Implied Volatility

Implied volatility is the number the whole options market is quoted in. Nobody trades an option by arguing about dollar prices; they argue about vol. Implied volatility is the single value of $\sigma$ that, plugged into Black-Scholes, reproduces the option's market price. It converts an opaque nonlinear price into one intuitive, comparable, forward-looking number, and the gap between it and realized volatility is one of the most persistent sources of edge in the market.

Definition and existence

The Black-Scholes call price $C_{\text{BS}}(\sigma)$, holding $S, K, r, T$ fixed, is a function of volatility alone. The implied volatility $\sigma_{\text{imp}}$ is the solution of

$C_{\text{BS}}(\sigma_{\text{imp}}) = C_{\text{market}}.$

That this equation has a unique solution follows from a single Greek: vega is strictly positive,

$\frac{\partial C_{\text{BS}}}{\partial\sigma} = S\,n(d_1)\sqrt T > 0.$

So $C_{\text{BS}}(\sigma)$ is strictly increasing in $\sigma$. As $\sigma\to 0$ the price tends to the discounted intrinsic value $\max(S - Ke^{-rT}, 0)$, and as $\sigma\to\infty$ it tends to $S$ (the call approaches the stock). Any arbitrage-free market price lies strictly between these bounds, so by the intermediate value theorem plus monotonicity there is exactly one $\sigma_{\text{imp}}$. The map price $\leftrightarrow$ vol is a bijection, which is why traders can move freely between the two languages.

Computing it

There is no closed form, so $\sigma_{\text{imp}}$ is found numerically. Newton-Raphson converges fast because vega is exactly the derivative needed:

$\sigma_{n+1} = \sigma_n - \frac{C_{\text{BS}}(\sigma_n) - C_{\text{market}}}{\mathcal V(\sigma_n)}.$

It converges quadratically for near-ATM options where vega is large and well-behaved. Deep in- or out-of-the-money, vega collapses toward zero and Newton becomes unstable (dividing by a tiny vega), so practitioners fall back to bisection or specialized methods (e.g. Jäckel's "Let's Be Rational"). A useful sanity check is the ATM approximation: for $r=0$, $K=S$, and small $\sigma\sqrt T$,

$C_{\text{ATM}} \approx 0.4\,S\,\sigma_{\text{imp}}\sqrt T \;\Longrightarrow\; \sigma_{\text{imp}} \approx \frac{C_{\text{ATM}}}{0.4\,S\sqrt T}.$

The constant is $1/\sqrt{2\pi} \approx 0.399$, the peak of the normal density, so an ATM straddle priced at 8% of spot for one year implies roughly $8/(0.4\times100) = 20\%$ vol.

The volatility surface

If Black-Scholes were literally true, every strike and maturity would return the same implied vol. It does not: plotting $\sigma_{\text{imp}}$ against strike gives the smile/skew, and against maturity gives the term structure. Together they form the implied-volatility surface $\sigma_{\text{imp}}(K, T)$, the market's fingerprint, and the object that local vol and stochastic vol models are built to fit. Implied vol is thus not a property of the stock but a quoting convention: a strike- and maturity-dependent parameter that makes a known-imperfect model reproduce prices.

Implied vs realized: the variance risk premium

Implied volatility is the market's risk-neutral, forward-looking estimate of volatility to expiry. Realized volatility is what the stock actually delivers, measured after the fact from returns:

$\sigma_{\text{real}}^2 = \frac{252}{n}\sum_{i=1}^{n}\Big(\ln\frac{S_i}{S_{i-1}}\Big)^2.$

Empirically, implied systematically exceeds realized, index options are, on average, "expensive." The wedge is the variance risk premium: sellers of options demand compensation for bearing the risk of volatility spikes and crash payoffs, so the risk-neutral (implied) expectation is biased above the physical (realized) one. This is not a free lunch, the premium is payment for a genuinely nasty, negatively-skewed exposure (short vol blows up in crises), but it is why systematic option-selling and variance-swap strategies have historically earned a premium, and why the delta-hedging P&L of a long option is on average negative for index options.

Worked example

A three-month ATM call ($T = 0.25$) on a $50 stock trades at $2.00, with $r = 0$. Using the approximation,

$\sigma_{\text{imp}} \approx \frac{2.00}{0.4\times 50\times\sqrt{0.25}} = \frac{2.00}{0.4\times50\times0.5} = \frac{2.00}{10} = 0.20.$

So the market implies 20% annualized vol. If the stock then realizes only 15% over the quarter, a delta-hedged long holder loses the 5-vol-point gap (dollar-gamma weighted), the variance risk premium accruing to the option seller.

What breaks in practice

• It is model-dependent quoting, not truth. Implied vol is defined through Black-Scholes; because BS is wrong, the number varies by strike. Comparing IVs across very different strikes without accounting for the smile is comparing apples to skewed oranges.
• Vega vanishes in the wings. For deep OTM options, tiny price changes map to large IV swings (low vega), so wing IVs are noisy and quote-sensitive, a one-cent price error can move implied vol by points.
• Discrete dividends, American features, borrow. The inversion assumes the clean European BS setup; American early-exercise, dividend timing, and hard-to-borrow financing all bias the naively-inverted IV, and desks strip these out first.
• Implied is risk-neutral. Reading implied vol as a forecast of realized vol ignores the variance risk premium, implied is a biased (upward) predictor.

In interviews

Be ready to explain why implied vol is unique (vega $> 0$ makes the BS price monotone in $\sigma$), how you'd compute it (Newton with vega, bisection in the wings), and the ATM shortcut $C_{\text{ATM}}\approx 0.4\,S\sigma\sqrt T$. The conceptual heavyweight is "implied vs realized": implied is the risk-neutral forward expectation, realized is ex-post, and implied usually exceeds realized because of the variance risk premium, sellers get paid for crash risk. A common trap: "is implied vol a prediction of future volatility?", only a risk-neutral one, biased above the physical forecast. This sets up the smile and variance swaps.