Realised volatility
sigma_real = sqrt(A) stdev(r_t)The sample standard deviation is scaled by the number A of return periods per year.
Foundations
Volatility (Realised vs Implied)
Realised volatility comes from past price moves; implied volatility is the market’s quote for future uncertainty.
Mental model
Realised volatility describes observed variation, while implied volatility is the market’s forward-looking price of uncertainty.
Realised volatility is computed from historical returns, for example as the annualised standard deviation over a recent window.
Implied volatility is the σ that, plugged into Black–Scholes, matches the current market price of an option.
Explore the mechanism
Why can the volatility observed in past returns differ from the volatility backed out of today's option price?
Realised volatility σrealised comes from past price moves; you estimate it from a return series. Implied volatility σimplied is the input to a pricing model (here Black–Scholes) that makes the model price match the market option price.
Numbers
Target slider volatility σtarget: 20%
Realised volatility σrealised from this path: 20.0% (annual)
Corresponding daily σ: 1.3% (≈ σ_realised / √252)
Implied volatility σimplied: 25.0% (annual)
Call price at σrealised: 5.64
Call price at σimplied: 7.04 (higher premium)
Parameters: S₀ = 100, K = 100, T = 0.5 years, r = 0.
Interpretation
The upper panel is the realised volatility engine: take daily returns rₜ, compute their sample standard deviation, and annualise it. Moving the “Target path volatility” slider stretches or compresses these bars; σrealised follows.
The lower panel is the option pricing engine: feed a volatility σ into Black–Scholes and read off C(σ). Marking σrealised and σimplied on the same curve shows how far the market’s volatility input can sit from what actually happened in the path. That gap is where volatility risk premium, jump risk and order-flow effects live in practice.
UNDERSTAND
What to notice
They differ because markets embed risk premia, jump risk, and supply–demand imbalances for options into implied volatility.
In calm periods, realised volatility can sit well below implied; in stress, both can spike, but not necessarily by the same amount or timing.
The quantitative structure
Notation
r_t- sampled return
sigma_real- annualized realised volatility
sigma_impl- implied volatility
Implied volatility
ModelPrice(sigma_impl) = MarketPriceImplied volatility is the model parameter that reproduces an observed option price.
This model assumes
- Realised-volatility sampling frequency and annualization are specified.
- The implied-volatility inversion uses a stated option-pricing model and consistent market inputs.
Where it appears
- option quoting and calibration
- measuring historical market variability
- comparing volatility risk premia
Where the intuition breaks
What this view leaves out
- Both measures depend on choices: realised volatility depends on sampling and window length, while implied volatility depends on the pricing model and option liquidity.
Common mistakes
“Implied volatility is the market's unbiased forecast of future realised volatility.”
Not quite.It is a model-implied price coordinate that can include risk premia and supply-demand effects.
Where this concept sits
Follow the strongest prerequisites, applications, extensions, and contrasts.
Connections for Volatility (Realised vs Implied)
Volatility (Realised vs Implied)YOU ARE HERE
Built onReturns and Log-ReturnsRealised volatility is estimated from return series, while log-returns are the natural scale for many continuous-time models.
Foundation ofImplied Volatility Smile and SurfaceThe smile and surface organise market-implied volatility across strike and maturity dimensions.
Used byBlack–Scholes Option PricingVolatility is the key uncertainty input in Black–Scholes and can be inverted from market option prices.
Used byGeometric Brownian Motion (GBM)Volatility is the diffusion-scale parameter controlling the dispersion of GBM paths.
Built on · Returns and Log-Returns
Realised volatility is estimated from return series, while log-returns are the natural scale for many continuous-time models.
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