Volatility and smiles

~9 min read

Interview occasional

Why it matters

Black–Scholes assumes one \(\sigma\). Markets do not: if you invert the BS formula at every strike you get a smile (or equity skew). Implied vol is how the market quotes options; the surface is the object vol desks actually trade. Interviewers use this to see whether you understand that BS is a quoting convention, and why OTM puts are “expensive” in implied-vol terms.

Core concepts

Two vols. Realized / historical vol is a statistic of past returns (close-to-close, Parkinson, etc.). Implied vol is the \(\sigma_{\text{imp}}(K,T)\) you plug into BS to match the market price of that option. They are not the same thing; the difference is a risk premium plus model error (see Black–Scholes intuition).
How you extract implied vol. Market price in, BS formula inverted numerically (Newton on vega). If the market were exactly BS with constant \(\sigma\), every strike would return the same number. It doesn’t.
Smile vs. skew. FX and some commodities: roughly U-shaped smile (OTM puts and calls richer). Equities: downward skew — low strikes (OTM puts) have higher implied vol than high strikes. The 1987 crash is the usual historical marker for persistent equity skew.
Why a skew exists (intuition, not a unique theorem).
- Crashophobia / demand for puts (portfolio insurance).
- Leverage: as \(S\) falls, equity-value vol of a levered firm rises.
- Jumps and fat left tails: a lognormal underprices left-tail payoffs, so those options must carry a higher BS \(\sigma\) to fit.
The surface, not a number. \(\sigma_{\text{imp}}(K,T)\) or in delta space \(\sigma_{\text{imp}}(\Delta,T)\). Term structure: short-dated events (earnings, FOMC) spike front-end vol; longer tenors are smoother. A sticky-strike vs. sticky-delta rule is a statement about how the surface moves when spot moves — it changes your delta (the “vol-gauntlet” / vanna-volga story).
Local vol vs. stochastic vol (one-level deeper). Dupire local vol \(\sigma_{\text{loc}}(S,t)\) is calibrated to today’s surface and is a perfect fit for vanillas, but it has specific (often too sticky) dynamics. Stochastic vol (Heston, SABR, SVI parametrizations) gives better forward smiles and vol-of-vol. Interviews: know why one extra process exists, not the PDE.
What implied vol is not. It is not a forecast of realized vol (though variance swaps and the VIX are related to a strip of vanillas). Selling a 20-delta put at 25% implied vs. 18% realized can still lose if the jump happens.
Put-call parity still holds. European \(C-P\) does not depend on vol. You cannot have a different implied vol for a European call and put of the same \(K,T\) — they share \(\sigma_{\text{imp}}\). If quotes disagree, that’s a data/American/early-exercise/borrow issue, not a new smile.

Mental model

  implied vol
    |
    | *                    *
    |   *                *
    |     *            *
    |       *  *  *  *          <-- FX-style smile
    |              *
    |  *  *                     <-- equity: extra left-tail skew
    |    *
    +------ low K ---- ATM ---- high K ----->

  Each point: the σ that makes BS(K,T; σ) = market price(K,T)
  The curve is a quoting language for the risk-neutral density's shape.

Breeden-Litzenberger: the second strike-derivative of a call price is the risk-neutral density of \(S_T\). A smile/skew is a non-lognormal density, written in BS units.

Interview questions

If BS were true with constant \(\sigma\), what would the implied-vol plot vs. strike look like? What do we actually see in SPX? Answer: A flat line. SPX: downward skew, OTM puts have higher \(\sigma_{\text{imp}}\) than OTM calls, and the level of the ATM vol itself moves over time.
Why can a European call and put with the same \(K,T\) not have different implied vols? Answer: Put-call parity fixes \(C-P\) independently of \(\sigma\). BS call and put with the same \(\sigma\) already satisfy parity, so inverting either market price (if parity holds) yields the same \(\sigma_{\text{imp}}\).
Give two economic reasons equity skew is typically downward-sloping. Answer: (1) Crash/put demand: investors bid up low-strike puts. (2) Leverage / inverse relationship of level and vol: down-moves come with higher vol, which a risk-neutral density with a fat left tail captures. Jumps in addition to diffusion produce the same qualitative skew.
You hedge a short OTM put using BS delta at the implied vol of that put. What can still hurt you? Answer: Realized path vs. implied (gamma/theta), changes in the surface (vega, vanna, volga), jumps over the strike, discrete hedging error, and the fact that the “correct” sticky rule (sticky strike vs. sticky delta) changes the effective delta. Implied-vol delta is not a complete hedge of a smile world.
What is the difference between local volatility and stochastic volatility in one sentence each? Answer: Local vol: \(\sigma\) is a deterministic function of \(S\) and \(t\), fitted to today’s vanillas (Dupire). Stochastic vol: \(\sigma\) itself is a random process, which produces vol-of-vol, a better forward smile, and extra risk you cannot hedge with spot alone.

Watch

Implied volatility — Khan Academy. Invert BS; implied vol as the market’s packaged view.
FRM: Implied volatility smile — Bionic Turtle. Smile vs. flat BS vol, and why quoted vols vary by strike.