3.4 Volatility Surfaces & Smiles

Basic Prerequisites: Black–Scholes–Merton and Stochastic Calculus & Itô’s Lemma. (these are the folder-level prerequisites for pages 02–06; page 01 states its own, smaller, entry requirements)

1. Intuition & Practical Objective

Black–Scholes–Merton has one free parameter, the volatility $σ$ . If the model were true, inverting every liquid option price back to its implied volatility would return the same number for every strike and every maturity. It never does. The inverted number traces a surface $σ_{BS} (K, T)$ - downward-sloping in strike for equities (“the skew”), U-shaped for FX (“the smile”), flattening with maturity. That surface is the empirical object and this folder is its topic-hub.

The one-sentence essence:

The volatility surface is the language in which the market quotes all of its options; a model is “right” only if it not only reproduces today’s surface (statics) but also moves it the way the market moves it (dynamics).

This folder is a hub: (a) the fast formula lookup below, and (b) six sub-pages from raw intuition through Dupire local volatility, SVI surface models, skew dynamics, failure modes, and the extensions (Heston/SABR/rough vol).

2. Mathematical Ground Truth & Derivations

Notation: $S$ spot, $F_{T} = S_{0} e^{(r - q) T}$ forward, $K$ strike, $T$ maturity, $y = ln (K / F_{T})$ log-moneyness, $w (y, T) = σ_{BS}^{2} (y, T) T$ total implied variance, $Γ_{BS} = \partial^{2} C_{BS} / \partial S^{2}$ BS gamma, $φ$ the normal density.

Quick-Reference Lookup (job #1). Every formula below is transcribed from the verified corpus (Gatheral, The Volatility Surface, ch 1–8; Bergomi, Stochastic Volatility Modeling, ch 2–7) and the numbers in the check column were re-executed from the scripts in §3 and the sub-pages.

Quantity	Formula	Verified check
Implied vol (definition)	solve $C_{BS} (S, K, T, r, σ_{BS}) = C_{mkt}$	skew slice $80/90/100/110/120$ recovered exactly (28/24/20/17/15%)
Breeden–Litzenberger density	$ϕ (K, T) = e^{r T} \frac{\partial ^{2} C}{\partial K ^{2}}$	min density $0.0178 > 0$ for an arbitrage-free SVI slice
Dupire local variance (strike form, eq 1.6 / 6.2)	$σ_{L}^{2} (K, T) = \frac{\partial C / \partial T}{\frac{1}{2} K ^{2} \partial ^{2} C / \partial K ^{2}}$	$y = 0$ : $0.042695$ ; flat vol: $= 0.04 = σ^{2}$
Dupire in implied total variance (eq 1.10)	$v_{L} = \frac{\partial w / \partial T}{1 - \frac{y}{w} \frac{\partial w}{\partial y} + \frac{1}{4} ( - \frac{1}{4} - \frac{1}{w} + \frac{y ^{2}}{w ^{2}} ) ( \frac{\partial w}{\partial y} ) ^{2} + \frac{1}{2} \frac{\partial ^{2} w}{\partial y ^{2}}}$	identical to the strike form (both $0.042695$ )
Local var = conditional expectation (eq 1.12)	$σ_{L}^{2} (K, T) = E [v_{T} ∣ S_{T} = K]$	(Gyöngy) structural identity
SVI (eq 3.20, Gatheral 2004)	$w (k) = a + b [ρ (k - m) + (k - m)^{2} + s^{2}]$	ATM $18.708%$ , ATM skew $- 0.075$
Variance swap / log contract (eq 4.21/5.17)	$σ_{V S, T}^{2} = \int_{- \infty}^{\infty} \frac{d y}{2 π} e^{- y^{2} /2} σ_{K (y) T}^{2}, y (K) = \frac{ln ( K / F _{T} )}{σ _{K T} T} - \frac{σ _{K T} T}{2}$	flat smile: exactly $0.04$
Implied var = gamma-weighted avg of local var (eq 2.32/3.5)	$σ_{BS}^{2} = \frac{E [ \int _{0}^{T} e ^{- r t} S _{t}^{2} Γ _{BS} σ _{L}^{2} d t ]}{E [ \int _{0}^{T} e ^{- r t} S _{t}^{2} Γ _{BS} d t ]}$	the structural bridge LV↔IV
ATMF skew ≈ path-average of local skew (eq 3.11/2.48)	$S_{T} = \frac{1}{T} \int_{0}^{T} \frac{t}{T} α (t) d t$ ; constant $α \Rightarrow S_{T} = α /2$	implied skew is half the local skew
Skew stickiness ratio (eq 2.61/2.64)	$R_{T} = \frac{1}{S _{T}} \frac{d σ ^ _{F_{T} T}}{d ln S _{0}} = 1 + \frac{1}{T} \int_{0}^{T} \frac{S _{t}}{S _{T}} d t$	decaying equity skew → $R_{T} \to 3$ ; constant skew → $R_{T} = 2$
Heston short-dated variance skew (eq 3.19/7.3)	$\partial_{k} σ_{BS}^{2}_{k = 0} \to \frac{ρ η}{2}$	computed $- 0.1388$ vs $ρ η /2 = - 0.1389$
Jump compensator skew (eq 5.10)	$\partial_{k} σ_{BS}^{2}_{k = 0} \approx - 2 μ_{J}$	additive with SV at $τ \to 0$
Rough/vol-of-vol benchmark (Bergomi eq 7.40)	$ν_{T} (t) = σ_{0} (\frac{τ _{0}}{T - t})^{α}$ , $α \approx 0.4$	power-law vol-of-vol term structure

Critical caveat (flagged in the corpus). Gatheral’s printed ATM term-structure formula (3.18), built from the unconditional expected-variance path, returns the long-run mean $\overset{v}{ˉ}$ as $T \to 0$ , not the current instantaneous variance $v_{0}$ . The physical short-dated limit is $v_{0}$ ; treat (3.18)‘s literal $T \to 0$ value accordingly (verified numerically: it returns $0.0354 = \overset{v}{ˉ}$ , not $0.0174 = v_{0}$ ).

3. Computational Implementation - the surface engine

Stdlib only (math, cmath) - no numpy/scipy. The recurring primitives are the BSM call, its vega, and a Newton implied-vol inverter. Every script in this folder reproduces the numbers in §2; the sub-pages carry the full listings.

4. Failure Modes & First-Principles Breakdowns

Hub signposts - the folder’s failure-mode analysis lives in 05 · Failure Modes & Practice. In one line each:

Static fit ≠ correct dynamics. Local volatility calibrates today’s surface exactly yet generates forward skews that are too flat - it misprices cliquets, barriers, digitals (Gatheral ch 8/10; Bergomi ch 2).
Arbitrage-free interpolation is hard. Unconstrained splines produce negative densities ( $\partial^{2} C / \partial K^{2} < 0$ , butterfly) or decreasing total variance ( $\partial_{T} w < 0$ , calendar).
The short-end smile needs jumps, not just SV. No time-homogeneous stochastic-vol model reproduces the observed $T \to 0$ skew decay; the compensator $- 2 μ_{J}$ of a jump component is required (Gatheral ch 5, 7).

5. References

Gatheral, Jim: The Volatility Surface: A Practitioner’s Guide (Wiley, 2006)
Bergomi, Lorenzo: Stochastic Volatility Modeling (CRC, 2016)
Hull, John C.: Options, Futures, and Other Derivatives (11th ed.)
Haug, Espen Gaarder: The Complete Guide to Option Pricing Formulas (2nd ed.)

6. Connected Graph Bridges

Foundational base: Stochastic Calculus & Itô · Probability & Measure Theory · Econometrics & Time Series
Sibling topic: Black–Scholes–Merton (the constant- $σ$ zero point this folder generalizes)
Related flat notes: Implied Volatility Surface & Smiles · Heston & SABR · The Greeks & Dynamic Hedging
Sub-pages (in-folder): 01 From Zero · 02 Implied vs Local Vol · 03 Surface Models · 04 Advanced Dynamics · 05 Failure Modes · 06 Advanced Extensions

Beginner: start at 01 · Practitioner: start at 05

Kwant Atlas

Explorer

3.4 Volatility Surfaces & Smiles

1. Intuition & Practical Objective

2. Mathematical Ground Truth & Derivations

3. Computational Implementation - the surface engine

4. Failure Modes & First-Principles Breakdowns

5. References

6. Connected Graph Bridges

3.4.1 The Volatility Surface from Zero

3.4.2 Implied vs Local Volatility

3.4.3 Surface Models

3.4.4 Advanced Dynamics

3.4.5 Failure Modes & Real-World Practice

3.4.6 Advanced Extensions