What is Implied Volatility (IV)?
Implied volatility is the value of σ you have to plug into a pricing model to reproduce the market price of an option. It is the market's consensus forecast of future volatility, expressed in price units.
Definition
Implied volatility (IV) is the annualised standard deviation of returns that, when inserted into a standard option pricing model, makes the model spit out the option's actual market price. It is not a forecast in the statistical sense — it is the trade-able number that lets options at different strikes and expiries be compared on a common scale. Every option on every underlying has its own IV; when you plot IV against strike for a given expiry, you get a "vol smile" or "vol skew"; when you plot it against expiry for a given strike, you get a "vol term structure"; when you stack all expiries together, you get a "vol surface." Trading volatility means trading these structures, not picking a direction on the underlying.
Why it matters & how it's calculated
Computationally, IV is the solution to BS(σ) = market_price where BS is your pricing model. There is no closed form — it is solved numerically (Newton-Raphson or Brent's method on the vega-positive interval). Several subtleties matter on a desk. First, IV depends on the model you use: a lognormal IV and a normal IV give different numbers for the same option, and the choice matters when comparing across rates (where normal frameworks are increasingly standard) and equity options. Second, IV for an option that is far OTM and barely worth anything has huge numerical instability — the bid-ask spread of a $0.05 wing option can swing IV by 10 vol points. Third, the relationship between IV and realised vol is the entire game: the volatility risk premium (VRP), defined as IV minus subsequent realised vol, is one of the most persistent positive expected-return signals in markets. Sellers of vol, on average, get paid the VRP; buyers pay it. Whether that's edge or compensation for tail risk is the whole question of vol strategy.
Formula
IV: solve BS(S, K, T, r, σ) = market_price for σ
Worked example
SPX is at 5,000, an at-the-money 30-day call trades at $50. You back-solve for σ in the BS formula and get σ = 0.16 (annualised). That option's implied vol is 16. Tomorrow the price rises to $55 with spot unchanged — implied vol rose to 17.5. The option became "more expensive" not because of direction, but because the market is now pricing more future movement.
Related concepts
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