What is implied volatility?
Implied volatility (IV) is the volatility parameter that makes the Black-Scholes price of an option equal to its observed market price. You solve the BS formula backward — given the market price, strike, spot, time to expiry, and risk-free rate, what σ makes the equation balance? That σ is the implied vol.
IV is forward-looking: it represents what the market collectively thinks volatility will be between now and expiry. It's derived from prices people are actually paying, not from historical observation. That makes it useful for option pricing (it's embedded in the price you trade against) and for sentiment reading (high IV = market expects turbulence; low IV = expects calm).
The volatility smile
Black-Scholes theory says IV should be constant across strikes and expiries for the same underlying. In reality it varies systematically:
- Smile: high IV at deep ITM and OTM strikes, lower at ATM. Typical for currency options.
- Smirk: high IV at OTM puts, low at OTM calls. Typical for equity indexes — the market prices crash protection at a premium.
- Surface: 2D extension across strikes AND expiries. What most options desks calibrate to.
The smile/smirk exists because real return distributions have fat tails and negative skew that the log-normal assumption in BS doesn't capture. Local-vol models (Dupire) and stochastic-vol models (Heston, SABR) explicitly model this surface.
What is historical volatility?
Historical volatility (HV) is the standard deviation of past returns, computed from close-to-close price observations over a window (typically 20, 30, 60, or 252 trading days).
HV = √(Σ(r_i − r̄)² / (n−1)) × √(periods per year)
Where r_i are periodic returns. HV is backward-looking and uses one observation per period. It's what every formula in classical portfolio theory assumes when it says "volatility" — the σ in Sharpe ratio, in Markowitz optimization, in parametric VaR, all refer to historical close-to-close stdev.
The trade-off is window size. Short windows (20-30 days) react quickly to regime changes but are noisy. Long windows (252 days) are smoother but slow to update. Most practitioners pick the window to match their holding horizon, or use multiple windows weighted together.
What is realized volatility?
Realized volatility (RV) is also backward-looking but uses high-frequency intraday data and specialized estimators that extract more volatility information than close-to-close stdev can see. The intraday range encodes volatility information that two daily closes don't.
The realized vol estimator family
Four estimators in order of sophistication:
- Close-to-close (HV): standard sample stdev. Uses only daily closes. About 1x efficient (baseline).
- Parkinson (1980): uses daily high-low range. Assumes continuous trading and no drift. ~5x more efficient than close-to-close — captures intraday vol that closes miss.
- Garman-Klass (1980): uses OHLC. Better than Parkinson on drift bias. ~7x more efficient.
- Yang-Zhang (2000): handles overnight gaps + drift bias. Minimum- variance unbiased estimator across all OHLC estimators. ~14x more efficient. The production gold standard.
For short-horizon vol forecasting (intraday risk, end-of-day position sizing), Yang-Zhang is the default. For longer horizons where the marginal efficiency gain doesn't justify the implementation complexity, close-to-close HV is fine.
A concrete example: same asset, three vol numbers
Suppose SPY is currently $480. The 30-day ATM option is trading at $8.40. The 30-day close-to-close HV is 13.8%. Realized vol (Yang-Zhang) over the past 30 days is 11.5%.
| Metric | Value | Interpretation |
|---|---|---|
| Implied (IV) | 16.2% | Market expects 16.2% annualized vol over the next 30 days |
| Historical (HV) | 13.8% | Last 30 days of closes had 13.8% annualized stdev |
| Realized YZ | 11.5% | Same 30 days, intraday-adjusted: actually 11.5% |
| IV − RV | +4.7pp | The volatility risk premium — options priced richer than realized |
That +4.7pp gap between IV and RV is the volatility risk premium. Systematic option- selling strategies (straddle selling, iron condors, vol-targeted writing) try to capture this. When the premium is wide and stable, they profit. When the premium collapses or inverts (IV < RV, which happens during regime breaks), they lose painfully — the short-vol blowups of 2018 and 2020 were both VRP-collapse events.
The decision rule
- Pricing an option, or evaluating an option's "richness" → Implied volatility. Use the Implied Volatility Calculator to solve for IV given a market price.
- Computing Sharpe, Sortino, Calmar, VaR, CVaR, or any portfolio metric → Historical volatility (close-to-close stdev). The Sharpe ratio calculator and VaR calculator both use this.
- Forecasting tomorrow's vol for intraday risk or short-horizon execution → Realized volatility with Yang-Zhang or Parkinson estimator.
- Backtesting an options strategy → Both: IV for entry pricing, HV for what the strategy actually realized in returns.
- Detecting vol regime change → Compare IV to recent RV. If IV ≪ RV, options may be underpriced (or a vol spike is coming). If IV ≫ RV, the VRP is wide (option selling has edge, but tail risk is elevated).
Common confusions
"Volatility" in Sharpe ratio is HV, not IV
When a hedge fund advertises a 1.5 Sharpe ratio, that's computed from realized returns and their sample stdev — historical volatility. Substituting IV gives a different (and theoretically incorrect) number. Don't mix the two when comparing strategies.
VIX is the IV of the SPX, not historical vol
The VIX index is computed from SPX options premiums via a specific formula (CBOE's 2003 methodology) and represents the market's implied vol expectation for the next 30 days. It's a forward-looking metric. The "realized VIX" (computed retroactively from actual SPX RV) almost always differs.
"Higher vol = more risk" only sometimes
For a single asset, HV captures most of what classical theory calls risk. But for strategies with skew (option selling, carry trades), HV understates risk because it treats upside and downside symmetrically. See VaR vs CVaR vs Max Drawdown for how tail-aware risk metrics handle this.
Related calculators
- Implied Volatility Calculator — solve for IV given a market option price (Newton-Raphson on Black-Scholes)
- Black-Scholes Calculator — uses σ as an input; pair with IV calculator for full round-trip
- Sharpe Ratio Calculator — uses HV in the denominator
- Value at Risk Calculator — uses HV for parametric VaR
- Monte Carlo Simulation Calculator — uses HV as the input vol parameter for GBM paths
References
- Black, F. & Scholes, M. (1973). "The pricing of options and corporate liabilities." Journal of Political Economy 81(3), 637-654.
- Parkinson, M. (1980). "The extreme value method for estimating the variance of the rate of return." Journal of Business 53(1), 61-65.
- Garman, M. B. & Klass, M. J. (1980). "On the estimation of security price volatilities from historical data." Journal of Business 53(1), 67-78.
- Yang, D. & Zhang, Q. (2000). "Drift-independent volatility estimation based on high, low, open, and close prices." Journal of Business 73(3), 477-491.
- Bollerslev, T., Tauchen, G., & Zhou, H. (2009). "Expected stock returns and variance risk premia." Review of Financial Studies 22(11), 4463-4492.