Solution: Why the Vol Smile Falsifies Black-Scholes

Exercise: Why the Vol Smile Falsifies Black-Scholes

Part 1: log-normal tail probabilities with ATM vol

Under $\mathbb{Q}$ with $\sigma = 0.18$ , $S_0 = 100$ , $T = 0.25$ , $r = 0$ :

d_2(K) = \frac{\ln(100/K) - 0.18^2 \cdot 0.25 / 2}{0.18 \sqrt{0.25}} = \frac{\ln(100/K) - 0.00405}{0.09}

Lower tail $K = 80$ .

\ln(100/80) = \ln(1.25) \approx 0.2231

d_2(80) = (0.2231 - 0.00405)/0.09 \approx 2.434

\mathbb{Q}(S_T < 80) = \Phi(-d_2(80)) = \Phi(-2.434) \approx 0.0075

So the ATM-implied log-normal assigns probability $\approx 0.75\%$ to the event $S_T < 80$ .

Upper tail $K = 120$ .

\ln(100/120) = -\ln(1.2) \approx -0.1823

d_2(120) = (-0.1823 - 0.00405)/0.09 \approx -2.071

\mathbb{Q}(S_T > 120) = \Phi(d_2(120)) = \Phi(-2.071) \approx 0.0192

The ATM-implied log-normal assigns $\approx 1.92\%$ to $S_T > 120$ .

Part 2: tail probabilities from each strike's own implied vol

At each strike, the market's implied log-normal uses that strike's own $\sigma_{\text{imp}}(K)$ .

Lower tail $K = 80$ , $\sigma_{\text{imp}}(80) = 0.28$ .

d_2(80) = (0.2231 - 0.28^2 \cdot 0.25/2)/(0.28\sqrt{0.25}) = (0.2231 - 0.0098)/0.14 \approx 1.524

\mathbb{Q}^{\text{market}}(S_T < 80) = \Phi(-1.524) \approx 0.0638

Upper tail $K = 120$ , $\sigma_{\text{imp}}(120) = 0.15$ .

d_2(120) = (-0.1823 - 0.15^2 \cdot 0.25 / 2)/(0.15\sqrt{0.25}) = (-0.1823 - 0.00281)/0.075 \approx -2.467

\mathbb{Q}^{\text{market}}(S_T > 120) = \Phi(-2.467) \approx 0.0068

Comparison.

Event	ATM log-normal	Market-implied
$S_T < 80$	$0.75\%$	$6.38\%$ — 8 $\times$ higher
$S_T > 120$	$1.92\%$	$0.68\%$ — 3 $\times$ lower

The market assigns much higher probability to large down-moves than the ATM log-normal predicts (fat left tail) and much lower probability to large up-moves (thin right tail). This is the empirical signature of the equity skew.

Part 3: risk-neutral density vs log-normal

Observation. A single log-normal distribution cannot simultaneously produce

\mathbb{Q}(S_T < 80) = 6.38\%

and

\mathbb{Q}(S_T > 120) = 0.68\%

with

S_0 = 100, T = 0.25, r = 0

. No

\sigma

makes both tail probabilities match. The market's implied distribution is not log-normal.

Shape interpretation.

\sigma_{\text{imp}}(K)

decreasing in

K

(the equity skew) is equivalent to:

Heavier left tail than log-normal: large down-moves are more likely
Lighter right tail than log-normal: large up-moves are less likely

Economically: the market is pricing crash risk (leverage effect + tail insurance demand), giving OTM puts a premium above log-normal fair value. Equivalently, investors are willing to pay more for left-tail protection than Black-Scholes would suggest, inflating the implied vol of low-strike puts.

Black-Scholes assumes a symmetric log-return distribution with no crash-risk premium. Any market that prices crash risk will exhibit a skew — the smile is not a model failure in the sense of being solvable with better estimation, it is a structural falsification of the log-normal assumption.

Part 4: Breeden-Litzenberger

Puts under risk-neutral pricing satisfy $P(K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(K - S_T)^+] = e^{-rT}\int_0^K (K - s)q(s)\,ds$ where $q(s)$ is the risk-neutral density of $S_T$ . Differentiating twice with respect to $K$ :

\frac{\partial P}{\partial K} = e^{-rT}\int_0^K q(s)\,ds = e^{-rT}\,\mathbb{Q}(S_T < K)

\frac{\partial^2 P}{\partial K^2} = e^{-rT} q(K)

So the risk-neutral density is the second derivative of the put-price curve in strike (up to the discount factor). In practice:

Observe the market implied-vol smile $\sigma_{\text{imp}}(K)$ .
Convert to market put prices $P_{\text{market}}(K) = P_{\text{BS}}(K, \sigma_{\text{imp}}(K))$ .
Take the second derivative of $P_{\text{market}}$ in $K$ (numerically or after fitting a smooth parameterisation like SVI).
Multiply by $e^{rT}$ to recover $q(K)$ .

This is how practitioners extract the market-implied risk-neutral density from the vol surface — a routine exercise in modern exotic-options pricing. The derived $q$ is then fed into Monte Carlo or PDE pricers for exotics consistent with the vanilla surface.

Takeaways

Flat implied vol $\Leftrightarrow$ log-normal risk-neutral distribution. Any departure from flatness is a departure from log-normality.
Equity skew $\Leftrightarrow$ fat left tail, thin right tail. Decreasing $\sigma_{\text{imp}}(K)$ means crash-risk pricing, not stochastic vol per se — though jump-diffusion and stochastic-vol models reproduce the skew for different reasons.
The smile is the market's risk-neutral density in disguise. Breeden-Litzenberger makes this precise: $q(K) \propto \partial^2 P/\partial K^2$ . Every pricing model in production is ultimately calibrated to reproduce this density.