3.11.3 Calibrating Local Volatility

Basic Prerequisites: 02 · The Calibration Problem and Implied Volatility Surfaces.

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1. Intuition & Practical Objective

The local-volatility model keeps the single-Brownian-driver world of BSM but lets volatility be a function of spot and time: $d{S}_t=(r-q)S_{t}dt+\sigma (t,S_t)S_{t}d{W}_t$. Its defining promise (Dupire 1994; Derman–Kani 1994): given all European prices, there is a unique local-volatility function that reproduces them. So local vol is the simplest “market model” - exactly calibratable to any arbitrage-free smile (Bergomi Ch 2 calls it exactly this).

The practical objective of this page: turn a market smile into a local-vol surface, and verify the surface really reproduces the smile. That round-trip is the whole job. You will see the two directions:

• Inversion (calibration): from implied (total) variance → local variance. The master tool is Gatheral’s eq (1.10), which writes local variance as a ratio of a calendar spread (time derivative of total variance) to a butterfly (strike derivatives).
• Forward (verification): price options under the recovered local vol and confirm their implied vols match the input smile.

Why this matters. Local vol “has no physical significance” - it is a by-product of representing vanilla prices as hedge instruments (Bergomi Ch 2). The model is meant to be recalibrated daily. Its cost is being one-factor: all implied vols are 100% correlated with spot, and the future skew is dictated by today’s smile (see 05 · Failure Modes).

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2. Mathematical Ground Truth & Derivations

Dupire in strike/maturity space. For market call prices $C(K,T)$ the risk-neutral density is ${\partial }^{2}C/\partial K^2$ (Breeden–Litzenberger), and the local variance is (Bergomi eq 2.3; Gatheral eq 1.4):

$$\boxed{{\sigma }_{\text{loc}}^2(K,T)=\frac{2(\frac{\partial C}{\partial T}+qC+(r-q)K\frac{\partial C}{\partial K})}{K^2\frac{{\partial }^{2}C}{\partial K^2}}}.$$

Equivalently, local variance is the risk-neutral expectation of instantaneous variance conditional on the final spot equalling the strike (Gatheral eq 1.12; Gyöngy’s theorem, Bergomi eq 2.6):

$${\sigma }^2(K,T,S_0)=\mathbb{E}[v_T\mid S_T=K].$$

The practitioner form - from implied total variance (Gatheral eq 1.10). Set log-moneyness $y=\ln (K/F_T)$ and total implied variance $w(y,T)={\sigma }_{\text{BS}}^2(K,T)T$. Then, with subscripts denoting partials in $y$,

$$\boxed{v_L(y,T)=\frac{\frac{\partial w}{\partial T}}{1-\frac{y}{w}{w}_y+\frac{1}{4}(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2})w_y^2+\frac{1}{2}{w}_{yy}}}.$$

The numerator is a calendar spread (vol rising in time); the denominator’s $\frac{1}{2}{w}_{yy}$ term is the butterfly (convexity in strike). The no-arbitrage conditions map onto these: strike arbitrage ⇔ ${\partial }^{2}C/\partial K^2<0$ (butterfly); maturity arbitrage ⇔ total implied variance $w$ increasing in $T$ at fixed moneyness (Bergomi eq 2.14–2.15).

Key scalings (Bergomi Ch 2, eqs 2.48–2.53). For a local vol $\sigma (t,S)=\sigma (t)+\alpha (t)x+\frac{1}{2}\beta (t)x^2$ in $x=\ln (S/F_t)$: the implied ATMF skew is $\frac{1}{2}$ the local skew (constant $\alpha$ ⇒ ${\mathcal{S}}_T=\alpha /2$), and implied curvature $\frac{1}{3}$ the local curvature. And for a power-law-decaying local skew $\alpha \propto t^{-\gamma }$, the implied skew decays with the same exponent $\gamma$ (equity: $\gamma \approx \frac{1}{2}$).

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3. Computational Implementation - calibrate the smile, then check it

We (1) define a smooth analytic “market” smile, (2) invert it to local variance via Gatheral (1.10), (3) simulate the resulting local-vol SDE by Monte Carlo, (4) back out implied vols and compare to the input smile. If the inversion is right, the round-trip closes. Stdlib only.

The round-trip closes: options priced under the calibrated local vol come back with implied vols matching the input smile within ~1–3 vol pts at short maturity (grid + Monte Carlo discretization error; the inversion itself is exact by construction - Gatheral (1.10) is a definition). Note the local vol is smile-shaped in strike (variance 0.064 on the put side vs 0.027 at the money) - that is how one function of $S$ encodes the whole smile. Deep-OTM/ITM strikes are dropped from the table because implied vol is ill-conditioned there under MC.

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4. Failure Modes & First-Principles Breakdowns

1. Ill-posedness of the inversion. Local vol requires second derivatives of prices ($w_{yy}$); differentiating noisy data amplifies bid/ask noise by ~$1/d{y}^2$. Tiny smile noise produces huge local-vol spikes or negative local variance (which must be clipped/regulated) - see 05 · Failure Modes for the numbers.
2. One-factor dynamics. All implied vols are perfectly correlated with each other and with spot; the model cannot represent independent vol-of-vol moves or a term structure of vol-of-vol (Bergomi Ch 2, Ch 6: Heston’s $\nu_T(t)\propto(1-e^{-kT})/(kT)$). This makes it wrong for forward-smile and vol-of-vol products.
3. The forward skew is dictated by today’s smile. Local vol implies future skews that are flatter than today’s; a desk selling forward-skew products (cliquets) with LV systematically misprices them (Bergomi Ch 2.6; Gatheral Ch 8: LV forward surfaces are flat relative to today’s, SV’s look like today’s).
4. Not enough to fit vanillas. Matching all European prices does not fix the dynamics; two models with the same smile can price exotics differently (Gatheral Ch 4: LV vs SV agree on Europeans, differ on exotics).

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Duffy, Finite Difference Methods in Financial Engineering

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6. Connected Graph Bridges

• Back: 02 · The Calibration Problem · Index Hub
• Forward: 04 · Calibrating Stochastic Vol · 05 · Failure Modes
• Sibling: Implied Volatility Surfaces · Heston & SABR