tau }d{\hat{\sigma }}_T^2(\tau ){⟩}_{0}d\tau ,$$

i.e. the ATMF skew is the (weighted) average of the instantaneous spot/vol covariance over the residual maturity - a statement first derived for local volatility and now shown to be model-independent at order 1. That is the cleanest form of the “skew is the covariance of spot with implied vol” mantra.

Short-maturity limit ($T\to 0$, order 2, ${\mu }_0\equiv \mu (0,0,{\xi }_0^0)$, ${\nu }_0\equiv \nu (0,0,0,{\xi }_0^0)$):

$$C^{x\xi }=\frac{T^2{\mu }_0}{2},C^{\xi \xi }=\frac{T^3{\nu }_0}{3},D=\frac{T^3{\mu }_{0}d{\mu }_0/d{\xi }_0^0}{6},S_0=\frac{{\mu }_0}{4({\xi }_0^0{)}^{3/2}},S_0=\frac{1}{2{\hat{\sigma }}_0^2}\frac{⟨d\ln Sd{\hat{\sigma }}_0⟩}{dt},$$

which reproduces (8.39)–(8.44): SABR $S_0=\rho \nu /2$, Heston $S_0=\rho \eta /(4\sqrt{V_0})$, vanishing correlation $S_0=0$ (all verified in 03 · SABR & Asymptotics).

Order-2 consequences worth remembering: at order 2 the skew is exactly quadratic in log-moneyness (hence “curvature” is meaningful only there); the truncated density can go negative at extreme strikes (use only near the money); and the triple $(C^{x\xi },C^{\xi \xi },D)$ is not enough to pin the dynamics - the SSR needs $\mu$ itself.

2.4 The gamma representation, and what the skew hedges

With ${\omega }_t={\hat{\sigma }}_T^2(t)$ and $Q=e^{-rt}{P}_{BS}$,

$$P=P_{BS}(0,S_0,{\hat{\sigma }}_T^2(0))+\mathbb{E}\left[{\int }_0^T{e}^{-rt}\right(\frac{{\partial }^2{P}_{BS}}{\partial S\partial {\hat{\sigma }}_T^2}d{S}_{t}d{\hat{\sigma }}_T^2(t)+\frac{1}{2}\frac{{\partial }^2{P}_{BS}}{\partial ({\hat{\sigma }}_T^2{)}^2}d{\hat{\sigma }}_T^2(t)d{\hat{\sigma }}_T^2(t)\left)\right],$$

whose order-1 truncation shows the materialising payoff for the spot/vol cross-gamma is ${\ln }^2(S_T/S_0)$ - a log-contract-like position with (signed) replication density $\rho (K)=\frac{2}{K}(1-\ln (K/S_0))$ (positive for $K\ll S_0$). This makes the implied integrated spot/vol covariance a model-free read-off from the market price of ${\ln }^2$, and it is the theoretical basis of the synthetic-skew trade (Bergomi §9.10: cross-gamma/theta P&L $=S{\hat{\sigma }}_0^2\frac{d^2\Pi }{dSd{\hat{\sigma }}_0}(R_T^{r,\text{short}}-2)\delta t$).

2.5 Forward-variance models: the fix (Bergomi ch 7)

Model $(S_t,\{{\xi }_t^T\})$ directly, with the pricing equation (7.4)

$$\frac{\partial P}{\partial t}+(r-q)S\frac{\partial P}{\partial S}+\frac{{\xi }_t^t}{2}{S}^2\frac{{\partial }^{2}P}{\partial S^2}+\frac{1}{2}\iint \nu \frac{{\delta }^{2}P}{\delta {\xi }^u\delta {\xi }^{u^{\prime }}}+\int \mu S\frac{{\delta }^{2}P}{\partial S\delta {\xi }^u}=rP,$$

subject to the cardinal rule that the break-even spot vol equals the instantaneous VS vol, $\sigma (t,S,\xi {)}^2={\xi }_t^t$ (no free theta). A Markov representation exists if and only if $\omega (u)=\omega e^{-ku}$, i.e. forward variances are driven by OU processes:

$${\xi }_t^T={\xi }_0^T\exp (\omega e^{-k(T-t)}{X}_t-\frac{{\omega }^2}{2}{e}^{-2k(T-t)}\mathbb{E}[X_t^2]),d{X}_t=-k{X}_{t}dt+d{W}_t,$$

which is exactly simulable. The two-factor workhorse (7.28–7.39) generalises this with two well-separated time scales and gives the vol-of-vol term structure

$${\nu }_T(t)=\nu {\alpha }_{\theta }\sqrt{{\sum }_{ij}{w}_i{w}_j{\rho }_{ij}I(k_i(T-t))I(k_j(T-t))},I(x)=\frac{1-e^{-x}}{x},$$

against the empirical power-law benchmark (7.40)

$${\nu }_T(t)={\sigma }_0(\frac{{\tau }_0}{T-t}{)}^{\alpha },\alpha \approx 0.4,{\tau }_0=3\text{m}.$$

This is the desk-grade answer to LV’s missing forward skew: a model that fits the VS term structure by construction and controls the vol-of-vol term structure with one or two extra factors. Note also the structural difference from Heston: ${\xi }_t^T$ is lognormal (so short vol is lognormal, not normal - the empirically preferred $\beta (v)\sim \sqrt{v}$ scaling).

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3. Computational Implementation - Heston’s forward-variance facts, the skew limits, and the vol-of-vol term structure

We (i) verify Heston’s VS-vol curve (6.4) and its short-maturity limit, (ii) verify the ATMF skew formula (6.20) against both of its exact limits, (iii) confirm the short-dated variance skew $\rho \eta /2$ from the characteristic-function pricer (an independent route to §03’s asymptotics), and (iv) compare the vol-of-vol term structures of Heston (6.9), the two-factor model (7.39) and the empirical power law (7.40). Stdlib only.

What the output establishes.

• The Heston forward-variance curve and its limit. ${\hat{\sigma }}_T$ runs $13.410\%\to 18.634\%$ as $T$ goes $0.05\to 20$ yr, and the short-maturity limit is exactly $\sqrt{v_0}=13.1909\%$. Note ${\hat{\sigma }}_T$ converges to $\sqrt{\bar{v}}$ for large $T$ - Heston’s forward-variance curve is a one-parameter exponential, which is why it cannot fit a rich VS term structure.
• The ATMF skew formula (6.20) is right, in both limits. At $T=0.01$ the formula gives $-0.367480$ against its exact short-maturity limit $-0.369105$ ($0.4\%$ off, and the gap shrinks as $T\to 0$); at $T=50$ it gives $-0.010972$ against the $1/T$ limit $-0.011140$ ($1.5\%$ off, again shrinking). Both asymptotics are reproduced.
• The short-dated variance skew is $-\rho \eta /2$ to three digits from a completely independent route. The characteristic-function pricer - no asymptotics anywhere - gives $-0.138331$/$-0.138404$ at $T=0.005$–$0.01$ versus the analytic $-\rho \eta /2=-0.138894$. (At $T=0.002$ with $h=2\times 10^{-4}$ the finite-difference window is no longer small compared with ${\sigma }_{BS}\sqrt{T}$, so the value drifts to $-0.120$ - a discretisation artefact of the test, not of the model.) This is the strongest single validation in the folder: the Gatheral (7.3) asymptote, the Bergomi (6.18b) limit and the exact Fourier price all agree.
• The vol-of-vol term structure test - Heston fails, the two-factor model passes. Normalising all three shapes at $T=0.25$: the two-factor model tracks the empirical power law within about $8\%$ over $0.1$–$5$ years ($0.916$–$1.000$), while Heston is $24\%$ low at 3 months ($0.763$), $13\%$ high at 1–2 years, and $41\%$ low at 5 years ($0.587$). Since ${\nu }_T(t)$ sets the price of anything depending on future vol-of-vol (VS options, forward-vol products), this is a pricing failure, not a cosmetic one.

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4. Failure Modes & First-Principles Breakdowns

1. Selecting a model by its static fit. Vanilla smiles barely constrain forward-skew-dependent products: Bergomi’s model-independent bounds leave a 95/105 forward call spread anywhere in $[1.6\%,7.7\%]$ from a flat 20% smile, narrowing only when congruent payoffs are added (Bergomi §3.1.7). Fitting today’s surface is not model validation.
2. Ignoring the direction of the SSR. LV and SV have the same $R_0=2$ and then diverge (LV up to $\approx 3$, SV down into $[1,2]$). A trader on the wrong side of $R_T$ has the wrong sign for the cross-gamma/theta P&L - the realised-SSR backtest (Euro Stoxx 50, 2007–2012) found $\approx 1.6$ realised against an implied $2$, a real materialisable P&L (Bergomi §9.10).
3. Market SSR can go negative, and it is not a model failure. Realised SSR was sharply negative in the Nikkei in 2012, traced to dealer autocall-vega hedging (Bergomi §9.11). Diagnosing this as a calibration error and “fixing” the model makes things worse.
4. Treating vol-of-vol as a single number. ${\nu }_T(t)$ is a term structure; Heston hard-wires it (6.9) and the mismatch is $2\times$ at 3 months and $1.7\times$ at 5 years against the benchmark. Any product whose vega is concentrated in the vol-of-vol (VS swaptions, forward-vol claims) is mispriced outright.
5. Using the order-2 expansion where it is not valid. The truncated density can go negative at extreme strikes; the implied variance is at most affine in $|k|$ (Lee). The Bergomi–Guyon expansion is a near-the-money tool. Exponential resummation (ch 8 App. C) improves tails but does not make it global.
6. Forgetting that vol-of-vol is what the vol-of-vol risk premium prices. ${\nu }_T$ calibrated to vanillas is a $\mathbb{Q}$ quantity incorporating a premium; using it as a forecast of realized vol-of-vol confuses measures (Bergomi ch 5).
7. Believing the two-factor model is “the” model. The two-factor fit to the benchmark (7.40) above is good but not exact; extra factors buy flexibility at the cost of unidentifiable parameters - and the variance curve (${\xi }_0^T$) must be treated as an input from the VS market, not as a calibration output.

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5. References

• Bergomi, Stochastic Volatility Modeling
• Gatheral, The Volatility Surface
• Bergomi & Guyon (2012), Stochastic volatility’s orderly smiles, Risk 25(5)
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 03 · SABR & Asymptotics
• Forward: 05 · Failure Modes & Practice · 06 · Advanced Extensions · Index Hub
• Cross-links: VS · 04 Advanced Dynamics (SSR and the LV/Heston skew, from the surface side) · The Greeks & Dynamic Hedging (cross-gamma/theta) · Econometrics & Time Series (estimating ${\nu }_T$ and the SSR from data)