What an Implied Volatility Smile Reveals About Black–Scholes

Black–Scholes assigns one constant volatility to the underlying, so its own option prices imply the same volatility across strikes and maturities. Market prices often require different implied volatilities. Local volatility, stochastic volatility, and jump models extend the stock dynamics in different ways to fit this surface and to express different assumptions about how it will move.

Implied volatility is the number you feed into Black–Scholes to reproduce a quoted premium. Compare several strikes of the same expiry. If out-of-the-money puts command a larger premium than the single-volatility model predicts, they have higher implied volatilities when quoted in Black–Scholes units. Equity index options often show such a downward strike skew. A smile describes a shape in which both wings are elevated relative to the middle.

Changing the implied volatility for one strike improves that single price. A model fitted jointly to strikes and maturities also needs to make sense of the relationship between quotes. Choosing a model for an exotic payoff adds another question. How does the model distribute future paths among prices that match the same European options today?

What one volatility cannot fit

In the zero-dividend case with constant

r

, Black–Scholes makes terminal log returns normal under the pricing measure, with mean

(r-\sigma^2/2)\tau

and variance

\sigma^2\tau

. A set of call prices across strikes encodes much more information than that one parameter. If the terminal stock has a smooth risk-neutral density

f_Q

, differentiating

C(K)=e^{-r\tau}\int_K^\infty(s-K)f_Q(s)\,ds

twice with respect to strike gives

C''(K)=e^{-r\tau}f_Q(K)

. The observed strike curvature can differ from the curvature implied by a single lognormal distribution.

Local volatility lets instantaneous volatility vary as a deterministic function of time and spot and can reproduce a sufficiently smooth arbitrage-free European surface under regularity assumptions. Heston makes variance itself random and lets the stock and variance shocks be correlated. Negative correlation is one mechanism for generating equity-style downside skew. Jump diffusion adds discontinuous moves and can help fit short-maturity wings. Each choice imposes distinct dynamics and hedging exposures even when today's vanilla prices are close.

A surface gives you a set of pricing constraints. At one expiry, a model should account for how call prices change as the strike moves through the distribution. Across expiries it must account for how those changes evolve with time. A stochastic-volatility model can fit some of that shape by allowing variance to move and correlate with spot. A jump model allows a large move to occur over a short interval without requiring a large diffusive volatility on every path. A good fit alone does not settle which dynamics you should use to hedge, especially away from the fitted strikes.

Take illustrative six-month call quotes on a non-dividend-paying stock at $100$ , with zero rates. Suppose implied volatilities at strikes $90$ , $100$ , and $110$ are $25\%$ , $20\%$ , and $18\%$ . The Black–Scholes prices for those three quotes are approximately $12.84$ , $5.64$ , and $1.75$ . At a common $20\%$ volatility, the three prices are about $11.77$ , $5.64$ , and $2.21$ . The low-strike quote exceeds the constant-volatility prediction, while the high-strike quote falls below it. This is a skew because the quoted volatility falls across those strikes. To compare multiple maturities, use a moneyness convention such as $\log(K/F)$ with the relevant forward $F$ .

Use the volatility surface tool to change skew, curvature and term slope separately and see what each does to the quoted surface.

A quick way to read one slice of the surface is to hold maturity fixed and imagine raising the strike. At every terminal stock value, the call payoff can only fall, and its rate of decrease flattens as strike passes that stock value. Positive discounted averaging preserves both the decreasing and convex shape in strike. A fitted surface that implies call prices breaking either property is suspect even if the volatilities look smooth. Check prices across strikes first, then use the implied-volatility plot to see where a flat-volatility model misses.

What an Implied Volatility Smile Reveals About Black–Scholes

What one volatility cannot fit

Move volatility and expiry in Black–Scholes

References and next steps