Implied Volatility vs Historical: Why the Smile Exists

Quant Finance Foundations, Part 10. Previously: The Greeks in Plain English.

Implied volatility and historical volatility share a name but describe two genuinely different things. One measures the past; the other prices the future. Confusing them, therefore, is the source of an enormous amount of muddled thinking.

This post assumes familiarity with the Black-Scholes inputs and with the option Greeks, vega in particular.

Historical volatility: a measurement

Historical (or realised) volatility is the standard deviation of past returns, annualised. Computing it is straightforward:

1. Take daily closing prices.
2. Compute log returns: ln(P_t / P_t−1).
3. Take the standard deviation of those returns.
4. Annualise by multiplying by √252 — the square-root-of-time scaling from Part 7, with 252 trading days in a year.

Simple, but with a choice buried in it: how far back do you look? A 30-day window responds quickly to changing conditions and is noisy. A 250-day window is stable and slow to react. Neither is correct; they answer different questions. Quote a historical volatility without stating the window and you have quoted nothing.

Two further wrinkles are worth knowing. Close-to-close estimation discards the intraday range, so estimators such as Parkinson and Garman-Klass use highs and lows to extract more information from the same data. And volatility clusters — turbulent days follow turbulent days — which is why GARCH-family models, which let today’s volatility depend on yesterday’s, fit financial data far better than a constant does.

Implied volatility: a price, not a measurement

Implied volatility runs the logic backwards. You take an option’s market price, then solve for the volatility input that makes Black-Scholes reproduce it. That number is what the market is charging.

The distinction matters enormously. Historical volatility is a fact about what happened. Implied volatility is a price for what might happen — set by supply and demand, and containing a risk premium on top of any forecast.

That premium is systematic and well documented: implied volatility tends to exceed subsequently realised volatility, most of the time. Sellers of options are being paid to bear the risk of a large move, and on average they collect. “On average” is carrying weight — the strategy of harvesting that premium produces steady small gains punctuated by severe losses, which is why it is often described as picking up coins in front of a steamroller.

The smile

Black-Scholes assumes one volatility for the underlying. If that were true, every option on the same underlying with the same expiry would imply the same volatility, and plotting implied volatility against strike would give a flat line.

It does not. The plot curves, and the shape has a name.

In currency markets the curve is roughly symmetric — out-of-the-money options on both sides imply higher volatility than at-the-money ones. That is the smile.

In equity markets it is lopsided. Low-strike puts imply substantially higher volatility than high-strike calls, producing a downward slope usually called the skew or smirk. On a major index the gap between a 90% strike and a 110% strike can easily exceed five volatility points.

The smile is not a curiosity. It is the market publicly disagreeing with the model it uses to quote prices.

Why it exists

Returns have fat tails. Black-Scholes assumes lognormal returns. Real returns produce extreme moves far more often than that allows. Options that only pay off in extreme scenarios are therefore worth more than the model says, and the market prices them accordingly. Forced through a formula that assumes thin tails, that extra value appears as elevated implied volatility.

Crashes are asymmetric. Equity markets fall faster than they rise, and volatility spikes when they fall. Downside protection is therefore genuinely worth more than upside participation. This is why equity skew is one-sided while FX — where a move is a fall for one currency and a rise for the other — is closer to symmetric.

Supply and demand. Institutional investors are structurally long equities and structurally buyers of downside protection. Persistent one-way demand for low-strike puts raises their price, and nothing forces it back.

The interesting historical note: before October 1987, equity index skew was close to flat. The crash reset how the market prices tail risk, and the skew has never gone away since. The smile is, in a real sense, a permanent memory of a single day.

The surface

Add maturity as a third axis and you get the volatility surface — implied volatility as a function of both strike and expiry. This is the actual object a derivatives desk maintains and marks daily.

The term structure has its own logic. Short-dated volatility responds violently to current conditions; long-dated volatility is anchored to long-run averages. Calm markets produce an upward-sloping surface; a crisis inverts it, with short-dated volatility spiking far above long-dated.

Building a usable surface is a real technical problem. It must be smooth enough to interpolate, and arbitrage-free — no butterfly arbitrage across strikes, no calendar arbitrage across maturities. Parametrisations such as SVI exist specifically to guarantee those constraints while fitting observed quotes.

What this means for the model

Practitioners handle the contradiction by demoting Black-Scholes to a quoting convention rather than a description of reality. Prices are converted to implied volatilities so they can be compared, interpolated and risk-managed — but nobody believes the single-volatility assumption underneath.

Models that take the smile seriously come in three families. Local volatility makes volatility a deterministic function of spot and time, fitting the surface exactly by construction. Stochastic volatility models such as Heston and SABR give volatility its own random process, generating smiles endogenously. Jump-diffusion adds discontinuous moves, which produces fat tails directly.

Each fits some part of the market well and none fits all of it. Choosing among them is an ordinary part of derivatives work, and the choice is documented, validated and defended.

Try it yourself

Build both measures side by side, in this order:

1. Download two years of daily prices for a liquid stock.
2. Compute log returns, then 30-day and 250-day rolling standard deviations.
3. Annualise each by multiplying by √252 and plot both series.
4. Pull the option chain for a single expiry on the same underlying.
5. Back out implied volatility per strike and plot it against strike.

Clustering will be immediately visible in the first chart, and the two windows will disagree sharply around every turning point.

Then pull the option chain for a single expiry, back out implied volatility for each strike using the Goal Seek method from Part 8, and plot implied volatility against strike. You will have constructed a smile from live market data — and seen, in one chart, the market’s own verdict on the model it quotes with.

Frequently asked questions

What is the difference between implied and historical volatility?

Historical volatility measures past returns. Implied volatility is backed out of current option prices and therefore includes a risk premium as well as a forecast.

Why does implied volatility usually exceed realised volatility?

Because option sellers are paid to bear the risk of a large move. On average they collect that premium, though the losses when it fails are severe.

What causes the volatility smile?

Fat-tailed returns, asymmetric crash risk and persistent demand for downside protection. Together these make extreme strikes worth more than the model assumes.

Why is equity skew one-sided while FX is symmetric?

Equity markets fall faster than they rise, so downside protection is worth more. In FX, by contrast, a move is a fall for one currency and a rise for the other.

Next in this series: building your first Monte Carlo pricer, and closing the loop by pricing one option three ways — with a binomial tree, with Black-Scholes, and by simulation.