Black-Scholes-Mertonintermediate

Implied Volatility

Implied volatility ( $\sigma_{\text{imp}}$ ) is the value of $\sigma$ that, when plugged into BSM, reproduces the observed market price of an option. There is no closed form for $\sigma_{\text{imp}}$ , so it is solved numerically with a root-finder such as Newton-Raphson or bisection (Hull, 2022, §15.11). Unlike historical volatility, which looks backward at realised returns, implied volatility is forward-looking and reflects what the options market expects future volatility to be.

Why it matters

BSM is a mapping from $\sigma$ to option price, monotonic in $\sigma$ . Inverting that map gives implied volatility. If you taste a finished dish and want to know how much salt went in, implied volatility is your answer for uncertainty. Quoting an option in volatility units rather than dollars also lets traders compare options across strikes and expiries on a common scale, which is why option desks quote in vol points.

Formulas

Definition

C_{\text{market}} = \text{BSM}(S_0,\, K,\, r,\, T,\, \sigma_{\text{imp}})

Implicitly defines

\sigma_{\text{imp}}

. The right-hand side is the standard BSM call (or put) price.

Newton-Raphson update

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

\mathcal{V}

is vega,

\partial C / \partial \sigma

. Usually converges in 3-5 iterations from a sensible starting guess.

Worked examples

Scenario

Market call price $C =$ A$5.00. $S_0 =$ A$50, $K =$ A$50, $r = 5\%$ , $T = 0.5$ . Solve for implied volatility.

Solution

Try $\sigma = 25\%$ , BSM gives roughly A$4.50, too low. Try $\sigma = 30\%$ , BSM gives roughly A$5.15, too high. Try $\sigma = 28.5\%$ , BSM gives roughly A$5.00. Implied volatility is about 28.5%, or in vol points, 28.5 vol.

Scenario

The Cboe Volatility Index (VIX) is a model-free implied volatility on S&P 500 options, the so-called fear gauge.

Solution

The VIX averaged about 15-20 over 2024-2025 calm periods and spiked above 65 during the August 2024 yen carry unwind. It reflects the market's 30-day forward-looking volatility expectation derived from a strip of out-of-the-money SPX option prices, not from any single BSM inversion.

Common mistakes

✗Implied volatility is the same as historical volatility. Historical volatility is the standard deviation of past returns. Implied volatility is the market's forward-looking estimate baked into option prices. The two can differ sharply, the gap (the volatility risk premium) is a well-known empirical regularity.
✗Implied volatility is constant across strikes. It is not. Plotting $\sigma_{\text{imp}}$ against strike traces out a volatility smile or skew, a direct violation of the constant-vol assumption that the original BSM paper made. The 1987 crash is widely credited with creating the equity-index skew.

Revision bullets

• $\sigma$ that makes BSM = market price
•Solved numerically, no closed form
•Forward-looking, not historical
•Varies with strike, the volatility smile
•VIX is a model-free implied vol on the S&P 500

Quick check

Implied volatility is best defined as

If implied volatility plotted against strike forms a U-shape with the lowest IV near the money, this pattern is called

Connected topics

Time Value BSM Intro

In learning paths

Pricing Models Path

Sources

Hull (2022), §15.11
Hull, John C. Options, Futures, and Other Derivatives. 11th ed. Pearson, 2022. ISBN 978-0-13-693997-9.
Standard treatment of implied volatility, numerical inversion of BSM, and the volatility smile.
Cboe VIX White Paper
Cboe Exchange, Inc. "Cboe Volatility Index (VIX) White Paper." Cboe, 2019.
Methodology for the VIX, the best-known model-free implied volatility benchmark derived from S&P 500 option prices.
Rubinstein (1985), JF
Rubinstein, Mark. "Nonparametric Tests of Alternative Option Pricing Models Using All Reported Trades and Quotes on the 30 Most Active CBOE Option Classes from August 23, 1976 through August 31, 1978." Journal of Finance 40, no. 2 (1985): 455-480.
Early documentation of empirical violations of constant volatility, including the implied volatility patterns that later became the smile and skew.

How to cite this page

Dr. Phil's Quant Lab. (2026). Implied Volatility. Derivatives Atlas. https://phucnguyenvan.com/concept/implied-volatility

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