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