From Binary Options to Leveraged Perpetuals

Part I: Binary Strips as Synthetic Spot

Part II: Extensions Toward Synthetic Leverage (WIP)

Part I shows that a continuous portfolio of binary (digital) call options spanning all strikes, evaluated in the

T → 0 limit, exactly replicates the spot price of the underlying asset (Breeden–Litzenberger spot identity).

Under the zero-basis assumption, the same portfolio replicates a perpetual future. This serves as the rigorous

base case.

Part II introduces research directions extending this base case toward leveraged synthetic instruments.

The central observation is that replacing the uniform binary weight with Kλ−1 yields a one-parameter family

spanning the entire leverage spectrum: uniform weight (λ = 1) recovers spot; linear weight (λ = 2) yields

squared (Squeeth-style) exposure; general λ yields an λ-leveraged power perpetual. Extensions cover leverage

cost under volatility skew, funding rate design via basis binaries, liquidation as a knock-out boundary, and

the P–Q binary gap as a model-free leverage premium estimator.

Work in progress — v6 (peer review revision). Comments welcome.