3.4.3 Surface Models

Basic Prerequisites: 02 · Implied vs Local Vol.

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1. Intuition & Practical Objective

You now know the surface is the market’s price object and that it uniquely pins a local-vol diffusion. The next practical problem is representation: how do you store, interpolate, extrapolate and risk-manage a surface without introducing arbitrage, and how do you extract the tradable quantities - the forward variances and variance-swap vol - that live inside it?

Three building blocks:

1. A parametric, arbitrage-free slice: SVI. Gatheral’s “stochastic-volatility-inspired” parametrization fits a whole maturity slice with five numbers and is built to avoid butterfly/calendar arbitrage (Gatheral eq 3.20).
2. Sticky rules. When the spot moves, what stays fixed - the fixed-strike vols (sticky-strike) or the smile translated with the spot (sticky-delta)? This single assumption changes every delta you hedge with (Bergomi §2.5; Hull §20.5 minimum-variance delta).
3. Forward variance / variance swaps. The tradable object that makes the surface’s dynamics concrete: a variance swap’s fair vol is almost exactly the log-contract implied vol, computable as a weighted integral of the smile (Bergomi ch 5; Gatheral §4.21/5.17).

One-line essence. “A surface model is a set of parameters + arbitrage bounds + a rule for how the surface moves; SVI supplies the parameters and bounds, the sticky rules supply the motion, and forward variance supplies the hedging instrument.”

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2. Mathematical Ground Truth & Derivations

2.1 SVI (Gatheral 2004, eq 3.20)

For a single maturity, with $k=\ln (K/F)$ log-strike,

$$\boxed{w(k)={\sigma }_{BS}^2(k)T=a+b\left[\rho (k-m)+\sqrt{(k-m{)}^2+s^2}\right]}$$

with parameters $a,b,\rho ,s,m$ ($b\ge 0$, $|\rho |<1$, $s>0$). As $|k|\to \infty$ the slice is asymptotically linear in $|k|$ with slopes $b(1\pm \rho )$ - matching Roger Lee’s regularity bound (implied variance grows at most linearly in $|k|$, with the gradients tied to the maximal finite moments of $S_T$; Gatheral §7.7, model-independent). The ATM variance skew is

$${\frac{\partial w}{\partial k}|}_{k=0}=b\left(\rho +\frac{-m}{\sqrt{m^2+s^2}}\right)\underset{m=0}{\to }b\rho .$$

The ATM level is $w(0)=a+b(\rho (-m)+\sqrt{m^2+s^2})$.

2.2 Sticky rules and the minimum-variance delta

Two benchmark behaviors of the surface as $S$ moves:

• Sticky-strike: $\partial {\sigma }_{BS}/\partial S=0$ - each fixed strike keeps its vol; the ATM point slides along the smile. Delta = BS delta.
• Sticky-delta (sticky-moneyness): ${\sigma }_{BS}=f(\ln (K/S))$ - the smile translates with the spot; each fixed log-moneyness keeps its vol.

The skew stickiness ratio $R_T$ interpolates them,

$$R_T=\frac{1}{{\mathcal{S}}_T}\frac{d{\hat{\sigma }}_{F_{T}T}}{d\ln S_0},R_T=1:\text{sticky-strike},R_T=0:\text{sticky-delta}$$

(Bergomi eq 2.61–2.62). The practical consequence is Hull’s minimum-variance delta: because vol moves with the spot,

$${\Delta }_{MV}={\Delta }_{BSM}+V_{BSM}\frac{\partial \mathbb{E}[{\sigma }_{imp}]}{\partial S}<{\Delta }_{BSM},$$

i.e. the naive BS delta over-hedges a short option position when the skew is downward-sloping (Hull §20.5).

2.3 Term structure and total variance (calendar no-arbitrage)

The surface’s maturity axis obeys the convex-order condition - total implied variance must increase with maturity at fixed moneyness (Bergomi eq 2.14/2.15):

$$T_1\le T_2⟹T_1{\hat{\sigma }}^2(k,T_1)\le T_2{\hat{\sigma }}^2(k,T_2)\text{for all fixed }k.$$

Equivalently, in $(y,T)$ coordinates the total-variance profiles must not cross. Slices are interpolated in $T$ by a monotone spline on $w$ (Gatheral uses Stineman monotonic).

2.4 Variance swaps, log contracts and forward variance

A variance swap pays realized variance; its fair strike is replicated by a log contract $f(S)=-2\ln S$. The weight $\rho (K)\propto 1/K^2$ is exactly the density whose vega is spot-independent ($\rho (K)K\varphi (S/K)$ constant; Bergomi eq 3.5). In any diffusive model calibrated to the smile,

$${\hat{\sigma }}_{VS,T}={\hat{\sigma }}_T\text{(log-contract and VS implied vols coincide)}$$

(Bergomi eq 5.13/5.14). The practical evaluation is a weighted integral of the smile (Gatheral/Chriss–Morokoff; Bergomi eq 4.21/5.17):

$$\boxed{{\hat{\sigma }}_{VS,T}^2={\int }_{-\infty }^{\infty }\frac{dy}{\sqrt{2\pi }}{e}^{-y^2/2}{\sigma }_{BS}^2(K(y),T),y(K)=\frac{\ln (K/F_T)}{{\sigma }_{KT}\sqrt{T}}-\frac{{\sigma }_{KT}\sqrt{T}}{2}}$$

For a flat smile this is exactly ${\sigma }_0^2$; for a skewed/convex smile the VS vol exceeds the ATM vol - a skew/convexity premium. Only in non-diffusive (jump/Lévy) models does ${\hat{\sigma }}_{VS,T}\ne {\hat{\sigma }}_T$; the gap measures implied short-return skewness, ${\hat{\sigma }}_{VS,T}^2-{\hat{\sigma }}_T^2\simeq -\frac{1}{3}\lambda ⟨J^3⟩$ (Bergomi eq 5.28/5.29).

Forward variance is the tradable state variable:

$${\xi }_t^T=\frac{d}{dT}[(T-t){\hat{\sigma }}_{VS,T}^2(t)],d{\xi }_t^T={\lambda }_t^{T}d{W}_t^T(\text{driftless}),$$

with the universal constraint ${\xi }_t^t={\sigma }_t^2$: the short end of the forward-variance curve is the instantaneous variance (Bergomi eq 5.6, 4.30). This is the foundation of forward-variance / Bergomi models (page 06).

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3. Computational Implementation - SVI, the VS integral, and the density

We evaluate an SVI slice, compute its ATM skew, extract the variance-swap implied variance via the log-contract integral (checking the flat-smile identity), and test the slice for butterfly arbitrage through the Breeden–Litzenberger density. Stdlib only.

Three verifications land at once: the flat-smile identity returns exactly $0.04={\sigma }_0^2$; the SVI slice’s butterfly density is everywhere positive ($\min =0.0178>0$), so it is static-arbitrage free; and the VS vol ($24.09\%$) exceeds the ATM vol ($18.71\%$) because the skewed/convex smile forces the log-contract to weight the fat left wing.

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4. Failure Modes & First-Principles Breakdowns

1. Parametric fit without arbitrage constraints. You can always fit five SVI numbers to one slice, but a family of slices fitted independently will violate calendar no-arbitrage. Fit all expirations simultaneously with a monotone total-variance interpolant in $T$ (Gatheral §3, ch 3).
2. The wrong sticky rule → the wrong delta. Using BS delta while the market is sticky-delta mis-hedges by the vega×skew term (Hull minimum-variance delta). The choice is not cosmetic; it is a statement about the surface’s motion, and it is directly measurable via $R_T$.
3. VS vol ≠ ATM vol is not “noise”. It is the convexity premium; treating the ATM implied vol as the variance-swap strike underprices the swap. Conversely, the residual gap ${\hat{\sigma }}_{VS,T}-{\hat{\sigma }}_T$ in a diffusive world should be ~$0$ - if your calibration produces a large non-zero gap, your surface extrapolation (not the market) is at fault.
4. Extrapolation drives the VS integral. The log-contract integral $\int dy{e}^{-y^2/2}{\sigma }_{K(y)}^2$ is very sensitive to the smile outside traded strikes (Bergomi §5.2). Practitioners invert the other way: use liquid VS quotes to pin the low-strike implied vols (Gatheral ch 3; Bergomi §5.5).
5. Term-structure kink at dividends. Total-variance continuity across dividend dates requires the Bos–Vandermark effective-dividend adjustment of $y$ (Bergomi eq 2.23/2.26); ignoring it creates spurious local-vol spikes.

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hull, Options, Futures, and Other Derivatives
• Lee, Roger: The Moment Formula for Implied Volatility at Extreme Strikes (2004)

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6. Connected Graph Bridges

• Back: 02 · Implied vs Local Vol
• Forward: 04 · Advanced Dynamics · Index Hub
• Sibling: Heston & SABR · Implied Volatility Surface & Smiles