3.11.4 Calibrating Stochastic Volatility

Basic Prerequisites: Heston & SABR and 02 · The Calibration Problem.

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1. Intuition & Practical Objective

Stochastic-volatility models add a second source of randomness - the volatility itself moves. Heston: $d{v}_t=-\lambda (v_t-\bar{v})dt+\eta \sqrt{v_t}d{Z}_2$. SABR: $d{\sigma }_t=\chi {\sigma }_{t}d{Z}_2$. Two extra facts make calibration structurally different from local vol:

1. Prices are no longer unique (the market is incomplete - one stock cannot hedge the vol driver), so the model parameters carry a risk-neutral choice about the price of vol risk. We work directly under $\mathbb{Q}$ (set the vol-risk price $\varphi =0$, Gatheral Ch 1) because we fit to option prices.
2. The parameters fight over the same features of the smile. The ATMF skew is (to first order) controlled by the combination $\rho \eta$ (Heston) or $\rho \chi$ (SABR), not by $\rho$ and $\eta$ separately. Two very different parameter sets can reproduce nearly the same smile.

The practical objective of this page is twofold: (a) run a real least-squares fit of SABR to a smile (it works, and recovers the truth), and (b) show the parameter-identification problem - that the smile fixes combinations of parameters, not the parameters themselves - and how a ridge penalty stabilizes the choice. This is the calibration subtlety that separates a working desk calibration from a naive one.

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2. Mathematical Ground Truth & Derivations

Heston’s fast-calibration structure (Gatheral Ch 3). In the Heston model the implied-variance skew is, to first order, $\rho \eta$-driven: short-dated skew $\to \rho \eta /2$, long-dated skew $\to \rho \eta /({\lambda }^{\prime }T)$ as $T\to \infty$ (Gatheral eq 3.19). This gives the practical recipe: two expirations fix ${\lambda }^{\prime }$ and $\rho \eta$; the ATM term structure fixes $\bar{v}$ and $v_0$; the skew curvature separates $\rho$ from $\eta$. But note the identifiability issue hidden here: the short- and long-dated skews each only pin the product $\rho \eta$ - and since $\rho$ and $\eta$ are anticorrelated in effect, they are hard to separate from a single slice.

SABR asymptotics (Gatheral Ch 7, Hagan–Kumar–Lesniewski–Woodward 2002). For $dS=\sigma S^{\beta }d{Z}_1,d\sigma =\chi \sigma d{Z}_2$:

$${\sigma }_{\text{BS}}(k)={\sigma }_0\frac{y}{f(y)}(1+\frac{1}{4}\rho \chi {\sigma }_0+\frac{2-3{\rho }^2}{24}{\chi }^{2}T+O(T^2)),y=-\chi \frac{k}{{\sigma }_0},f(y)=\ln \frac{\sqrt{1-2\rho y+y^2}+y-\rho }{1-\rho },$$

with ATM skew $\partial {\sigma }_{\text{BS}}/\partial k{|}_{k=0}=\rho \chi /2$. The skew is set by $\rho$; the curvature (wings) by $\chi$ (vol-of-vol); $\beta$ controls the backbone (how ATM vol depends on level). Because $\beta$ and $\rho$ trade off - a lower $\beta$ (more normal dynamics) can mimic the effect of a more negative $\rho$ - the pair is not identifiable from one slice: the smile pins the combination, and different $(\beta ,\rho )$ with adjusted $(\alpha ,\chi )$ fit equally well.

The $\kappa /\eta$ ridge. In Heston-style fitting, the mean-reversion speed $\lambda$ (or $\kappa$) and the vol-of-vol $\eta$ are highly collinear through the skew term $\rho \eta$ and the short/long skew ratio; the literature routinely reports that the speed of mean reversion is poorly identified from a single snapshot. The first-principles remedy is the same ridge as in 02: add $+\lambda \Vert \theta -{\theta }_0{\Vert }^2$ so the optimizer does not wander along the flat (near-degenerate) direction of the objective.

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3. Computational Implementation - a real SABR fit + the identification problem

Fit the 4-parameter Hagan SABR formula to a smile generated from a known truth (so we can see both the recovery and the degeneracy). Then freeze $\beta$ at different values and refit - if many $\beta$‘s fit equally well, $\beta$ is not identifiable. Then add a ridge on $\beta$. Stdlib only (Nelder–Mead least squares).

Two lessons jump out. (1) The free fit recovers the truth ($\beta =0.51,\rho =-0.30$ vs true $0.5,-0.3$), because the data was generated by SABR. (2) Non-identifiability: freezing $\beta$ at $0.3$, $0.5$, or $1.0$ gives statistically identical fits (RMSE 0.00138 / 0.00137 / 0.00141) yet very different $\rho$ ($-0.27,-0.30,-0.37$) and $\alpha$. The smile cannot tell them apart - but they carry different hedges. The ridge is the fix: with $\lambda =1$ the fit pins $\beta$ to the prior $0.6$ while losing nothing measurable in fit (RMSE 0.00138 vs 0.00137). You cannot fit your way out of an identification problem - you must regularize it.

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4. Failure Modes & First-Principles Breakdowns

1. Non-identifiability of $\beta /\rho$ (SABR) and $\kappa /\eta$ (Heston). The smile pins products like $\rho \eta$, not the parameters. Different sets fit equally but hedge differently. Regularize or fix one parameter (the desk’s convention fixes $\beta$, e.g. $\beta =0.5$ equity, $\beta =1$ FX).
2. One-slice blindness. A single maturity fits $\rho$ and $\chi$ but cannot separate mean reversion; term structure (multiple maturities) is required for $\lambda$, and even then $\lambda$ is weakly identified. Gatheral’s recipe uses the term structure of skew to separate parameters.
3. No time-homogeneous SV model fits the market short end. Gatheral Ch 3: the observed short-dated skew rises faster than Heston allows - a pure diffusion SV model cannot match the far short-end smile; jumps are needed (see Heston & SABR). A calibration that “fits” by contorting parameters there is overfitting.
4. Heston’s structural deficiencies. Bergomi Ch 6: Heston hard-wires skew $\propto 1/\hat{\sigma }$ (inverse to vol level - reality shows the opposite), has a single vol-of-vol time scale $\propto (1-e^{-kT})/(kT)$, and cannot fit a general variance-swap term structure. Calibrating it to a wide surface forces compromises.
5. Incomplete-market pricing choice. The $\varphi =0$ choice (no vol-risk premium) is a modeling assumption; it affects exotic prices even when the vanilla fit is identical.

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hagan–Kumar–Lesniewski–Woodward (2002), “Managing Smile Risk”
• Hull, Options, Futures, and Other Derivatives

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6. Connected Graph Bridges

• Back: 03 · Calibrating Local Vol · Index Hub
• Forward: 05 · Failure Modes · 06 · Advanced Extensions
• Sibling: Heston & SABR · Implied Volatility Surfaces