Visualize and compare all Black-Scholes Greeks (Delta, Gamma, Theta, Vega, Rho, Epsilon) with interactive charts, multi-option comparison, and delta-gamma hedging tools.
Black-Scholes Greeks
Stock price (S)
Strike (K)
Time to maturity (T, years)
Risk-free rate (r, %)
Dividend yield (q, %)
Volatility (σ, %)
Call/Put identical for this Greek (chart unchanged)
Opt 1 ATM
Opt 2 ATM
Opt 3 ATM
Price
—
—
—
Δ Delta
—
—
—
Γ Gamma
—
—
—
Θ Theta
—
—
—
ν Vega
—
—
—
ρ Rho
—
—
—
ε Epsilon
—
—
—
Option 2
S
K
T
r%
q%
σ%
Option 3
S
K
T
r%
q%
σ%
Plot vs:
Theta Decay: Option price vs Time remaining
Delta Hedging / Leverage Calculator
Option 1 Delta—
Option Price—
Shares owned
Stock price (S)—
Contracts (x100)
Options (units)—
Stock portfolio value—
Options cost—
Stock delta—
Options delta—
—
⚠ Counterintuitive: Hedging with OTM options produces the highest portfolio gamma, while ITM options produce the lowest even though gamma per option is highest ATM.
Why? To hedge a fixed stock position, you need \(n = \frac{\text{shares}}{\Delta \times 100}\) contracts.
Portfolio gamma becomes \(\Gamma_{\text{portfolio}} = n \times \Gamma \times 100 = \text{shares} \times \frac{\Gamma}{\Delta}\).
What matters is the Γ/Δ ratio, not absolute Γ:
• ITM: high Δ, moderate Γ → few contracts → lowest portfolio Γ
• ATM: medium Δ, highest Γ → moderate contracts → middle portfolio Γ
• OTM: low Δ, low Γ → many contracts needed → highest portfolio Γ
Use ITM options for a stable hedge (low rebalancing), OTM for maximum gamma exposure per dollar of delta hedged.
Delta-Gamma Hedge Calculator
Uses Option 1 and Option 2 to simultaneously neutralize both delta and gamma. Add Option 2 if not visible.
Shares owned
Stock price (S)—
Opt 1 Δ/Γ—
Opt 2 Δ/Γ—
—
Video Walkthrough
For more details on the documentation and methodology, follow the Python script.
Disclaimer: Educational purposes only. Black-Scholes assumes constant volatility, continuous trading, and no transaction costs.
Written and Modelled by
David Arias, CFA
Licensed portfolio manager with 4+ years of experience, specializing in emerging markets private debt, derivatives, and quantitative finance.
