Implied volatility and the smile

Implied volatility

Every input to Black–Scholes is observable except volatility. Turn the formula around: given the market price of an option, find the $\sigma$ that makes the formula match. That number is the implied volatility (IV).

There is no closed form, but the call price is strictly increasing in $\sigma$ , so a few steps of Newton’s method using vega converge quickly:

\sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C_{\text{market}}}{\mathcal{V}(\sigma_n)}

IV is the market’s price of future volatility, including a risk premium. On average, implied volatility of index options sits above subsequently realised volatility — the variance risk premium — which is why systematically selling options earns money most of the time and loses badly occasionally.

The smile and the skew

If Black–Scholes were exactly right, every option on the same asset and expiry would have the same IV. Plot IV against strike and you instead see:

Equity indices: a downward skew. Low-strike puts have much higher IV than high-strike calls, because crashes are more feared (and more frequent) than melt-ups, and because investors pay up for portfolio insurance. This skew became pronounced after the 1987 crash.
FX: a more symmetric smile, since either currency can crash against the other.
Crypto: a smile whose tilt swings with sentiment — call skew in euphoric rallies, put skew in sell-offs.

The smile is the market’s correction to lognormality: it prices fat tails and jumps that Black–Scholes ignores.

The term structure

Plot ATM IV against expiry. In calm markets it usually slopes upward (contango): more time, more uncertainty priced. In a panic it inverts: short-dated IV spikes above long-dated, because the stress is now. The VIX term structure is the canonical example.

The volatility surface

Strike × expiry × IV forms the volatility surface. Desks fit it with parametric models (SVI is common) so they can price any option consistently and avoid arbitrage, such as calendar spreads with negative value or butterflies with negative price.

Ways traders use it

Realised vs implied. Compare IV to a forecast of realised volatility; buy options (and hedge) when IV looks cheap, sell when rich. See the gamma/theta trade-off in the Greeks.
Skew trades. Risk reversals (long one wing, short the other) bet on the skew steepening or flattening.
Regime signal. An inverted term structure or exploding skew is a stress indicator usable even by non-option strategies, for example to cut leverage.

Common mistake

Selling high-IV options because “IV is high”. IV is high when the market expects large moves, and it is frequently right. The edge is in IV relative to what will be realised, net of the tail risk you are warehousing.