Option Pricing Generators

We characterize a class of option pricing models by their algebraic structure. Option prices are monoids, that is operators endowed with the commutativity and associativity property and an identity element. If the price of the underlying asset is bounded, the operator corresponds to the concept of t-conorm, while if it is defined on the positive real line the operator is a pseudo-addition. These operators have the same no-arbitrage properties as the classical option pricing models, but are also associative. Each model in this class is characterized by a univariate increasing function that is defined the generator of the model. The generator encodes a synthetic representation of the probability structure of the underlying asset. We provide no arbitrage conditions for the generators and practical guidelines to construct them.