Additive Logistic Processes in Option Pricing

In option pricing it is customary to specify first a stochastic underlying model and then extract from it valuation equations. Still, it is possible to reverse this paradigm: starting from an arbitrage-free valuation formula, one could derive a family of risk-neutral probabilities and a corresponding risk-neutral underlying asset process. In this paper we start from two simple arbitrage-free valuation formulae, one supported by a real-valued spot price and inspired by the log-sum-exponential function, and the other supported on positive spot prices and given by an l^p vector norm. We observe that such formulae lead respectively to Standard Logistic and Dagum (or "log-skew logistic'') risk-neutral distributions for the underlying, and proceed to exhibit supporting stochastic processes of additive type for a market underlying having as time marginals two such distributions. By construction, the underlying price processes analyzed produce closed-forms for option prices which are even simpler of those of the Bachelier and Black-Scholes models. Additionally a single time-normalized moneyness curve fully characterizes the option value. Logistic additive processes provide parsimonious and simple option pricing models capturing various important stylized facts such as returns kurtosis, skewness and self-similarity, at the minimum price of a single market observable input.