3.4.2 Implied vs Local Volatility

Basic Prerequisites: BSM PDE & Derivation and 01 · From Zero.

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1. Intuition & Practical Objective

Implied volatility is a quote: one number per option, defined by inverting the BSM formula. It is not a model of how the stock moves. Local volatility is a model: a function ${\sigma }_L(S,t)$ that says “the stock’s instantaneous vol at spot $S$ and time $t$ is this.” Two different objects.

The miracle that connects them is Dupire’s theorem (1994): given the prices of all European calls $C(K,T)$ - i.e. the entire implied-vol surface - there is a unique diffusion

$$d{S}_t={\mu }_t{S}_{t}dt+{\sigma }_L(S_t,t)S_{t}d{W}_t$$

that reproduces exactly those prices. So the surface is not just a quote convention: it uniquely determines a one-factor diffusion (Gatheral §1; Bergomi ch 2). Two complementary constructions of the same object:

• The local vol from the surface (Dupire’s formula): a closed-form expression built from derivatives of call prices.
• The surface from the local vol (Gyöngy’s theorem / averaging): the BS implied variance is a gamma-weighted average of local variance along the paths.

The practical objective: understand that local volatility is a one-factor, complete, exactly-calibratable model - the simplest honest way to make a diffusion match the smile - and that its fatal flaw is not the fit but the dynamics (next pages).

One-line essence. “Local vol is the unique diffusion whose instantaneous variance, averaged along the paths that end at the strike with the right gamma weights, reproduces the market’s implied variance today - but it is not what a stochastic-volatility trader would call ‘the vol’.“

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2. Mathematical Ground Truth & Derivations

2.1 Dupire’s equation (strike form)

Under the risk-neutral measure, the undiscounted call satisfies

$$\frac{\partial C}{\partial T}=\frac{1}{2}{\sigma }^2{K}^2\frac{{\partial }^{2}C}{\partial K^2}+\mu \left(C-K\frac{\partial C}{\partial K}\right),\mu =r-D$$

(Gatheral eq 1.4). Inverting it for ${\sigma }^2$ gives the canonical definition

$$\boxed{{\sigma }_L^2(K,T)=\frac{\partial C/\partial T}{\frac{1}{2}{K}^2{\partial }^{2}C/\partial K^2}}$$

(Bergomi eq 2.3 with dividends; Gatheral eq 1.6; identical to the forward-moneyness form $\partial C/\partial T=\frac{1}{2}{v}_L{K}^2{\partial }^{2}C/\partial K^2$). The denominator’s meaning is Breeden–Litzenberger: ${\partial }^{2}C/\partial K^2=e^{-rT}\varphi (K,T)\ge 0$, so a negative butterfly implies negative local variance - an arbitrage.

Sketch of the derivation (Gatheral §1.2): write $C={\int }_K^{\infty }d{S}_T\varphi (S_T,T)(S_T-K)$; let the risk-neutral density $\varphi$ evolve by the Fokker–Planck equation $\frac{1}{2}{\partial }_{S_T}^2[{\sigma }^2{S}_T^2\varphi ]-{\partial }_{S_T}[\mu S_T\varphi ]={\partial }_T\varphi$; differentiate $C$ in $T$ and integrate by parts twice.

2.2 Dupire in implied total variance - the workhorse form

Write total implied variance $w(y,T)={\sigma }_{BS}^2(y,T)T$ with $y=\ln (K/F_T)$. Then (Gatheral eq 1.10; Bergomi eq 2.19)

$$\boxed{v_L(y,T)=\frac{\frac{\partial w}{\partial T}}{1-\frac{y}{w}\frac{\partial w}{\partial y}+\frac{1}{4}\left(-\frac{1}{4}-\frac{1}{w}+\frac{y^2}{w^2}\right){\left(\frac{\partial w}{\partial y}\right)}^2+\frac{1}{2}\frac{{\partial }^{2}w}{\partial y^2}}}$$

The two forms agree identically (Bergomi’s version expands exactly to this). No-skew special case: if $\partial w/\partial y=0$ then $v_L={\partial }_{T}w$ - local variance is just the forward implied variance, $w(T)={\int }_0^T{v}_L(t)dt$. The denominator quantifies how skew and curvature of the smile pull local variance above/below the ATM forward variance.

2.3 The bridge both ways

Local variance = risk-neutral conditional expectation of instantaneous variance (Dupire 1996; Derman–Kani 1998; Gatheral eq 1.12):

$${\sigma }_L^2(K,T)=\mathbb{E}\left[v_T|S_T=K\right].$$

Implied variance = gamma-weighted average of local variance (Gatheral eq 3.5; Bergomi eq 2.32):

$${\sigma }_{BS}^2(K,T)=\frac{\mathbb{E}\left[{\int }_0^T{e}^{-rt}{S}_t^2{\Gamma }_{BS}{\sigma }_L^2(S_t,t)dt\right]}{\mathbb{E}\left[{\int }_0^T{e}^{-rt}{S}_t^2{\Gamma }_{BS}dt\right]},{\Gamma }_{BS}=\frac{{\partial }^2{C}_{BS}}{\partial S_t^2}.$$

A quadratic expansion about the “Brownian-bridge” most-probable path gives the path-integral approximation

$${\sigma }_{BS}^2(K,T)\approx \frac{1}{T}{\int }_0^T{v}_L({\tilde{x}}_t)dt,{\tilde{x}}_t=\frac{t}{T}\ln \frac{K}{F_T}(\text{const-vol}),$$

i.e. implied variance ≈ the average of local variance along the most probable path to the strike (Gatheral eq 3.11; Bergomi eq 2.42/2.43). This single picture explains why the implied skew is roughly half the local skew (§2.4).

2.4 Skew mapping

Parametrize the local skew linearly, $\sigma (t,S)=\sigma (t)+\alpha (t)x+\frac{\beta (t)}{2}{x}^2$, $x=\ln (S/F_t)$, with constant $\alpha$ (Bergomi §2.4). Then

$${\mathcal{S}}_T\equiv {\frac{d{\hat{\sigma }}_{KT}}{d\ln K}|}_{\text{ATMF}}=\frac{1}{T}{\int }_0^T\frac{t}{T}\alpha (t)dt\underset{\text{const }\alpha }{\to }\frac{\alpha }{2},$$ $$\text{curvature}=\frac{1}{T}{\int }_0^T{\left(\frac{t}{T}\right)}^2\beta (t)dt\underset{\text{const }\beta }{\to }\frac{\beta }{3}.$$

The implied skew is half the local skew; the implied curvature is a third of the local curvature (Bergomi eq 2.50a,b). The Berestycki–Busca–Florent short-maturity limit is a harmonic average, $1/\hat{\sigma }(0,K)=\frac{1}{\ln (K/S)}{\int }_S^K\frac{1}{\sigma (0,S)}\frac{dS}{S}$ (Bergomi eq 2.54) - there is no temporal averaging as $T\to 0$.

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3. Computational Implementation - Dupire by two independent routes

We build an implied surface ${\sigma }_{BS}^2(y)={\sigma }_0^2+\beta y$ (a linear variance skew), then compute local variance two ways: (i) from the definition, ${\partial }_{T}C/(\frac{1}{2}{K}^2{\partial }_{KK}C)$ with the calls priced off the surface; (ii) from the implied-variance formula §2.2. They must agree. Stdlib only.

The two constructions agree to six decimals at every strike - the numerical proof that Dupire’s strike formula and the implied-variance formula are the same object. Note the skew leverage: where the smile slopes down (low $y$), local variance is pushed up ($0.0920$ vs ATM $0.0427$); on the upper wing it collapses ($0.0094$). And on a flat surface the formula returns ${\sigma }^2$ exactly - the no-skew case $v_L={\partial }_{T}w$.

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4. Failure Modes & First-Principles Breakdowns

1. Local vol is an “effective theory”, not instantaneous variance. ${\sigma }_L^2(K,T)=\mathbb{E}[v_T|S_T=K]$ is a conditional average; Dumas–Fleming–Whaley showed constant local vols are inconsistent with observed surface dynamics. The function has “no physical significance” (Bergomi ch 2) - it is a by-product of a Markov representation in $(t,S)$, meant to be recalibrated daily.
2. Differentiation amplifies noise. Both Dupire forms take second derivatives of a surface you only observe sparsely; finite differences of raw market prices explode. Work in $w(y,T)$ with a smooth arbitrage-free interpolant (SVI, §03), not on raw quotes.
3. Butterfly/calendar violations produce negative local variance. ${\partial }_K^{2}C<0$ (butterfly) or ${\partial }_{T}w<0$ (calendar) make ${\sigma }_L^2$ negative or undefined - the formula is telling you the surface is impossible, not that the math failed.
4. One-factor completeness is a straitjacket. Local vol makes all implied vols perfectly correlated with the spot and with each other; the whole future surface is dictated by today’s smile. Fitting today’s prices perfectly says nothing about path-dependent exotics - that is the failure mode quantified in 04 · Advanced Dynamics and 05 · Failure Modes.

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5. References

• Gatheral, The Volatility Surface
• Bergomi, Stochastic Volatility Modeling
• Hull, Options, Futures, and Other Derivatives
• Dupire, Bruno: Pricing with a Smile, Risk 7 (1994), 18–20. Derman–Kani (1994, binomial) and Breeden–Litzenberger (1978) for the density link.

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6. Connected Graph Bridges

• Back: 01 · From Zero · BSM PDE
• Forward: 03 · Surface Models · Index Hub
• Sibling: Implied Volatility Surface & Smiles · Heston & SABR