Four kinds of volatility, the empirical surface, and the families of model — deterministic, stochastic, uncertain — that try to explain what one number can't.
Black–Scholes asks for "the" volatility as if it were a single number. The market does not oblige. Wilmott opens chapter 9 by sorting out the four distinct things people mean by the word.
The instantaneous standard deviation of returns of the real asset at the present moment. Not observable, not even well-defined except as a limit. The thing models try to describe.
A statistic computed from the past path of $S$: $\hat\sigma$ from sample standard deviation of log-returns over a window. Backward-looking, observable, noisy.
The number you plug into Black–Scholes to recover the observed market price of an option. Forward-looking, derived, contract-specific — the market's opinion.
The volatility that, between two future dates $T_1$ and $T_2$, is consistent with the implied vols out to each. Important for forward-starting and cliquet-style contracts.
All four would be equal in a world where Black–Scholes were exactly true. In the real world they disagree, and the disagreement is structured — in $K$ (smile), in $T$ (term structure), and over time. The rest of the chapter is a tour of model families that try to absorb that structure.
The simplest estimator. Take $n$ consecutive log-returns from daily closes,
$$r_i = \ln\!\left(\frac{S_i}{S_{i-1}}\right),$$
and compute the sample standard deviation:
$$\hat\sigma^2 = \frac{1}{n-1}\sum_{i=1}^{n}\left(r_i - \bar r\right)^2 \cdot \frac{1}{\Delta t}.$$
The factor $1/\Delta t$ annualises the variance — if $\Delta t = 1/252$ (one trading day), multiply by $252$. The annualised standard deviation $\hat\sigma$ is what you'd quote.
Re-estimate using only the last $n$ days: a rolling window. Long windows give a stable but slow estimate; short windows are reactive but noisy. Typical: $n = 30$ to $100$ days.
Give the most recent return more weight: $w_i \propto \lambda^{n-i}$ with $\lambda \in (0,1)$. RiskMetrics famously uses $\lambda = 0.94$ for daily data — a half-life of about $11$ days.
Either way, historical vol tells you about the past. The implicit assumption is that the next $n$ days will look like the last $n$ — an assumption the market doesn't share, which is why historical and implied vols often disagree.
The data have an empirical regularity: large moves cluster together. A turbulent day is more likely than average to be followed by another turbulent day. A constant-$\sigma$ model can't reproduce this. Two simple time-series specifications can.
The exponentially weighted moving average variance:
$$\hat\sigma_t^2 = \lambda\, \hat\sigma_{t-1}^2 + (1-\lambda)\, r_{t-1}^2.$$
One parameter $\lambda$. The recursion gives an updating rule rather than a window: each new return refines the estimate.
Generalised autoregressive conditional heteroscedasticity — Bollerslev (1986). One step richer:
$$\sigma_t^2 = \omega + \alpha\, r_{t-1}^2 + \beta\, \sigma_{t-1}^2.$$
Three parameters. There is a long-run variance $\omega / (1 - \alpha - \beta)$, an autoregressive part $\beta$ (variance persists) and a shock part $\alpha$ (recent returns matter). EWMA is the special case $\omega = 0$, $\alpha + \beta = 1$.
If $\sigma$ varies through time, the option price strictly depends on the distribution of $\sigma$ over $[t, T]$. GARCH suggests how to extrapolate $\sigma$ forward conditional on recent history — useful for option strategies whose payoff depends on the realised vol path (variance swaps, vol-targeting funds).
Close-to-close returns throw away most of the day's information. Each day's high and low contain news the close doesn't. Two classical estimators exploit them.
From the daily high $H$ and low $L$ only:
$$\hat\sigma^2_{\text{P}} = \frac{1}{4\ln 2}\left[\ln(H/L)\right]^2.$$
Variance over one day; multiply by $252$ to annualise. Roughly $5\times$ more efficient than close-to-close on a GBM.
Uses open $O$, high $H$, low $L$, close $C$:
$$\hat\sigma^2_{\text{GK}} = \tfrac{1}{2}\!\left[\ln(H/L)\right]^2 - (2\ln 2 - 1)\!\left[\ln(C/O)\right]^2.$$
Even more efficient — about $7$ to $8\times$ close-to-close.
Both assume zero drift and continuous sampling. They become biased when the price has gaps (overnight news, opening jumps) or when trading is illiquid — cases where the printed high or low understates the true range.
Given a market call price $C^{\text{mkt}}$ at strike $K$ and maturity $T$, the implied volatility $\sigma_{\text{imp}}$ is the unique solution of
$$C_{\text{BS}}(S, K, T, r, \sigma_{\text{imp}}) = C^{\text{mkt}}.$$
There is no closed-form inverse. $C_{\text{BS}}(\sigma)$ is strictly increasing in $\sigma$, so bisection always works; Newton's method is faster:
$$\sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C^{\text{mkt}}}{\mathcal{V}(\sigma_n)},$$
where $\mathcal{V} = \partial C / \partial \sigma$ is the vega:
$$\mathcal{V} = S\sqrt{T}\, N'(d_1), \qquad d_1 = \frac{\ln(S/K) + (r + \tfrac{1}{2}\sigma^2)T}{\sigma\sqrt{T}}.$$
Newton can fail if the initial guess is far from the root, or if the option is deeply in/out of the money — vega vanishes and the step blows up. A robust solver brackets the root with bisection first, then switches to Newton when the interval is small. Slide 10's solver does exactly this.
The market quotes options in implied vols, not prices. The Black–Scholes formula is treated as a price-to-vol converter — even by traders who don't believe its assumptions.
Plot implied vol against strike for fixed $T$. You don't get a flat line. You get a smile (FX, post-1987 indices) or a skew (equity indices: low strikes have higher implied vol).
Real returns have fatter tails than the lognormal. Out-of-the-money options are worth more than Black–Scholes says, so their implied vol is higher.
Post-1987, equity indices show a steep skew: low-strike puts trade at much higher implied vol — the market prices crash risk explicitly.
Demand for protective puts (hedgers) pushes their price up. Implied vol is partly the BS-invert of an inventory imbalance, not a forecast.
The implied-vol surface $\sigma_{\text{imp}}(K, T)$ is the two-dimensional version: implied vol as a function of strike and maturity. Its shape varies across markets and over time. Calibrating a model means choosing parameters so its implied-vol surface matches the market's.
The smile is a symptom — it tells you that the constant-vol lognormal model is wrong. It does not itself tell you which richer model is right. That's why Wilmott devotes the rest of chapter 9 to model families.
The simplest extension: keep one Brownian, let the volatility depend on the spot and time. Replace $\sigma$ with $\sigma(S,t)$:
$$dS = \mu S\, dt + \sigma(S, t)\, S\, dW.$$
If $\sigma(S,t)$ is a known function, the market is complete and Black–Scholes-style PDE pricing still works — with a non-constant diffusion coefficient. The big result, due to Dupire (1994), is that there is a unique $\sigma(K, T)$ that exactly reproduces the market's call surface $C(K,T)$:
$$\sigma^2_{\text{loc}}(K,T) = \frac{2\,\dfrac{\partial C}{\partial T} + 2(r-q)K\dfrac{\partial C}{\partial K} + 2qC}{K^2\,\dfrac{\partial^2 C}{\partial K^2}}.$$
So a perfect calibrating model exists, in closed form — if you can differentiate the call surface twice in $K$ and once in $T$. That "if" is fatal in practice: market quotes are sparse and noisy, and double differentiation amplifies the noise. Dupire is theoretically elegant and practically delicate.
Give volatility its own Brownian motion. The general spec:
$$dS = \mu S\, dt + \sigma S\, dW_1,$$
$$d\sigma = \alpha(S, \sigma, t)\, dt + \beta(S, \sigma, t)\, dW_2,$$
$$\mathrm{corr}(dW_1, dW_2) = \rho.$$
Now there are two sources of randomness and the market is incomplete: you can't hedge $S$ and $\sigma$ with $S$ alone. To price you need a market price of volatility risk $\lambda$ — an extra parameter the market itself has to imply.
Take the variance $v = \sigma^2$ to follow a mean-reverting square-root process:
$$dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2.$$
Four parameters — $\kappa$ (mean reversion), $\theta$ (long-run variance), $\xi$ (vol-of-vol), $\rho$ (correlation) — plus the initial $v_0$. Famous because it admits a semi-closed-form via characteristic functions and Fourier inversion. Standard benchmark.
Geometric Brownian variance — analytically tractable but no mean reversion.
$dF = \sigma F^\beta\,dW_1$, $d\sigma = \nu\sigma\,dW_2$. Industry standard for swaption smiles.
Volatility driven by fractional Brownian motion with Hurst $H < 1/2$. Fits empirical roughness better than diffusions.
An alternative philosophy — Avellaneda, Levy and Parás (1995): don't try to specify a probability law for $\sigma$ at all. Just say it lives in a band,
$$\sigma_{\min} \le \sigma(t) \le \sigma_{\max},$$
and choose hedging parameters to bound the value of the portfolio in the worst case.
The seller-side worst-case price satisfies a nonlinear PDE — the Black–Scholes–Barenblatt equation:
$$V_t + \tfrac{1}{2}\Sigma(V_{SS})^2 S^2 V_{SS} + rSV_S - rV = 0,$$
where $\Sigma(V_{SS}) = \sigma_{\max}$ if $V_{SS} > 0$ and $\sigma_{\min}$ if $V_{SS} < 0$. The "worst" $\sigma$ depends on the option's gamma at each $(S,t)$.
The lowest possible no-arbitrage price — you'd be over-paying if the seller offered less. Use $\sigma_{\max}$ where $V_{SS} < 0$, $\sigma_{\min}$ where $V_{SS} > 0$.
The highest possible no-arbitrage price — quote at least this and you can't be picked off. Swap the conditions above.
The result is a bid–ask spread generated entirely from model uncertainty. Nonlinearity also means portfolios do not price additively — a vertical spread costs less worst-case than the two legs separately. This is the deep payoff of the framework: it rewards hedged structures.
Probabilistic models say "here's the distribution of $\sigma$." Uncertain-vol says "I refuse to make that claim — here's the range I'm prepared to defend." The first gives a unique number with implicit certainty; the second gives an interval with explicit honesty.
A parametric quadratic smile in log-moneyness:
$$\sigma_{\text{imp}}(K) = \sigma_0 + a\,m + b\,m^2, \qquad m = \frac{\ln(K/S_0)}{\sqrt{T}}.$$
The left of the canvas plots $\sigma_{\text{imp}}$ against strike. The right plots the resulting Black–Scholes call prices — the smile creates an entire term structure of derivative prices from a few coefficients.
Pick a target ATM call price; the solver finds the implied vol that reproduces it, iterating until $|C(\sigma) - C^{\text{tgt}}| < 10^{-6}$. Each row is one Newton step.
| iter | $\sigma_n$ | $C(\sigma_n)$ | $C - C^{\text{tgt}}$ | vega |
|---|
Newton converges quadratically when started near the root; from far away it can wander, especially when vega is small. Try $\sigma_0 = 1.5$ for a deep OTM-style target — you'll see the step shrink and the iteration overshoot then correct.