Chapter 20: Volatility Smiles and Volatility Surfaces

Implied Volatilities of Calls and Puts

If the Black–Scholes–Merton model were perfectly correct, all options on the same underlying with the same maturity would have the same implied volatility. In reality, implied volatility varies with strike price — this pattern is the volatility smile (or skew).

Note: Put–call parity implies that European calls and puts with the same strike and maturity must have the same implied volatility, even if the smile exists.

Volatility Smile for Foreign Currency Options

For foreign currency options, the volatility smile is roughly a U-shape (a smile):

$\text{Implied volatility }{\sigma }_{\text{imp}}\text{ is higher at low and high strikes than at-the-money}$

Explanation: The market prices in the possibility of extreme exchange rate moves — currencies can jump due to central bank interventions, trade disputes, etc. These tail risks are not captured by the lognormal assumption.

Why a Smile?

The lognormal distribution assigns very low probability to extreme moves (thin tails). The market compensates for tail risk by pricing out-of-the-money and in-the-money options at higher implied volatilities.

Volatility Smile for Equity Options

For equity options, the pattern is typically a downward-sloping skew:

$\text{Higher implied volatility for lower strikes (OTM puts), lower for higher strikes (OTM calls)}$

This "skew" or "smirk" pattern has been observed since the 1987 crash. Explanations:

1. Crashophobia: Investors are willing to pay a premium for downside protection (OTM puts) because crashes happen
2. Leverage effect: As stock prices fall, a company's debt-to-equity ratio increases, making the stock riskier → volatility increases when prices fall

Pre-1987 vs. Post-1987

Before the 1987 crash, the volatility smile for equity options was nearly flat. The crash changed market psychology permanently — investors now price in crash risk.

The Volatility Term Structure

Volatility also varies with maturity. The volatility term structure shows how at-the-money implied volatility changes with time to expiration. Patterns:

• "Volatility spike": Short-term implied vol jumps above long-term vol during crises
• Normal: Somewhat upward or flat slope

Volatility Surfaces

A volatility surface combines the smile (across strikes) and the term structure (across maturities):

${\sigma }_{\text{imp}}=f(K,T)$

Traders use the volatility surface to price exotic options and to find relative value opportunities. Different models produce different surface shapes.

Minimum Variance Delta

When volatility depends on the strike (smile/skew), the standard Black–Scholes–Merton delta:

$\Delta =N(d_1)$

is not correct for hedging. The minimum variance delta adjusts for the correlation between volatility and the stock price:

${\Delta }_{\text{MV}}={\Delta }_{\text{BSM}}+\nu \cdot \frac{\partial {\sigma }_{\text{imp}}}{\partial S}$

where $\nu$ is vega and $\partial {\sigma }_{\text{imp}}/\partial S$ captures the smile dynamics (typically negative for equities — the "sticky delta" rule).

The Role of the Model

The volatility smile means the Black–Scholes–Merton model is wrong (the underlying is not lognormal). But traders still use it as a quoting convention — they quote implied volatility rather than price. The smile becomes a tool for interpolation and relative value assessment rather than an assumption violation.

When a Single Large Jump is Anticipated

If the market expects a large price jump (e.g., earnings announcement, merger decision), the implied volatility for short-dated options can exhibit a "W-shaped" smile — high vol at extreme strikes, low in between, because the market prices in a bimodal outcome distribution.