Option Dual Gamma
Computes the dual gamma of a call option: the second derivative of the price with respect to the strike, ∂²C/∂K². While gamma measures the price's curvature relative to the asset, dual gamma measures the curvature relative to the strike, and it's tied to the risk-neutral probability density of the future price — in fact, dual gamma is exactly that density discounted. It's used to extract implied price distributions from option prices. Enter the spot price, the strike, the rate, the volatility and the term.
Result
—
Option Dual Gamma
Computes the dual gamma of a call option: the second derivative of the price with respect to the strike, ∂²C/∂K². While gamma measures the price's curvature relative to the asset, dual gamma measures the curvature relative to the strike, and it's tied to the risk-neutral probability density of the future price — in fact, dual gamma is exactly that density discounted. It's used to extract implied price distributions from option prices. Enter the spot price, the strike, the rate, the volatility and the term.
The price curvature with respect to the strike
Every well-known Greek measures the option price's sensitivity to something in the market: the asset, time, volatility. Dual gamma looks in an unusual direction: the strike. It's the second derivative of the option price with respect to the strike, the curvature of the price as you vary the strike, not the asset.
This seemingly abstract number has a deep meaning. Dual gamma is, up to the discount factor, exactly the risk-neutral probability density of the asset price at expiry. It's the famous Breeden-Litzenberger result: the implied price distribution is hidden in the second derivative of option prices with respect to the strike.
Enter the spot price, the strike, the interest rate, the volatility and the term. The tool returns the dual gamma. Those who extract implied distributions from the options market use exactly this relationship, computing dual gamma across many strikes to reconstruct what the market thinks about future prices.
Related Tools
Call Dual Delta
Computes the dual delta of a call option: the premium's sensitivity to the strike price, that is, dC/dK. While ordinary delta measures the reaction to the underlying's price, dual delta measures how much the option would change if the strike were slightly different. For a call it equals −e^(−rT)·N(d2) and has a practical reading: in absolute terms it approximates the risk-neutral probability of the option finishing in the money. Enter the spot price, the strike, the interest rate, the term in years and the volatility.
Option Color (Gamma Decay)
Computes the color of an option, the third-order Greek that shows how much the gamma changes with each passing day, all else equal. Since gamma measures how fast the delta moves, color tells you whether that speed is accelerating or slowing as expiry approaches, handy for anyone managing gamma positions near the exercise date, where an at-the-money option's gamma spikes. The result comes per year and per day, computed for a call with no dividends. Enter the spot price, the strike, the interest rate, the term in years and the volatility.
Option Theta (Black-Scholes)
Computes the theta of a European call option under Black-Scholes, the Greek that measures how much premium the option bleeds with each unit of time that passes. The formula pairs the decay of extrinsic value, −S·φ(d1)·σ/(2√T), with the strike-discount effect, −r·K·e^(−rT)·N(d2). For a call with no dividends theta is always negative: time works against the buyer. The result comes as annual theta and per day (÷365). Enter the spot price, the strike, the interest rate, the term in years and the volatility.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.