Implied volatility & the surface

M2.2 ended on an asymmetry. Of the five Black–Scholes inputs, four are observable — spot, strike, expiry, rate — and one, σ, describes the future and cannot be looked up anywhere.

The market's response to this is the most pragmatic move in all of finance: stop trying to supply σ, and start solving for it instead.

Running the formula backwards

Options trade at prices set by supply and demand, not by anybody's model. Take an observed price and ask the inverse question:

   forward  (what the textbook does):
        S, K, T, r,  σ = 20%    ──▶  Black–Scholes  ──▶  price = £3.99

   backward (what the market does):
        S, K, T, r,  price = £4.50  ──▶  solve  ──▶  σ = 22.6%
                                                      ▲
                                          IMPLIED VOLATILITY

Implied volatility is the number you must feed Black–Scholes to reproduce the price the market is actually charging. Nothing more. Because the price is strictly increasing in σ, the inversion is unique and well behaved, and any root-finder does it in microseconds.

This reframes what the model is for. Black–Scholes is not being used to predict prices — it is being used as a translation layer. It converts a price, which depends on strike, spot, expiry and rates in ways that make two options hard to compare, into a single number that is roughly comparable across contracts. That is why traders quote in vol, negotiate in vol, and think in vol.

The famous line — the wrong model everyone uses correctly — means precisely this. Nobody needs the model's assumptions to be true. They need everyone to agree on the same invertible function, and Black–Scholes is the one everyone agreed on.

The test the model fails

Here is the sharp consequence. Black–Scholes assumes σ is a property of the stock — one number, constant, describing how that asset moves. If that were true, then every option on the same underlying, at every strike and every expiry, would imply the same σ.

Plot the implied vols and they are not the same. They are not close.

   implied vol by strike, one expiry, an equity index

     σᵢ
   30%│●
      │ ●                                    EQUITY SKEW
   25%│   ●                                  downside puts imply
      │     ●                                much higher vol than
   20%│        ●                             upside calls
      │            ●   ●
   15%│                    ●   ●   ●
      └────┬────┬────┬────┬────┬────┬────▶ strike
          80   90   100  110  120  130
                     ▲ spot

For equity indices the pattern is a skew (or smirk): implied vol falls monotonically as strike rises. In FX it is closer to a symmetric smile, both wings above the middle. Add the maturity dimension — implied vol also varies with time to expiry, in a term structure — and the whole object is a two-dimensional volatility surface over strike and maturity, which moves every day.

The surface is not a curiosity or a market inefficiency. It has been there since 1987, in every liquid options market, and it has a clear meaning:

The surface is a map of Black–Scholes' errors. Where the market pays more than the model says, it is pricing something the model left out.

What the shape is telling you

Three omissions account for most of it.

Fat tails. Black–Scholes assumes lognormal prices, so a −20% day is essentially impossible. Reality disagrees — decisively, and this is the entire subject of M3. Out-of-the-money puts pay off precisely in those "impossible" states, so the market charges more for them than a Gaussian model thinks they are worth. Higher price, higher implied vol, and the wing lifts.

Crash asymmetry, and why equities skew rather than smile. Equity markets fall faster than they rise: volatility spikes in selloffs and subsides in rallies. Two standard explanations, both partly true — the leverage effect (a falling equity price raises a firm's debt-to-equity ratio, making the remaining equity genuinely riskier) and plain demand (portfolio insurance is bought by people who own stock and fear a crash, so downside protection is structurally bid). Either way the risk is one-sided, and so is the curve. The skew's steepening after the 1987 crash is one of the cleanest natural experiments in finance — before it, the surface was much flatter.

Volatility is itself random. Black–Scholes takes σ as constant. Actual volatility clusters, mean-reverts, and jumps. An option is convex in σ, so uncertainty about volatility raises value — most at the wings, which is smile-shaped.

The serious models exist to fit exactly these: local volatility (Dupire — make σ a deterministic function of spot and time, fitting today's surface exactly), stochastic volatility (Heston, SABR — give σ its own random process, which produces smiles naturally), and jump-diffusion (Merton — add discrete jumps, which produce fat tails directly). All are in a real sense repairs to the assumption list at the end of M2.2.

Vol as the tradeable object

Once implied vol is the price, it becomes something to have a view on — and M2.3 already gave you the machinery. Buy an option, delta-hedge it, and you are long realized vol and short implied vol:

   your view                          the trade
   ────────────────────────────────────────────────────────────────
   "the market will move MORE than    buy options, delta-hedge
    22% implied"                       → long gamma, short theta
                                       → wins if realized > 22%

   "implied vol itself will RISE"     buy options, hold
                                       → long vega
                                       → wins on repricing, no
                                         movement required

   "the skew is too steep"            sell puts, buy calls, hedge
                                       → a view on the SHAPE, not
                                         the level

That third row is where volatility trading stops being a bet on one number. Desks take positions on the level of the surface, its slope (skew), its curvature (smile), and its term structure — each a distinct exposure with its own drivers.

One empirical regularity worth knowing, because it explains a whole industry: implied volatility usually exceeds subsequently realized volatility. The variance risk premium is persistently positive, meaning option sellers earn on average. This is not free money — you are being paid to hold exactly the risk that hurts most when everything else does, which is why the strategy's return profile looks like short gamma, because it is. M0.8 warned about this shape; here is where it gets a price.

Everything above rests on the claim that returns are far from Gaussian. M3 is where that claim gets tested properly — and where the "impossible" days turn out to be routine.

Source: Hull ch.20 on volatility smiles. For the surface as a practitioner object, Jim Gatheral, The Volatility Surface, is the standard — dense but the first two chapters repay the effort. Euan Sinclair, Volatility Trading, for the trading views. Dupire (1994) and Heston (1993) are the two papers behind most of what desks actually run.