Pricing rides as option contracts: guarantees and memberships under travel-time uncertainty

RESUMO

Modern ride-share platforms must commit to a price before a trip is taken, yet the realized fare depends on travel time that is uncertain at the moment of sale, which can occur days in advance. The upfront price thus decomposes into the expected fare and the premium on an insurance claim whose payoff is the shortfall between the realized and promised prices. This work computes the fair premium for such a claim, following from contingent-claim pricing. Recent advances in travel-time distribution modeling cast the asymptotic distribution of travel time as a Brownian motion on a metric graph carrying a cyclostationary travel-time process. We thus derive closed-form premiums under both a population and a route-conditional travel-time law. A single pricing equation then covers products of increasing informational difficulty, such as on-demand and scheduled rides and capped-membership and commuter products. To price on sparse data, we develop route and travel-time samplers and validate them out of sample on 2014 Quebec City GPS trips, where they reproduce the empirical trip-time distribution. We show that guarantee premiums are inexpensive, typically under 5% of the fare. Conditioning on the realized route roughly halves the premium while reducing tail risk. Analyzing membership abuse, we show that adverse selection acts precisely along the dimension the price fails to condition on, yielding a design principle: subsidies additive in the fare, such as a fixed per-ride discount, leave marginal incentives unchanged and are abuse-proof by construction, with deterministic closed-form cost.