9.5 The smile and the skew

9.5.1The risk-neutral density#

Proof

$C(K)=e^{-rT}\int_K^\infty(s-K)q(s)\,ds$. Differentiate under the integral (Leibniz): $\partial_KC=-e^{-rT}\int_K^\infty q(s)\,ds$, and again: $\partial^2_KC=e^{-rT}q(K)$.∎

A skew means $q$ has a fatter left tail than the lognormal. A flat smile at 13% says the market prices a 10% Nifty fall in a week as essentially impossible; a skewed one prices it as rare but real.

9.5.2Sources of the skew#

Why the skew exists

1. Crash risk and jumps. Index returns are negatively skewed with fat left tails (M02). A downward jump moves OTM puts far more than calls. Pricing them at the at-the-money vol would make them too cheap.
2. The leverage effect. Volatility rises when prices fall. In a stochastic-vol model with negative spot–vol correlation (Heston below with $\rho<0$), low strikes are priced under higher expected vol.
3. Supply and demand. Institutions buy index puts for portfolio insurance and sell calls to write covered calls. Dealers who absorb this net demand charge for the inventory risk. Bollen and Whaley (2004), "Does Net Buying Pressure Affect the Shape of Implied Volatility Functions?", JF 59(2): 711–753, and Gârleanu, Pedersen and Poteshman (2009), "Demand-Based Option Pricing", RFS 22(10): 4259–4299, show that net demand moves implied vols.

9.5.3Quoting and moving the smile#

Quoting the smile Desks quote in delta, not strike, so that quotes stay comparable as spot moves. With $\sigma_{25C}$ and $\sigma_{25P}$ the implied vols of the 25-delta call and put:

The risk reversal measures skew (negative for equity indices). The butterfly measures curvature, the extra price of both wings. A 25-delta strangle is quoted at $\sigma_{ATM}+BF_{25}$.

Sticky strike vs sticky delta (Derman 1999, "Regimes of Volatility", Risk 12(4)) When spot moves, what happens to the surface?

• Sticky strike: $\sigma_i(K)$ stays fixed for each strike. The ATM vol then falls as spot rises (you move to lower-vol strikes). Delta is the Black–Scholes delta.
• Sticky delta (sticky moneyness): $\sigma_i$ is a fixed function of $K/S$. The ATM vol is unchanged as spot moves, and each strike's vol changes: with $\sigma=f(K/S)$, $\partial\sigma/\partial S=-(K/S)\,\partial\sigma/\partial K$. Delta becomes

which, with a downward skew ($\partial\sigma/\partial K<0$), is larger than $\Delta_{BS}$ for a call.

Neither rule is a law. In a sell-off, Nifty implied vols typically rise across strikes, faster than sticky strike implies. The choice changes your hedge ratio, and therefore your gamma P&L, by a few delta points (Problem 13).

9.5.4Local and stochastic volatility#

Local volatility (orientation only) Dupire (1994), "Pricing with a Smile", Risk 7(1): 18–20, shows that a diffusion $dS=(r-q)S\,dt+\sigma_{loc}(S,t)S\,dW$ reproduces every European price on the surface if and only if

It fits today's surface exactly but implies unrealistic dynamics (the smile flattens as time passes).

Stochastic volatility (orientation only) Heston (1993), RFS 6(2): 327–343:

Variance mean-reverts to $\theta$ at speed $\kappa$, with vol-of-vol $\xi$; $\rho<0$ generates the skew and $\xi$ the curvature. European prices have a semi-closed form via the characteristic function. Gatheral (2006) chs. 2–4 is the reference for both models.

9.5.5The Nifty skew#

Nifty data point M08 · 8.10 backs out implied vols for the December 2026 Nifty expiry from the 5 Nov 2026 settlement prices (Black-76 on the future at 24,719.10, OTM options): roughly 13.4% at the 24,000 put, 12.6% at 24,500, 12.2% at the 25,000 call and 12.0% at 25,500. That is a downward skew of about 1.4 vol points across a 6% range of strikes, the pattern this section explains.