Replicating market makers

We present a method for constructing constant function market makers (CFMMs) whose portfolio value functions match a desired payoff. More specifically, we show that the space of concave, nonnegative, nondecreasing, 1-homogeneous payoff functions and the space of convex CFMMs are equivalent; in other words, every CFMM has a concave, nonnegative, nondecreasing, 1-homogeneous payoff function, and every payoff function with these properties has a corresponding convex CFMM. We demonstrate a simple method for recovering a CFMM trading function that produces this desired payoff. This method uses only basic tools from convex analysis and is intimately related to Fenchel conjugacy. We demonstrate our result by constructing trading functions corresponding to basic payoffs, as well as standard financial derivatives, such as options and swaps.