Pricing average price advertisement options when underlying spot market prices are discontinuous

Advertisement (ad) options have been recently studied as a novel guaranteed delivery (GD) system in online advertising. In essence, an ad option is a contract that gives an advertiser a right but not obligation to enter into transactions to purchase ad inventories such as page views or link clicks from a specific slot at one or multiple pre-specified prices in a specific future period. Compared to guaranteed contracts, the advertiser pays a lower upfront fee but can have greater flexibility and more control in advertising. So far ad option studies have been restricted to the situations where the option payoff is determined by the underlying auction payment price at a specific time point and the price evolution over time is assumed to be continuous. The former leads to a biased option payoff calculation and the latter is invalid empirically for many ad slots. This paper discusses a new option pricing framework which can be applied to a general situation. The option payoff is calculated based on the average price over a specific future period. As we use the general mean, our framework contains different payoff functions as special cases. Further, we use jump-diffusion stochastic models to describe the auction payment price movement, which have Markov and price discontinuity properties, and those properties are validated by our statistical investigation of ad auctions from different datasets. In the paper, we propose a general option pricing solution based on Monte Carlo simulation and also give an explicit pricing formula for a special case. The latter is also a generalisation of the option pricing models in some other recent developments.