Gamma Capture provides a mathematically superior foundation for 0-DTE options.
Call Premium = PV * Σ i∈{−range,+range} : MAX[ (S + b * x_net_i) − K, 0 ] × Prob_Skellam( x_net_i ; μ+, μ− )
Put Premium = PV * Σ i∈{−range,+range} : MAX[ K - (S + b * x_net_i), 0 ] × Prob_Skellam( x_net_i ; μ+, μ− )
Gamma Capture Crossing Intensity per Strike (K)
λ(K) = λ₀ + K_add_c + n * φ (K > ATM)
λ(K) = λ₀ + K_add_p + n/φ (K < ATM)
Gamma Capture is similar to asking how many bricks λ(K) and what cost per brick (b) does it take to build a brick wall? How many buy and sell limit orders (barrier widths) does the price need to cross to sum to an options premium? Gamma Capture is the mathematical expression for delta hedging an option. The formula measures the realized volatility that a vol trader earns by placing bids and offers (up/down barriers) into the market. Put-Call Parity and Monotonicity hold.
What is the fundamental driver of price volatility? Time or crossing buy and sell orders? Change relative to nearest neighbor or distance traveled? PDF or PMF?
Geometric Brownian Motion
Var[ log price change ] = Var[ΔlogS] = σ² * time
Gamma Capture Motion
Var[ dollar price change ] = Var[ΔS] = barrier width² * Number of crossings
Exp[ΔS] = b * (μ+ − μ−)
Geometric Brownian Motion accumulates variance continuously over time. Gamma Capture Motion, however, earns variance by crossing discrete barriers. No crossings, no variance. Time decay happens because as time passes without a crossing, the variance window narrows. The probability of reaching Out-the-Money (OTM) strikes shrinks. In traditional 0-DTE options pricing models, gamma explodes and theta crashes as expiration approaches. The Gamma Capture model turns risk into reality by counting barrier crossings.
What makes the score go up in a basketball game, the clock running or teams making baskets?
Making buckets.
The clock can run for a full quarter, and if nobody scores, the score doesn't change at all. Points only show up on the board when the ball goes through the hoop. But the clock doesn't score points. It just limits how many chances each team gets.
Geometric Brownian Motion (GBM), acts like the score creeps up a little with every tick of the clock, even when nobody takes a shot. Gamma Capture works like a real game: the score moves only when a basket is made. Gamma Capture asks how many baskets are likely before the buzzer, and how many it would take to reach a given score.
Reference:
"From Barrier Crossings to Terminal Distributions: A Skellam-Based Options Pricing Framework for 0-DTE Markets" (January 06, 2026).
Available on SSRN: https://ssrn.com/abstract=7029658