the Black-Scholes assumptions are

Volatility smile

Implied volatility varies with strike price, contradicting Black-Scholes

the Black-Scholes formula prices

Black-Scholes formula: C = S*N(d1) - X*e^(-rT)*N(d2), P = X*e^(-rT)*N(-d2) - S*N(-d1)

implied volatility tells you

Implied volatility indicates the market's expectation of future price movement

d₁ and d₂ are in Black-Scholes: d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T), d₂ = d₁ - σ√T

d₁ = [ln(S/K) + (r + σ²/2)T] / (σ√T), d₂ = d₁ - σ√T

VIX

VIX measures 30-day S&P 500 volatility

dollar-cost averaging achieves

Dollar-cost averaging smooths out volatility by investing fixed amounts regularly