3.4.4 Advanced Dynamics

Basic Prerequisites: 03 · Surface Models.

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1. Intuition & Practical Objective

All stochastic-volatility-with-jumps models produce essentially the same surface shape (Gatheral ch 7, §7.8). So the shape cannot tell you which model is right. What separates models - and what costs money - is the dynamics: how the surface moves when the spot moves and when time passes. This page is the “why dynamics decides” page.

Two distinct questions:

1. Statics (today’s shape): answered by the skew level, ${\partial }_k{\sigma }_{BS}^2$. Every SV model gives about $\rho \eta \beta (v)/2$ short-dated and $\rho \eta \beta (v)/({\lambda }^{\prime }T)$ long-dated, and jumps add $-2{\mu }_J$ - so the shape is (to first order) model-independent (Gatheral §7.1–7.3).
2. Dynamics (how it moves): answered by the skew stickiness ratio $R_T$ and the forward skew ${\mathcal{S}}_{\theta }(\tau )$. Here models diverge sharply: local vol generates $R_T\to \approx 2$–$3$ and too-flat future skews; time-homogeneous SV keeps future skews equal to today’s.

The practical objective: know that vanilla smiles barely constrain cliquet/forward-start prices, and that the skew-level independence and the empirical lognormal-variance scaling point to a specific dynamics (lognormal variance, $\beta (v)\sim \sqrt{v}$, not a square-root process).

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2. Mathematical Ground Truth & Derivations

2.1 Skew stickiness ratio in local volatility (Bergomi eq 2.64)

The regression definition is $R_T=\frac{1}{{\mathcal{S}}_T}\frac{⟨d{\hat{\sigma }}_{F_{T}T}d\ln S_0⟩}{⟨(d\ln S_0{)}^2⟩}$ (Bergomi 2.62), with $R_T=1$ sticky-strike, $R_T=0$ sticky-delta. In local volatility

$$R_T=1+\frac{1}{T}{\int }_0^T\frac{{\mathcal{S}}_t}{{\mathcal{S}}_T}dt,$$

and the $R=2$ rule: if the skew $\mathcal{S}$ (equivalently $\alpha$) is maturity-independent, $R_T=2$ for all $T$; moreover ${\lim }_{T\to 0}{R}_T=2$ for any smooth LV (exact, via the backward/forward symmetry ${\hat{\sigma }}_{ST}(K)={\hat{\sigma }}_{KT}(S)$, zero rates). For a power-law skew ${\mathcal{S}}_T\propto T^{-\gamma }$,

$$R_T\to \frac{2-\gamma }{1-\gamma }(T\to \infty ):\gamma =\frac{1}{2}(\text{equity})\Rightarrow R_{\infty }=3;\gamma =1\Rightarrow R_T\propto \ln T.$$

The vol-of-vol implied by LV is $\mathrm{vol}({\hat{\sigma }}_{F_{T}T})\to 2{\mathcal{S}}_T$ as $T\to 0$ (Bergomi eq 2.85) - a large, structural number.

2.2 Skew dynamics under SV vs LV

• SV: future surfaces look like today’s - the skew is time-homogeneous. The empirical fact that the skew slope is roughly independent of the volatility level translates to $\beta (v)\sim \sqrt{v}$, i.e. variance is approximately lognormal, not square-root (Gatheral §8.1).
• LV: future (forward) surfaces are substantially flatter than today’s, because the forward local skews are flatter: from (2.90)/(2.91) with today’s skew ${\mathcal{S}}_t=\alpha (t/{\tau }_0{)}^{-\gamma }$,

$${\mathcal{S}}_{\theta }(\tau )={\mathcal{S}}_{\tau +\theta }-\frac{\tau }{\theta }\left(\frac{1}{\theta }{\int }_{\tau }^{\tau +\theta }{\mathcal{S}}_{t}dt-{\mathcal{S}}_{\tau +\theta }\right),{\mathcal{S}}_{\theta }(\tau )\propto {\left(\frac{\theta }{\tau }\right)}^{\gamma }{\mathcal{S}}_{\theta }\ll {\mathcal{S}}_{\theta }.$$

• Consequence: a wrong SV model (off $\sim 1.5\times$ if vol doubles) still beats LV, which generates almost no forward skew (Gatheral §8.1–8.2). LV sellers of forward-skew products win the deal and lose money (Gatheral ch 8/10).

2.3 Heston skew and term structure

Heston (Gatheral §2.2, §3.4; Bergomi ch 6) with ${\lambda }^{\prime }=\lambda -\rho \eta /2$, ${\bar{v}}^{\prime }=\bar{v}\lambda /{\lambda }^{\prime }$:

$${\hat{\sigma }}_{BS}^2{|}_{K=F_T}=\frac{(\bar{v}-{\bar{v}}^{\prime })(1-e^{-{\lambda }^{\prime }T})}{{\lambda }^{\prime }T}+{\bar{v}}^{\prime },$$ $${\sigma }_{BS}^2\approx {\hat{w}}_T^{\prime }/T+\rho \eta \frac{x_T}{{\lambda }^{\prime }T}\left(1-\frac{1-e^{-{\lambda }^{\prime }T}}{{\lambda }^{\prime }T}\right).$$

• Short-dated ATM skew $\to \frac{\rho \eta }{2}$ - independent of $\lambda$ and $T$ (confirmed numerically below, $-0.1388$ vs $-0.1389$).
• Long-dated ATM skew $\to \frac{\rho \eta }{{\lambda }^{\prime }T}$ - decays as $1/T$.
• The skew is independent of the variance level $v_0,\bar{v}$ (approximately true even for $v_0\ne \bar{v}$); increasing $|\rho |$ or $\eta$ steepens it; $\eta$ also sets curvature (kurtosis).
• Heston hard-wires ${\mathcal{S}}_T\propto 1/{\hat{\sigma }}_{F_{T}T}$, which reality does not show (Bergomi §6 criticism).

2.4 Jumps and the SABR/SV asymptotics

Adding jumps (Gatheral ch 5, §7.3). For short $\Delta T$,

$${\sigma }_{BS}^2\text{ skew}{|}_{k=0}\approx -2{\mu }_J,{\mu }_J=\lambda \mathbb{E}[J-1]\text{(compensator)}.$$

Jumps and stochastic vol contribute additively to the ATM variance skew at $\tau =0$:

$${\frac{\partial v_{BS}}{\partial k}|}_{k=0}\to \rho b(\sigma )-2{\mu }_J.$$

The jump compensator drives the short-expiry skew; the expected jump size drives its decay (Gatheral §5.4). The SABR model

$$d{S}_t={\sigma }_t{S}_t^{\beta }d{Z}_1,d{\sigma }_t=\chi {\sigma }_{t}d{Z}_2,d{Z}_{1}d{Z}_2=\rho dt$$

(Hagan et al. 2002) has no mean reversion, so it is a short-expiration tool, but it has an exact $\tau \to 0$ smile formula that factorizes: ${\sigma }_{BS}(k)={\sigma }_0\frac{y}{f(y)}(1+\frac{1}{4}\rho \chi {\sigma }_0+\frac{2-3{\rho }^2}{24}{\chi }^2\tau +\cdots )$ with $y=-\chi k/{\sigma }_0$ (Gatheral eq 7.7). It implies ${\partial }_k{\sigma }_{BS}{|}_{k=0}=\rho /2$ - the special case of (7.6) - and the Medvedev–Scaillet small-time expansion (Gatheral eq 7.4–7.6) reproduces it, proving ${\partial }_{k}I{|}_{k=0}\to \rho b(\sigma )/(2\sigma )$.

Long expirations (Fouque–Papanicolaou–Sircar): for log-OU volatility the skew ${\partial }_x{\sigma }_{BS}\approx \rho \xi /(\lambda T)$ (Gatheral eq 7.10), matching Heston for large $\lambda T$. The natural interpolation between the two limits is Gatheral eq 7.11 (not Bergomi’s), which Lewis’s small-$\eta$ expansion proves exact to $O(\eta )$ (Gatheral §7.6, eq 7.11–7.12). Extreme strikes: Lee’s moment formula, ${\beta }^{\ast }=g(q^{\ast })$, $g(x)=2-4(\sqrt{x^2+x}-x)$ - model-independent (Gatheral §7.7).

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3. Computational Implementation - SSR and the Heston skew limits

We (i) verify the LV skew-stickiness formula on a realistic decaying equity skew (and the $R=2$ rule), and (ii) verify the Heston ATM term structure and the short-dated-skew limit $\rho \eta /2$. Stdlib only.

The LV skew stickiness ratio climbs toward $R=2$–$3$ exactly as the power-law asymptotics predict (a $1/\sqrt{T}$ skew → $R_{\infty }=3$), and the constant-skew case gives the exact $R=2$ rule. The Heston skew converges to $\rho \eta /2=-0.1389$ as $T\to 0$ (computed $-0.1388$ at $T=0.001$) and decays toward $1/T$ at long maturities - the same $1/T$ decay that is too slow to fit the observed short-end smile (Gatheral Table 3.2 / Fig 3.6). Note also the printed ATM variance $\to 0.0354=\bar{v}$ as $T\to 0$ - the flagged notation subtlety: the physical short limit should be $v_0=0.0174$, revealing (3.18)‘s unconditional-path construction.

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4. Failure Modes & First-Principles Breakdowns

1. Judging 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. Local volatility’s missing forward skew. LV generates future skews $\propto (\theta /\tau {)}^{\gamma }$ times today’s - systematically flat. Digitally-capped cliquets, barriers and digitals are all mispriced (Gatheral ch 8–10: a 1-year ATM digital mispriced by $\sim 12\%$ of notional if the skew term is dropped).
3. Square-root variance is the wrong scaling. Empirically the skew slope is ~independent of vol level, implying $\beta (v)\sim \sqrt{v}$ (lognormal variance), not Heston’s $\beta =1$. Models with $R_T$ and vol-of-vol term structure of the wrong shape misprice variance swaptions and cliquets even with matched vanillas (Gatheral §8.1; Bergomi ch 6–7).
4. No time-homogeneous SV fits the short end. The observed short-dated skew rises faster than any SV model allows; jumps are required (Gatheral ch 3 conclusion, ch 5). Fitting Heston to the whole surface forces a compromise that is too flat short-dated.
5. SABR is short-dated only. No mean reversion ⇒ SABR does not reproduce the long-dated skew term structure; using it for long expirations is a first-principles error (Gatheral §7.2).

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hagan, Kumar, Lesniewski, Woodward: Managing Smile Risk (Wilmott, 2002)

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6. Connected Graph Bridges

• Back: 03 · Surface Models
• Forward: 05 · Failure Modes · Index Hub
• Sibling: Heston & SABR · 06 · Advanced Extensions