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Quantitative Finance

Option Greeks

The Greeks measure sensitivity of option price to parameters: Delta (price), Gamma (delta change), Theta (time decay), Vega (volatility), Rho (rates). Essential for risk management and hedging.

Delta (Δ)
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Gamma (Γ)
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Theta (Θ)
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Vega (ν)
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Rho (ρ)
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The Formulas

Δ = N(d₁) (call) or N(d₁)-1 (put)
Γ = N'(d₁) / (Sσ√T)
Θ = -S N'(d₁)σ / (2√T) - rKe⁻ʳᵀN(d₂) (call)
Vega = S√T N'(d₁) / 100
Rho = KTe⁻ʳᵀN(d₂) / 100 (call)

How It Works

// Calculate d1 and d2
const d1 = (Math.log(S/K) + (r + sigma*sigma/2)*T) / (sigma * Math.sqrt(T));
const d2 = d1 - sigma * Math.sqrt(T);

// Delta: rate of change of price wrt underlying
const delta = isCall ? normalCDF(d1) : normalCDF(d1) - 1;

// Gamma: rate of change of delta
const gamma = normalPDF(d1) / (S * sigma * Math.sqrt(T));

// Theta: time decay per day
const theta = (-S * normalPDF(d1) * sigma / (2 * Math.sqrt(T)) 
               - r * K * Math.exp(-r*T) * normalCDF(d2)) / 365;

// Vega: sensitivity to volatility (per 1% change)
const vega = S * Math.sqrt(T) * normalPDF(d1) / 100;

// Rho: sensitivity to interest rates
const rho = K * T * Math.exp(-r*T) * normalCDF(d2) / 100;