Model-free implied-volatility index, computed as the square root of the 30-day variance swap rate replicated from the strip of out-of-the-money S&P 500 options: VIX^2 ≈ dfrac2Tsum_i dfracΔ K_iK_i^2,e^rT,Q(K_i) - dfrac1Tleft(dfracFK_0 - 1right)^2. Unlike Black-Scholes implied vol, no model assumption — only static replication via Carr-Madan. Empirically negatively correlated with SPX returns (ρ ≈ -0.7 to -0.8), hence the 'fear gauge'. VIX futures and ETPs let traders express forward variance directly.