Implied Volatility and the Volatility Smile/Skew
Why one Black–Scholes number cannot fit every strike, and what the resulting curve tells you about the market’s view of tails.
Leads to: 15.3–15.5 build models (local vol, stochastic vol) that reproduce the smile.
Learning Objectives
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- Define implied volatility as the σ that inverts the Black–Scholes formula to a market price.
- Explain why a single Black–Scholes σ cannot simultaneously fit options of different strikes.
- Distinguish the volatility smile from the equity skew and connect each to the implied risk-neutral density.
- Implement a Newton–Raphson implied-volatility solver using vega and analyze its convergence.
- Interpret smile steepness in terms of the tails of the risk-neutral distribution.
Key Vocabulary
- Implied volatility
- The value of \(\sigma\) that makes the Black–Scholes price equal the observed market price of an option.
- Volatility smile
- Implied vol plotted against strike/moneyness that is higher for OTM puts and calls than ATM - a U shape.
- Volatility skew
- An asymmetric smile (equities): implied vol rises for low strikes (OTM puts), reflecting crash fear.
- Moneyness
- A normalized strike, e.g. \(K/S\), \(\ln(K/S)\), or \(\ln(K/F)/(\sigma\sqrt{T})\) (standardized).
- Risk-neutral density
- The pricing distribution \(q(S_T)\) implied by option prices; recovered via Breeden–Litzenberger.
- Vega
- \(\partial C/\partial\sigma\gt 0\): because price is strictly increasing in \(\sigma\), implied vol is unique when it exists.
- Newton–Raphson
- A root-finder \(\sigma_{n+1}=\sigma_n-(C(\sigma_n)-C_{mkt})/\text{vega}(\sigma_n)\) exploiting the known derivative.
Intuition & Motivation
Definition: inverting the formula
Given a market price \(C_{\text{mkt}}\) for a call at strike \(K\), maturity \(T\), the implied volatility \(\sigma_{\text{imp}}(K,T)\) is the solution of
Because \(\partial C_{\text{BS}}/\partial\sigma=\text{vega}\gt 0\) for \(T\gt 0\), the map \(\sigma\mapsto C_{\text{BS}}\) is strictly increasing, so the solution is unique whenever the price is between its no-arbitrage bounds. This is what makes implied vol a legitimate quoting convention.
Why one σ cannot fit all strikes
Suppose you calibrate a single \(\sigma\) to the ATM option. Black–Scholes then predicts the prices of all other strikes via the same lognormal density. Empirically those predictions are wrong: OTM puts trade richer (higher implied vol) than the ATM number, and OTM calls often do too. If a single \(\sigma\) fit every strike, the implied-vol curve would be flat. It is not - therefore no single constant volatility reproduces the market, and the deviation is the smile.
Smile versus skew, and the implied density
The shape carries information. By Breeden–Litzenberger the risk-neutral density is the second strike-derivative of the call price,
so the curvature of the price curve (hence the smile) is the density itself, which must be non-negative. Two canonical shapes:
| Market | Typical shape | What it encodes |
|---|---|---|
| FX (symmetric) | Smile (U-shaped) | Fat tails on both sides; up and down moves both feared |
| Equity index | Skew (down-sloping) | Rich OTM puts: crash fear, negative spot–vol correlation |
| Short-dated | Steeper | Tails dominate; jump risk large relative to diffusion |
| Long-dated | Flatter | Averages out toward a central vol level |
Interactive: solve for implied volatility (Newton–Raphson)
Vega is the derivative we need, so Newton’s method converges quadratically from a sensible start. Implement the solver and confirm it recovers the \(\sigma\) that generated a price.
- Calling implied vol a forecast of future realized vol: it is a risk-neutral quoting device that also embeds risk premia.
- Comparing raw option premiums across strikes instead of implied vols - premiums confound moneyness with vol.
- Running naked Newton for deep ITM/OTM options where vega → 0: the update explodes. Bracket with bisection.
- Expecting a flat line: a non-flat smile is the normal state of the world, not a data error.
- Quote and interpolate in vol space, not price space - vol is far smoother across strikes and time.
- Use log-moneyness \(k=\ln(K/F)\) or a standardized \(k/(\sigma\sqrt{T})\) on the x-axis so smiles are comparable across maturities.
- Seed the solver with a good guess (Brenner–Subrahmanyam ATM approximation) to speed convergence and avoid the flat-vega tails.
- Watch the ATM skew slope \(\partial\sigma_{imp}/\partial k\): it is the single most-quoted risk-reversal signal.
Knowledge Check
Practical Exercise
You observe three one-year calls on a \(100\)-spot index (\(r=2\%\)): the \(90\)-strike implies \(24\%\), the \(100\)-strike implies \(20\%\), and the \(110\)-strike implies \(19\%\). (a) Sketch and name the shape. (b) What does it say about the risk-neutral density versus lognormal? (c) A colleague proposes pricing all three with the single ATM \(20\%\). What goes wrong, and for which option is the error worst?
(a) Implied vol falls as strike rises (24% → 20% → 19%): a down-sloping skew, the classic equity-index shape.
(b) The high implied vol at the low strike means the market prices the \(90\)-strike put/call region as if large downward moves are more likely than lognormal - a fat left tail. The density is left-skewed relative to the Black–Scholes lognormal.
(c) Using a flat \(20\%\) would under-price the \(90\)-strike (true implied \(24\%\gt 20\%\)) - the worst error, because the skew is steepest there and the market demands a premium for downside protection - and slightly over-price the \(110\)-strike (true \(19\%\lt 20\%\)). A single \(\sigma\) cannot match the cross-section; you need a smile-consistent model (local or stochastic vol).
Lesson Summary
Formula Sheet Additions
- Did I invert in vol space, not compare raw premiums?
- Did I guard Newton against vega→0 in the deep tails?
- Did I read smile height as a tail statement, not a pricing error?
- Did I keep risk-neutral and physical interpretations distinct?
Retrieval Practice
Close the lesson and answer from memory before checking. This is deliberate, effortful recall - the single highest-yield study action.
A: Because the Black–Scholes price is strictly increasing in \(\sigma\) (vega \(\gt 0\)); the monotone map has a unique inverse whenever the price sits inside its no-arbitrage bounds.
A: Market prices imply different \(\sigma\) at different strikes (a non-flat smile/skew), which a single lognormal density with one \(\sigma\) cannot reproduce.
A: Via \(q(K)=e^{rT}\partial^2C/\partial K^2\): higher implied vol at a strike fattens the density’s tail there; equity skew means a fat left (downside) tail.
Flashcards
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Completion Checklist
- I can explain the core ideas in my own words
- I worked the derivations/examples by hand
- I completed the interactive workbench(es)
- I passed the knowledge check
Source References
This lesson synthesizes and paraphrases concepts from the sources below. No copyrighted text, problem sets, or solutions are reproduced. Return to the originals for full depth.
- The Volatility Surface (Jim Gatheral, 2006) foundational - Ch. 1–2 - Ch. 1–2: implied volatility, the smile, and its link to the risk-neutral density.
- Stochastic Calculus for Finance II (Steven Shreve, 2004) foundational - Ch. 4 - Ch. 4: Black–Scholes formula used as the inversion target for implied vol.
- Methods of Mathematical Finance (Karatzas & Shreve, 1998) foundational - Ch. 2 - Ch. 2: European claim pricing and the density recovered from call prices.