Option Greeks Formula Reference
The Greek formulas you actually need, grouped and ready to use — with the d₁/d₂ terms they depend on and the rupee-scaling maths for the Indian lot system.
Option Greeks Formula Reference: This reference collects every option Greek formula — the Black-Scholes Delta, Gamma, Theta, Vega and Rho, the d₁ and d₂ building blocks, and the rupee-scaling maths that converts per-share Greeks into Nifty lot exposure.
The Greeks (Black-Scholes)
| Delta | Call Δ = N(d₁) · Put Δ = N(d₁) − 1 |
|---|---|
| Gamma | Γ = n(d₁) / (S · σ · √T) |
| Theta | Θ_call = −(S·n(d₁)·σ)/(2√T) − r·K·e^(−rT)·N(d₂) (per year; ÷365 per day) |
| Vega | ν = S · n(d₁) · √T (per 1.00 vol; ÷100 for per 1%) |
| Rho | ρ_call = K · T · e^(−rT) · N(d₂) · ρ_put = −K · T · e^(−rT) · N(−d₂) |
| Vanna | Vanna = ∂Δ/∂σ = ∂ν/∂S = −n(d₁) · d₂ / σ |
| Charm | Charm = ∂Δ/∂t = −n(d₁) · (2rT − d₂σ√T) / (2Tσ√T) |
| Vomma | Vomma = ν · (d₁ · d₂) / σ |
| Color | Color = ∂Gamma/∂t = ∂³V/∂S²∂t = −n(d₁)/(2·S·T·σ√T) · [2rT + 1 + d₁·(2rT − d₂σ√T)/(σ√T)] |
| Speed | Speed = ∂Gamma/∂S = ∂³V/∂S³ = −Gamma/S · (d₁/(σ√T) + 1) |
| Ultima | Ultima = ∂Vomma/∂σ = ∂³V/∂σ³ = (−Vega/σ²) · [d₁·d₂·(1 − d₁·d₂) + d₁² + d₂²] |
| Lambda | λ = Δ × (S ÷ V) = (∂V/∂S) × (S ÷ V) |
| Zomma | Zomma = ∂Γ/∂σ = Γ · ((d₁·d₂ − 1) / σ) |
| Veta | Veta = ∂ν/∂t = −S·n(d₁)·√T · ( r·d₁/(σ√T) − (1+d₁·d₂)/(2T) ) |
The building blocks: d₁ and d₂
| d₁ | [ ln(S/K) + (r + σ²/2)·T ] / (σ·√T) |
|---|---|
| d₂ | d₁ − σ·√T |
| Call price | S·N(d₁) − K·e^(−rT)·N(d₂) |
| Put price | K·e^(−rT)·N(−d₂) − S·N(−d₁) |
Position & rupee scaling
| Position Delta | Σ (leg Δ × lots × lot size) |
|---|---|
| Rupee P&L from a move | Δ × move × lot size × lots |
| Daily rupee decay | Θ × lot size × lots |
| Rupee Vega exposure | Vega × ΔIV(%) × lot size × lots |
S = underlying (Nifty) spot, K = strike, T = time to expiry in years, r = risk-free rate, σ = implied volatility, N() = standard-normal CDF. See our Methodology for models and assumptions, and put these to work in the Greeks calculator.
Frequently asked questions
What is the formula for Delta?
For a call, Delta = N(d₁); for a put, Delta = N(d₁) − 1, where N is the standard-normal CDF and d₁ comes from Black-Scholes. Delta lies between 0 and 1 for calls and 0 and −1 for puts.
What is the formula for Gamma?
Gamma = n(d₁) / (S·σ·√T), where n is the standard-normal probability density. It is identical for a call and a put at the same strike and is always positive for long options.
How do I convert a Greek into rupees for Nifty?
Multiply the per-share Greek by the lot size (75 for Nifty) and the number of lots. For example, a position Theta of −8 per share on 2 lots decays about 8 × 75 × 2 = ₹1,200 per day.
Last reviewed 7 July 2026. Educational content only — not investment advice.