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Volatility Smile and Skew

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Overview

Black-Scholes assumes a single, constant volatility per underlying. In practice, the implied volatility extracted from market option prices varies systematically across strikes and expirations — producing the volatility smile (or, in equities, the volatility skew). Understanding this surface is a prerequisite for pricing, hedging, and trading options.

What the Market Shows

ATM OTM put OTM call IV smile (fx, commodities) ATM OTM put OTM call IV demand for crash protection skew (equity index)
smile = symmetric (fat tails both sides) · skew = asymmetric (puts pricier than calls)

For a given expiration, plot implied vol on the y-axis against strike (or moneyness) on the x-axis. Three canonical shapes:

Shape Where Seen Typical Cause
Smile (symmetric U) FX, commodities Fat-tailed returns; symmetric crash insurance
Skew / smirk (down-sloping) Equity indices Demand for OTM puts (downside hedging) > OTM calls
Reverse skew (up-sloping) Some commodities (oil, ag) Supply shock asymmetry 
       Implied Vol
        │
   skew │\
        │ \
        │  \____
        │       ‾‾‾‾‾_____
        │                  ‾‾
        └────────────────────────── Strike
       OTM put       ATM        OTM call

Why Vol Isn't Flat

Black-Scholes assumes log-normal returns. Real returns:

  1. Fat tails — large moves happen more often than the normal predicts; markets price OTM options accordingly.
  2. Jumps — discontinuous moves around events; jump-diffusion models (Merton) produce smiles naturally.
  3. Stochastic volatility — vol itself moves randomly; vol-of-vol gives smile curvature (Heston, SABR).
  4. Supply/demand imbalances — institutional hedging programs (e.g., pension funds buying SPX puts) keep skew elevated.
  5. Leverage effect — equity vol rises when price falls (Black, 1976); reinforces downside skew.

Measuring Smile and Skew

Skew

Skew_25Δ = IV(25Δ put) − IV(25Δ call)

where: IV(25Δ put) implied vol of the 25-delta put · IV(25Δ call) implied vol of the 25-delta call · delta-quoted strikes are the FX-desk convention. does: measures asymmetry of the smile at a constant moneyness offset. Positive in equity indices (puts richer than calls) — tells you the market is paying up for downside protection; rising skew signals growing crash anxiety, flattening signals complacency.

Sometimes quoted as the slope of IV vs. log-moneyness (d IV / d log(K/S)).

Convexity (Butterfly)

Butterfly_25Δ = (IV(25Δ put) + IV(25Δ call)) / 2 − IV(ATM)

where: average of IV(25Δ put) and IV(25Δ call) minus IV(ATM) — the convexity proxy on the IV surface. does: measures smile curvature — how much richer the wings price vs the body. Positive butterfly tells you the market expects fatter tails / higher vol-of-vol than a flat surface implies; rising butterfly signals demand for tail insurance on both sides.

Positive butterfly = "smiley" — wings priced richer than the body.

Term Structure

Plot ATM IV vs. expiration. Contango (upward sloping) is normal; backwardation (downward sloping) usually signals near-term stress.

The Volatility Surface

A 2-D function IV(K, T) over strikes and expirations. Required for:

  • Consistent pricing of exotics (barrier, lookback, Asian)
  • Smile-aware Greeks (vanna, volga)
  • Calibration of stochastic-vol models

Trading the Smile

Trade Construction View
Long skew / risk reversal Long OTM call, short OTM put (or vice versa) Skew will steepen / flatten
Long butterfly (vol-of-vol) Long wings, short body Smile will steepen
Calendar spread Long longer-dated, short shorter-dated, same strike Term structure will flatten / steepen
Variance swap / VIX trade Direct vol exposure Realized vol vs. implied

Practical Implications

  1. Don't price exotics with flat vol. Even ATM-only quoting misses convexity.
  2. Greeks change with smile. Vanna and volga matter for FX and longer-dated equity books.
  3. Skew is sticky. "Sticky strike" vs. "sticky moneyness" assumptions change hedge ratios.
  4. Reading the skew is a sentiment gauge. Elevated put skew = elevated crash anxiety; flat skew = complacency.
  5. Calibration ≠ truth. Surface fits are sensitive to noise in deep-OTM points; trim or downweight illiquid quotes.

Models That Produce Smiles

Model Mechanism Used For
Heston Stochastic vol with mean reversion FX, equity index
SABR Stochastic vol + CEV beta Swaptions, FX
Local vol (Dupire) σ(S, t) calibrated to surface Exotics on a fixed underlying
Jump-diffusion (Merton) Adds Poisson jumps to GBM Equity tails
Bates Heston + jumps FX, commodities

References: 1. Gatheral, The Volatility Surface: A Practitioner's Guide, Wiley, 2006. 2. Dupire, "Pricing with a Smile", Risk, 1994. 3. Heston, "A Closed-Form Solution for Options with Stochastic Volatility", Review of Financial Studies, 1993.

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